diff --git a/404.html b/404.html
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+++ b/404.html
@@ -312,6 +312,63 @@
           
         
       
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="/chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
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@@ -327,7 +384,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -345,14 +402,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -458,7 +515,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -476,14 +533,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -593,7 +650,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -611,14 +668,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -770,7 +827,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -788,14 +845,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -947,7 +1004,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -965,14 +1022,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1130,7 +1187,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1148,14 +1205,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1284,7 +1341,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1302,14 +1359,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1442,7 +1499,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1460,14 +1517,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1638,7 +1695,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1656,14 +1713,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1792,7 +1849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1810,14 +1867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1950,7 +2007,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1968,14 +2025,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2160,7 +2217,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2178,14 +2235,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2463,7 +2520,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2481,14 +2538,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2640,7 +2697,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2821,7 +2878,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3040,7 +3097,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3213,7 +3270,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
           </label>
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+          <label class="md-nav__title" for="__nav_20">
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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--- a/chapter_appendix/contribution/index.html
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
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+          </ul>
+        </nav>
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+    </li>
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
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-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
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-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2200,14 +2257,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2485,7 +2542,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2503,14 +2560,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2662,7 +2719,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
@@ -2680,14 +2737,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2843,7 +2900,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
@@ -2861,14 +2918,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3062,7 +3119,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
@@ -3080,14 +3137,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3237,7 +3294,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" checked>
         
           
           
@@ -3255,14 +3312,14 @@
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-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3429,7 +3486,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3486,7 +3543,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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     纸质书
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-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -3840,7 +3897,7 @@ aria-label="页脚"
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_appendix/index.html b/chapter_appendix/index.html
index 3bcaf9fe3..cadd4f8f3 100644
--- a/chapter_appendix/index.html
+++ b/chapter_appendix/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
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+      
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+          
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2200,14 +2257,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2485,7 +2542,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2503,14 +2560,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2662,7 +2719,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
@@ -2680,14 +2737,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2843,7 +2900,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
@@ -2861,14 +2918,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3062,7 +3119,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
@@ -3080,14 +3137,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3237,7 +3294,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" checked>
         
           
           
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-              <label class="md-nav__link md-nav__link--active" for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link md-nav__link--active" for="__nav_18" id="__nav_18_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3362,7 +3419,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3419,7 +3476,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19 2-5 4.5v11l5-4.5V2M6.5 5C4.55 5 2.45 5.4 1 6.5v14.66c0 .25.25.5.5.5.1 0 .15-.07.25-.07 1.35-.65 3.3-1.09 4.75-1.09 1.95 0 4.05.4 5.5 1.5 1.35-.85 3.8-1.5 5.5-1.5 1.65 0 3.35.31 4.75 1.06.1.05.15.03.25.03.25 0 .5-.25.5-.5V6.5c-.6-.45-1.25-.75-2-1V19c-1.1-.35-2.3-.5-3.5-.5-1.7 0-4.15.65-5.5 1.5V6.5C10.55 5.4 8.45 5 6.5 5Z"/></svg>
   
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     纸质书
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -3728,7 +3785,7 @@ aria-label="页脚"
       <div class="md-copyright">
   
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_appendix/installation/index.html b/chapter_appendix/installation/index.html
index 3c13e6caa..844163dba 100644
--- a/chapter_appendix/installation/index.html
+++ b/chapter_appendix/installation/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
         
       
         
@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2200,14 +2257,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2485,7 +2542,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2503,14 +2560,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
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@@ -2662,7 +2719,7 @@
         
         
         
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+          <label class="md-nav__title" for="__nav_15">
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+          <label class="md-nav__title" for="__nav_16">
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-          <label class="md-nav__title" for="__nav_16">
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+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
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-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
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@@ -3564,7 +3621,7 @@
         
         
         
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+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_appendix/terminology/index.html b/chapter_appendix/terminology/index.html
index 5ddf55ae1..eeae890c8 100644
--- a/chapter_appendix/terminology/index.html
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@@ -334,6 +334,63 @@
           
         
       
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
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+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
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+            </a>
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+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
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+  
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
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-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
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-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
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+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
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-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
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             </a>
             
               
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+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2200,14 +2257,14 @@
             </a>
             
               
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             第 15 章 &nbsp; 贪心
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             参考文献
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+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_array_and_linkedlist/array/index.html b/chapter_array_and_linkedlist/array/index.html
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+      
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 6 章 &nbsp; 哈希表
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             第 8 章 &nbsp; 堆
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             第 16 章 &nbsp; 附录
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 <!-- Page content -->
 <h1 id="41">4.1 &nbsp; 数组<a class="headerlink" href="#41" title="Permanent link">&para;</a></h1>
-<p>「数组 array」是一种线性数据结构，其将相同类型的元素存储在连续的内存空间中。我们将元素在数组中的位置称为该元素的「索引 index」。图 4-1 展示了数组的主要概念和存储方式。</p>
+<p><u>数组（array）</u>是一种线性数据结构，其将相同类型的元素存储在连续的内存空间中。我们将元素在数组中的位置称为该元素的<u>索引（index）</u>。图 4-1 展示了数组的主要概念和存储方式。</p>
 <p><a class="glightbox" href="../array.assets/array_definition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="数组定义与存储方式" class="animation-figure" src="../array.assets/array_definition.png" /></a></p>
 <p align="center"> 图 4-1 &nbsp; 数组定义与存储方式 </p>
 
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+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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             第 5 章 &nbsp; 栈与队列
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             第 6 章 &nbsp; 哈希表
           </label>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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@@ -1662,7 +1719,7 @@
         
         
         
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           </label>
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           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2184,7 +2241,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2487,7 +2544,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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             第 13 章 &nbsp; 回溯
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+          <label class="md-nav__title" for="__nav_16">
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             第 14 章 &nbsp; 动态规划
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@@ -3064,7 +3121,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
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@@ -3419,7 +3476,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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index f2a3f99fa..4eb16bdd9 100644
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+++ b/chapter_array_and_linkedlist/linked_list/index.html
@@ -334,6 +334,63 @@
           
         
       
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-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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             第 0 章 &nbsp; 前言
           </label>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
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             第 1 章 &nbsp; 初识算法
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-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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 <!-- Page content -->
 <h1 id="42">4.2 &nbsp; 链表<a class="headerlink" href="#42" title="Permanent link">&para;</a></h1>
 <p>内存空间是所有程序的公共资源，在一个复杂的系统运行环境下，空闲的内存空间可能散落在内存各处。我们知道，存储数组的内存空间必须是连续的，而当数组非常大时，内存可能无法提供如此大的连续空间。此时链表的灵活性优势就体现出来了。</p>
-<p>「链表 linked list」是一种线性数据结构，其中的每个元素都是一个节点对象，各个节点通过“引用”相连接。引用记录了下一个节点的内存地址，通过它可以从当前节点访问到下一个节点。</p>
+<p><u>链表（linked list）</u>是一种线性数据结构，其中的每个元素都是一个节点对象，各个节点通过“引用”相连接。引用记录了下一个节点的内存地址，通过它可以从当前节点访问到下一个节点。</p>
 <p>链表的设计使得各个节点可以分散存储在内存各处，它们的内存地址无须连续。</p>
 <p><a class="glightbox" href="../linked_list.assets/linkedlist_definition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="链表定义与存储方式" class="animation-figure" src="../linked_list.assets/linkedlist_definition.png" /></a></p>
 <p align="center"> 图 4-5 &nbsp; 链表定义与存储方式 </p>
 
-<p>观察图 4-5 ，链表的组成单位是「节点 node」对象。每个节点都包含两项数据：节点的“值”和指向下一节点的“引用”。</p>
+<p>观察图 4-5 ，链表的组成单位是<u>节点（node）</u>对象。每个节点都包含两项数据：节点的“值”和指向下一节点的“引用”。</p>
 <ul>
 <li>链表的首个节点被称为“头节点”，最后一个节点被称为“尾节点”。</li>
 <li>尾节点指向的是“空”，它在 Java、C++ 和 Python 中分别被记为 <code>null</code>、<code>nullptr</code> 和 <code>None</code> 。</li>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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             第 0 章 &nbsp; 前言
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 <!-- Page content -->
 <h1 id="43">4.3 &nbsp; 列表<a class="headerlink" href="#43" title="Permanent link">&para;</a></h1>
-<p>「列表 list」是一个抽象的数据结构概念，它表示元素的有序集合，支持元素访问、修改、添加、删除和遍历等操作，无须使用者考虑容量限制的问题。列表可以基于链表或数组实现。</p>
+<p><u>列表（list）</u>是一个抽象的数据结构概念，它表示元素的有序集合，支持元素访问、修改、添加、删除和遍历等操作，无须使用者考虑容量限制的问题。列表可以基于链表或数组实现。</p>
 <ul>
 <li>链表天然可以看作一个列表，其支持元素增删查改操作，并且可以灵活动态扩容。</li>
 <li>数组也支持元素增删查改，但由于其长度不可变，因此只能看作一个具有长度限制的列表。</li>
 </ul>
 <p>当使用数组实现列表时，<strong>长度不可变的性质会导致列表的实用性降低</strong>。这是因为我们通常无法事先确定需要存储多少数据，从而难以选择合适的列表长度。若长度过小，则很可能无法满足使用需求；若长度过大，则会造成内存空间浪费。</p>
-<p>为解决此问题，我们可以使用「动态数组 dynamic array」来实现列表。它继承了数组的各项优点，并且可以在程序运行过程中进行动态扩容。</p>
+<p>为解决此问题，我们可以使用<u>动态数组（dynamic array）</u>来实现列表。它继承了数组的各项优点，并且可以在程序运行过程中进行动态扩容。</p>
 <p>实际上，<strong>许多编程语言中的标准库提供的列表是基于动态数组实现的</strong>，例如 Python 中的 <code>list</code> 、Java 中的 <code>ArrayList</code> 、C++ 中的 <code>vector</code> 和 C# 中的 <code>List</code> 等。在接下来的讨论中，我们将把“列表”和“动态数组”视为等同的概念。</p>
 <h2 id="431">4.3.1 &nbsp; 列表常用操作<a class="headerlink" href="#431" title="Permanent link">&para;</a></h2>
 <h3 id="1">1. &nbsp; 初始化列表<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
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+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_array_and_linkedlist/ram_and_cache/index.html b/chapter_array_and_linkedlist/ram_and_cache/index.html
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+            序
+          </label>
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+          </ul>
+        </nav>
+      
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
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+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
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+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
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+          <label class="md-nav__title" for="__nav_6">
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             第 4 章 &nbsp; 数组与链表
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+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
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             第 6 章 &nbsp; 哈希表
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             第 7 章 &nbsp; 树
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             第 8 章 &nbsp; 堆
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             第 10 章 &nbsp; 搜索
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             第 11 章 &nbsp; 排序
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             第 16 章 &nbsp; 附录
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     纸质书
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -3630,7 +3687,7 @@
 <p>在本章的前两节中，我们探讨了数组和链表这两种基础且重要的数据结构，它们分别代表了“连续存储”和“分散存储”两种物理结构。</p>
 <p>实际上，<strong>物理结构在很大程度上决定了程序对内存和缓存的使用效率</strong>，进而影响算法程序的整体性能。</p>
 <h2 id="441">4.4.1 &nbsp; 计算机存储设备<a class="headerlink" href="#441" title="Permanent link">&para;</a></h2>
-<p>计算机中包括三种类型的存储设备：「硬盘 hard disk」、「内存 random-access memory, RAM」、「缓存 cache memory」。表 4-2 展示了它们在计算机系统中的不同角色和性能特点。</p>
+<p>计算机中包括三种类型的存储设备：<u>硬盘（hard disk）</u>、<u>内存（random-access memory, RAM）</u>、<u>缓存（cache memory）</u>。表 4-2 展示了它们在计算机系统中的不同角色和性能特点。</p>
 <p align="center"> 表 4-2 &nbsp; 计算机的存储设备 </p>
 
 <div class="center-table">
@@ -3699,8 +3756,8 @@
 <p>一方面，<strong>内存是有限的，且同一块内存不能被多个程序共享</strong>，因此我们希望数据结构能够尽可能高效地利用空间。数组的元素紧密排列，不需要额外的空间来存储链表节点间的引用（指针），因此空间效率更高。然而，数组需要一次性分配足够的连续内存空间，这可能导致内存浪费，数组扩容也需要额外的时间和空间成本。相比之下，链表以“节点”为单位进行动态内存分配和回收，提供了更大的灵活性。</p>
 <p>另一方面，在程序运行时，<strong>随着反复申请与释放内存，空闲内存的碎片化程度会越来越高</strong>，从而导致内存的利用效率降低。数组由于其连续的存储方式，相对不容易导致内存碎片化。相反，链表的元素是分散存储的，在频繁的插入与删除操作中，更容易导致内存碎片化。</p>
 <h2 id="443">4.4.3 &nbsp; 数据结构的缓存效率<a class="headerlink" href="#443" title="Permanent link">&para;</a></h2>
-<p>缓存虽然在空间容量上远小于内存，但它比内存快得多，在程序执行速度上起着至关重要的作用。由于缓存的容量有限，只能存储一小部分频繁访问的数据，因此当 CPU 尝试访问的数据不在缓存中时，就会发生「缓存未命中 cache miss」，此时 CPU 不得不从速度较慢的内存中加载所需数据。</p>
-<p>显然，<strong>“缓存未命中”越少，CPU 读写数据的效率就越高</strong>，程序性能也就越好。我们将 CPU 从缓存中成功获取数据的比例称为「缓存命中率 cache hit rate」，这个指标通常用来衡量缓存效率。</p>
+<p>缓存虽然在空间容量上远小于内存，但它比内存快得多，在程序执行速度上起着至关重要的作用。由于缓存的容量有限，只能存储一小部分频繁访问的数据，因此当 CPU 尝试访问的数据不在缓存中时，就会发生<u>缓存未命中（cache miss）</u>，此时 CPU 不得不从速度较慢的内存中加载所需数据。</p>
+<p>显然，<strong>“缓存未命中”越少，CPU 读写数据的效率就越高</strong>，程序性能也就越好。我们将 CPU 从缓存中成功获取数据的比例称为<u>缓存命中率（cache hit rate）</u>，这个指标通常用来衡量缓存效率。</p>
 <p>为了尽可能达到更高的效率，缓存会采取以下数据加载机制。</p>
 <ul>
 <li><strong>缓存行</strong>：缓存不是单个字节地存储与加载数据，而是以缓存行为单位。相比于单个字节的传输，缓存行的传输形式更加高效。</li>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
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+            序
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 2 章 &nbsp; 复杂度分析
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             第 3 章 &nbsp; 数据结构
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             第 4 章 &nbsp; 数组与链表
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             第 5 章 &nbsp; 栈与队列
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             第 6 章 &nbsp; 哈希表
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             第 7 章 &nbsp; 树
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
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+          <label class="md-nav__title" for="__nav_14">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
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             第 13 章 &nbsp; 回溯
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@@ -2903,7 +2960,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3122,7 +3179,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
           </label>
@@ -3295,7 +3352,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3420,7 +3477,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
           </label>
@@ -3477,7 +3534,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
           </label>
@@ -3839,7 +3896,7 @@ aria-label="页脚"
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_backtracking/backtracking_algorithm/index.html b/chapter_backtracking/backtracking_algorithm/index.html
index 71f441015..8825a229a 100644
--- a/chapter_backtracking/backtracking_algorithm/index.html
+++ b/chapter_backtracking/backtracking_algorithm/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
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+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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             第 10 章 &nbsp; 搜索
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             第 11 章 &nbsp; 排序
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+          <label class="md-nav__title" for="__nav_14">
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             第 12 章 &nbsp; 分治
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+          <label class="md-nav__title" for="__nav_15">
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
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             参考文献
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+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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@@ -3673,7 +3730,7 @@
 
 <!-- Page content -->
 <h1 id="131">13.1 &nbsp; 回溯算法<a class="headerlink" href="#131" title="Permanent link">&para;</a></h1>
-<p>「回溯算法 backtracking algorithm」是一种通过穷举来解决问题的方法，它的核心思想是从一个初始状态出发，暴力搜索所有可能的解决方案，当遇到正确的解则将其记录，直到找到解或者尝试了所有可能的选择都无法找到解为止。</p>
+<p><u>回溯算法（backtracking algorithm）</u>是一种通过穷举来解决问题的方法，它的核心思想是从一个初始状态出发，暴力搜索所有可能的解决方案，当遇到正确的解则将其记录，直到找到解或者尝试了所有可能的选择都无法找到解为止。</p>
 <p>回溯算法通常采用“深度优先搜索”来遍历解空间。在“二叉树”章节中，我们提到前序、中序和后序遍历都属于深度优先搜索。接下来，我们利用前序遍历构造一个回溯问题，逐步了解回溯算法的工作原理。</p>
 <div class="admonition question">
 <p class="admonition-title">例题一</p>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+          <div class="md-nav__link md-nav__container">
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+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
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+        </nav>
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 2 章 &nbsp; 复杂度分析
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             第 3 章 &nbsp; 数据结构
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             第 4 章 &nbsp; 数组与链表
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+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
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+          <label class="md-nav__title" for="__nav_8">
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             第 6 章 &nbsp; 哈希表
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
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           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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             第 11 章 &nbsp; 排序
           </label>
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             第 13 章 &nbsp; 回溯
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             第 15 章 &nbsp; 贪心
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+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
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@@ -3419,7 +3476,7 @@
         
         
         
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_backtracking/n_queens_problem/index.html b/chapter_backtracking/n_queens_problem/index.html
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+++ b/chapter_backtracking/n_queens_problem/index.html
@@ -334,6 +334,63 @@
           
         
       
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-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
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           </label>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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             第 1 章 &nbsp; 初识算法
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-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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             第 5 章 &nbsp; 栈与队列
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+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
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           </label>
@@ -1660,7 +1717,7 @@
         
         
         
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           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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           </label>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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             第 11 章 &nbsp; 排序
           </label>
@@ -2485,7 +2542,7 @@
         
         
         
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           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" checked>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="true">
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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@@ -3131,7 +3188,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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             第 15 章 &nbsp; 贪心
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@@ -3304,7 +3361,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
           </label>
@@ -3429,7 +3486,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
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             参考文献
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@@ -3486,7 +3543,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_backtracking/permutations_problem/index.html b/chapter_backtracking/permutations_problem/index.html
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@@ -334,6 +334,63 @@
           
         
       
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+          
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+          <label class="md-nav__title" for="__nav_1">
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+            序
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+        </nav>
+      
+    </li>
+  
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
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-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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             第 1 章 &nbsp; 初识算法
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
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-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
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-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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             第 3 章 &nbsp; 数据结构
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             </a>
             
               
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+          <label class="md-nav__title" for="__nav_6">
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             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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@@ -1324,14 +1381,14 @@
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
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+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2485,7 +2542,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" checked>
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="true">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_15">
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             第 13 章 &nbsp; 回溯
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
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             第 14 章 &nbsp; 动态规划
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
           </label>
@@ -3352,7 +3409,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
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             参考文献
           </label>
@@ -3534,7 +3591,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_backtracking/subset_sum_problem/index.html b/chapter_backtracking/subset_sum_problem/index.html
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+++ b/chapter_backtracking/subset_sum_problem/index.html
@@ -334,6 +334,63 @@
           
         
       
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+          
+          <div class="md-nav__link md-nav__container">
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
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+    
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
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-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
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-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2485,7 +2542,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
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-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2664,7 +2721,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" checked>
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2960,7 +3017,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
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-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
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             第 14 章 &nbsp; 动态规划
           </label>
@@ -3179,7 +3236,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
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@@ -3352,7 +3409,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
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@@ -3477,7 +3534,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
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@@ -3534,7 +3591,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+            序
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 2 章 &nbsp; 复杂度分析
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             第 3 章 &nbsp; 数据结构
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             第 4 章 &nbsp; 数组与链表
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             第 5 章 &nbsp; 栈与队列
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+          <label class="md-nav__title" for="__nav_8">
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             第 6 章 &nbsp; 哈希表
           </label>
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             第 7 章 &nbsp; 树
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             第 8 章 &nbsp; 堆
           </label>
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             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
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-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
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             第 12 章 &nbsp; 分治
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
           </label>
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             第 16 章 &nbsp; 附录
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             参考文献
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
         
       
         
@@ -349,7 +406,7 @@
         
         
         
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -617,7 +674,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" checked>
         
           
           
@@ -635,14 +692,14 @@
             </a>
             
               
-              <label class="md-nav__link md-nav__link--active" for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link md-nav__link--active" for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -794,7 +851,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -812,14 +869,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -971,7 +1028,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -989,14 +1046,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1154,7 +1211,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1308,7 +1365,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1326,14 +1383,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1466,7 +1523,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1662,7 +1719,7 @@
         
         
         
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1816,7 +1873,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1974,7 +2031,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2184,7 +2241,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2487,7 +2544,7 @@
         
         
         
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
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-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
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-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
           </label>
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+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
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             参考文献
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     纸质书
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+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_computational_complexity/iteration_and_recursion/index.html b/chapter_computational_complexity/iteration_and_recursion/index.html
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 3 章 &nbsp; 数据结构
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             第 4 章 &nbsp; 数组与链表
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             第 5 章 &nbsp; 栈与队列
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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             第 6 章 &nbsp; 哈希表
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             第 7 章 &nbsp; 树
           </label>
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             第 8 章 &nbsp; 堆
           </label>
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             第 9 章 &nbsp; 图
           </label>
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             第 10 章 &nbsp; 搜索
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             第 11 章 &nbsp; 排序
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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             参考文献
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@@ -3753,7 +3810,7 @@
 <h1 id="22">2.2 &nbsp; 迭代与递归<a class="headerlink" href="#22" title="Permanent link">&para;</a></h1>
 <p>在算法中，重复执行某个任务是很常见的，它与复杂度分析息息相关。因此，在介绍时间复杂度和空间复杂度之前，我们先来了解如何在程序中实现重复执行任务，即两种基本的程序控制结构：迭代、递归。</p>
 <h2 id="221">2.2.1 &nbsp; 迭代<a class="headerlink" href="#221" title="Permanent link">&para;</a></h2>
-<p>「迭代 iteration」是一种重复执行某个任务的控制结构。在迭代中，程序会在满足一定的条件下重复执行某段代码，直到这个条件不再满足。</p>
+<p><u>迭代（iteration）</u>是一种重复执行某个任务的控制结构。在迭代中，程序会在满足一定的条件下重复执行某段代码，直到这个条件不再满足。</p>
 <h3 id="1-for">1. &nbsp; for 循环<a class="headerlink" href="#1-for" title="Permanent link">&para;</a></h3>
 <p><code>for</code> 循环是最常见的迭代形式之一，<strong>适合在预先知道迭代次数时使用</strong>。</p>
 <p>以下函数基于 <code>for</code> 循环实现了求和 <span class="arithmatex">\(1 + 2 + \dots + n\)</span> ，求和结果使用变量 <code>res</code> 记录。需要注意的是，Python 中 <code>range(a, b)</code> 对应的区间是“左闭右开”的，对应的遍历范围为 <span class="arithmatex">\(a, a + 1, \dots, b-1\)</span> ：</p>
@@ -4573,7 +4630,7 @@
 <p>在这种情况下，函数的操作数量与 <span class="arithmatex">\(n^2\)</span> 成正比，或者说算法运行时间和输入数据大小 <span class="arithmatex">\(n\)</span> 成“平方关系”。</p>
 <p>我们可以继续添加嵌套循环，每一次嵌套都是一次“升维”，将会使时间复杂度提高至“立方关系”“四次方关系”，以此类推。</p>
 <h2 id="222">2.2.2 &nbsp; 递归<a class="headerlink" href="#222" title="Permanent link">&para;</a></h2>
-<p>「递归 recursion」是一种算法策略，通过函数调用自身来解决问题。它主要包含两个阶段。</p>
+<p><u>递归（recursion）</u>是一种算法策略，通过函数调用自身来解决问题。它主要包含两个阶段。</p>
 <ol>
 <li><strong>递</strong>：程序不断深入地调用自身，通常传入更小或更简化的参数，直到达到“终止条件”。</li>
 <li><strong>归</strong>：触发“终止条件”后，程序从最深层的递归函数开始逐层返回，汇聚每一层的结果。</li>
@@ -4793,7 +4850,7 @@
 
 <p>在实际中，编程语言允许的递归深度通常是有限的，过深的递归可能导致栈溢出错误。</p>
 <h3 id="2">2. &nbsp; 尾递归<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
-<p>有趣的是，<strong>如果函数在返回前的最后一步才进行递归调用</strong>，则该函数可以被编译器或解释器优化，使其在空间效率上与迭代相当。这种情况被称为「尾递归 tail recursion」。</p>
+<p>有趣的是，<strong>如果函数在返回前的最后一步才进行递归调用</strong>，则该函数可以被编译器或解释器优化，使其在空间效率上与迭代相当。这种情况被称为<u>尾递归（tail recursion）</u>。</p>
 <ul>
 <li><strong>普通递归</strong>：当函数返回到上一层级的函数后，需要继续执行代码，因此系统需要保存上一层调用的上下文。</li>
 <li><strong>尾递归</strong>：递归调用是函数返回前的最后一个操作，这意味着函数返回到上一层级后，无须继续执行其他操作，因此系统无须保存上一层函数的上下文。</li>
@@ -5162,7 +5219,7 @@
 <p><div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20fib%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%96%90%E6%B3%A2%E9%82%A3%E5%A5%91%E6%95%B0%E5%88%97%EF%BC%9A%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E7%BB%88%E6%AD%A2%E6%9D%A1%E4%BB%B6%20f%281%29%20%3D%200,%20f%282%29%20%3D%201%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20n%20-%201%0A%20%20%20%20%23%20%E9%80%92%E5%BD%92%E8%B0%83%E7%94%A8%20f%28n%29%20%3D%20f%28n-1%29%20%2B%20f%28n-2%29%0A%20%20%20%20res%20%3D%20fib%28n%20-%201%29%20%2B%20fib%28n%20-%202%29%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E7%BB%93%E6%9E%9C%20f%28n%29%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20fib%28n%29%0A%20%20%20%20print%28f%22%5Cn%E6%96%90%E6%B3%A2%E9%82%A3%E5%A5%91%E6%95%B0%E5%88%97%E7%9A%84%E7%AC%AC%20%7Bn%7D%20%E9%A1%B9%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
 <div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20fib%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%96%90%E6%B3%A2%E9%82%A3%E5%A5%91%E6%95%B0%E5%88%97%EF%BC%9A%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E7%BB%88%E6%AD%A2%E6%9D%A1%E4%BB%B6%20f%281%29%20%3D%200,%20f%282%29%20%3D%201%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20n%20-%201%0A%20%20%20%20%23%20%E9%80%92%E5%BD%92%E8%B0%83%E7%94%A8%20f%28n%29%20%3D%20f%28n-1%29%20%2B%20f%28n-2%29%0A%20%20%20%20res%20%3D%20fib%28n%20-%201%29%20%2B%20fib%28n%20-%202%29%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E7%BB%93%E6%9E%9C%20f%28n%29%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20fib%28n%29%0A%20%20%20%20print%28f%22%5Cn%E6%96%90%E6%B3%A2%E9%82%A3%E5%A5%91%E6%95%B0%E5%88%97%E7%9A%84%E7%AC%AC%20%7Bn%7D%20%E9%A1%B9%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 &gt;</a></div></p>
 </details>
-<p>观察以上代码，我们在函数内递归调用了两个函数，<strong>这意味着从一个调用产生了两个调用分支</strong>。如图 2-6 所示，这样不断递归调用下去，最终将产生一棵层数为 <span class="arithmatex">\(n\)</span> 的「递归树 recursion tree」。</p>
+<p>观察以上代码，我们在函数内递归调用了两个函数，<strong>这意味着从一个调用产生了两个调用分支</strong>。如图 2-6 所示，这样不断递归调用下去，最终将产生一棵层数为 <span class="arithmatex">\(n\)</span> 的<u>递归树（recursion tree）</u>。</p>
 <p><a class="glightbox" href="../iteration_and_recursion.assets/recursion_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="斐波那契数列的递归树" class="animation-figure" src="../iteration_and_recursion.assets/recursion_tree.png" /></a></p>
 <p align="center"> 图 2-6 &nbsp; 斐波那契数列的递归树 </p>
 
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
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+          <ul class="md-nav__list" data-md-scrollfix>
+            
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+          </ul>
+        </nav>
+      
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 2 章 &nbsp; 复杂度分析
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             第 3 章 &nbsp; 数据结构
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             第 4 章 &nbsp; 数组与链表
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             第 5 章 &nbsp; 栈与队列
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 <p>一方面，<strong>难以排除测试环境的干扰因素</strong>。硬件配置会影响算法的性能。比如在某台计算机中，算法 <code>A</code> 的运行时间比算法 <code>B</code> 短；但在另一台配置不同的计算机中，可能得到相反的测试结果。这意味着我们需要在各种机器上进行测试，统计平均效率，而这是不现实的。</p>
 <p>另一方面，<strong>展开完整测试非常耗费资源</strong>。随着输入数据量的变化，算法会表现出不同的效率。例如，在输入数据量较小时，算法 <code>A</code> 的运行时间比算法 <code>B</code> 短；而在输入数据量较大时，测试结果可能恰恰相反。因此，为了得到有说服力的结论，我们需要测试各种规模的输入数据，而这需要耗费大量的计算资源。</p>
 <h2 id="212">2.1.2 &nbsp; 理论估算<a class="headerlink" href="#212" title="Permanent link">&para;</a></h2>
-<p>由于实际测试具有较大的局限性，因此我们可以考虑仅通过一些计算来评估算法的效率。这种估算方法被称为「渐近复杂度分析 asymptotic complexity analysis」，简称「复杂度分析」。</p>
+<p>由于实际测试具有较大的局限性，因此我们可以考虑仅通过一些计算来评估算法的效率。这种估算方法被称为<u>渐近复杂度分析（asymptotic complexity analysis）</u>，简称<u>复杂度分析</u>。</p>
 <p>复杂度分析能够体现算法运行所需的时间和空间资源与输入数据大小之间的关系。<strong>它描述了随着输入数据大小的增加，算法执行所需时间和空间的增长趋势</strong>。这个定义有些拗口，我们可以将其分为三个重点来理解。</p>
 <ul>
-<li>“时间和空间资源”分别对应「时间复杂度 time complexity」和「空间复杂度 space complexity」。</li>
+<li>“时间和空间资源”分别对应<u>时间复杂度（time complexity）</u>和<u>空间复杂度（space complexity）</u>。</li>
 <li>“随着输入数据大小的增加”意味着复杂度反映了算法运行效率与输入数据体量之间的关系。</li>
 <li>“时间和空间的增长趋势”表示复杂度分析关注的不是运行时间或占用空间的具体值，而是时间或空间增长的“快慢”。</li>
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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 <!-- Page content -->
 <h1 id="24">2.4 &nbsp; 空间复杂度<a class="headerlink" href="#24" title="Permanent link">&para;</a></h1>
-<p>「空间复杂度 space complexity」用于衡量算法占用内存空间随着数据量变大时的增长趋势。这个概念与时间复杂度非常类似，只需将“运行时间”替换为“占用内存空间”。</p>
+<p><u>空间复杂度（space complexity）</u>用于衡量算法占用内存空间随着数据量变大时的增长趋势。这个概念与时间复杂度非常类似，只需将“运行时间”替换为“占用内存空间”。</p>
 <h2 id="241">2.4.1 &nbsp; 算法相关空间<a class="headerlink" href="#241" title="Permanent link">&para;</a></h2>
 <p>算法在运行过程中使用的内存空间主要包括以下几种。</p>
 <ul>
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+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
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+          </ul>
+        </nav>
+      
+    </li>
+  
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             第 0 章 &nbsp; 前言
           </label>
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             第 1 章 &nbsp; 初识算法
           </label>
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+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
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+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
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+          <label class="md-nav__title" for="__nav_6">
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             第 4 章 &nbsp; 数组与链表
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             第 5 章 &nbsp; 栈与队列
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             第 6 章 &nbsp; 哈希表
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             第 7 章 &nbsp; 树
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             第 8 章 &nbsp; 堆
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             第 9 章 &nbsp; 图
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             第 10 章 &nbsp; 搜索
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             第 11 章 &nbsp; 排序
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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             参考文献
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     纸质书
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 <p><strong>Q</strong>：尾递归的空间复杂度是 <span class="arithmatex">\(O(1)\)</span> 吗？</p>
 <p>理论上，尾递归函数的空间复杂度可以优化至 <span class="arithmatex">\(O(1)\)</span> 。不过绝大多数编程语言（例如 Java、Python、C++、Go、C# 等）不支持自动优化尾递归，因此通常认为空间复杂度是 <span class="arithmatex">\(O(n)\)</span> 。</p>
 <p><strong>Q</strong>：函数和方法这两个术语的区别是什么？</p>
-<p>「函数 function」可以被独立执行，所有参数都以显式传递。「方法 method」与一个对象关联，被隐式传递给调用它的对象，能够对类的实例中包含的数据进行操作。</p>
+<p><u>函数（function）</u>可以被独立执行，所有参数都以显式传递。<u>方法（method）</u>与一个对象关联，被隐式传递给调用它的对象，能够对类的实例中包含的数据进行操作。</p>
 <p>下面以几种常见的编程语言为例来说明。</p>
 <ul>
 <li>C 语言是过程式编程语言，没有面向对象的概念，所以只有函数。但我们可以通过创建结构体（struct）来模拟面向对象编程，与结构体相关联的函数就相当于其他编程语言中的方法。</li>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+            序
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+        </nav>
+      
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 2 章 &nbsp; 复杂度分析
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             第 3 章 &nbsp; 数据结构
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             第 4 章 &nbsp; 数组与链表
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             第 5 章 &nbsp; 栈与队列
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             第 6 章 &nbsp; 哈希表
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             第 7 章 &nbsp; 树
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             第 8 章 &nbsp; 堆
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             第 10 章 &nbsp; 搜索
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             第 16 章 &nbsp; 附录
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-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -4470,7 +4527,7 @@
 T(n) = 3 + 2n
 \]</div>
 <p><span class="arithmatex">\(T(n)\)</span> 是一次函数，说明其运行时间的增长趋势是线性的，因此它的时间复杂度是线性阶。</p>
-<p>我们将线性阶的时间复杂度记为 <span class="arithmatex">\(O(n)\)</span> ，这个数学符号称为「大 <span class="arithmatex">\(O\)</span> 记号 big-<span class="arithmatex">\(O\)</span> notation」，表示函数 <span class="arithmatex">\(T(n)\)</span> 的「渐近上界 asymptotic upper bound」。</p>
+<p>我们将线性阶的时间复杂度记为 <span class="arithmatex">\(O(n)\)</span> ，这个数学符号称为<u>大（<span class="arithmatex">\(O\)</span> 记号 big-<span class="arithmatex">\(O\)</span> notation）</u>，表示函数 <span class="arithmatex">\(T(n)\)</span> 的<u>渐近上界（asymptotic upper bound）</u>。</p>
 <p>时间复杂度分析本质上是计算“操作数量 <span class="arithmatex">\(T(n)\)</span>”的渐近上界，它具有明确的数学定义。</p>
 <div class="admonition abstract">
 <p class="admonition-title">函数渐近上界</p>
@@ -7379,7 +7436,7 @@ aria-label="页脚"
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
diff --git a/chapter_data_structure/basic_data_types/index.html b/chapter_data_structure/basic_data_types/index.html
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+++ b/chapter_data_structure/basic_data_types/index.html
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+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
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+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -794,7 +851,7 @@
         
         
         
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+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
           </label>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_6">
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             第 4 章 &nbsp; 数组与链表
           </label>
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+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1672,7 +1729,7 @@
         
         
         
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+          <label class="md-nav__title" for="__nav_10">
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             第 8 章 &nbsp; 堆
           </label>
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             第 9 章 &nbsp; 图
           </label>
@@ -1984,7 +2041,7 @@
         
         
         
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             </a>
             
               
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2194,7 +2251,7 @@
         
         
         
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             第 11 章 &nbsp; 排序
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             第 12 章 &nbsp; 分治
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             第 13 章 &nbsp; 回溯
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+          <label class="md-nav__title" for="__nav_16">
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             第 14 章 &nbsp; 动态规划
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+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
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@@ -3429,7 +3486,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
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+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_data_structure/character_encoding/index.html b/chapter_data_structure/character_encoding/index.html
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--- a/chapter_data_structure/character_encoding/index.html
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@@ -334,6 +334,63 @@
           
         
       
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
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+  
+
+            </a>
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+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
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+    
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+  
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
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-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
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-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -794,7 +851,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" checked>
         
           
           
@@ -812,14 +869,14 @@
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-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1239,7 +1296,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1257,14 +1314,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1393,7 +1450,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1411,14 +1468,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1551,7 +1608,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1569,14 +1626,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1747,7 +1804,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1901,7 +1958,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1919,14 +1976,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -2059,7 +2116,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -2077,14 +2134,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2269,7 +2326,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2287,14 +2344,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2572,7 +2629,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2590,14 +2647,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2749,7 +2806,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
@@ -2767,14 +2824,14 @@
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 <h1 id="34">3.4 &nbsp; 字符编码 *<a class="headerlink" href="#34" title="Permanent link">&para;</a></h1>
 <p>在计算机中，所有数据都是以二进制数的形式存储的，字符 <code>char</code> 也不例外。为了表示字符，我们需要建立一套“字符集”，规定每个字符和二进制数之间的一一对应关系。有了字符集之后，计算机就可以通过查表完成二进制数到字符的转换。</p>
 <h2 id="341-ascii">3.4.1 &nbsp; ASCII 字符集<a class="headerlink" href="#341-ascii" title="Permanent link">&para;</a></h2>
-<p>「ASCII 码」是最早出现的字符集，其全称为 American Standard Code for Information Interchange（美国标准信息交换代码）。它使用 7 位二进制数（一个字节的低 7 位）表示一个字符，最多能够表示 128 个不同的字符。如图 3-6 所示，ASCII 码包括英文字母的大小写、数字 0 ~ 9、一些标点符号，以及一些控制字符（如换行符和制表符）。</p>
+<p><u>ASCII（码）</u>是最早出现的字符集，其全称为 American Standard Code for Information Interchange（美国标准信息交换代码）。它使用 7 位二进制数（一个字节的低 7 位）表示一个字符，最多能够表示 128 个不同的字符。如图 3-6 所示，ASCII 码包括英文字母的大小写、数字 0 ~ 9、一些标点符号，以及一些控制字符（如换行符和制表符）。</p>
 <p><a class="glightbox" href="../character_encoding.assets/ascii_table.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="ASCII 码" class="animation-figure" src="../character_encoding.assets/ascii_table.png" /></a></p>
 <p align="center"> 图 3-6 &nbsp; ASCII 码 </p>
 
-<p>然而，<strong>ASCII 码仅能够表示英文</strong>。随着计算机的全球化，诞生了一种能够表示更多语言的「EASCII」字符集。它在 ASCII 的 7 位基础上扩展到 8 位，能够表示 256 个不同的字符。</p>
+<p>然而，<strong>ASCII 码仅能够表示英文</strong>。随着计算机的全球化，诞生了一种能够表示更多语言的<u>EASCII</u>字符集。它在 ASCII 的 7 位基础上扩展到 8 位，能够表示 256 个不同的字符。</p>
 <p>在世界范围内，陆续出现了一批适用于不同地区的 EASCII 字符集。这些字符集的前 128 个字符统一为 ASCII 码，后 128 个字符定义不同，以适应不同语言的需求。</p>
 <h2 id="342-gbk">3.4.2 &nbsp; GBK 字符集<a class="headerlink" href="#342-gbk" title="Permanent link">&para;</a></h2>
-<p>后来人们发现，<strong>EASCII 码仍然无法满足许多语言的字符数量要求</strong>。比如汉字有近十万个，光日常使用的就有几千个。中国国家标准总局于 1980 年发布了「GB2312」字符集，其收录了 6763 个汉字，基本满足了汉字的计算机处理需要。</p>
-<p>然而，GB2312 无法处理部分罕见字和繁体字。「GBK」字符集是在 GB2312 的基础上扩展得到的，它共收录了 21886 个汉字。在 GBK 的编码方案中，ASCII 字符使用一个字节表示，汉字使用两个字节表示。</p>
+<p>后来人们发现，<strong>EASCII 码仍然无法满足许多语言的字符数量要求</strong>。比如汉字有近十万个，光日常使用的就有几千个。中国国家标准总局于 1980 年发布了<u>GB2312</u>字符集，其收录了 6763 个汉字，基本满足了汉字的计算机处理需要。</p>
+<p>然而，GB2312 无法处理部分罕见字和繁体字。<u>GBK</u>字符集是在 GB2312 的基础上扩展得到的，它共收录了 21886 个汉字。在 GBK 的编码方案中，ASCII 字符使用一个字节表示，汉字使用两个字节表示。</p>
 <h2 id="343-unicode">3.4.3 &nbsp; Unicode 字符集<a class="headerlink" href="#343-unicode" title="Permanent link">&para;</a></h2>
 <p>随着计算机技术的蓬勃发展，字符集与编码标准百花齐放，而这带来了许多问题。一方面，这些字符集一般只定义了特定语言的字符，无法在多语言环境下正常工作。另一方面，同一种语言存在多种字符集标准，如果两台计算机使用的是不同的编码标准，则在信息传递时就会出现乱码。</p>
 <p>那个时代的研究人员就在想：<strong>如果推出一个足够完整的字符集，将世界范围内的所有语言和符号都收录其中，不就可以解决跨语言环境和乱码问题了吗</strong>？在这种想法的驱动下，一个大而全的字符集 Unicode 应运而生。</p>
-<p>「Unicode」的中文名称为“统一码”，理论上能容纳 100 多万个字符。它致力于将全球范围内的字符纳入统一的字符集之中，提供一种通用的字符集来处理和显示各种语言文字，减少因为编码标准不同而产生的乱码问题。</p>
+<p><u>Unicode</u>的中文名称为“统一码”，理论上能容纳 100 多万个字符。它致力于将全球范围内的字符纳入统一的字符集之中，提供一种通用的字符集来处理和显示各种语言文字，减少因为编码标准不同而产生的乱码问题。</p>
 <p>自 1991 年发布以来，Unicode 不断扩充新的语言与字符。截至 2022 年 9 月，Unicode 已经包含 149186 个字符，包括各种语言的字符、符号甚至表情符号等。在庞大的 Unicode 字符集中，常用的字符占用 2 字节，有些生僻的字符占用 3 字节甚至 4 字节。</p>
 <p>Unicode 是一种通用字符集，本质上是给每个字符分配一个编号（称为“码点”），<strong>但它并没有规定在计算机中如何存储这些字符码点</strong>。我们不禁会问：当多种长度的 Unicode 码点同时出现在一个文本中时，系统如何解析字符？例如给定一个长度为 2 字节的编码，系统如何确认它是一个 2 字节的字符还是两个 1 字节的字符？</p>
 <p>对于以上问题，<strong>一种直接的解决方案是将所有字符存储为等长的编码</strong>。如图 3-7 所示，“Hello”中的每个字符占用 1 字节，“算法”中的每个字符占用 2 字节。我们可以通过高位填 0 将“Hello 算法”中的所有字符都编码为 2 字节长度。这样系统就可以每隔 2 字节解析一个字符，恢复这个短语的内容了。</p>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
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             第 0 章 &nbsp; 前言
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             第 4 章 &nbsp; 数组与链表
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             第 5 章 &nbsp; 栈与队列
           </label>
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             第 6 章 &nbsp; 哈希表
           </label>
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           </label>
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           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
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             第 10 章 &nbsp; 搜索
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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             第 11 章 &nbsp; 排序
           </label>
@@ -2545,7 +2602,7 @@
         
         
         
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           </label>
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             第 13 章 &nbsp; 回溯
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             第 15 章 &nbsp; 贪心
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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             第 16 章 &nbsp; 附录
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             参考文献
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_data_structure/index.html b/chapter_data_structure/index.html
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+      
+    </li>
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-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 2 章 &nbsp; 复杂度分析
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             第 3 章 &nbsp; 数据结构
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             第 4 章 &nbsp; 数组与链表
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@@ -1154,7 +1211,7 @@
         
         
         
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             第 5 章 &nbsp; 栈与队列
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@@ -1308,7 +1365,7 @@
         
         
         
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             第 6 章 &nbsp; 哈希表
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           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
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             第 10 章 &nbsp; 搜索
           </label>
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             第 11 章 &nbsp; 排序
           </label>
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             第 13 章 &nbsp; 回溯
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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             第 0 章 &nbsp; 前言
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             第 16 章 &nbsp; 附录
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-    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><!--! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M96 0C43 0 0 43 0 96v320c0 53 43 96 96 96h320c17.7 0 32-14.3 32-32s-14.3-32-32-32v-64c17.7 0 32-14.3 32-32V32c0-17.7-14.3-32-32-32H96zm0 384h256v64H96c-17.7 0-32-14.3-32-32s14.3-32 32-32zm32-240c0-8.8 7.2-16 16-16h192c8.8 0 16 7.2 16 16s-7.2 16-16 16H144c-8.8 0-16-7.2-16-16zm16 48h192c8.8 0 16 7.2 16 16s-7.2 16-16 16H144c-8.8 0-16-7.2-16-16s7.2-16 16-16z"/></svg>
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19 2-5 4.5v11l5-4.5V2M6.5 5C4.55 5 2.45 5.4 1 6.5v14.66c0 .25.25.5.5.5.1 0 .15-.07.25-.07 1.35-.65 3.3-1.09 4.75-1.09 1.95 0 4.05.4 5.5 1.5 1.35-.85 3.8-1.5 5.5-1.5 1.65 0 3.35.31 4.75 1.06.1.05.15.03.25.03.25 0 .5-.25.5-.5V6.5c-.6-.45-1.25-.75-2-1V19c-1.1-.35-2.3-.5-3.5-.5-1.7 0-4.15.65-5.5 1.5V6.5C10.55 5.4 8.45 5 6.5 5Z"/></svg>
   
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 <p><a class="glightbox" href="../number_encoding.assets/1s_2s_complement.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="原码、反码与补码之间的相互转换" class="animation-figure" src="../number_encoding.assets/1s_2s_complement.png" /></a></p>
 <p align="center"> 图 3-4 &nbsp; 原码、反码与补码之间的相互转换 </p>
 
-<p>「原码 sign-magnitude」虽然最直观，但存在一些局限性。一方面，<strong>负数的原码不能直接用于运算</strong>。例如在原码下计算 <span class="arithmatex">\(1 + (-2)\)</span> ，得到的结果是 <span class="arithmatex">\(-3\)</span> ，这显然是不对的。</p>
+<p><u>原码（sign-magnitude）</u>虽然最直观，但存在一些局限性。一方面，<strong>负数的原码不能直接用于运算</strong>。例如在原码下计算 <span class="arithmatex">\(1 + (-2)\)</span> ，得到的结果是 <span class="arithmatex">\(-3\)</span> ，这显然是不对的。</p>
 <div class="arithmatex">\[
 \begin{aligned}
 &amp; 1 + (-2) \newline
@@ -3626,7 +3683,7 @@
 &amp; \rightarrow -3
 \end{aligned}
 \]</div>
-<p>为了解决此问题，计算机引入了「反码 1's complement」。如果我们先将原码转换为反码，并在反码下计算 <span class="arithmatex">\(1 + (-2)\)</span> ，最后将结果从反码转换回原码，则可得到正确结果 <span class="arithmatex">\(-1\)</span> 。</p>
+<p>为了解决此问题，计算机引入了<u>反码（1's complement）</u>。如果我们先将原码转换为反码，并在反码下计算 <span class="arithmatex">\(1 + (-2)\)</span> ，最后将结果从反码转换回原码，则可得到正确结果 <span class="arithmatex">\(-1\)</span> 。</p>
 <div class="arithmatex">\[
 \begin{aligned}
 &amp; 1 + (-2) \newline
@@ -3644,7 +3701,7 @@
 -0 &amp; \rightarrow 1000 \; 0000
 \end{aligned}
 \]</div>
-<p>与原码一样，反码也存在正负零歧义问题，因此计算机进一步引入了「补码 2's complement」。我们先来观察一下负零的原码、反码、补码的转换过程：</p>
+<p>与原码一样，反码也存在正负零歧义问题，因此计算机进一步引入了<u>补码（2's complement）</u>。我们先来观察一下负零的原码、反码、补码的转换过程：</p>
 <div class="arithmatex">\[
 \begin{aligned}
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
diff --git a/chapter_data_structure/summary/index.html b/chapter_data_structure/summary/index.html
index 503556faf..8a063c2fa 100644
--- a/chapter_data_structure/summary/index.html
+++ b/chapter_data_structure/summary/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
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+      
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+        
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
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+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
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+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
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+    
+      
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+  
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
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@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
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+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
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             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1212,7 +1269,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1366,7 +1423,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1384,14 +1441,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1524,7 +1581,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1542,14 +1599,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1720,7 +1777,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1738,14 +1795,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1874,7 +1931,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1892,14 +1949,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -2032,7 +2089,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -2050,14 +2107,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2242,7 +2299,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2545,7 +2602,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
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             第 12 章 &nbsp; 分治
           </label>
@@ -2722,7 +2779,7 @@
         
         
         
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-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
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             第 13 章 &nbsp; 回溯
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-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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+          <label class="md-nav__title" for="__nav_16">
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             第 14 章 &nbsp; 动态规划
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-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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             参考文献
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19 2-5 4.5v11l5-4.5V2M6.5 5C4.55 5 2.45 5.4 1 6.5v14.66c0 .25.25.5.5.5.1 0 .15-.07.25-.07 1.35-.65 3.3-1.09 4.75-1.09 1.95 0 4.05.4 5.5 1.5 1.35-.85 3.8-1.5 5.5-1.5 1.65 0 3.35.31 4.75 1.06.1.05.15.03.25.03.25 0 .5-.25.5-.5V6.5c-.6-.45-1.25-.75-2-1V19c-1.1-.35-2.3-.5-3.5-.5-1.7 0-4.15.65-5.5 1.5V6.5C10.55 5.4 8.45 5 6.5 5Z"/></svg>
   
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     纸质书
@@ -3504,8 +3561,8 @@
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -3811,7 +3868,7 @@ aria-label="页脚"
       <div class="md-copyright">
   
     <div class="md-copyright__highlight">
-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
diff --git a/chapter_divide_and_conquer/binary_search_recur/index.html b/chapter_divide_and_conquer/binary_search_recur/index.html
index 06ee2350a..d84abd2f2 100644
--- a/chapter_divide_and_conquer/binary_search_recur/index.html
+++ b/chapter_divide_and_conquer/binary_search_recur/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
         
       
         
@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2200,14 +2257,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2487,7 +2544,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" checked>
         
           
           
@@ -2505,14 +2562,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2713,7 +2770,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
@@ -2731,14 +2788,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2894,7 +2951,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
@@ -2912,14 +2969,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3113,7 +3170,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
@@ -3131,14 +3188,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3286,7 +3343,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
@@ -3304,14 +3361,14 @@
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@@ -3468,7 +3525,7 @@
         
         
         
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_divide_and_conquer/build_binary_tree_problem/index.html b/chapter_divide_and_conquer/build_binary_tree_problem/index.html
index e3c3e4d12..a1f6bfa87 100644
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+++ b/chapter_divide_and_conquer/build_binary_tree_problem/index.html
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+  
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+            序
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+          </ul>
+        </nav>
+      
+    </li>
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-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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             第 3 章 &nbsp; 数据结构
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             第 4 章 &nbsp; 数组与链表
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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             第 5 章 &nbsp; 栈与队列
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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             第 6 章 &nbsp; 哈希表
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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             </a>
             
               
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             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
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-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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             第 10 章 &nbsp; 搜索
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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+          <label class="md-nav__title" for="__nav_13">
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             第 11 章 &nbsp; 排序
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@@ -2487,7 +2544,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" checked>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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             第 13 章 &nbsp; 回溯
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-          <label class="md-nav__title" for="__nav_15">
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+          <label class="md-nav__title" for="__nav_16">
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-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
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@@ -3313,7 +3370,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
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@@ -3495,7 +3552,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
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+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_divide_and_conquer/divide_and_conquer/index.html b/chapter_divide_and_conquer/divide_and_conquer/index.html
index 72da1bb97..e351b1d4c 100644
--- a/chapter_divide_and_conquer/divide_and_conquer/index.html
+++ b/chapter_divide_and_conquer/divide_and_conquer/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
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+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
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+  </span>
+  
+
+            </a>
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+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
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+  
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
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-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
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-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2200,14 +2257,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2487,7 +2544,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" checked>
         
           
           
@@ -2505,14 +2562,14 @@
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-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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             纸质书
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 <!-- Page content -->
 <h1 id="121">12.1 &nbsp; 分治算法<a class="headerlink" href="#121" title="Permanent link">&para;</a></h1>
-<p>「分治 divide and conquer」，全称分而治之，是一种非常重要且常见的算法策略。分治通常基于递归实现，包括“分”和“治”两个步骤。</p>
+<p><u>分治（divide and conquer）</u>，全称分而治之，是一种非常重要且常见的算法策略。分治通常基于递归实现，包括“分”和“治”两个步骤。</p>
 <ol>
 <li><strong>分（划分阶段）</strong>：递归地将原问题分解为两个或多个子问题，直至到达最小子问题时终止。</li>
 <li><strong>治（合并阶段）</strong>：从已知解的最小子问题开始，从底至顶地将子问题的解进行合并，从而构建出原问题的解。</li>
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+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_divide_and_conquer/hanota_problem/index.html b/chapter_divide_and_conquer/hanota_problem/index.html
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             第 0 章 &nbsp; 前言
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             第 3 章 &nbsp; 数据结构
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             第 5 章 &nbsp; 栈与队列
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             第 6 章 &nbsp; 哈希表
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             第 7 章 &nbsp; 树
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             第 10 章 &nbsp; 搜索
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@@ -2182,7 +2239,7 @@
         
         
         
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
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             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
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             第 13 章 &nbsp; 回溯
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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+          <label class="md-nav__title" for="__nav_16">
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             第 14 章 &nbsp; 动态规划
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@@ -3131,7 +3188,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
           </label>
@@ -3304,7 +3361,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
           </label>
@@ -3486,7 +3543,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_divide_and_conquer/index.html b/chapter_divide_and_conquer/index.html
index e9c427aaf..534624eb2 100644
--- a/chapter_divide_and_conquer/index.html
+++ b/chapter_divide_and_conquer/index.html
@@ -334,6 +334,63 @@
           
         
       
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+      
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+        
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_hello_algo/" class="md-nav__link ">
+              
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+
+            </a>
+            
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+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
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+  
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
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-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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             第 3 章 &nbsp; 数据结构
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
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-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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             第 4 章 &nbsp; 数组与链表
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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             参考文献
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@@ -3419,7 +3476,7 @@
         
         
         
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_divide_and_conquer/summary/index.html b/chapter_divide_and_conquer/summary/index.html
index 9c223a45b..78e91b6b2 100644
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+++ b/chapter_divide_and_conquer/summary/index.html
@@ -334,6 +334,63 @@
           
         
       
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-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
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+          <label class="md-nav__title" for="__nav_2">
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             第 5 章 &nbsp; 栈与队列
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             第 6 章 &nbsp; 哈希表
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           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
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             第 10 章 &nbsp; 搜索
           </label>
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             第 11 章 &nbsp; 排序
           </label>
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
           </label>
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             参考文献
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@@ -3429,7 +3486,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_dynamic_programming/dp_problem_features/index.html b/chapter_dynamic_programming/dp_problem_features/index.html
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--- a/chapter_dynamic_programming/dp_problem_features/index.html
+++ b/chapter_dynamic_programming/dp_problem_features/index.html
@@ -334,6 +334,63 @@
           
         
       
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+          
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+  
+
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+        
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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             第 1 章 &nbsp; 初识算法
           </label>
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-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
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-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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             第 3 章 &nbsp; 数据结构
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-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
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             第 6 章 &nbsp; 哈希表
           </label>
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           </label>
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           </label>
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             第 11 章 &nbsp; 排序
           </label>
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_dynamic_programming/dp_solution_pipeline/index.html b/chapter_dynamic_programming/dp_solution_pipeline/index.html
index bc51b5491..a6de4f69f 100644
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+++ b/chapter_dynamic_programming/dp_solution_pipeline/index.html
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-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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             第 4 章 &nbsp; 数组与链表
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           </label>
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             第 11 章 &nbsp; 排序
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             第 16 章 &nbsp; 附录
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+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 2 章 &nbsp; 复杂度分析
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             第 3 章 &nbsp; 数据结构
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             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
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             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2485,7 +2542,7 @@
         
         
         
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           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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             第 0 章 &nbsp; 前言
           </label>
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+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
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+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
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             </a>
             
               
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
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             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
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-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
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-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
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-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
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-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2485,7 +2542,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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             第 12 章 &nbsp; 分治
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3064,7 +3121,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
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+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
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             参考文献
           </label>
@@ -3419,7 +3476,7 @@
         
         
         
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+++ b/chapter_dynamic_programming/intro_to_dynamic_programming/index.html
@@ -334,6 +334,63 @@
           
         
       
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+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
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+  
+
+            </a>
+            
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+        
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
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+          </ul>
+        </nav>
+      
+    </li>
+  
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+    
+      
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+  
+  
+  
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
           </label>
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+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
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-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
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             第 4 章 &nbsp; 数组与链表
           </label>
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+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
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-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
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+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
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+          <label class="md-nav__title" for="__nav_10">
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             第 8 章 &nbsp; 堆
           </label>
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-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
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             第 9 章 &nbsp; 图
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-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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+          <label class="md-nav__title" for="__nav_13">
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             第 11 章 &nbsp; 排序
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             第 12 章 &nbsp; 分治
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+          <label class="md-nav__title" for="__nav_15">
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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             参考文献
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     纸质书
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
           </label>
@@ -3637,7 +3694,7 @@
 
 <!-- Page content -->
 <h1 id="141">14.1 &nbsp; 初探动态规划<a class="headerlink" href="#141" title="Permanent link">&para;</a></h1>
-<p>「动态规划 dynamic programming」是一个重要的算法范式，它将一个问题分解为一系列更小的子问题，并通过存储子问题的解来避免重复计算，从而大幅提升时间效率。</p>
+<p><u>动态规划（dynamic programming）</u>是一个重要的算法范式，它将一个问题分解为一系列更小的子问题，并通过存储子问题的解来避免重复计算，从而大幅提升时间效率。</p>
 <p>在本节中，我们从一个经典例题入手，先给出它的暴力回溯解法，观察其中包含的重叠子问题，再逐步导出更高效的动态规划解法。</p>
 <div class="admonition question">
 <p class="admonition-title">爬楼梯</p>
@@ -4902,9 +4959,9 @@ dp[i] = dp[i-1] + dp[i-2]
 <p>与回溯算法一样，动态规划也使用“状态”概念来表示问题求解的特定阶段，每个状态都对应一个子问题以及相应的局部最优解。例如，爬楼梯问题的状态定义为当前所在楼梯阶数 <span class="arithmatex">\(i\)</span> 。</p>
 <p>根据以上内容，我们可以总结出动态规划的常用术语。</p>
 <ul>
-<li>将数组 <code>dp</code> 称为「<span class="arithmatex">\(dp\)</span> 表」，<span class="arithmatex">\(dp[i]\)</span> 表示状态 <span class="arithmatex">\(i\)</span> 对应子问题的解。</li>
-<li>将最小子问题对应的状态（第 <span class="arithmatex">\(1\)</span> 阶和第 <span class="arithmatex">\(2\)</span> 阶楼梯）称为「初始状态」。</li>
-<li>将递推公式 <span class="arithmatex">\(dp[i] = dp[i-1] + dp[i-2]\)</span> 称为「状态转移方程」。</li>
+<li>将数组 <code>dp</code> 称为<u><span class="arithmatex">\(dp\)</span>（表）</u>，<span class="arithmatex">\(dp[i]\)</span> 表示状态 <span class="arithmatex">\(i\)</span> 对应子问题的解。</li>
+<li>将最小子问题对应的状态（第 <span class="arithmatex">\(1\)</span> 阶和第 <span class="arithmatex">\(2\)</span> 阶楼梯）称为<u>初始状态</u>。</li>
+<li>将递推公式 <span class="arithmatex">\(dp[i] = dp[i-1] + dp[i-2]\)</span> 称为<u>状态转移方程</u>。</li>
 </ul>
 <h2 id="1414">14.1.4 &nbsp; 空间优化<a class="headerlink" href="#1414" title="Permanent link">&para;</a></h2>
 <p>细心的读者可能发现了，<strong>由于 <span class="arithmatex">\(dp[i]\)</span> 只与 <span class="arithmatex">\(dp[i-1]\)</span> 和 <span class="arithmatex">\(dp[i-2]\)</span> 有关，因此我们无须使用一个数组 <code>dp</code> 来存储所有子问题的解</strong>，而只需两个变量滚动前进即可。代码如下所示：</p>
@@ -5303,7 +5360,7 @@ aria-label="页脚"
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
diff --git a/chapter_dynamic_programming/knapsack_problem/index.html b/chapter_dynamic_programming/knapsack_problem/index.html
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+              
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
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+            
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+          </ul>
+        </nav>
+      
+    </li>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
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             第 1 章 &nbsp; 初识算法
           </label>
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+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
           </label>
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             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
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-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
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-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
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-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
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             第 7 章 &nbsp; 树
           </label>
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             第 8 章 &nbsp; 堆
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             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
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-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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+          <label class="md-nav__title" for="__nav_12">
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             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
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             第 11 章 &nbsp; 排序
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             第 12 章 &nbsp; 分治
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             第 13 章 &nbsp; 回溯
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+          <label class="md-nav__title" for="__nav_16">
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@@ -3140,7 +3197,7 @@
         
         
         
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-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
           </label>
@@ -3313,7 +3370,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3438,7 +3495,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3495,7 +3552,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
           </label>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_dynamic_programming/summary/index.html b/chapter_dynamic_programming/summary/index.html
index b61746491..05a92716b 100644
--- a/chapter_dynamic_programming/summary/index.html
+++ b/chapter_dynamic_programming/summary/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
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+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
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+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
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+        
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
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-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2200,14 +2257,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2485,7 +2542,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2503,14 +2560,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
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             第 12 章 &nbsp; 分治
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             参考文献
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+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_dynamic_programming/unbounded_knapsack_problem/index.html b/chapter_dynamic_programming/unbounded_knapsack_problem/index.html
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-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 3 章 &nbsp; 数据结构
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             第 4 章 &nbsp; 数组与链表
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             第 5 章 &nbsp; 栈与队列
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             第 6 章 &nbsp; 哈希表
           </label>
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             第 7 章 &nbsp; 树
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             第 8 章 &nbsp; 堆
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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             第 9 章 &nbsp; 图
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             第 10 章 &nbsp; 搜索
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
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@@ -2662,7 +2719,7 @@
         
         
         
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_16">
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@@ -3230,7 +3287,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
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@@ -3403,7 +3460,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3528,7 +3585,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
           </label>
@@ -3585,7 +3642,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_graph/graph/index.html b/chapter_graph/graph/index.html
index 10f2f86e1..43d44216c 100644
--- a/chapter_graph/graph/index.html
+++ b/chapter_graph/graph/index.html
@@ -334,6 +334,63 @@
           
         
       
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+      
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+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
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+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
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+          
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
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-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
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-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
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             </a>
             
               
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+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1816,7 +1873,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" checked>
         
           
           
@@ -1834,14 +1891,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19 2-5 4.5v11l5-4.5V2M6.5 5C4.55 5 2.45 5.4 1 6.5v14.66c0 .25.25.5.5.5.1 0 .15-.07.25-.07 1.35-.65 3.3-1.09 4.75-1.09 1.95 0 4.05.4 5.5 1.5 1.35-.85 3.8-1.5 5.5-1.5 1.65 0 3.35.31 4.75 1.06.1.05.15.03.25.03.25 0 .5-.25.5-.5V6.5c-.6-.45-1.25-.75-2-1V19c-1.1-.35-2.3-.5-3.5-.5-1.7 0-4.15.65-5.5 1.5V6.5C10.55 5.4 8.45 5 6.5 5Z"/></svg>
   
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 <!-- Page content -->
 <h1 id="91">9.1 &nbsp; 图<a class="headerlink" href="#91" title="Permanent link">&para;</a></h1>
-<p>「图 graph」是一种非线性数据结构，由「顶点 vertex」和「边 edge」组成。我们可以将图 <span class="arithmatex">\(G\)</span> 抽象地表示为一组顶点 <span class="arithmatex">\(V\)</span> 和一组边 <span class="arithmatex">\(E\)</span> 的集合。以下示例展示了一个包含 5 个顶点和 7 条边的图。</p>
+<p><u>图（graph）</u>是一种非线性数据结构，由<u>顶点（vertex）</u>和<u>边（edge）</u>组成。我们可以将图 <span class="arithmatex">\(G\)</span> 抽象地表示为一组顶点 <span class="arithmatex">\(V\)</span> 和一组边 <span class="arithmatex">\(E\)</span> 的集合。以下示例展示了一个包含 5 个顶点和 7 条边的图。</p>
 <div class="arithmatex">\[
 \begin{aligned}
 V &amp; = \{ 1, 2, 3, 4, 5 \} \newline
@@ -3680,7 +3737,7 @@ G &amp; = \{ V, E \} \newline
 <p align="center"> 图 9-1 &nbsp; 链表、树、图之间的关系 </p>
 
 <h2 id="911">9.1.1 &nbsp; 图的常见类型与术语<a class="headerlink" href="#911" title="Permanent link">&para;</a></h2>
-<p>根据边是否具有方向，可分为「无向图 undirected graph」和「有向图 directed graph」，如图 9-2 所示。</p>
+<p>根据边是否具有方向，可分为<u>无向图（undirected graph）</u>和<u>有向图（directed graph）</u>，如图 9-2 所示。</p>
 <ul>
 <li>在无向图中，边表示两顶点之间的“双向”连接关系，例如微信或 QQ 中的“好友关系”。</li>
 <li>在有向图中，边具有方向性，即 <span class="arithmatex">\(A \rightarrow B\)</span> 和 <span class="arithmatex">\(A \leftarrow B\)</span> 两个方向的边是相互独立的，例如微博或抖音上的“关注”与“被关注”关系。</li>
@@ -3688,7 +3745,7 @@ G &amp; = \{ V, E \} \newline
 <p><a class="glightbox" href="../graph.assets/directed_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="有向图与无向图" class="animation-figure" src="../graph.assets/directed_graph.png" /></a></p>
 <p align="center"> 图 9-2 &nbsp; 有向图与无向图 </p>
 
-<p>根据所有顶点是否连通，可分为「连通图 connected graph」和「非连通图 disconnected graph」，如图 9-3 所示。</p>
+<p>根据所有顶点是否连通，可分为<u>连通图（connected graph）</u>和<u>非连通图（disconnected graph）</u>，如图 9-3 所示。</p>
 <ul>
 <li>对于连通图，从某个顶点出发，可以到达其余任意顶点。</li>
 <li>对于非连通图，从某个顶点出发，至少有一个顶点无法到达。</li>
@@ -3696,20 +3753,20 @@ G &amp; = \{ V, E \} \newline
 <p><a class="glightbox" href="../graph.assets/connected_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="连通图与非连通图" class="animation-figure" src="../graph.assets/connected_graph.png" /></a></p>
 <p align="center"> 图 9-3 &nbsp; 连通图与非连通图 </p>
 
-<p>我们还可以为边添加“权重”变量，从而得到如图 9-4 所示的「有权图 weighted graph」。例如在《王者荣耀》等手游中，系统会根据共同游戏时间来计算玩家之间的“亲密度”，这种亲密度网络就可以用有权图来表示。</p>
+<p>我们还可以为边添加“权重”变量，从而得到如图 9-4 所示的<u>有权图（weighted graph）</u>。例如在《王者荣耀》等手游中，系统会根据共同游戏时间来计算玩家之间的“亲密度”，这种亲密度网络就可以用有权图来表示。</p>
 <p><a class="glightbox" href="../graph.assets/weighted_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="有权图与无权图" class="animation-figure" src="../graph.assets/weighted_graph.png" /></a></p>
 <p align="center"> 图 9-4 &nbsp; 有权图与无权图 </p>
 
 <p>图数据结构包含以下常用术语。</p>
 <ul>
-<li>「邻接 adjacency」：当两顶点之间存在边相连时，称这两顶点“邻接”。在图 9-4 中，顶点 1 的邻接顶点为顶点 2、3、5。</li>
-<li>「路径 path」：从顶点 A 到顶点 B 经过的边构成的序列被称为从 A 到 B 的“路径”。在图 9-4 中，边序列 1-5-2-4 是顶点 1 到顶点 4 的一条路径。</li>
-<li>「度 degree」：一个顶点拥有的边数。对于有向图，「入度 in-degree」表示有多少条边指向该顶点，「出度 out-degree」表示有多少条边从该顶点指出。</li>
+<li><u>邻接（adjacency）</u>：当两顶点之间存在边相连时，称这两顶点“邻接”。在图 9-4 中，顶点 1 的邻接顶点为顶点 2、3、5。</li>
+<li><u>路径（path）</u>：从顶点 A 到顶点 B 经过的边构成的序列被称为从 A 到 B 的“路径”。在图 9-4 中，边序列 1-5-2-4 是顶点 1 到顶点 4 的一条路径。</li>
+<li><u>度（degree）</u>：一个顶点拥有的边数。对于有向图，<u>入度（in-degree）</u>表示有多少条边指向该顶点，<u>出度（out-degree）</u>表示有多少条边从该顶点指出。</li>
 </ul>
 <h2 id="912">9.1.2 &nbsp; 图的表示<a class="headerlink" href="#912" title="Permanent link">&para;</a></h2>
 <p>图的常用表示方式包括“邻接矩阵”和“邻接表”。以下使用无向图进行举例。</p>
 <h3 id="1">1. &nbsp; 邻接矩阵<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
-<p>设图的顶点数量为 <span class="arithmatex">\(n\)</span> ，「邻接矩阵 adjacency matrix」使用一个 <span class="arithmatex">\(n \times n\)</span> 大小的矩阵来表示图，每一行（列）代表一个顶点，矩阵元素代表边，用 <span class="arithmatex">\(1\)</span> 或 <span class="arithmatex">\(0\)</span> 表示两个顶点之间是否存在边。</p>
+<p>设图的顶点数量为 <span class="arithmatex">\(n\)</span> ，<u>邻接矩阵（adjacency matrix）</u>使用一个 <span class="arithmatex">\(n \times n\)</span> 大小的矩阵来表示图，每一行（列）代表一个顶点，矩阵元素代表边，用 <span class="arithmatex">\(1\)</span> 或 <span class="arithmatex">\(0\)</span> 表示两个顶点之间是否存在边。</p>
 <p>如图 9-5 所示，设邻接矩阵为 <span class="arithmatex">\(M\)</span>、顶点列表为 <span class="arithmatex">\(V\)</span> ，那么矩阵元素 <span class="arithmatex">\(M[i, j] = 1\)</span> 表示顶点 <span class="arithmatex">\(V[i]\)</span> 到顶点 <span class="arithmatex">\(V[j]\)</span> 之间存在边，反之 <span class="arithmatex">\(M[i, j] = 0\)</span> 表示两顶点之间无边。</p>
 <p><a class="glightbox" href="../graph.assets/adjacency_matrix.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="图的邻接矩阵表示" class="animation-figure" src="../graph.assets/adjacency_matrix.png" /></a></p>
 <p align="center"> 图 9-5 &nbsp; 图的邻接矩阵表示 </p>
@@ -3722,7 +3779,7 @@ G &amp; = \{ V, E \} \newline
 </ul>
 <p>使用邻接矩阵表示图时，我们可以直接访问矩阵元素以获取边，因此增删查改操作的效率很高，时间复杂度均为 <span class="arithmatex">\(O(1)\)</span> 。然而，矩阵的空间复杂度为 <span class="arithmatex">\(O(n^2)\)</span> ，内存占用较多。</p>
 <h3 id="2">2. &nbsp; 邻接表<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
-<p>「邻接表 adjacency list」使用 <span class="arithmatex">\(n\)</span> 个链表来表示图，链表节点表示顶点。第 <span class="arithmatex">\(i\)</span> 个链表对应顶点 <span class="arithmatex">\(i\)</span> ，其中存储了该顶点的所有邻接顶点（与该顶点相连的顶点）。图 9-6 展示了一个使用邻接表存储的图的示例。</p>
+<p><u>邻接表（adjacency list）</u>使用 <span class="arithmatex">\(n\)</span> 个链表来表示图，链表节点表示顶点。第 <span class="arithmatex">\(i\)</span> 个链表对应顶点 <span class="arithmatex">\(i\)</span> ，其中存储了该顶点的所有邻接顶点（与该顶点相连的顶点）。图 9-6 展示了一个使用邻接表存储的图的示例。</p>
 <p><a class="glightbox" href="../graph.assets/adjacency_list.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="图的邻接表表示" class="animation-figure" src="../graph.assets/adjacency_list.png" /></a></p>
 <p align="center"> 图 9-6 &nbsp; 图的邻接表表示 </p>
 
@@ -3951,7 +4008,7 @@ aria-label="页脚"
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
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+  
+
+            </a>
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+        
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
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+          </ul>
+        </nav>
+      
+    </li>
+  
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+      
+      
+  
+  
+  
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
           </label>
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+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
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-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
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             第 8 章 &nbsp; 堆
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_11">
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             第 9 章 &nbsp; 图
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
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+          <label class="md-nav__title" for="__nav_13">
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             第 11 章 &nbsp; 排序
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             第 12 章 &nbsp; 分治
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+          <label class="md-nav__title" for="__nav_15">
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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             参考文献
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     纸质书
@@ -3513,8 +3570,8 @@
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -6056,7 +6113,7 @@ aria-label="页脚"
       <div class="md-copyright">
   
     <div class="md-copyright__highlight">
-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
diff --git a/chapter_graph/graph_traversal/index.html b/chapter_graph/graph_traversal/index.html
index 1146d8df4..e00c7fb94 100644
--- a/chapter_graph/graph_traversal/index.html
+++ b/chapter_graph/graph_traversal/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
         
       
         
@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1816,7 +1873,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" checked>
         
           
           
@@ -1834,14 +1891,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -2080,7 +2137,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -2098,14 +2155,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2290,7 +2347,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2308,14 +2365,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2593,7 +2650,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2611,14 +2668,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2770,7 +2827,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
@@ -2788,14 +2845,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2951,7 +3008,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
@@ -2969,14 +3026,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3170,7 +3227,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
@@ -3188,14 +3245,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3343,7 +3400,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3468,7 +3525,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3525,7 +3582,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
@@ -3533,7 +3590,7 @@
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     纸质书
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -3698,7 +3755,7 @@
 <!-- Page content -->
 <h1 id="93">9.3 &nbsp; 图的遍历<a class="headerlink" href="#93" title="Permanent link">&para;</a></h1>
 <p>树代表的是“一对多”的关系，而图则具有更高的自由度，可以表示任意的“多对多”关系。因此，我们可以把树看作图的一种特例。显然，<strong>树的遍历操作也是图的遍历操作的一种特例</strong>。</p>
-<p>图和树都需要应用搜索算法来实现遍历操作。图的遍历方式也可分为两种：「广度优先遍历」和「深度优先遍历」。</p>
+<p>图和树都需要应用搜索算法来实现遍历操作。图的遍历方式也可分为两种：<u>广度优先遍历</u>和<u>深度优先遍历</u>。</p>
 <h2 id="931">9.3.1 &nbsp; 广度优先遍历<a class="headerlink" href="#931" title="Permanent link">&para;</a></h2>
 <p><strong>广度优先遍历是一种由近及远的遍历方式，从某个节点出发，始终优先访问距离最近的顶点，并一层层向外扩张</strong>。如图 9-9 所示，从左上角顶点出发，首先遍历该顶点的所有邻接顶点，然后遍历下一个顶点的所有邻接顶点，以此类推，直至所有顶点访问完毕。</p>
 <p><a class="glightbox" href="../graph_traversal.assets/graph_bfs.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="图的广度优先遍历" class="animation-figure" src="../graph_traversal.assets/graph_bfs.png" /></a></p>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
diff --git a/chapter_graph/index.html b/chapter_graph/index.html
index 508e3e348..c371cde53 100644
--- a/chapter_graph/index.html
+++ b/chapter_graph/index.html
@@ -334,6 +334,63 @@
           
         
       
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+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
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+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
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+  
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
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             第 8 章 &nbsp; 堆
           </label>
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" checked>
         
           
           
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-              <label class="md-nav__link md-nav__link--active" for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link md-nav__link--active" for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="true">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1974,7 +2031,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2184,7 +2241,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2487,7 +2544,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
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             第 12 章 &nbsp; 分治
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
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             第 13 章 &nbsp; 回溯
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-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3064,7 +3121,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
@@ -3082,14 +3139,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3237,7 +3294,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
@@ -3255,14 +3312,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3362,7 +3419,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3419,7 +3476,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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     纸质书
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -3734,7 +3791,7 @@ aria-label="页脚"
       <div class="md-copyright">
   
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
diff --git a/chapter_graph/summary/index.html b/chapter_graph/summary/index.html
index f6c764c28..1246ff56b 100644
--- a/chapter_graph/summary/index.html
+++ b/chapter_graph/summary/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1816,7 +1873,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" checked>
         
           
           
@@ -1834,14 +1891,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -2032,7 +2089,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -2050,14 +2107,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2242,7 +2299,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2260,14 +2317,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2545,7 +2602,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2563,14 +2620,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
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@@ -2722,7 +2779,7 @@
         
         
         
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+          <label class="md-nav__title" for="__nav_15">
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
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             第 14 章 &nbsp; 动态规划
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@@ -3122,7 +3179,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
           </label>
@@ -3477,7 +3534,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_greedy/fractional_knapsack_problem/index.html b/chapter_greedy/fractional_knapsack_problem/index.html
index b0b4a68c0..e5c9697bb 100644
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+++ b/chapter_greedy/fractional_knapsack_problem/index.html
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+            </a>
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
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+    
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
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-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
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-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
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-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
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-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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+          <label class="md-nav__title" for="__nav_10">
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             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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+          <label class="md-nav__title" for="__nav_11">
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             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2200,14 +2257,14 @@
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_greedy/greedy_algorithm/index.html b/chapter_greedy/greedy_algorithm/index.html
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+      
+    </li>
+  
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-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 3 章 &nbsp; 数据结构
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             第 4 章 &nbsp; 数组与链表
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             第 6 章 &nbsp; 哈希表
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             第 7 章 &nbsp; 树
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             第 8 章 &nbsp; 堆
           </label>
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             纸质书
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 <!-- Page content -->
 <h1 id="151">15.1 &nbsp; 贪心算法<a class="headerlink" href="#151" title="Permanent link">&para;</a></h1>
-<p>「贪心算法 greedy algorithm」是一种常见的解决优化问题的算法，其基本思想是在问题的每个决策阶段，都选择当前看起来最优的选择，即贪心地做出局部最优的决策，以期获得全局最优解。贪心算法简洁且高效，在许多实际问题中有着广泛的应用。</p>
+<p><u>贪心算法（greedy algorithm）</u>是一种常见的解决优化问题的算法，其基本思想是在问题的每个决策阶段，都选择当前看起来最优的选择，即贪心地做出局部最优的决策，以期获得全局最优解。贪心算法简洁且高效，在许多实际问题中有着广泛的应用。</p>
 <p>贪心算法和动态规划都常用于解决优化问题。它们之间存在一些相似之处，比如都依赖最优子结构性质，但工作原理不同。</p>
 <ul>
 <li>动态规划会根据之前阶段的所有决策来考虑当前决策，并使用过去子问题的解来构建当前子问题的解。</li>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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             第 5 章 &nbsp; 栈与队列
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
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-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
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-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2485,7 +2542,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2662,7 +2719,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2843,7 +2900,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3064,7 +3121,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" checked>
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3237,7 +3294,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3362,7 +3419,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
           </label>
@@ -3419,7 +3476,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_greedy/max_capacity_problem/index.html b/chapter_greedy/max_capacity_problem/index.html
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--- a/chapter_greedy/max_capacity_problem/index.html
+++ b/chapter_greedy/max_capacity_problem/index.html
@@ -334,6 +334,63 @@
           
         
       
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+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
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+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
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+    
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+  
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
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-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_9">
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           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
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             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2485,7 +2542,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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             第 12 章 &nbsp; 分治
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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             第 13 章 &nbsp; 回溯
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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             第 14 章 &nbsp; 动态规划
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" checked>
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+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3429,7 +3486,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
           </label>
@@ -3486,7 +3543,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_greedy/max_product_cutting_problem/index.html b/chapter_greedy/max_product_cutting_problem/index.html
index 3bffa762b..0a83d9e10 100644
--- a/chapter_greedy/max_product_cutting_problem/index.html
+++ b/chapter_greedy/max_product_cutting_problem/index.html
@@ -334,6 +334,63 @@
           
         
       
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+          
+          <div class="md-nav__link md-nav__container">
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+
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+        
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
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+    
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
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             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2485,7 +2542,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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             第 13 章 &nbsp; 回溯
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
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             参考文献
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@@ -3486,7 +3543,7 @@
         
         
         
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+          <label class="md-nav__title" for="__nav_1">
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+            序
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+        </nav>
+      
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
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-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
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@@ -480,7 +537,7 @@
         
         
         
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2200,14 +2257,14 @@
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2485,7 +2542,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2503,14 +2560,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2662,7 +2719,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
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-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2843,7 +2900,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
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-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3064,7 +3121,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" checked>
         
           
           
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-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3247,7 +3304,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3372,7 +3429,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3429,7 +3486,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
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+        </nav>
+      
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@@ -349,7 +406,7 @@
         
         
         
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
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@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1308,7 +1365,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" checked>
         
           
           
@@ -1326,14 +1383,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1542,7 +1599,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1560,14 +1617,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1738,7 +1795,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1756,14 +1813,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1892,7 +1949,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -2050,7 +2107,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -2068,14 +2125,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2260,7 +2317,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2278,14 +2335,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
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           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2563,7 +2620,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2581,14 +2638,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2740,7 +2797,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2921,7 +2978,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
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-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3140,7 +3197,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
@@ -3158,14 +3215,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3313,7 +3370,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3438,7 +3495,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3495,7 +3552,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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     纸质书
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-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+++ b/chapter_hashing/hash_collision/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
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+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
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+  
+
+            </a>
+            
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+        
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
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+          </ul>
+        </nav>
+      
+    </li>
+  
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+    
+      
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+  
+  
+  
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
           </label>
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+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
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             第 4 章 &nbsp; 数组与链表
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="true">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
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+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
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+          <label class="md-nav__title" for="__nav_10">
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             第 8 章 &nbsp; 堆
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
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             第 9 章 &nbsp; 图
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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+          <label class="md-nav__title" for="__nav_13">
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             第 11 章 &nbsp; 排序
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             第 12 章 &nbsp; 分治
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+          <label class="md-nav__title" for="__nav_15">
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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             参考文献
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     纸质书
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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@@ -3693,7 +3750,7 @@
 </ol>
 <p>哈希表的结构改良方法主要包括“链式地址”和“开放寻址”。</p>
 <h2 id="621">6.2.1 &nbsp; 链式地址<a class="headerlink" href="#621" title="Permanent link">&para;</a></h2>
-<p>在原始哈希表中，每个桶仅能存储一个键值对。「链式地址 separate chaining」将单个元素转换为链表，将键值对作为链表节点，将所有发生冲突的键值对都存储在同一链表中。图 6-5 展示了一个链式地址哈希表的例子。</p>
+<p>在原始哈希表中，每个桶仅能存储一个键值对。<u>链式地址（separate chaining）</u>将单个元素转换为链表，将键值对作为链表节点，将所有发生冲突的键值对都存储在同一链表中。图 6-5 展示了一个链式地址哈希表的例子。</p>
 <p><a class="glightbox" href="../hash_collision.assets/hash_table_chaining.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="链式地址哈希表" class="animation-figure" src="../hash_collision.assets/hash_table_chaining.png" /></a></p>
 <p align="center"> 图 6-5 &nbsp; 链式地址哈希表 </p>
 
@@ -5097,7 +5154,7 @@
 </details>
 <p>值得注意的是，当链表很长时，查询效率 <span class="arithmatex">\(O(n)\)</span> 很差。<strong>此时可以将链表转换为“AVL 树”或“红黑树”</strong>，从而将查询操作的时间复杂度优化至 <span class="arithmatex">\(O(\log n)\)</span> 。</p>
 <h2 id="622">6.2.2 &nbsp; 开放寻址<a class="headerlink" href="#622" title="Permanent link">&para;</a></h2>
-<p>「开放寻址 open addressing」不引入额外的数据结构，而是通过“多次探测”来处理哈希冲突，探测方式主要包括线性探测、平方探测和多次哈希等。</p>
+<p><u>开放寻址（open addressing）</u>不引入额外的数据结构，而是通过“多次探测”来处理哈希冲突，探测方式主要包括线性探测、平方探测和多次哈希等。</p>
 <p>下面以线性探测为例，介绍开放寻址哈希表的工作机制。</p>
 <h3 id="1">1. &nbsp; 线性探测<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
 <p>线性探测采用固定步长的线性搜索来进行探测，其操作方法与普通哈希表有所不同。</p>
@@ -5114,7 +5171,7 @@
 <p><a class="glightbox" href="../hash_collision.assets/hash_table_open_addressing_deletion.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="在开放寻址中删除元素导致的查询问题" class="animation-figure" src="../hash_collision.assets/hash_table_open_addressing_deletion.png" /></a></p>
 <p align="center"> 图 6-7 &nbsp; 在开放寻址中删除元素导致的查询问题 </p>
 
-<p>为了解决该问题，我们可以采用「懒删除 lazy deletion」机制：它不直接从哈希表中移除元素，<strong>而是利用一个常量 <code>TOMBSTONE</code> 来标记这个桶</strong>。在该机制下，<code>None</code> 和 <code>TOMBSTONE</code> 都代表空桶，都可以放置键值对。但不同的是，线性探测到 <code>TOMBSTONE</code> 时应该继续遍历，因为其之下可能还存在键值对。</p>
+<p>为了解决该问题，我们可以采用<u>懒删除（lazy deletion）</u>机制：它不直接从哈希表中移除元素，<strong>而是利用一个常量 <code>TOMBSTONE</code> 来标记这个桶</strong>。在该机制下，<code>None</code> 和 <code>TOMBSTONE</code> 都代表空桶，都可以放置键值对。但不同的是，线性探测到 <code>TOMBSTONE</code> 时应该继续遍历，因为其之下可能还存在键值对。</p>
 <p>然而，<strong>懒删除可能会加速哈希表的性能退化</strong>。这是因为每次删除操作都会产生一个删除标记，随着 <code>TOMBSTONE</code> 的增加，搜索时间也会增加，因为线性探测可能需要跳过多个 <code>TOMBSTONE</code> 才能找到目标元素。</p>
 <p>为此，考虑在线性探测中记录遇到的首个 <code>TOMBSTONE</code> 的索引，并将搜索到的目标元素与该 <code>TOMBSTONE</code> 交换位置。这样做的好处是当每次查询或添加元素时，元素会被移动至距离理想位置（探测起始点）更近的桶，从而优化查询效率。</p>
 <p>以下代码实现了一个包含懒删除的开放寻址（线性探测）哈希表。为了更加充分地使用哈希表的空间，我们将哈希表看作一个“环形数组”，当越过数组尾部时，回到头部继续遍历。</p>
@@ -6921,7 +6978,7 @@ aria-label="页脚"
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
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+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
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             </a>
             
               
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 2 章 &nbsp; 复杂度分析
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             第 3 章 &nbsp; 数据结构
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             第 4 章 &nbsp; 数组与链表
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             第 5 章 &nbsp; 栈与队列
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             第 6 章 &nbsp; 哈希表
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             第 7 章 &nbsp; 树
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 <!-- Page content -->
 <h1 id="61">6.1 &nbsp; 哈希表<a class="headerlink" href="#61" title="Permanent link">&para;</a></h1>
-<p>「哈希表 hash table」，又称「散列表」，它通过建立键 <code>key</code> 与值 <code>value</code> 之间的映射，实现高效的元素查询。具体而言，我们向哈希表中输入一个键 <code>key</code> ，则可以在 <span class="arithmatex">\(O(1)\)</span> 时间内获取对应的值 <code>value</code> 。</p>
+<p><u>哈希表（hash table）</u>，又称<u>散列表</u>，它通过建立键 <code>key</code> 与值 <code>value</code> 之间的映射，实现高效的元素查询。具体而言，我们向哈希表中输入一个键 <code>key</code> ，则可以在 <span class="arithmatex">\(O(1)\)</span> 时间内获取对应的值 <code>value</code> 。</p>
 <p>如图 6-1 所示，给定 <span class="arithmatex">\(n\)</span> 个学生，每个学生都有“姓名”和“学号”两项数据。假如我们希望实现“输入一个学号，返回对应的姓名”的查询功能，则可以采用图 6-1 所示的哈希表来实现。</p>
 <p><a class="glightbox" href="../hash_map.assets/hash_table_lookup.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="哈希表的抽象表示" class="animation-figure" src="../hash_map.assets/hash_table_lookup.png" /></a></p>
 <p align="center"> 图 6-1 &nbsp; 哈希表的抽象表示 </p>
@@ -4120,8 +4177,8 @@
 <div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%93%88%E5%B8%8C%E8%A1%A8%0A%20%20%20%20hmap%20%3D%20%7B%7D%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%B7%BB%E5%8A%A0%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%9C%A8%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E6%B7%BB%E5%8A%A0%E9%94%AE%E5%80%BC%E5%AF%B9%20%28key,%20value%29%0A%20%20%20%20hmap%5B12836%5D%20%3D%20%22%E5%B0%8F%E5%93%88%22%0A%20%20%20%20hmap%5B15937%5D%20%3D%20%22%E5%B0%8F%E5%95%B0%22%0A%20%20%20%20hmap%5B16750%5D%20%3D%20%22%E5%B0%8F%E7%AE%97%22%0A%20%20%20%20hmap%5B13276%5D%20%3D%20%22%E5%B0%8F%E6%B3%95%22%0A%20%20%20%20hmap%5B10583%5D%20%3D%20%22%E5%B0%8F%E9%B8%AD%22%0A%20%20%20%20%0A%20%20%20%20%23%20%E9%81%8D%E5%8E%86%E5%93%88%E5%B8%8C%E8%A1%A8%0A%20%20%20%20%23%20%E9%81%8D%E5%8E%86%E9%94%AE%E5%80%BC%E5%AF%B9%20key-%3Evalue%0A%20%20%20%20for%20key,%20value%20in%20hmap.items%28%29%3A%0A%20%20%20%20%20%20%20%20print%28key,%20%22-%3E%22,%20value%29%0A%20%20%20%20%23%20%E5%8D%95%E7%8B%AC%E9%81%8D%E5%8E%86%E9%94%AE%20key%0A%20%20%20%20for%20key%20in%20hmap.keys%28%29%3A%0A%20%20%20%20%20%20%20%20print%28key%29%0A%20%20%20%20%23%20%E5%8D%95%E7%8B%AC%E9%81%8D%E5%8E%86%E5%80%BC%20value%0A%20%20%20%20for%20value%20in%20hmap.values%28%29%3A%0A%20%20%20%20%20%20%20%20print%28value%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=8&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 &gt;</a></div></p>
 </details>
 <h2 id="612">6.1.2 &nbsp; 哈希表简单实现<a class="headerlink" href="#612" title="Permanent link">&para;</a></h2>
-<p>我们先考虑最简单的情况，<strong>仅用一个数组来实现哈希表</strong>。在哈希表中，我们将数组中的每个空位称为「桶 bucket」，每个桶可存储一个键值对。因此，查询操作就是找到 <code>key</code> 对应的桶，并在桶中获取 <code>value</code> 。</p>
-<p>那么，如何基于 <code>key</code> 定位对应的桶呢？这是通过「哈希函数 hash function」实现的。哈希函数的作用是将一个较大的输入空间映射到一个较小的输出空间。在哈希表中，输入空间是所有 <code>key</code> ，输出空间是所有桶（数组索引）。换句话说，输入一个 <code>key</code> ，<strong>我们可以通过哈希函数得到该 <code>key</code> 对应的键值对在数组中的存储位置</strong>。</p>
+<p>我们先考虑最简单的情况，<strong>仅用一个数组来实现哈希表</strong>。在哈希表中，我们将数组中的每个空位称为<u>桶（bucket）</u>，每个桶可存储一个键值对。因此，查询操作就是找到 <code>key</code> 对应的桶，并在桶中获取 <code>value</code> 。</p>
+<p>那么，如何基于 <code>key</code> 定位对应的桶呢？这是通过<u>哈希函数（hash function）</u>实现的。哈希函数的作用是将一个较大的输入空间映射到一个较小的输出空间。在哈希表中，输入空间是所有 <code>key</code> ，输出空间是所有桶（数组索引）。换句话说，输入一个 <code>key</code> ，<strong>我们可以通过哈希函数得到该 <code>key</code> 对应的键值对在数组中的存储位置</strong>。</p>
 <p>输入一个 <code>key</code> ，哈希函数的计算过程分为以下两步。</p>
 <ol>
 <li>通过某种哈希算法 <code>hash()</code> 计算得到哈希值。</li>
@@ -5415,7 +5472,7 @@
 <div class="highlight"><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="m">12836</span><span class="w"> </span>%<span class="w"> </span><span class="nv">100</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">36</span>
 <a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="m">20336</span><span class="w"> </span>%<span class="w"> </span><span class="nv">100</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">36</span>
 </code></pre></div>
-<p>如图 6-3 所示，两个学号指向了同一个姓名，这显然是不对的。我们将这种多个输入对应同一输出的情况称为「哈希冲突 hash collision」。</p>
+<p>如图 6-3 所示，两个学号指向了同一个姓名，这显然是不对的。我们将这种多个输入对应同一输出的情况称为<u>哈希冲突（hash collision）</u>。</p>
 <p><a class="glightbox" href="../hash_map.assets/hash_collision.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="哈希冲突示例" class="animation-figure" src="../hash_map.assets/hash_collision.png" /></a></p>
 <p align="center"> 图 6-3 &nbsp; 哈希冲突示例 </p>
 
@@ -5425,7 +5482,7 @@
 <p align="center"> 图 6-4 &nbsp; 哈希表扩容 </p>
 
 <p>类似于数组扩容，哈希表扩容需将所有键值对从原哈希表迁移至新哈希表，非常耗时；并且由于哈希表容量 <code>capacity</code> 改变，我们需要通过哈希函数来重新计算所有键值对的存储位置，这进一步增加了扩容过程的计算开销。为此，编程语言通常会预留足够大的哈希表容量，防止频繁扩容。</p>
-<p>「负载因子 load factor」是哈希表的一个重要概念，其定义为哈希表的元素数量除以桶数量，用于衡量哈希冲突的严重程度，<strong>也常作为哈希表扩容的触发条件</strong>。例如在 Java 中，当负载因子超过 <span class="arithmatex">\(0.75\)</span> 时，系统会将哈希表扩容至原先的 <span class="arithmatex">\(2\)</span> 倍。</p>
+<p><u>负载因子（load factor）</u>是哈希表的一个重要概念，其定义为哈希表的元素数量除以桶数量，用于衡量哈希冲突的严重程度，<strong>也常作为哈希表扩容的触发条件</strong>。例如在 Java 中，当负载因子超过 <span class="arithmatex">\(0.75\)</span> 时，系统会将哈希表扩容至原先的 <span class="arithmatex">\(2\)</span> 倍。</p>
 
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             第 0 章 &nbsp; 前言
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@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1308,7 +1365,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" checked>
         
           
           
@@ -1326,14 +1383,14 @@
             </a>
             
               
-              <label class="md-nav__link md-nav__link--active" for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link md-nav__link--active" for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1466,7 +1523,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1484,14 +1541,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1662,7 +1719,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1680,14 +1737,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1816,7 +1873,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1834,14 +1891,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1974,7 +2031,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1992,14 +2049,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2184,7 +2241,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2202,14 +2259,14 @@
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2487,7 +2544,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2664,7 +2721,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2845,7 +2902,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3064,7 +3121,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
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-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
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@@ -3237,7 +3294,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
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             参考文献
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@@ -3419,7 +3476,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
         
       
         
@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1308,7 +1365,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" checked>
         
           
           
@@ -1326,14 +1383,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1524,7 +1581,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1542,14 +1599,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1720,7 +1777,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1738,14 +1795,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1874,7 +1931,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1892,14 +1949,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -2032,7 +2089,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -2050,14 +2107,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2242,7 +2299,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2260,14 +2317,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2545,7 +2602,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2563,14 +2620,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2722,7 +2779,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
@@ -2740,14 +2797,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2903,7 +2960,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
@@ -2921,14 +2978,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3122,7 +3179,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
@@ -3140,14 +3197,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3295,7 +3352,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
@@ -3313,14 +3370,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3420,7 +3477,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
@@ -3439,8 +3496,8 @@
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3477,7 +3534,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
@@ -3485,7 +3542,7 @@
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     纸质书
@@ -3504,8 +3561,8 @@
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -3819,7 +3876,7 @@ aria-label="页脚"
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
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index 6dc812f3f..2a308ab21 100644
--- a/chapter_heap/build_heap/index.html
+++ b/chapter_heap/build_heap/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
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+  
+
+            </a>
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+        
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
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+          </ul>
+        </nav>
+      
+    </li>
+  
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+    
+      
+      
+  
+  
+  
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+    
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+        
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
           </label>
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+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
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             第 4 章 &nbsp; 数组与链表
           </label>
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+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_10">
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             第 8 章 &nbsp; 堆
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
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             第 9 章 &nbsp; 图
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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+          <label class="md-nav__title" for="__nav_13">
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             第 11 章 &nbsp; 排序
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             第 12 章 &nbsp; 分治
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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             参考文献
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     纸质书
@@ -3513,8 +3570,8 @@
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
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index d09fb5f93..878bce753 100644
--- a/chapter_heap/heap/index.html
+++ b/chapter_heap/heap/index.html
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
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+          </ul>
+        </nav>
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+    </li>
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
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+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1662,7 +1719,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" checked>
         
           
           
@@ -1680,14 +1737,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1925,7 +1982,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1943,14 +2000,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -2083,7 +2140,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -2101,14 +2158,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2293,7 +2350,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2311,14 +2368,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2596,7 +2653,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2614,14 +2671,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2773,7 +2830,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
@@ -2791,14 +2848,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2954,7 +3011,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3173,7 +3230,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
           </label>
@@ -3346,7 +3403,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
           </label>
@@ -3528,7 +3585,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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     纸质书
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@@ -3703,10 +3760,10 @@
 
 <!-- Page content -->
 <h1 id="81">8.1 &nbsp; 堆<a class="headerlink" href="#81" title="Permanent link">&para;</a></h1>
-<p>「堆 heap」是一种满足特定条件的完全二叉树，主要可分为两种类型，如图 8-1 所示。</p>
+<p><u>堆（heap）</u>是一种满足特定条件的完全二叉树，主要可分为两种类型，如图 8-1 所示。</p>
 <ul>
-<li>「小顶堆 min heap」：任意节点的值 <span class="arithmatex">\(\leq\)</span> 其子节点的值。</li>
-<li>「大顶堆 max heap」：任意节点的值 <span class="arithmatex">\(\geq\)</span> 其子节点的值。</li>
+<li><u>小顶堆（min heap）</u>：任意节点的值 <span class="arithmatex">\(\leq\)</span> 其子节点的值。</li>
+<li><u>大顶堆（max heap）</u>：任意节点的值 <span class="arithmatex">\(\geq\)</span> 其子节点的值。</li>
 </ul>
 <p><a class="glightbox" href="../heap.assets/min_heap_and_max_heap.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="小顶堆与大顶堆" class="animation-figure" src="../heap.assets/min_heap_and_max_heap.png" /></a></p>
 <p align="center"> 图 8-1 &nbsp; 小顶堆与大顶堆 </p>
@@ -3718,7 +3775,7 @@
 <li>对于大顶堆（小顶堆），堆顶元素（根节点）的值是最大（最小）的。</li>
 </ul>
 <h2 id="811">8.1.1 &nbsp; 堆的常用操作<a class="headerlink" href="#811" title="Permanent link">&para;</a></h2>
-<p>需要指出的是，许多编程语言提供的是「优先队列 priority queue」，这是一种抽象的数据结构，定义为具有优先级排序的队列。</p>
+<p>需要指出的是，许多编程语言提供的是<u>优先队列（priority queue）</u>，这是一种抽象的数据结构，定义为具有优先级排序的队列。</p>
 <p>实际上，<strong>堆通常用于实现优先队列，大顶堆相当于元素按从大到小的顺序出队的优先队列</strong>。从使用角度来看，我们可以将“优先队列”和“堆”看作等价的数据结构。因此，本书对两者不做特别区分，统一称作“堆”。</p>
 <p>堆的常用操作见表 8-1 ，方法名需要根据编程语言来确定。</p>
 <p align="center"> 表 8-1 &nbsp; 堆的操作效率 </p>
@@ -4471,7 +4528,7 @@
 <div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20MaxHeap%3A%0A%20%20%20%20%22%22%22%E5%A4%A7%E9%A1%B6%E5%A0%86%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22%E6%9E%84%E9%80%A0%E6%96%B9%E6%B3%95%22%22%22%0A%20%20%20%20%20%20%20%20%23%20%E5%B0%86%E5%88%97%E8%A1%A8%E5%85%83%E7%B4%A0%E5%8E%9F%E5%B0%81%E4%B8%8D%E5%8A%A8%E6%B7%BB%E5%8A%A0%E8%BF%9B%E5%A0%86%0A%20%20%20%20%20%20%20%20self.max_heap%20%3D%20nums%0A%0A%20%20%20%20def%20left%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22%E8%8E%B7%E5%8F%96%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E7%9A%84%E7%B4%A2%E5%BC%95%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20*%20i%20%2B%201%0A%0A%20%20%20%20def%20right%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22%E8%8E%B7%E5%8F%96%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E7%9A%84%E7%B4%A2%E5%BC%95%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20*%20i%20%2B%202%0A%0A%20%20%20%20def%20parent%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22%E8%8E%B7%E5%8F%96%E7%88%B6%E8%8A%82%E7%82%B9%E7%9A%84%E7%B4%A2%E5%BC%95%22%22%22%0A%20%20%20%20%20%20%20%20return%20%28i%20-%201%29%20//%202%20%20%23%20%E5%90%91%E4%B8%8B%E6%95%B4%E9%99%A4%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22%E8%8E%B7%E5%8F%96%E5%A0%86%E5%A4%A7%E5%B0%8F%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self.max_heap%29%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22%E5%88%A4%E6%96%AD%E5%A0%86%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%22%22%22%0A%20%20%20%20%20%20%20%20return%20self.size%28%29%20%3D%3D%200%0A%0A%20%20%20%20def%20peek%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22%E8%AE%BF%E9%97%AE%E5%A0%86%E9%A1%B6%E5%85%83%E7%B4%A0%22%22%22%0A%20%20%20%20%20%20%20%20return%20self.max_heap%5B0%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%A4%A7%E9%A1%B6%E5%A0%86%0A%20%20%20%20%23%20%E8%AF%B7%E6%B3%A8%E6%84%8F%EF%BC%8C%E8%BE%93%E5%85%A5%E6%95%B0%E7%BB%84%E5%B7%B2%E7%BB%8F%E6%98%AF%E4%B8%80%E4%B8%AA%E5%B7%B2%E7%BB%8F%E6%98%AF%E4%B8%80%E4%B8%AA%E5%90%88%E6%B3%95%E7%9A%84%E5%A0%86%20%0A%20%20%20%20max_heap%20%3D%20MaxHeap%28%5B9,%208,%206,%206,%207,%205,%202,%201,%204,%203,%206,%202%5D%29%0A%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%A0%86%E9%A1%B6%E5%85%83%E7%B4%A0%0A%20%20%20%20peek%20%3D%20max_heap.peek%28%29%0A%20%20%20%20print%28f%22%5Cn%E5%A0%86%E9%A1%B6%E5%85%83%E7%B4%A0%E4%B8%BA%20%7Bpeek%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 &gt;</a></div></p>
 </details>
 <h3 id="3">3. &nbsp; 元素入堆<a class="headerlink" href="#3" title="Permanent link">&para;</a></h3>
-<p>给定元素 <code>val</code> ，我们首先将其添加到堆底。添加之后，由于 <code>val</code> 可能大于堆中其他元素，堆的成立条件可能已被破坏，<strong>因此需要修复从插入节点到根节点的路径上的各个节点</strong>，这个操作被称为「堆化 heapify」。</p>
+<p>给定元素 <code>val</code> ，我们首先将其添加到堆底。添加之后，由于 <code>val</code> 可能大于堆中其他元素，堆的成立条件可能已被破坏，<strong>因此需要修复从插入节点到根节点的路径上的各个节点</strong>，这个操作被称为<u>堆化（heapify）</u>。</p>
 <p>考虑从入堆节点开始，<strong>从底至顶执行堆化</strong>。如图 8-3 所示，我们比较插入节点与其父节点的值，如果插入节点更大，则将它们交换。然后继续执行此操作，从底至顶修复堆中的各个节点，直至越过根节点或遇到无须交换的节点时结束。</p>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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             第 0 章 &nbsp; 前言
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+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1662,7 +1719,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" checked>
         
           
           
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             </a>
             
               
-              <label class="md-nav__link md-nav__link--active" for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1816,7 +1873,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1974,7 +2031,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1992,14 +2049,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2184,7 +2241,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2487,7 +2544,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2664,7 +2721,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
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-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2845,7 +2902,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
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-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3064,7 +3121,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3237,7 +3294,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3362,7 +3419,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3419,7 +3476,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_heap/summary/index.html b/chapter_heap/summary/index.html
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--- a/chapter_heap/summary/index.html
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@@ -334,6 +334,63 @@
           
         
       
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
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+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
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+            序
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+        </nav>
+      
+    </li>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
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-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
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@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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@@ -810,14 +867,14 @@
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-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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             第 3 章 &nbsp; 数据结构
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@@ -969,7 +1026,7 @@
         
         
         
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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@@ -1482,14 +1539,14 @@
             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1662,7 +1719,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" checked>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="true">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1874,7 +1931,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -2032,7 +2089,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
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-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2242,7 +2299,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2545,7 +2602,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
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+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2722,7 +2779,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
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-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
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-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3122,7 +3179,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
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+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3420,7 +3477,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
           </label>
@@ -3477,7 +3534,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
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+          
+          <div class="md-nav__link md-nav__container">
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+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
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+    
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+  
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
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+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
           </label>
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             </a>
             
               
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" checked>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="true">
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+          <label class="md-nav__title" for="__nav_10">
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             第 8 章 &nbsp; 堆
           </label>
@@ -1883,7 +1940,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_11">
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             第 9 章 &nbsp; 图
           </label>
@@ -2041,7 +2098,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_12">
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             第 10 章 &nbsp; 搜索
           </label>
@@ -2251,7 +2308,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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             第 11 章 &nbsp; 排序
           </label>
@@ -2554,7 +2611,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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             第 12 章 &nbsp; 分治
           </label>
@@ -2731,7 +2788,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_15">
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             第 13 章 &nbsp; 回溯
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
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+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
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@@ -3486,7 +3543,7 @@
         
         
         
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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new file mode 100644
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@@ -0,0 +1,3805 @@
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_preface/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M21 4H3a2 2 0 0 0-2 2v13a2 2 0 0 0 2 2h18a2 2 0 0 0 2-2V6a2 2 0 0 0-2-2M3 19V6h8v13H3m18 0h-8V6h8v13m-7-9.5h6V11h-6V9.5m0 2.5h6v1.5h-6V12m0 2.5h6V16h-6v-1.5Z"/></svg>
+  
+  <span class="md-ellipsis">
+    第 0 章 &nbsp; 前言
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
+            <span class="md-nav__icon md-icon"></span>
+            第 0 章 &nbsp; 前言
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_preface/about_the_book/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    0.1 &nbsp; 关于本书
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_preface/suggestions/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    0.2 &nbsp; 如何使用本书
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_preface/summary/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    0.3 &nbsp; 小结
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_introduction/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2m0 16H5V5h14v14M6.2 7.7h5v1.5h-5V7.7m6.8 8.1h5v1.5h-5v-1.5m0-2.6h5v1.5h-5v-1.5M8 18h1.5v-2h2v-1.5h-2v-2H8v2H6V16h2v2m6.1-7.1 1.4-1.4 1.4 1.4 1.1-1-1.4-1.4L18 7.1 16.9 6l-1.4 1.4L14.1 6 13 7.1l1.4 1.4L13 9.9l1.1 1Z"/></svg>
+  
+  <span class="md-ellipsis">
+    第 1 章 &nbsp; 初识算法
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
+            <span class="md-nav__icon md-icon"></span>
+            第 1 章 &nbsp; 初识算法
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_introduction/algorithms_are_everywhere/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    1.1 &nbsp; 算法无处不在
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_introduction/what_is_dsa/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    1.2 &nbsp; 算法是什么
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_introduction/summary/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    1.3 &nbsp; 小结
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_computational_complexity/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M6 2h12v6l-4 4 4 4v6H6v-6l4-4-4-4V2m10 14.5-4-4-4 4V20h8v-3.5m-4-5 4-4V4H8v3.5l4 4M10 6h4v.75l-2 2-2-2V6Z"/></svg>
+  
+  <span class="md-ellipsis">
+    第 2 章 &nbsp; 复杂度分析
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
+            <span class="md-nav__icon md-icon"></span>
+            第 2 章 &nbsp; 复杂度分析
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_computational_complexity/performance_evaluation/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    2.1 &nbsp; 算法效率评估
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_computational_complexity/iteration_and_recursion/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    2.2 &nbsp; 迭代与递归
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_computational_complexity/time_complexity/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    2.3 &nbsp; 时间复杂度
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_computational_complexity/space_complexity/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    2.4 &nbsp; 空间复杂度
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_computational_complexity/summary/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    2.5 &nbsp; 小结
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_data_structure/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8h8m-2 2H5v4h4v-4M12 2l5.5 9h-11L12 2m0 3.86L10.08 9h3.84L12 5.86M17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5Z"/></svg>
+  
+  <span class="md-ellipsis">
+    第 3 章 &nbsp; 数据结构
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
+            <span class="md-nav__icon md-icon"></span>
+            第 3 章 &nbsp; 数据结构
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    3.1 &nbsp; 数据结构分类
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_data_structure/basic_data_types/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    3.2 &nbsp; 基本数据类型
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_data_structure/number_encoding/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    3.3 &nbsp; 数字编码 *
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_data_structure/character_encoding/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    3.4 &nbsp; 字符编码 *
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_data_structure/summary/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    3.5 &nbsp; 小结
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_array_and_linkedlist/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 5v14h17V5H3m4 2v2H5V7h2m-2 6v-2h2v2H5m0 2h2v2H5v-2m13 2H9v-2h9v2m0-4H9v-2h9v2m0-4H9V7h9v2Z"/></svg>
+  
+  <span class="md-ellipsis">
+    第 4 章 &nbsp; 数组与链表
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
+            <span class="md-nav__icon md-icon"></span>
+            第 4 章 &nbsp; 数组与链表
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_array_and_linkedlist/array/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    4.1 &nbsp; 数组
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    4.2 &nbsp; 链表
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_array_and_linkedlist/list/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    4.3 &nbsp; 列表
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_array_and_linkedlist/ram_and_cache/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    4.4 &nbsp; 内存与缓存 *
+  </span>
+  
+    
+  
+  
+    <span class="md-status md-status--new" title="最近添加">
+    </span>
+  
+
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_array_and_linkedlist/summary/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    4.5 &nbsp; 小结
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_stack_and_queue/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
+  
+  <span class="md-ellipsis">
+    第 5 章 &nbsp; 栈与队列
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
+            <span class="md-nav__icon md-icon"></span>
+            第 5 章 &nbsp; 栈与队列
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_stack_and_queue/stack/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    5.1 &nbsp; 栈
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_stack_and_queue/queue/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    5.2 &nbsp; 队列
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_stack_and_queue/deque/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    5.3 &nbsp; 双向队列
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_stack_and_queue/summary/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    5.4 &nbsp; 小结
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_hashing/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
+  
+  <span class="md-ellipsis">
+    第 6 章 &nbsp; 哈希表
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
+            <span class="md-nav__icon md-icon"></span>
+            第 6 章 &nbsp; 哈希表
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_hashing/hash_map/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    6.1 &nbsp; 哈希表
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_hashing/hash_collision/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    6.2 &nbsp; 哈希冲突
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_hashing/hash_algorithm/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    6.3 &nbsp; 哈希算法
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_hashing/summary/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    6.4 &nbsp; 小结
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_tree/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.5 17c-.14 0-.26 0-.39.04L17.5 13.8c.45-.45.75-1.09.75-1.8a2.5 2.5 0 0 0-2.5-2.5c-.14 0-.25 0-.4.04L13.74 6.3c.47-.46.76-1.09.76-1.8a2.5 2.5 0 0 0-5 0c0 .7.29 1.34.76 1.79L8.65 9.54c-.15-.04-.26-.04-.4-.04a2.5 2.5 0 0 0-2.5 2.5c0 .71.29 1.34.75 1.79l-1.61 3.25C4.76 17 4.64 17 4.5 17a2.5 2.5 0 0 0 0 5A2.5 2.5 0 0 0 7 19.5c0-.7-.29-1.34-.76-1.79l1.62-3.25c.14.04.26.04.39.04s.25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0A2.5 2.5 0 0 0 12 17c-.13 0-.26 0-.39.04L10 13.8c.45-.45.75-1.09.75-1.8 0-.7-.29-1.33-.75-1.79l1.61-3.25c.13.04.26.04.39.04s.26 0 .39-.04L14 10.21a2.5 2.5 0 0 0 1.75 4.29c.13 0 .25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0 2.5 2.5 0 0 0-2.5-2.5m-15 3.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1m8.5-1c0 .55-.45 1-1 1s-1-.45-1-1 .45-1 1-1 1 .45 1 1M7.25 12c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1M11 4.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m3.75 7.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m4.75 8.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1Z"/></svg>
+  
+  <span class="md-ellipsis">
+    第 7 章 &nbsp; 树
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
+            <span class="md-nav__icon md-icon"></span>
+            第 7 章 &nbsp; 树
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_tree/binary_tree/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    7.1 &nbsp; 二叉树
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_tree/binary_tree_traversal/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    7.2 &nbsp; 二叉树遍历
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_tree/array_representation_of_tree/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    7.3 &nbsp; 二叉树数组表示
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_tree/binary_search_tree/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    7.4 &nbsp; 二叉搜索树
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_tree/avl_tree/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    7.5 &nbsp; AVL 树 *
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_tree/summary/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    7.6 &nbsp; 小结
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_heap/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 1a2.5 2.5 0 0 0-2.5 2.5A2.5 2.5 0 0 0 11 5.79V7H7a2 2 0 0 0-2 2v.71A2.5 2.5 0 0 0 3.5 12 2.5 2.5 0 0 0 5 14.29V15H4a2 2 0 0 0-2 2v1.21A2.5 2.5 0 0 0 .5 20.5 2.5 2.5 0 0 0 3 23a2.5 2.5 0 0 0 2.5-2.5A2.5 2.5 0 0 0 4 18.21V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 9 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2H7v-.71A2.5 2.5 0 0 0 8.5 12 2.5 2.5 0 0 0 7 9.71V9h10v.71A2.5 2.5 0 0 0 15.5 12a2.5 2.5 0 0 0 1.5 2.29V15h-1a2 2 0 0 0-2 2v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 15 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 21 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2h-1v-.71A2.5 2.5 0 0 0 20.5 12 2.5 2.5 0 0 0 19 9.71V9a2 2 0 0 0-2-2h-4V5.79a2.5 2.5 0 0 0 1.5-2.29A2.5 2.5 0 0 0 12 1m0 1.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M6 11a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m12 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M3 19.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1Z"/></svg>
+  
+  <span class="md-ellipsis">
+    第 8 章 &nbsp; 堆
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
+            <span class="md-nav__icon md-icon"></span>
+            第 8 章 &nbsp; 堆
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_heap/heap/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    8.1 &nbsp; 堆
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_heap/build_heap/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    8.2 &nbsp; 建堆操作
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_heap/top_k/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    8.3 &nbsp; Top-k 问题
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_heap/summary/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    8.4 &nbsp; 小结
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_graph/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59-.44.06M6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43.43 0 .83.16 1.12.43l4.28-2.53H6.6M12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68.93 0 1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25 5.51-9.64m2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24-4.5-2.58Z"/></svg>
+  
+  <span class="md-ellipsis">
+    第 9 章 &nbsp; 图
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
+            <span class="md-nav__icon md-icon"></span>
+            第 9 章 &nbsp; 图
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_graph/graph/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    9.1 &nbsp; 图
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_graph/graph_operations/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    9.2 &nbsp; 图基础操作
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_graph/graph_traversal/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    9.3 &nbsp; 图的遍历
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_graph/summary/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    9.4 &nbsp; 小结
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_searching/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z"/></svg>
+  
+  <span class="md-ellipsis">
+    第 10 章 &nbsp; 搜索
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
+            <span class="md-nav__icon md-icon"></span>
+            第 10 章 &nbsp; 搜索
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_searching/binary_search/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    10.1 &nbsp; 二分查找
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_searching/binary_search_insertion/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    10.2 &nbsp; 二分查找插入点
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_searching/binary_search_edge/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    10.3 &nbsp; 二分查找边界
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    10.4 &nbsp; 哈希优化策略
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    10.5 &nbsp; 重识搜索算法
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_searching/summary/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    10.6 &nbsp; 小结
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_sorting/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 17h3l-4 4-4-4h3V3h2M2 17h10v2H2M6 5v2H2V5m0 6h7v2H2v-2Z"/></svg>
+  
+  <span class="md-ellipsis">
+    第 11 章 &nbsp; 排序
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
+            <span class="md-nav__icon md-icon"></span>
+            第 11 章 &nbsp; 排序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/sorting_algorithm/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    11.1 &nbsp; 排序算法
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/selection_sort/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    11.2 &nbsp; 选择排序
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/bubble_sort/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    11.3 &nbsp; 冒泡排序
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/insertion_sort/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    11.4 &nbsp; 插入排序
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/quick_sort/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    11.5 &nbsp; 快速排序
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/merge_sort/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    11.6 &nbsp; 归并排序
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/heap_sort/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    11.7 &nbsp; 堆排序
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/bucket_sort/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    11.8 &nbsp; 桶排序
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/counting_sort/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    11.9 &nbsp; 计数排序
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/radix_sort/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    11.10 &nbsp; 基数排序
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/summary/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    11.11 &nbsp; 小结
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_divide_and_conquer/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17 7v2h5V7h-5M2 9v6h5V9H2m10 0v2H9v2h3v2l3-3-3-3m5 2v2h5v-2h-5m0 4v2h5v-2h-5Z"/></svg>
+  
+  <span class="md-ellipsis">
+    第 12 章 &nbsp; 分治
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
+            <span class="md-nav__icon md-icon"></span>
+            第 12 章 &nbsp; 分治
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_divide_and_conquer/divide_and_conquer/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    12.1 &nbsp; 分治算法
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_divide_and_conquer/binary_search_recur/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    12.2 &nbsp; 分治搜索策略
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_divide_and_conquer/build_binary_tree_problem/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    12.3 &nbsp; 构建树问题
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_divide_and_conquer/hanota_problem/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    12.4 &nbsp; 汉诺塔问题
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_divide_and_conquer/summary/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    12.5 &nbsp; 小结
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_backtracking/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18 15a3 3 0 0 1 3 3 3 3 0 0 1-3 3 2.99 2.99 0 0 1-2.83-2H14v-2h1.17c.41-1.17 1.52-2 2.83-2m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m0-9a1.43 1.43 0 0 0 1.43-1.43 1.43 1.43 0 1 0-2.86 0A1.43 1.43 0 0 0 18 8m0-5.43a4 4 0 0 1 4 4C22 9.56 18 14 18 14s-4-4.44-4-7.43a4 4 0 0 1 4-4M8.83 17H10v2H8.83A2.99 2.99 0 0 1 6 21a3 3 0 0 1-3-3c0-1.31.83-2.42 2-2.83V14h2v1.17c.85.3 1.53.98 1.83 1.83M6 17a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1M6 3a3 3 0 0 1 3 3c0 1.31-.83 2.42-2 2.83V10H5V8.83A2.99 2.99 0 0 1 3 6a3 3 0 0 1 3-3m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m5 14v-2h2v2h-2m-4-6H5v-2h2v2Z"/></svg>
+  
+  <span class="md-ellipsis">
+    第 13 章 &nbsp; 回溯
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
+            <span class="md-nav__icon md-icon"></span>
+            第 13 章 &nbsp; 回溯
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    13.1 &nbsp; 回溯算法
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_backtracking/permutations_problem/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    13.2 &nbsp; 全排列问题
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    13.3 &nbsp; 子集和问题
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_backtracking/n_queens_problem/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    13.4 &nbsp; N 皇后问题
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_backtracking/summary/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    13.5 &nbsp; 小结
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_dynamic_programming/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M22 15h-2v3c0 1.11-.89 2-2 2h-3v2l-3-3 3-3v2h3v-3h-2l3-3 3 3m0-11v4c0 1.1-.9 2-2 2H10v10c0 1.1-.9 2-2 2H4c-1.1 0-2-.9-2-2V4c0-1.1.9-2 2-2h16c1.1 0 2 .9 2 2M4 8h4V4H4v4m0 2v4h4v-4H4m4 10v-4H4v4h4m6-12V4h-4v4h4m6-4h-4v4h4V4Z"/></svg>
+  
+  <span class="md-ellipsis">
+    第 14 章 &nbsp; 动态规划
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
+            <span class="md-nav__icon md-icon"></span>
+            第 14 章 &nbsp; 动态规划
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_dynamic_programming/intro_to_dynamic_programming/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    14.1 &nbsp; 初探动态规划
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_dynamic_programming/dp_problem_features/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    14.2 &nbsp; DP 问题特性
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_dynamic_programming/dp_solution_pipeline/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    14.3 &nbsp; DP 解题思路
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_dynamic_programming/knapsack_problem/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    14.4 &nbsp; 0-1 背包问题
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_dynamic_programming/unbounded_knapsack_problem/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    14.5 &nbsp; 完全背包问题
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_dynamic_programming/edit_distance_problem/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    14.6 &nbsp; 编辑距离问题
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../chapter_dynamic_programming/summary/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    14.7 &nbsp; 小结
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
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+<h1 id="_1">序<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
+<p>几年前，我在力扣上分享了“剑指 Offer”系列题解，受到了许多读者的鼓励和支持。在与读者交流期间，我最常被问的一个问题是“如何入门算法”。逐渐地，我对这个问题产生了浓厚的兴趣。</p>
+<p>两眼一抹黑地刷题似乎是最受欢迎的方法，简单、直接且有效。然而刷题就如同玩“扫雷”游戏，自学能力强的人能够顺利将地雷逐个排掉，而基础不足的人很可能被炸得满头是包，并在挫折中步步退缩。通读教材也是一种常见做法，但对于面向求职的人来说，毕业论文、投递简历、准备笔试和面试已经消耗了大部分精力，啃厚重的书往往变成了一项艰巨的挑战。</p>
+<p>如果你也面临类似的困扰，那么很幸运这本书“找”到了你。本书是我对这个问题给出的答案，即使不是最优解，也至少是一次积极的尝试。本书虽然不足以让你直接拿到 Offer，但会引导你探索数据结构与算法的“知识地图”，带你了解不同“地雷”的形状、大小和分布位置，让你掌握各种“排雷方法”。有了这些本领，相信你可以更加自如地刷题和阅读文献，逐步构建起完整的知识体系。</p>
+<p>我深深赞同费曼教授所言：“Knowledge isn't free. You have to pay attention.”从这个意义上看，这本书并非完全“免费”。为了不辜负你为本书所付出的宝贵“注意力”，我会竭尽所能，投入最大的“注意力”来完成本书的创作。</p>
+<p>本人自知学疏才浅，书中内容虽然已经过一段时间的打磨，但一定仍有许多错误，恳请各位老师和同学批评指正。</p>
+<p><a class="glightbox" href="../assets/covers/chapter_hello_algo.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Hello 算法" class="cover-image" src="../assets/covers/chapter_hello_algo.jpg" /></a></p>
+<div style="text-align: center;">
+    <h2 style="margin-top: 0.8em; margin-bottom: 0.8em;">Hello，算法！</h2>
+</div>
+
+<p>计算机的出现给世界带来了巨大变革，它凭借高速的计算能力和出色的可编程性，成为了执行算法与处理数据的理想媒介。无论是电子游戏的逼真画面、自动驾驶的智能决策，还是 AlphaGo 的精彩棋局、ChatGPT 的自然交互，这些应用都是算法在计算机上的精妙演绎。</p>
+<p>事实上，在计算机问世之前，算法和数据结构就已经存在于世界的各个角落。早期的算法相对简单，例如古代的计数方法和工具制作步骤等。随着文明的进步，算法逐渐变得更加精细和复杂。从巧夺天工的匠人技艺、到解放生产力的工业产品、再到宇宙运行的科学规律，几乎每一件平凡或令人惊叹的事物背后，都隐藏着精妙的算法思想。</p>
+<p>同样，数据结构无处不在：大到社会网络，小到地铁线路，许多系统都可以建模为“图”；大到一个国家，小到一个家庭，社会的主要组织形式呈现出“树”的特征；冬天的衣服就像“栈”，最先穿上的最后才能脱下；羽毛球筒则如同“队列”，一端放入、另一端取出；字典就像一个“哈希表”，能够快速查找目标词条。</p>
+<p>本书旨在通过清晰易懂的动画插图和可运行的代码示例，使读者理解算法和数据结构的核心概念，并能够通过编程来实现它们。在此基础上，本书致力于揭示算法在复杂世界中的生动体现，展现算法之美。希望本书能够帮助到你！</p>
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+</html>
\ No newline at end of file
diff --git a/chapter_introduction/algorithms_are_everywhere/index.html b/chapter_introduction/algorithms_are_everywhere/index.html
index 22b44382d..a1e0dd362 100644
--- a/chapter_introduction/algorithms_are_everywhere/index.html
+++ b/chapter_introduction/algorithms_are_everywhere/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
         
       
         
@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -482,7 +539,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" checked>
         
           
           
@@ -500,14 +557,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -627,7 +684,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -645,14 +702,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -804,7 +861,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -822,14 +879,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -981,7 +1038,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -999,14 +1056,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1164,7 +1221,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1182,14 +1239,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1318,7 +1375,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1336,14 +1393,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1476,7 +1533,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1494,14 +1551,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1672,7 +1729,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1690,14 +1747,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1826,7 +1883,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1844,14 +1901,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1984,7 +2041,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -2002,14 +2059,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2194,7 +2251,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2212,14 +2269,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2497,7 +2554,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
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+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2674,7 +2731,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2855,7 +2912,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3074,7 +3131,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
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-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3247,7 +3304,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3372,7 +3429,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3429,7 +3486,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_introduction/index.html b/chapter_introduction/index.html
index 1e285dfc3..5c4c680fd 100644
--- a/chapter_introduction/index.html
+++ b/chapter_introduction/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
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+  <span class="md-ellipsis">
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+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -482,7 +539,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" checked>
         
           
           
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-              <label class="md-nav__link md-nav__link--active" for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link md-nav__link--active" for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -617,7 +674,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -794,7 +851,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -812,14 +869,14 @@
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-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -971,7 +1028,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -989,14 +1046,14 @@
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-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1154,7 +1211,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1172,14 +1229,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1308,7 +1365,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1326,14 +1383,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1466,7 +1523,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1484,14 +1541,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
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             第 14 章 &nbsp; 动态规划
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             第 16 章 &nbsp; 附录
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             纸质书
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 <p>一位少女翩翩起舞，与数据交织在一起，裙摆上飘扬着算法的旋律。</p>
 <p>她邀请你共舞，请紧跟她的步伐，踏入充满逻辑与美感的算法世界。</p>
 </div>
-<p>计算机的出现给世界带来了巨大变革，它凭借高速的计算能力和出色的可编程性，成为了执行算法与处理数据的理想媒介。无论是电子游戏的逼真画面、自动驾驶的智能决策，还是 AlphaGo 的精彩棋局、ChatGPT 的自然交互，这些应用都是算法在计算机上的精妙演绎。</p>
-<p>事实上，在计算机问世之前，算法和数据结构就已经存在于世界的各个角落。早期的算法相对简单，例如古代的计数方法和工具制作步骤等。随着文明的进步，算法逐渐变得更加精细和复杂。从巧夺天工的匠人技艺、到解放生产力的工业产品、再到宇宙运行的科学规律，几乎每一件平凡或令人惊叹的事物背后，都隐藏着精妙的算法思想。</p>
-<p>同样，数据结构无处不在：大到社会网络，小到地铁线路，许多系统都可以建模为“图”；大到一个国家，小到一个家庭，社会的主要组织形式呈现出“树”的特征；冬天的衣服就像“栈”，最先穿上的最后才能脱下；羽毛球筒则如同“队列”，一端放入、另一端取出；字典就像一个“哈希表”，能够快速查找目标词条。</p>
-<p>那么，什么是算法，它在我们日常生活中扮演着怎样的角色？接下来，我们将就这些问题展开探讨。</p>
 <h2 id="_1">本章内容<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h2>
 <ul>
 <li><a href="https://www.hello-algo.com/chapter_introduction/algorithms_are_everywhere/">1.1 &nbsp; 算法无处不在</a></li>
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+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1164,7 +1221,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1318,7 +1375,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1476,7 +1533,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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           </label>
@@ -1672,7 +1729,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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           </label>
@@ -1826,7 +1883,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
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           </label>
@@ -1984,7 +2041,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2194,7 +2251,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2497,7 +2554,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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             第 13 章 &nbsp; 回溯
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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             第 15 章 &nbsp; 贪心
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
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             参考文献
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@@ -3429,7 +3486,7 @@
         
         
         
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_introduction/what_is_dsa/index.html b/chapter_introduction/what_is_dsa/index.html
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+          
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+        </nav>
+      
+    </li>
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-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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@@ -684,7 +741,7 @@
         
         
         
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             第 4 章 &nbsp; 数组与链表
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             第 10 章 &nbsp; 搜索
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             第 13 章 &nbsp; 回溯
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 <!-- Page content -->
 <h1 id="12">1.2 &nbsp; 算法是什么<a class="headerlink" href="#12" title="Permanent link">&para;</a></h1>
 <h2 id="121">1.2.1 &nbsp; 算法定义<a class="headerlink" href="#121" title="Permanent link">&para;</a></h2>
-<p>「算法 algorithm」是在有限时间内解决特定问题的一组指令或操作步骤，它具有以下特性。</p>
+<p><u>算法（algorithm）</u>是在有限时间内解决特定问题的一组指令或操作步骤，它具有以下特性。</p>
 <ul>
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-<p>「数据结构 data structure」是计算机中组织和存储数据的方式，具有以下设计目标。</p>
+<p><u>数据结构（data structure）</u>是计算机中组织和存储数据的方式，具有以下设计目标。</p>
 <ul>
 <li>空间占用尽量少，以节省计算机内存。</li>
 <li>数据操作尽可能快速，涵盖数据访问、添加、删除、更新等。</li>
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+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+            序
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+    </li>
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 2 章 &nbsp; 复杂度分析
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             第 3 章 &nbsp; 数据结构
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             第 4 章 &nbsp; 数组与链表
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             第 5 章 &nbsp; 栈与队列
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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             第 6 章 &nbsp; 哈希表
           </label>
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             第 8 章 &nbsp; 堆
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             第 9 章 &nbsp; 图
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
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             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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             第 11 章 &nbsp; 排序
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             第 12 章 &nbsp; 分治
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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             参考文献
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+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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@@ -325,6 +325,63 @@
       
   
   
+  
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+          
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+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
     
   
   
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="true">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
           </label>
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             </a>
             
               
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
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             第 4 章 &nbsp; 数组与链表
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
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-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
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             第 6 章 &nbsp; 哈希表
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+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
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             第 8 章 &nbsp; 堆
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
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             第 9 章 &nbsp; 图
           </label>
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-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
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             第 11 章 &nbsp; 排序
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+          <label class="md-nav__title" for="__nav_14">
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             第 12 章 &nbsp; 分治
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+          <label class="md-nav__title" for="__nav_15">
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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             参考文献
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     纸质书
@@ -3513,8 +3570,8 @@
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_preface/index.html b/chapter_preface/index.html
index 6f9d01247..0a9ee49fc 100644
--- a/chapter_preface/index.html
+++ b/chapter_preface/index.html
@@ -15,6 +15,8 @@
         <link rel="canonical" href="https://www.hello-algo.com/chapter_preface/">
       
       
+        <link rel="prev" href="../chapter_hello_algo/">
+      
       
         <link rel="next" href="about_the_book/">
       
@@ -323,6 +325,63 @@
       
   
   
+  
+  
+    
+    
+      
+        
+          
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+      
+    
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
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+            
+          </ul>
+        </nav>
+      
+    </li>
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@@ -349,7 +408,7 @@
         
         
         
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             </a>
             
               
-              <label class="md-nav__link md-nav__link--active" for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link md-nav__link--active" for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
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+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +851,7 @@
         
         
         
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
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+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
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@@ -987,14 +1046,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1211,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1229,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1523,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1719,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1737,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1873,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1891,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2031,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2049,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2241,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2200,14 +2259,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2485,7 +2544,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2503,14 +2562,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2662,7 +2721,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
@@ -2680,14 +2739,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2843,7 +2902,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
@@ -2861,14 +2920,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3062,7 +3121,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
@@ -3080,14 +3139,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3235,7 +3294,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
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@@ -3360,7 +3419,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3417,7 +3476,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
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+          <label class="md-nav__title" for="__nav_20">
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@@ -3538,10 +3597,6 @@
 <p>算法犹如美妙的交响乐，每一行代码都像韵律般流淌。</p>
 <p>愿这本书在你的脑海中轻轻响起，留下独特而深刻的旋律。</p>
 </div>
-<p>两年前，我在力扣上分享了“剑指 Offer”系列题解，受到了许多读者的鼓励和支持。在与读者交流期间，我最常被问的一个问题是“如何入门算法”。逐渐地，我对这个问题产生了浓厚的兴趣。</p>
-<p>两眼一抹黑地刷题似乎是最受欢迎的方法，简单、直接且有效。然而刷题就如同玩“扫雷”游戏，自学能力强的人能够顺利将地雷逐个排掉，而基础不足的人很可能被炸得满头是包，并在挫折中步步退缩。通读教材也是一种常见做法，但对于面向求职的人来说，毕业论文、投递简历、准备笔试和面试已经消耗了大部分精力，啃厚重的书往往变成了一项艰巨的挑战。</p>
-<p>如果你也面临类似的困扰，那么很幸运这本书“找”到了你。本书是我对这个问题给出的答案，即使不是最优解，也至少是一次积极的尝试。本书虽然不足以让你直接拿到 Offer，但会引导你探索数据结构与算法的“知识地图”，带你了解不同“地雷”的形状、大小和分布位置，让你掌握各种“排雷方法”。有了这些本领，相信你可以更加自如地刷题和阅读文献，逐步构建起完整的知识体系。</p>
-<p>我深深赞同费曼教授所言：“Knowledge isn't free. You have to pay attention.”从这个意义上看，这本书并非完全“免费”。为了不辜负你为本书所付出的宝贵“注意力”，我会竭尽所能，投入最大的“注意力”来完成本书的创作。</p>
 <h2 id="_1">本章内容<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h2>
 <ul>
 <li><a href="https://www.hello-algo.com/chapter_preface/about_the_book/">0.1 &nbsp; 关于本书</a></li>
@@ -3566,6 +3621,27 @@ aria-label="页脚"
 
 <!-- Link to previous page -->
 
+  
+  <a
+    href="../chapter_hello_algo/"
+    class="md-footer__link md-footer__link--prev"
+    aria-label="上一页: 序"
+    rel="prev"
+  >
+    <div class="md-footer__button md-icon">
+      
+      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11h12Z"/></svg>
+    </div>
+    <div class="md-footer__title">
+      <span class="md-footer__direction">
+        上一页
+      </span>
+      <div class="md-ellipsis">
+        序
+      </div>
+    </div>
+  </a>
+
 
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@@ -3673,6 +3749,22 @@ aria-label="页脚"
       
       <nav class="md-footer__inner md-grid" aria-label="页脚" >
         
+          
+          <a href="../chapter_hello_algo/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 序">
+            <div class="md-footer__button md-icon">
+              
+              <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11h12Z"/></svg>
+            </div>
+            <div class="md-footer__title">
+              <span class="md-footer__direction">
+                上一页
+              </span>
+              <div class="md-ellipsis">
+                序
+              </div>
+            </div>
+          </a>
+        
         
           
           <a href="about_the_book/" class="md-footer__link md-footer__link--next" aria-label="下一页: 0.1 &amp;nbsp; 关于本书">
@@ -3698,7 +3790,7 @@ aria-label="页脚"
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     <div class="md-copyright__highlight">
-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
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+            序
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+            
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+          </ul>
+        </nav>
+      
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@@ -351,7 +408,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" checked>
         
           
           
@@ -369,14 +426,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -567,7 +624,7 @@
         
         
         
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             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -702,7 +759,7 @@
         
         
         
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             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -879,7 +936,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
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-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
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+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1393,7 +1450,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
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-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
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             第 10 章 &nbsp; 搜索
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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             第 11 章 &nbsp; 排序
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 <h2 id="021">0.2.1 &nbsp; 行文风格约定<a class="headerlink" href="#021" title="Permanent link">&para;</a></h2>
 <ul>
 <li>标题后标注 <code>*</code> 的是选读章节，内容相对困难。如果你的时间有限，可以先跳过。</li>
-<li>重要专有名词及其英文翻译会用 <code>「 」</code> 括号标注，例如 <code>「数组 array」</code> 。建议记住它们，以便阅读文献。</li>
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-<li>重要名词、重点内容和总结性语句会 <strong>加粗</strong> ，这类文字值得特别关注。</li>
+<li>专业术语会使用黑体（纸质版和 PDF 版）或添加下划线（网页版），例如<u>数组（array）</u>。建议记住它们，以便阅读文献。</li>
+<li>重点内容和总结性语句会 <strong>加粗</strong>，这类文字值得特别关注。</li>
+<li>有特指含义的词句会使用“引号”标注，以避免歧义。</li>
 <li>当涉及编程语言之间不一致的名词时，本书均以 Python 为准，例如使用 <code>None</code> 来表示“空”。</li>
 <li>本书部分放弃了编程语言的注释规范，以换取更加紧凑的内容排版。注释主要分为三种类型：标题注释、内容注释、多行注释。</li>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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             第 0 章 &nbsp; 前言
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             第 7 章 &nbsp; 树
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
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           </label>
@@ -1984,7 +2041,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2194,7 +2251,7 @@
         
         
         
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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             第 11 章 &nbsp; 排序
           </label>
@@ -2497,7 +2554,7 @@
         
         
         
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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           </label>
@@ -2674,7 +2731,7 @@
         
         
         
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             第 13 章 &nbsp; 回溯
           </label>
@@ -2855,7 +2912,7 @@
         
         
         
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
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             第 14 章 &nbsp; 动态规划
           </label>
@@ -3074,7 +3131,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
           </label>
@@ -3247,7 +3304,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
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             参考文献
           </label>
@@ -3429,7 +3486,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_reference/index.html b/chapter_reference/index.html
index 13ba13f4c..ea384520f 100644
--- a/chapter_reference/index.html
+++ b/chapter_reference/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
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+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_hello_algo/" class="md-nav__link ">
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+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
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+
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+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
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+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
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+            
+          </ul>
+        </nav>
+      
+    </li>
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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             第 6 章 &nbsp; 哈希表
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@@ -1482,14 +1539,14 @@
             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
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+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2182,7 +2239,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2485,7 +2542,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
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-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
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+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2662,7 +2719,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
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-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_15">
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             第 13 章 &nbsp; 回溯
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
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-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3235,7 +3292,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3362,7 +3419,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" checked>
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
           </label>
@@ -3419,7 +3476,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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index f862428a0..275c49e42 100644
--- a/chapter_searching/binary_search/index.html
+++ b/chapter_searching/binary_search/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
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+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
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+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
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+          <ul class="md-nav__list" data-md-scrollfix>
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+      
+    </li>
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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             第 5 章 &nbsp; 栈与队列
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+          <label class="md-nav__title" for="__nav_8">
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             第 6 章 &nbsp; 哈希表
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           </label>
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+          <label class="md-nav__title" for="__nav_12">
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             第 10 章 &nbsp; 搜索
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             第 11 章 &nbsp; 排序
           </label>
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             第 12 章 &nbsp; 分治
           </label>
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             第 13 章 &nbsp; 回溯
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+          <label class="md-nav__title" for="__nav_16">
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             第 14 章 &nbsp; 动态规划
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+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
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+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
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             参考文献
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             纸质书
           </label>
@@ -3601,7 +3658,7 @@
 
 <!-- Page content -->
 <h1 id="101">10.1 &nbsp; 二分查找<a class="headerlink" href="#101" title="Permanent link">&para;</a></h1>
-<p>「二分查找 binary search」是一种基于分治策略的高效搜索算法。它利用数据的有序性，每轮缩小一半搜索范围，直至找到目标元素或搜索区间为空为止。</p>
+<p><u>二分查找（binary search）</u>是一种基于分治策略的高效搜索算法。它利用数据的有序性，每轮缩小一半搜索范围，直至找到目标元素或搜索区间为空为止。</p>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
 <p>给定一个长度为 <span class="arithmatex">\(n\)</span> 的数组 <code>nums</code> ，元素按从小到大的顺序排列且不重复。请查找并返回元素 <code>target</code> 在该数组中的索引。若数组不包含该元素，则返回 <span class="arithmatex">\(-1\)</span> 。示例如图 10-1 所示。</p>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1974,7 +2031,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" checked>
         
           
           
@@ -1992,14 +2049,14 @@
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-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2266,7 +2323,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2284,14 +2341,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2569,7 +2626,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2587,14 +2644,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2746,7 +2803,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2927,7 +2984,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3146,7 +3203,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
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-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3319,7 +3376,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3444,7 +3501,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3501,7 +3558,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_searching/binary_search_insertion/index.html b/chapter_searching/binary_search_insertion/index.html
index 9245f4aec..c041f7e55 100644
--- a/chapter_searching/binary_search_insertion/index.html
+++ b/chapter_searching/binary_search_insertion/index.html
@@ -334,6 +334,63 @@
           
         
       
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+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
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+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
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+        
+          
+        
+      
         
       
         
@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 2 章 &nbsp; 复杂度分析
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             第 3 章 &nbsp; 数据结构
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             第 4 章 &nbsp; 数组与链表
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+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
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+          <label class="md-nav__title" for="__nav_8">
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             第 6 章 &nbsp; 哈希表
           </label>
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@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" checked>
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="true">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="true">
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             第 10 章 &nbsp; 搜索
           </label>
@@ -2242,7 +2299,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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             第 11 章 &nbsp; 排序
           </label>
@@ -2545,7 +2602,7 @@
         
         
         
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             第 13 章 &nbsp; 回溯
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
         
       
         
@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1974,7 +2031,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" checked>
         
           
           
@@ -1992,14 +2049,14 @@
             </a>
             
               
-              <label class="md-nav__link md-nav__link--active" for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link md-nav__link--active" for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2184,7 +2241,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2202,14 +2259,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2487,7 +2544,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2505,14 +2562,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2664,7 +2721,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
@@ -2682,14 +2739,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2845,7 +2902,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
@@ -2863,14 +2920,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3064,7 +3121,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
@@ -3082,14 +3139,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3237,7 +3294,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
@@ -3255,14 +3312,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3362,7 +3419,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
@@ -3381,8 +3438,8 @@
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3419,7 +3476,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
@@ -3427,7 +3484,7 @@
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     纸质书
@@ -3446,8 +3503,8 @@
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -3736,7 +3793,7 @@ aria-label="页脚"
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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index f4eddf795..b862a66f4 100644
--- a/chapter_searching/replace_linear_by_hashing/index.html
+++ b/chapter_searching/replace_linear_by_hashing/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
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+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
           </label>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
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             第 4 章 &nbsp; 数组与链表
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
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-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
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             第 6 章 &nbsp; 哈希表
           </label>
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+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
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             第 8 章 &nbsp; 堆
           </label>
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-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
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             第 9 章 &nbsp; 图
           </label>
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-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="true">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
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             第 11 章 &nbsp; 排序
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+          <label class="md-nav__title" for="__nav_14">
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             第 12 章 &nbsp; 分治
           </label>
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+          <label class="md-nav__title" for="__nav_15">
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             第 13 章 &nbsp; 回溯
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+          <label class="md-nav__title" for="__nav_16">
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             第 14 章 &nbsp; 动态规划
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+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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             参考文献
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     纸质书
@@ -3504,8 +3561,8 @@
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
diff --git a/chapter_searching/searching_algorithm_revisited/index.html b/chapter_searching/searching_algorithm_revisited/index.html
index 08898eeb6..935826483 100644
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+++ b/chapter_searching/searching_algorithm_revisited/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
+    
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+    
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+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
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+          
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+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
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+  
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+            </a>
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+          </div>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
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+    
+      
+      
+  
+  
+  
+  
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1974,7 +2031,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" checked>
         
           
           
@@ -1992,14 +2049,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2251,7 +2308,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2269,14 +2326,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2554,7 +2611,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2572,14 +2629,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2731,7 +2788,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2912,7 +2969,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3131,7 +3188,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3304,7 +3361,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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     纸质书
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-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -3619,7 +3676,7 @@
 
 <!-- Page content -->
 <h1 id="105">10.5 &nbsp; 重识搜索算法<a class="headerlink" href="#105" title="Permanent link">&para;</a></h1>
-<p>「搜索算法 searching algorithm」用于在数据结构（例如数组、链表、树或图）中搜索一个或一组满足特定条件的元素。</p>
+<p><u>搜索算法（searching algorithm）</u>用于在数据结构（例如数组、链表、树或图）中搜索一个或一组满足特定条件的元素。</p>
 <p>搜索算法可根据实现思路分为以下两类。</p>
 <ul>
 <li><strong>通过遍历数据结构来定位目标元素</strong>，例如数组、链表、树和图的遍历等。</li>
@@ -3926,7 +3983,7 @@ aria-label="页脚"
       <div class="md-copyright">
   
     <div class="md-copyright__highlight">
-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
diff --git a/chapter_searching/summary/index.html b/chapter_searching/summary/index.html
index c01cdac63..8924cffad 100644
--- a/chapter_searching/summary/index.html
+++ b/chapter_searching/summary/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
+    
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+    <li class="md-nav__item md-nav__item--nested">
+      
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+        
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
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+    
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+      
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+          
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1974,7 +2031,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" checked>
         
           
           
@@ -1992,14 +2049,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2194,7 +2251,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2212,14 +2269,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2497,7 +2554,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2515,14 +2572,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2674,7 +2731,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
@@ -2692,14 +2749,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2855,7 +2912,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
@@ -2873,14 +2930,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3074,7 +3131,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
@@ -3092,14 +3149,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3247,7 +3304,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
@@ -3265,14 +3322,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3372,7 +3429,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3429,7 +3486,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
@@ -3437,7 +3494,7 @@
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19 2-5 4.5v11l5-4.5V2M6.5 5C4.55 5 2.45 5.4 1 6.5v14.66c0 .25.25.5.5.5.1 0 .15-.07.25-.07 1.35-.65 3.3-1.09 4.75-1.09 1.95 0 4.05.4 5.5 1.5 1.35-.85 3.8-1.5 5.5-1.5 1.65 0 3.35.31 4.75 1.06.1.05.15.03.25.03.25 0 .5-.25.5-.5V6.5c-.6-.45-1.25-.75-2-1V19c-1.1-.35-2.3-.5-3.5-.5-1.7 0-4.15.65-5.5 1.5V6.5C10.55 5.4 8.45 5 6.5 5Z"/></svg>
   
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     纸质书
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -3722,7 +3779,7 @@ aria-label="页脚"
       <div class="md-copyright">
   
     <div class="md-copyright__highlight">
-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
diff --git a/chapter_sorting/bubble_sort/index.html b/chapter_sorting/bubble_sort/index.html
index 23e6bc7ed..7593c73bc 100644
--- a/chapter_sorting/bubble_sort/index.html
+++ b/chapter_sorting/bubble_sort/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
         
       
         
@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1678,14 +1735,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2184,7 +2241,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" checked>
         
           
           
@@ -2202,14 +2259,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2554,7 +2611,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2572,14 +2629,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2731,7 +2788,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
@@ -2749,14 +2806,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
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             纸质书
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 <!-- Page content -->
 <h1 id="113">11.3 &nbsp; 冒泡排序<a class="headerlink" href="#113" title="Permanent link">&para;</a></h1>
-<p>「冒泡排序 bubble sort」通过连续地比较与交换相邻元素实现排序。这个过程就像气泡从底部升到顶部一样，因此得名冒泡排序。</p>
+<p><u>冒泡排序（bubble sort）</u>通过连续地比较与交换相邻元素实现排序。这个过程就像气泡从底部升到顶部一样，因此得名冒泡排序。</p>
 <p>如图 11-4 所示，冒泡过程可以利用元素交换操作来模拟：从数组最左端开始向右遍历，依次比较相邻元素大小，如果“左元素 &gt; 右元素”就交换二者。遍历完成后，最大的元素会被移动到数组的最右端。</p>
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+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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             第 0 章 &nbsp; 前言
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             第 1 章 &nbsp; 初识算法
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             第 5 章 &nbsp; 栈与队列
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             第 6 章 &nbsp; 哈希表
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             第 15 章 &nbsp; 贪心
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 <!-- Page content -->
 <h1 id="118">11.8 &nbsp; 桶排序<a class="headerlink" href="#118" title="Permanent link">&para;</a></h1>
 <p>前述几种排序算法都属于“基于比较的排序算法”，它们通过比较元素间的大小来实现排序。此类排序算法的时间复杂度无法超越 <span class="arithmatex">\(O(n \log n)\)</span> 。接下来，我们将探讨几种“非比较排序算法”，它们的时间复杂度可以达到线性阶。</p>
-<p>「桶排序 bucket sort」是分治策略的一个典型应用。它通过设置一些具有大小顺序的桶，每个桶对应一个数据范围，将数据平均分配到各个桶中；然后，在每个桶内部分别执行排序；最终按照桶的顺序将所有数据合并。</p>
+<p><u>桶排序（bucket sort）</u>是分治策略的一个典型应用。它通过设置一些具有大小顺序的桶，每个桶对应一个数据范围，将数据平均分配到各个桶中；然后，在每个桶内部分别执行排序；最终按照桶的顺序将所有数据合并。</p>
 <h2 id="1181">11.8.1 &nbsp; 算法流程<a class="headerlink" href="#1181" title="Permanent link">&para;</a></h2>
 <p>考虑一个长度为 <span class="arithmatex">\(n\)</span> 的数组，其元素是范围 <span class="arithmatex">\([0, 1)\)</span> 内的浮点数。桶排序的流程如图 11-13 所示。</p>
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+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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             参考文献
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             纸质书
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 <!-- Page content -->
 <h1 id="119">11.9 &nbsp; 计数排序<a class="headerlink" href="#119" title="Permanent link">&para;</a></h1>
-<p>「计数排序 counting sort」通过统计元素数量来实现排序，通常应用于整数数组。</p>
+<p><u>计数排序（counting sort）</u>通过统计元素数量来实现排序，通常应用于整数数组。</p>
 <h2 id="1191">11.9.1 &nbsp; 简单实现<a class="headerlink" href="#1191" title="Permanent link">&para;</a></h2>
 <p>先来看一个简单的例子。给定一个长度为 <span class="arithmatex">\(n\)</span> 的数组 <code>nums</code> ，其中的元素都是“非负整数”，计数排序的整体流程如图 11-16 所示。</p>
 <ol>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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             第 0 章 &nbsp; 前言
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             第 2 章 &nbsp; 复杂度分析
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             第 3 章 &nbsp; 数据结构
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             第 13 章 &nbsp; 回溯
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             第 16 章 &nbsp; 附录
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             纸质书
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@@ -3605,7 +3662,7 @@
 <p class="admonition-title">Tip</p>
 <p>阅读本节前，请确保已学完“堆“章节。</p>
 </div>
-<p>「堆排序 heap sort」是一种基于堆数据结构实现的高效排序算法。我们可以利用已经学过的“建堆操作”和“元素出堆操作”实现堆排序。</p>
+<p><u>堆排序（heap sort）</u>是一种基于堆数据结构实现的高效排序算法。我们可以利用已经学过的“建堆操作”和“元素出堆操作”实现堆排序。</p>
 <ol>
 <li>输入数组并建立小顶堆，此时最小元素位于堆顶。</li>
 <li>不断执行出堆操作，依次记录出堆元素，即可得到从小到大排序的序列。</li>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+        </nav>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2184,7 +2241,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" checked>
         
           
           
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-              <label class="md-nav__link md-nav__link--active" for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link md-nav__link--active" for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2487,7 +2544,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2664,7 +2721,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
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-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2845,7 +2902,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
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-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
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             第 14 章 &nbsp; 动态规划
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
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             参考文献
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@@ -3419,7 +3476,7 @@
         
         
         
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+            序
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             第 4 章 &nbsp; 数组与链表
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             第 14 章 &nbsp; 动态规划
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             纸质书
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 <!-- Page content -->
 <h1 id="114">11.4 &nbsp; 插入排序<a class="headerlink" href="#114" title="Permanent link">&para;</a></h1>
-<p>「插入排序 insertion sort」是一种简单的排序算法，它的工作原理与手动整理一副牌的过程非常相似。</p>
+<p><u>插入排序（insertion sort）</u>是一种简单的排序算法，它的工作原理与手动整理一副牌的过程非常相似。</p>
 <p>具体来说，我们在未排序区间选择一个基准元素，将该元素与其左侧已排序区间的元素逐一比较大小，并将该元素插入到正确的位置。</p>
 <p>图 11-6 展示了数组插入元素的操作流程。设基准元素为 <code>base</code> ，我们需要将从目标索引到 <code>base</code> 之间的所有元素向右移动一位，然后将 <code>base</code> 赋值给目标索引。</p>
 <p><a class="glightbox" href="../insertion_sort.assets/insertion_operation.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="单次插入操作" class="animation-figure" src="../insertion_sort.assets/insertion_operation.png" /></a></p>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
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+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
           </label>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
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             第 4 章 &nbsp; 数组与链表
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
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-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
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             第 6 章 &nbsp; 哈希表
           </label>
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+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
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             第 8 章 &nbsp; 堆
           </label>
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-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
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             第 9 章 &nbsp; 图
           </label>
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-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_13">
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             第 11 章 &nbsp; 排序
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+          <label class="md-nav__title" for="__nav_14">
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             第 12 章 &nbsp; 分治
           </label>
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+          <label class="md-nav__title" for="__nav_15">
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             第 13 章 &nbsp; 回溯
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+          <label class="md-nav__title" for="__nav_16">
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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             参考文献
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     纸质书
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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 <!-- Page content -->
 <h1 id="116">11.6 &nbsp; 归并排序<a class="headerlink" href="#116" title="Permanent link">&para;</a></h1>
-<p>「归并排序 merge sort」是一种基于分治策略的排序算法，包含图 11-10 所示的“划分”和“合并”阶段。</p>
+<p><u>归并排序（merge sort）</u>是一种基于分治策略的排序算法，包含图 11-10 所示的“划分”和“合并”阶段。</p>
 <ol>
 <li><strong>划分阶段</strong>：通过递归不断地将数组从中点处分开，将长数组的排序问题转换为短数组的排序问题。</li>
 <li><strong>合并阶段</strong>：当子数组长度为 1 时终止划分，开始合并，持续地将左右两个较短的有序数组合并为一个较长的有序数组，直至结束。</li>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
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+
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
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+  
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@@ -349,7 +406,7 @@
         
         
         
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@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
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+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
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+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
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+          <label class="md-nav__title" for="__nav_6">
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             第 4 章 &nbsp; 数组与链表
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+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
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+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
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+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
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+          <label class="md-nav__title" for="__nav_10">
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             第 8 章 &nbsp; 堆
           </label>
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             第 9 章 &nbsp; 图
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+          <label class="md-nav__title" for="__nav_12">
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             第 10 章 &nbsp; 搜索
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             第 11 章 &nbsp; 排序
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             第 12 章 &nbsp; 分治
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
           </label>
@@ -3504,7 +3561,7 @@
         
         
         
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -3655,7 +3712,7 @@
 
 <!-- Page content -->
 <h1 id="115">11.5 &nbsp; 快速排序<a class="headerlink" href="#115" title="Permanent link">&para;</a></h1>
-<p>「快速排序 quick sort」是一种基于分治策略的排序算法，运行高效，应用广泛。</p>
+<p><u>快速排序（quick sort）</u>是一种基于分治策略的排序算法，运行高效，应用广泛。</p>
 <p>快速排序的核心操作是“哨兵划分”，其目标是：选择数组中的某个元素作为“基准数”，将所有小于基准数的元素移到其左侧，而大于基准数的元素移到其右侧。具体来说，哨兵划分的流程如图 11-8 所示。</p>
 <ol>
 <li>选取数组最左端元素作为基准数，初始化两个指针 <code>i</code> 和 <code>j</code> 分别指向数组的两端。</li>
@@ -5150,7 +5207,7 @@ aria-label="页脚"
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     <div class="md-copyright__highlight">
-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
diff --git a/chapter_sorting/radix_sort/index.html b/chapter_sorting/radix_sort/index.html
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+    
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+    
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+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
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+  
+
+            </a>
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+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
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+  
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
           </label>
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+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
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+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
           </label>
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             第 3 章 &nbsp; 数据结构
           </label>
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             第 4 章 &nbsp; 数组与链表
           </label>
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             第 5 章 &nbsp; 栈与队列
           </label>
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+          <label class="md-nav__title" for="__nav_8">
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             第 6 章 &nbsp; 哈希表
           </label>
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+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
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             第 8 章 &nbsp; 堆
           </label>
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             第 9 章 &nbsp; 图
           </label>
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             </a>
             
               
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             第 10 章 &nbsp; 搜索
           </label>
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             第 11 章 &nbsp; 排序
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             第 12 章 &nbsp; 分治
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             第 13 章 &nbsp; 回溯
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             纸质书
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 <!-- Page content -->
 <h1 id="1110">11.10 &nbsp; 基数排序<a class="headerlink" href="#1110" title="Permanent link">&para;</a></h1>
 <p>上一节介绍了计数排序，它适用于数据量 <span class="arithmatex">\(n\)</span> 较大但数据范围 <span class="arithmatex">\(m\)</span> 较小的情况。假设我们需要对 <span class="arithmatex">\(n = 10^6\)</span> 个学号进行排序，而学号是一个 <span class="arithmatex">\(8\)</span> 位数字，这意味着数据范围 <span class="arithmatex">\(m = 10^8\)</span> 非常大，使用计数排序需要分配大量内存空间，而基数排序可以避免这种情况。</p>
-<p>「基数排序 radix sort」的核心思想与计数排序一致，也通过统计个数来实现排序。在此基础上，基数排序利用数字各位之间的递进关系，依次对每一位进行排序，从而得到最终的排序结果。</p>
+<p><u>基数排序（radix sort）</u>的核心思想与计数排序一致，也通过统计个数来实现排序。在此基础上，基数排序利用数字各位之间的递进关系，依次对每一位进行排序，从而得到最终的排序结果。</p>
 <h2 id="11101">11.10.1 &nbsp; 算法流程<a class="headerlink" href="#11101" title="Permanent link">&para;</a></h2>
 <p>以学号数据为例，假设数字的最低位是第 <span class="arithmatex">\(1\)</span> 位，最高位是第 <span class="arithmatex">\(8\)</span> 位，基数排序的流程如图 11-18 所示。</p>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
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             第 5 章 &nbsp; 栈与队列
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             第 6 章 &nbsp; 哈希表
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             第 7 章 &nbsp; 树
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             第 9 章 &nbsp; 图
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             第 10 章 &nbsp; 搜索
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+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2536,7 +2593,7 @@
         
         
         
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           </label>
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             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
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+          <label class="md-nav__title" for="__nav_16">
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             第 14 章 &nbsp; 动态规划
           </label>
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+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
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+          <label class="md-nav__title" for="__nav_19">
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             参考文献
           </label>
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+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -3583,7 +3640,7 @@
 
 <!-- Page content -->
 <h1 id="112">11.2 &nbsp; 选择排序<a class="headerlink" href="#112" title="Permanent link">&para;</a></h1>
-<p>「选择排序 selection sort」的工作原理非常简单：开启一个循环，每轮从未排序区间选择最小的元素，将其放到已排序区间的末尾。</p>
+<p><u>选择排序（selection sort）</u>的工作原理非常简单：开启一个循环，每轮从未排序区间选择最小的元素，将其放到已排序区间的末尾。</p>
 <p>设数组的长度为 <span class="arithmatex">\(n\)</span> ，选择排序的算法流程如图 11-2 所示。</p>
 <ol>
 <li>初始状态下，所有元素未排序，即未排序（索引）区间为 <span class="arithmatex">\([0, n-1]\)</span> 。</li>
@@ -4069,7 +4126,7 @@ aria-label="页脚"
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
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             </a>
             
               
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             第 0 章 &nbsp; 前言
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             第 2 章 &nbsp; 复杂度分析
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             第 3 章 &nbsp; 数据结构
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             第 4 章 &nbsp; 数组与链表
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             第 5 章 &nbsp; 栈与队列
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             第 6 章 &nbsp; 哈希表
           </label>
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             第 7 章 &nbsp; 树
           </label>
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@@ -3601,16 +3658,16 @@
 
 <!-- Page content -->
 <h1 id="111">11.1 &nbsp; 排序算法<a class="headerlink" href="#111" title="Permanent link">&para;</a></h1>
-<p>「排序算法 sorting algorithm」用于对一组数据按照特定顺序进行排列。排序算法有着广泛的应用，因为有序数据通常能够被更高效地查找、分析和处理。</p>
+<p><u>排序算法（sorting algorithm）</u>用于对一组数据按照特定顺序进行排列。排序算法有着广泛的应用，因为有序数据通常能够被更高效地查找、分析和处理。</p>
 <p>如图 11-1 所示，排序算法中的数据类型可以是整数、浮点数、字符或字符串等。排序的判断规则可根据需求设定，如数字大小、字符 ASCII 码顺序或自定义规则。</p>
 <p><a class="glightbox" href="../sorting_algorithm.assets/sorting_examples.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="数据类型和判断规则示例" class="animation-figure" src="../sorting_algorithm.assets/sorting_examples.png" /></a></p>
 <p align="center"> 图 11-1 &nbsp; 数据类型和判断规则示例 </p>
 
 <h2 id="1111">11.1.1 &nbsp; 评价维度<a class="headerlink" href="#1111" title="Permanent link">&para;</a></h2>
 <p><strong>运行效率</strong>：我们期望排序算法的时间复杂度尽量低，且总体操作数量较少（时间复杂度中的常数项变小）。对于大数据量的情况，运行效率显得尤为重要。</p>
-<p><strong>就地性</strong>：顾名思义，「原地排序」通过在原数组上直接操作实现排序，无须借助额外的辅助数组，从而节省内存。通常情况下，原地排序的数据搬运操作较少，运行速度也更快。</p>
-<p><strong>稳定性</strong>：「稳定排序」在完成排序后，相等元素在数组中的相对顺序不发生改变。</p>
-<p>稳定排序是多级排序场景的必要条件。假设我们有一个存储学生信息的表格，第 1 列和第 2 列分别是姓名和年龄。在这种情况下，「非稳定排序」可能导致输入数据的有序性丧失：</p>
+<p><strong>就地性</strong>：顾名思义，<u>原地排序</u>通过在原数组上直接操作实现排序，无须借助额外的辅助数组，从而节省内存。通常情况下，原地排序的数据搬运操作较少，运行速度也更快。</p>
+<p><strong>稳定性</strong>：<u>稳定排序</u>在完成排序后，相等元素在数组中的相对顺序不发生改变。</p>
+<p>稳定排序是多级排序场景的必要条件。假设我们有一个存储学生信息的表格，第 1 列和第 2 列分别是姓名和年龄。在这种情况下，<u>非稳定排序</u>可能导致输入数据的有序性丧失：</p>
 <div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># 输入数据是按照姓名排序好的</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="c1"># (name, age)</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="w">  </span><span class="o">(</span><span class="s1">&#39;A&#39;</span>,<span class="w"> </span><span class="m">19</span><span class="o">)</span>
@@ -3628,9 +3685,9 @@
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="w">  </span><span class="o">(</span><span class="s1">&#39;C&#39;</span>,<span class="w"> </span><span class="m">21</span><span class="o">)</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="w">  </span><span class="o">(</span><span class="s1">&#39;E&#39;</span>,<span class="w"> </span><span class="m">23</span><span class="o">)</span>
 </code></pre></div>
-<p><strong>自适应性</strong>：「自适应排序」的时间复杂度会受输入数据的影响，即最佳时间复杂度、最差时间复杂度、平均时间复杂度并不完全相等。</p>
+<p><strong>自适应性</strong>：<u>自适应排序</u>的时间复杂度会受输入数据的影响，即最佳时间复杂度、最差时间复杂度、平均时间复杂度并不完全相等。</p>
 <p>自适应性需要根据具体情况来评估。如果最差时间复杂度差于平均时间复杂度，说明排序算法在某些数据下性能可能劣化，因此被视为负面属性；而如果最佳时间复杂度优于平均时间复杂度，则被视为正面属性。</p>
-<p><strong>是否基于比较</strong>：「基于比较的排序」依赖比较运算符（<span class="arithmatex">\(&lt;\)</span>、<span class="arithmatex">\(=\)</span>、<span class="arithmatex">\(&gt;\)</span>）来判断元素的相对顺序，从而排序整个数组，理论最优时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。而「非比较排序」不使用比较运算符，时间复杂度可达 <span class="arithmatex">\(O(n)\)</span> ，但其通用性相对较差。</p>
+<p><strong>是否基于比较</strong>：<u>基于比较的排序</u>依赖比较运算符（<span class="arithmatex">\(&lt;\)</span>、<span class="arithmatex">\(=\)</span>、<span class="arithmatex">\(&gt;\)</span>）来判断元素的相对顺序，从而排序整个数组，理论最优时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。而<u>非比较排序</u>不使用比较运算符，时间复杂度可达 <span class="arithmatex">\(O(n)\)</span> ，但其通用性相对较差。</p>
 <h2 id="1112">11.1.2 &nbsp; 理想排序算法<a class="headerlink" href="#1112" title="Permanent link">&para;</a></h2>
 <p><strong>运行快、原地、稳定、正向自适应、通用性好</strong>。显然，迄今为止尚未发现兼具以上所有特性的排序算法。因此，在选择排序算法时，需要根据具体的数据特点和问题需求来决定。</p>
 <p>接下来，我们将共同学习各种排序算法，并基于上述评价维度对各个排序算法的优缺点进行分析。</p>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1464,7 +1521,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1482,14 +1539,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1660,7 +1717,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1814,7 +1871,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1832,14 +1889,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1972,7 +2029,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -1990,14 +2047,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2184,7 +2241,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" checked>
         
           
           
@@ -2202,14 +2259,14 @@
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2545,7 +2602,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2722,7 +2779,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
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-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2903,7 +2960,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
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-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3122,7 +3179,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
@@ -3140,14 +3197,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3295,7 +3352,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3420,7 +3477,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3477,7 +3534,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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     纸质书
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-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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@@ -3819,7 +3876,7 @@ aria-label="页脚"
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+++ b/chapter_stack_and_queue/deque/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
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+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
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+  
+  <span class="md-ellipsis">
+    序
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+  
+
+            </a>
+            
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+        
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
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+          </ul>
+        </nav>
+      
+    </li>
+  
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+    
+      
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+  
+  
+  
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
           </label>
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+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
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             第 4 章 &nbsp; 数组与链表
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+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
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-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
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+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
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             第 8 章 &nbsp; 堆
           </label>
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-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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+          <label class="md-nav__title" for="__nav_11">
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             第 9 章 &nbsp; 图
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
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+          <label class="md-nav__title" for="__nav_13">
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             第 11 章 &nbsp; 排序
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             第 12 章 &nbsp; 分治
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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             参考文献
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     纸质书
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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@@ -3667,7 +3724,7 @@
 
 <!-- Page content -->
 <h1 id="53">5.3 &nbsp; 双向队列<a class="headerlink" href="#53" title="Permanent link">&para;</a></h1>
-<p>在队列中，我们仅能删除头部元素或在尾部添加元素。如图 5-7 所示，「双向队列 double-ended queue」提供了更高的灵活性，允许在头部和尾部执行元素的添加或删除操作。</p>
+<p>在队列中，我们仅能删除头部元素或在尾部添加元素。如图 5-7 所示，<u>双向队列（double-ended queue）</u>提供了更高的灵活性，允许在头部和尾部执行元素的添加或删除操作。</p>
 <p><a class="glightbox" href="../deque.assets/deque_operations.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="双向队列的操作" class="animation-figure" src="../deque.assets/deque_operations.png" /></a></p>
 <p align="center"> 图 5-7 &nbsp; 双向队列的操作 </p>
 
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+          
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+            <a href="../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
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+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
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+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
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+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
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+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
           </label>
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             第 4 章 &nbsp; 数组与链表
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+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
           </label>
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-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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             第 6 章 &nbsp; 哈希表
           </label>
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+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
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+          <label class="md-nav__title" for="__nav_10">
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             第 8 章 &nbsp; 堆
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+          <label class="md-nav__title" for="__nav_11">
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             第 9 章 &nbsp; 图
           </label>
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-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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+          <label class="md-nav__title" for="__nav_12">
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             第 10 章 &nbsp; 搜索
           </label>
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             第 11 章 &nbsp; 排序
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             第 12 章 &nbsp; 分治
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
@@ -3381,8 +3438,8 @@
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3419,7 +3476,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
@@ -3427,7 +3484,7 @@
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19 2-5 4.5v11l5-4.5V2M6.5 5C4.55 5 2.45 5.4 1 6.5v14.66c0 .25.25.5.5.5.1 0 .15-.07.25-.07 1.35-.65 3.3-1.09 4.75-1.09 1.95 0 4.05.4 5.5 1.5 1.35-.85 3.8-1.5 5.5-1.5 1.65 0 3.35.31 4.75 1.06.1.05.15.03.25.03.25 0 .5-.25.5-.5V6.5c-.6-.45-1.25-.75-2-1V19c-1.1-.35-2.3-.5-3.5-.5-1.7 0-4.15.65-5.5 1.5V6.5C10.55 5.4 8.45 5 6.5 5Z"/></svg>
   
   <span class="md-ellipsis">
     纸质书
@@ -3446,8 +3503,8 @@
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -3734,7 +3791,7 @@ aria-label="页脚"
       <div class="md-copyright">
   
     <div class="md-copyright__highlight">
-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
diff --git a/chapter_stack_and_queue/queue/index.html b/chapter_stack_and_queue/queue/index.html
index dc1d15501..a40280c6e 100644
--- a/chapter_stack_and_queue/queue/index.html
+++ b/chapter_stack_and_queue/queue/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
         
       
         
@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1154,7 +1211,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" checked>
         
           
           
@@ -1172,14 +1229,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1399,7 +1456,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1417,14 +1474,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1557,7 +1614,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
         
           
           
@@ -1575,14 +1632,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1753,7 +1810,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1771,14 +1828,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1907,7 +1964,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1925,14 +1982,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -2065,7 +2122,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -2083,14 +2140,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2275,7 +2332,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2293,14 +2350,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2578,7 +2635,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2596,14 +2653,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2755,7 +2812,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
@@ -2773,14 +2830,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2936,7 +2993,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
@@ -2954,14 +3011,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3155,7 +3212,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
@@ -3173,14 +3230,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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             参考文献
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             纸质书
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@@ -3667,7 +3724,7 @@
 
 <!-- Page content -->
 <h1 id="52">5.2 &nbsp; 队列<a class="headerlink" href="#52" title="Permanent link">&para;</a></h1>
-<p>「队列 queue」是一种遵循先入先出规则的线性数据结构。顾名思义，队列模拟了排队现象，即新来的人不断加入队列尾部，而位于队列头部的人逐个离开。</p>
+<p><u>队列（queue）</u>是一种遵循先入先出规则的线性数据结构。顾名思义，队列模拟了排队现象，即新来的人不断加入队列尾部，而位于队列头部的人逐个离开。</p>
 <p>如图 5-4 所示，我们将队列头部称为“队首”，尾部称为“队尾”，将把元素加入队尾的操作称为“入队”，删除队首元素的操作称为“出队”。</p>
 <p><a class="glightbox" href="../queue.assets/queue_operations.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="队列的先入先出规则" class="animation-figure" src="../queue.assets/queue_operations.png" /></a></p>
 <p align="center"> 图 5-4 &nbsp; 队列的先入先出规则 </p>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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             第 0 章 &nbsp; 前言
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             第 2 章 &nbsp; 复杂度分析
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             第 3 章 &nbsp; 数据结构
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             第 5 章 &nbsp; 栈与队列
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             第 6 章 &nbsp; 哈希表
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             第 7 章 &nbsp; 树
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             第 8 章 &nbsp; 堆
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             第 10 章 &nbsp; 搜索
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             第 11 章 &nbsp; 排序
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             第 16 章 &nbsp; 附录
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             参考文献
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             纸质书
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@@ -3685,7 +3742,7 @@
 
 <!-- Page content -->
 <h1 id="51">5.1 &nbsp; 栈<a class="headerlink" href="#51" title="Permanent link">&para;</a></h1>
-<p>「栈 stack」是一种遵循先入后出逻辑的线性数据结构。</p>
+<p><u>栈（stack）</u>是一种遵循先入后出逻辑的线性数据结构。</p>
 <p>我们可以将栈类比为桌面上的一摞盘子，如果想取出底部的盘子，则需要先将上面的盘子依次移走。我们将盘子替换为各种类型的元素（如整数、字符、对象等），就得到了栈这种数据结构。</p>
 <p>如图 5-1 所示，我们把堆叠元素的顶部称为“栈顶”，底部称为“栈底”。将把元素添加到栈顶的操作叫作“入栈”，删除栈顶元素的操作叫作“出栈”。</p>
 <p><a class="glightbox" href="../stack.assets/stack_operations.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="栈的先入后出规则" class="animation-figure" src="../stack.assets/stack_operations.png" /></a></p>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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             第 0 章 &nbsp; 前言
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             第 3 章 &nbsp; 数据结构
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             第 5 章 &nbsp; 栈与队列
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             第 6 章 &nbsp; 哈希表
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             第 7 章 &nbsp; 树
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             第 8 章 &nbsp; 堆
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
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+          <label class="md-nav__title" for="__nav_11">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2242,7 +2299,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
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             第 11 章 &nbsp; 排序
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@@ -2545,7 +2602,7 @@
         
         
         
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             第 13 章 &nbsp; 回溯
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
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             第 14 章 &nbsp; 动态规划
           </label>
@@ -3122,7 +3179,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
           </label>
@@ -3295,7 +3352,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3420,7 +3477,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
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             参考文献
           </label>
@@ -3477,7 +3534,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_tree/array_representation_of_tree/index.html b/chapter_tree/array_representation_of_tree/index.html
index 7adc63c5e..d9358de1c 100644
--- a/chapter_tree/array_representation_of_tree/index.html
+++ b/chapter_tree/array_representation_of_tree/index.html
@@ -334,6 +334,63 @@
           
         
       
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+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
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+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
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+  
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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             第 3 章 &nbsp; 数据结构
           </label>
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-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1466,7 +1523,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" checked>
         
           
           
@@ -1484,14 +1541,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="true">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_11">
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2251,7 +2308,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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             第 11 章 &nbsp; 排序
           </label>
@@ -2554,7 +2611,7 @@
         
         
         
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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           </label>
@@ -2731,7 +2788,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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+          <label class="md-nav__title" for="__nav_15">
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             第 13 章 &nbsp; 回溯
           </label>
@@ -2912,7 +2969,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
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             第 14 章 &nbsp; 动态规划
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@@ -3131,7 +3188,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
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+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
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@@ -3304,7 +3361,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3429,7 +3486,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
@@ -3486,7 +3543,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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diff --git a/chapter_tree/avl_tree/index.html b/chapter_tree/avl_tree/index.html
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--- a/chapter_tree/avl_tree/index.html
+++ b/chapter_tree/avl_tree/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
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+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
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           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
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-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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             第 3 章 &nbsp; 数据结构
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
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             第 4 章 &nbsp; 数组与链表
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
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-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
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-    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><!--! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M96 0C43 0 0 43 0 96v320c0 53 43 96 96 96h320c17.7 0 32-14.3 32-32s-14.3-32-32-32v-64c17.7 0 32-14.3 32-32V32c0-17.7-14.3-32-32-32H96zm0 384h256v64H96c-17.7 0-32-14.3-32-32s14.3-32 32-32zm32-240c0-8.8 7.2-16 16-16h192c8.8 0 16 7.2 16 16s-7.2 16-16 16H144c-8.8 0-16-7.2-16-16zm16 48h192c8.8 0 16 7.2 16 16s-7.2 16-16 16H144c-8.8 0-16-7.2-16-16s7.2-16 16-16z"/></svg>
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 <p><a class="glightbox" href="../avl_tree.assets/avltree_degradation_from_inserting_node.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="AVL 树在插入节点后发生退化" class="animation-figure" src="../avl_tree.assets/avltree_degradation_from_inserting_node.png" /></a></p>
 <p align="center"> 图 7-25 &nbsp; AVL 树在插入节点后发生退化 </p>
 
-<p>1962 年 G. M. Adelson-Velsky 和 E. M. Landis 在论文“An algorithm for the organization of information”中提出了「AVL 树」。论文中详细描述了一系列操作，确保在持续添加和删除节点后，AVL 树不会退化，从而使得各种操作的时间复杂度保持在 <span class="arithmatex">\(O(\log n)\)</span> 级别。换句话说，在需要频繁进行增删查改操作的场景中，AVL 树能始终保持高效的数据操作性能，具有很好的应用价值。</p>
+<p>1962 年 G. M. Adelson-Velsky 和 E. M. Landis 在论文“An algorithm for the organization of information”中提出了<u>AVL（树）</u>。论文中详细描述了一系列操作，确保在持续添加和删除节点后，AVL 树不会退化，从而使得各种操作的时间复杂度保持在 <span class="arithmatex">\(O(\log n)\)</span> 级别。换句话说，在需要频繁进行增删查改操作的场景中，AVL 树能始终保持高效的数据操作性能，具有很好的应用价值。</p>
 <h2 id="751-avl">7.5.1 &nbsp; AVL 树常见术语<a class="headerlink" href="#751-avl" title="Permanent link">&para;</a></h2>
-<p>AVL 树既是二叉搜索树，也是平衡二叉树，同时满足这两类二叉树的所有性质，因此是一种「平衡二叉搜索树 balanced binary search tree」。</p>
+<p>AVL 树既是二叉搜索树，也是平衡二叉树，同时满足这两类二叉树的所有性质，因此是一种<u>平衡二叉搜索树（balanced binary search tree）</u>。</p>
 <h3 id="1">1. &nbsp; 节点高度<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
 <p>由于 AVL 树的相关操作需要获取节点高度，因此我们需要为节点类添加 <code>height</code> 变量：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:14"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><input id="__tabbed_1_14" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label><label for="__tabbed_1_14">Zig</label></div>
@@ -4268,7 +4325,7 @@
 </div>
 </div>
 <h3 id="2">2. &nbsp; 节点平衡因子<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
-<p>节点的「平衡因子 balance factor」定义为节点左子树的高度减去右子树的高度，同时规定空节点的平衡因子为 <span class="arithmatex">\(0\)</span> 。我们同样将获取节点平衡因子的功能封装成函数，方便后续使用：</p>
+<p>节点的<u>平衡因子（balance factor）</u>定义为节点左子树的高度减去右子树的高度，同时规定空节点的平衡因子为 <span class="arithmatex">\(0\)</span> 。我们同样将获取节点平衡因子的功能封装成函数，方便后续使用：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:14"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><input id="__tabbed_3_14" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label><label for="__tabbed_3_14">Zig</label></div>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
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+  
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+            </a>
+            
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+        
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+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
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+          </ul>
+        </nav>
+      
+    </li>
+  
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+    
+      
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+  
+  
+  
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             </a>
             
               
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
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             第 0 章 &nbsp; 前言
           </label>
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+          <label class="md-nav__title" for="__nav_3">
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             第 1 章 &nbsp; 初识算法
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
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             第 2 章 &nbsp; 复杂度分析
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-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
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             第 3 章 &nbsp; 数据结构
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
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             第 4 章 &nbsp; 数组与链表
           </label>
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+          <label class="md-nav__title" for="__nav_7">
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             第 5 章 &nbsp; 栈与队列
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-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
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+          <label class="md-nav__title" for="__nav_9">
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             第 7 章 &nbsp; 树
           </label>
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+          <label class="md-nav__title" for="__nav_10">
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             第 8 章 &nbsp; 堆
           </label>
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
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             第 9 章 &nbsp; 图
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
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-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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+          <label class="md-nav__title" for="__nav_13">
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             第 11 章 &nbsp; 排序
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             第 12 章 &nbsp; 分治
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             第 13 章 &nbsp; 回溯
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             第 14 章 &nbsp; 动态规划
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             第 15 章 &nbsp; 贪心
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             第 16 章 &nbsp; 附录
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             参考文献
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     纸质书
@@ -3555,8 +3612,8 @@
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
             <span class="md-nav__icon md-icon"></span>
             纸质书
           </label>
@@ -3703,7 +3760,7 @@
 
 <!-- Page content -->
 <h1 id="74">7.4 &nbsp; 二叉搜索树<a class="headerlink" href="#74" title="Permanent link">&para;</a></h1>
-<p>如图 7-16 所示，「二叉搜索树 binary search tree」满足以下条件。</p>
+<p>如图 7-16 所示，<u>二叉搜索树（binary search tree）</u>满足以下条件。</p>
 <ol>
 <li>对于根节点，左子树中所有节点的值 <span class="arithmatex">\(&lt;\)</span> 根节点的值 <span class="arithmatex">\(&lt;\)</span> 右子树中所有节点的值。</li>
 <li>任意节点的左、右子树也是二叉搜索树，即同样满足条件 <code>1.</code> 。</li>
@@ -5441,7 +5498,7 @@ aria-label="页脚"
       <div class="md-copyright">
   
     <div class="md-copyright__highlight">
-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
diff --git a/chapter_tree/binary_tree/index.html b/chapter_tree/binary_tree/index.html
index cdd6215f0..aca263ab0 100644
--- a/chapter_tree/binary_tree/index.html
+++ b/chapter_tree/binary_tree/index.html
@@ -334,6 +334,63 @@
           
         
       
+    
+    
+    
+    
+    <li class="md-nav__item md-nav__item--nested">
+      
+        
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
         
       
         
@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1466,7 +1523,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" checked>
         
           
           
@@ -1484,14 +1541,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1804,7 +1861,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1822,14 +1879,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1958,7 +2015,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1976,14 +2033,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -2116,7 +2173,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -2134,14 +2191,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2326,7 +2383,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2344,14 +2401,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2629,7 +2686,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2647,14 +2704,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2806,7 +2863,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
@@ -2824,14 +2881,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2987,7 +3044,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
@@ -3005,14 +3062,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_15">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3206,7 +3263,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
@@ -3224,14 +3281,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_16">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3379,7 +3436,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
           
@@ -3397,14 +3454,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_17">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3504,7 +3561,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
         
           
           
@@ -3523,8 +3580,8 @@
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
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 <!-- Page content -->
 <h1 id="71">7.1 &nbsp; 二叉树<a class="headerlink" href="#71" title="Permanent link">&para;</a></h1>
-<p>「二叉树 binary tree」是一种非线性数据结构，代表“祖先”与“后代”之间的派生关系，体现了“一分为二”的分治逻辑。与链表类似，二叉树的基本单元是节点，每个节点包含值、左子节点引用和右子节点引用。</p>
+<p><u>二叉树（binary tree）</u>是一种非线性数据结构，代表“祖先”与“后代”之间的派生关系，体现了“一分为二”的分治逻辑。与链表类似，二叉树的基本单元是节点，每个节点包含值、左子节点引用和右子节点引用。</p>
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@@ -3942,7 +3999,7 @@
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-<p>每个节点都有两个引用（指针），分别指向「左子节点 left-child node」和「右子节点 right-child node」，该节点被称为这两个子节点的「父节点 parent node」。当给定一个二叉树的节点时，我们将该节点的左子节点及其以下节点形成的树称为该节点的「左子树 left subtree」，同理可得「右子树 right subtree」。</p>
+<p>每个节点都有两个引用（指针），分别指向<u>左子节点（left-child node）</u>和<u>右子节点（right-child node）</u>，该节点被称为这两个子节点的<u>父节点（parent node）</u>。当给定一个二叉树的节点时，我们将该节点的左子节点及其以下节点形成的树称为该节点的<u>左子树（left subtree）</u>，同理可得<u>右子树（right subtree）</u>。</p>
 <p><strong>在二叉树中，除叶节点外，其他所有节点都包含子节点和非空子树</strong>。如图 7-1 所示，如果将“节点 2”视为父节点，则其左子节点和右子节点分别是“节点 4”和“节点 5”，左子树是“节点 4 及其以下节点形成的树”，右子树是“节点 5 及其以下节点形成的树”。</p>
 <p><a class="glightbox" href="../binary_tree.assets/binary_tree_definition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="父节点、子节点、子树" class="animation-figure" src="../binary_tree.assets/binary_tree_definition.png" /></a></p>
 <p align="center"> 图 7-1 &nbsp; 父节点、子节点、子树 </p>
@@ -3950,14 +4007,14 @@
 <h2 id="711">7.1.1 &nbsp; 二叉树常见术语<a class="headerlink" href="#711" title="Permanent link">&para;</a></h2>
 <p>二叉树的常用术语如图 7-2 所示。</p>
 <ul>
-<li>「根节点 root node」：位于二叉树顶层的节点，没有父节点。</li>
-<li>「叶节点 leaf node」：没有子节点的节点，其两个指针均指向 <code>None</code> 。</li>
-<li>「边 edge」：连接两个节点的线段，即节点引用（指针）。</li>
-<li>节点所在的「层 level」：从顶至底递增，根节点所在层为 1 。</li>
-<li>节点的「度 degree」：节点的子节点的数量。在二叉树中，度的取值范围是 0、1、2 。</li>
-<li>二叉树的「高度 height」：从根节点到最远叶节点所经过的边的数量。</li>
-<li>节点的「深度 depth」：从根节点到该节点所经过的边的数量。</li>
-<li>节点的「高度 height」：从距离该节点最远的叶节点到该节点所经过的边的数量。</li>
+<li><u>根节点（root node）</u>：位于二叉树顶层的节点，没有父节点。</li>
+<li><u>叶节点（leaf node）</u>：没有子节点的节点，其两个指针均指向 <code>None</code> 。</li>
+<li><u>边（edge）</u>：连接两个节点的线段，即节点引用（指针）。</li>
+<li>节点所在的<u>层（level）</u>：从顶至底递增，根节点所在层为 1 。</li>
+<li>节点的<u>度（degree）</u>：节点的子节点的数量。在二叉树中，度的取值范围是 0、1、2 。</li>
+<li>二叉树的<u>高度（height）</u>：从根节点到最远叶节点所经过的边的数量。</li>
+<li>节点的<u>深度（depth）</u>：从根节点到该节点所经过的边的数量。</li>
+<li>节点的<u>高度（height）</u>：从距离该节点最远的叶节点到该节点所经过的边的数量。</li>
 </ul>
 <p><a class="glightbox" href="../binary_tree.assets/binary_tree_terminology.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉树的常用术语" class="animation-figure" src="../binary_tree.assets/binary_tree_terminology.png" /></a></p>
 <p align="center"> 图 7-2 &nbsp; 二叉树的常用术语 </p>
@@ -4306,26 +4363,26 @@
 </div>
 <h2 id="713">7.1.3 &nbsp; 常见二叉树类型<a class="headerlink" href="#713" title="Permanent link">&para;</a></h2>
 <h3 id="1_1">1. &nbsp; 完美二叉树<a class="headerlink" href="#1_1" title="Permanent link">&para;</a></h3>
-<p>如图 7-4 所示，「完美二叉树 perfect binary tree」所有层的节点都被完全填满。在完美二叉树中，叶节点的度为 <span class="arithmatex">\(0\)</span> ，其余所有节点的度都为 <span class="arithmatex">\(2\)</span> ；若树的高度为 <span class="arithmatex">\(h\)</span> ，则节点总数为 <span class="arithmatex">\(2^{h+1} - 1\)</span> ，呈现标准的指数级关系，反映了自然界中常见的细胞分裂现象。</p>
+<p>如图 7-4 所示，<u>完美二叉树（perfect binary tree）</u>所有层的节点都被完全填满。在完美二叉树中，叶节点的度为 <span class="arithmatex">\(0\)</span> ，其余所有节点的度都为 <span class="arithmatex">\(2\)</span> ；若树的高度为 <span class="arithmatex">\(h\)</span> ，则节点总数为 <span class="arithmatex">\(2^{h+1} - 1\)</span> ，呈现标准的指数级关系，反映了自然界中常见的细胞分裂现象。</p>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>请注意，在中文社区中，完美二叉树常被称为「满二叉树」。</p>
+<p>请注意，在中文社区中，完美二叉树常被称为<u>满二叉树</u>。</p>
 </div>
 <p><a class="glightbox" href="../binary_tree.assets/perfect_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完美二叉树" class="animation-figure" src="../binary_tree.assets/perfect_binary_tree.png" /></a></p>
 <p align="center"> 图 7-4 &nbsp; 完美二叉树 </p>
 
 <h3 id="2_1">2. &nbsp; 完全二叉树<a class="headerlink" href="#2_1" title="Permanent link">&para;</a></h3>
-<p>如图 7-5 所示，「完全二叉树 complete binary tree」只有最底层的节点未被填满，且最底层节点尽量靠左填充。</p>
+<p>如图 7-5 所示，<u>完全二叉树（complete binary tree）</u>只有最底层的节点未被填满，且最底层节点尽量靠左填充。</p>
 <p><a class="glightbox" href="../binary_tree.assets/complete_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完全二叉树" class="animation-figure" src="../binary_tree.assets/complete_binary_tree.png" /></a></p>
 <p align="center"> 图 7-5 &nbsp; 完全二叉树 </p>
 
 <h3 id="3">3. &nbsp; 完满二叉树<a class="headerlink" href="#3" title="Permanent link">&para;</a></h3>
-<p>如图 7-6 所示，「完满二叉树 full binary tree」除了叶节点之外，其余所有节点都有两个子节点。</p>
+<p>如图 7-6 所示，<u>完满二叉树（full binary tree）</u>除了叶节点之外，其余所有节点都有两个子节点。</p>
 <p><a class="glightbox" href="../binary_tree.assets/full_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完满二叉树" class="animation-figure" src="../binary_tree.assets/full_binary_tree.png" /></a></p>
 <p align="center"> 图 7-6 &nbsp; 完满二叉树 </p>
 
 <h3 id="4">4. &nbsp; 平衡二叉树<a class="headerlink" href="#4" title="Permanent link">&para;</a></h3>
-<p>如图 7-7 所示，「平衡二叉树 balanced binary tree」中任意节点的左子树和右子树的高度之差的绝对值不超过 1 。</p>
+<p>如图 7-7 所示，<u>平衡二叉树（balanced binary tree）</u>中任意节点的左子树和右子树的高度之差的绝对值不超过 1 。</p>
 <p><a class="glightbox" href="../binary_tree.assets/balanced_binary_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="平衡二叉树" class="animation-figure" src="../binary_tree.assets/balanced_binary_tree.png" /></a></p>
 <p align="center"> 图 7-7 &nbsp; 平衡二叉树 </p>
 
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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             第 1 章 &nbsp; 初识算法
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-    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><!--! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M96 0C43 0 0 43 0 96v320c0 53 43 96 96 96h320c17.7 0 32-14.3 32-32s-14.3-32-32-32v-64c17.7 0 32-14.3 32-32V32c0-17.7-14.3-32-32-32H96zm0 384h256v64H96c-17.7 0-32-14.3-32-32s14.3-32 32-32zm32-240c0-8.8 7.2-16 16-16h192c8.8 0 16 7.2 16 16s-7.2 16-16 16H144c-8.8 0-16-7.2-16-16zm16 48h192c8.8 0 16 7.2 16 16s-7.2 16-16 16H144c-8.8 0-16-7.2-16-16s7.2-16 16-16z"/></svg>
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 <p>从物理结构的角度来看，树是一种基于链表的数据结构，因此其遍历方式是通过指针逐个访问节点。然而，树是一种非线性数据结构，这使得遍历树比遍历链表更加复杂，需要借助搜索算法来实现。</p>
 <p>二叉树常见的遍历方式包括层序遍历、前序遍历、中序遍历和后序遍历等。</p>
 <h2 id="721">7.2.1 &nbsp; 层序遍历<a class="headerlink" href="#721" title="Permanent link">&para;</a></h2>
-<p>如图 7-9 所示，「层序遍历 level-order traversal」从顶部到底部逐层遍历二叉树，并在每一层按照从左到右的顺序访问节点。</p>
-<p>层序遍历本质上属于「广度优先遍历 breadth-first traversal」，也称「广度优先搜索 breadth-first search, BFS」，它体现了一种“一圈一圈向外扩展”的逐层遍历方式。</p>
+<p>如图 7-9 所示，<u>层序遍历（level-order traversal）</u>从顶部到底部逐层遍历二叉树，并在每一层按照从左到右的顺序访问节点。</p>
+<p>层序遍历本质上属于<u>广度优先遍历（breadth-first traversal）</u>，也称<u>广度优先搜索（breadth-first search, BFS）</u>，它体现了一种“一圈一圈向外扩展”的逐层遍历方式。</p>
 <p><a class="glightbox" href="../binary_tree_traversal.assets/binary_tree_bfs.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉树的层序遍历" class="animation-figure" src="../binary_tree_traversal.assets/binary_tree_bfs.png" /></a></p>
 <p align="center"> 图 7-9 &nbsp; 二叉树的层序遍历 </p>
 
@@ -4024,7 +4081,7 @@
 <li><strong>空间复杂度为 <span class="arithmatex">\(O(n)\)</span></strong> ：在最差情况下，即满二叉树时，遍历到最底层之前，队列中最多同时存在 <span class="arithmatex">\((n + 1) / 2\)</span> 个节点，占用 <span class="arithmatex">\(O(n)\)</span> 空间。</li>
 </ul>
 <h2 id="722">7.2.2 &nbsp; 前序、中序、后序遍历<a class="headerlink" href="#722" title="Permanent link">&para;</a></h2>
-<p>相应地，前序、中序和后序遍历都属于「深度优先遍历 depth-first traversal」，也称「深度优先搜索 depth-first search, DFS」，它体现了一种“先走到尽头，再回溯继续”的遍历方式。</p>
+<p>相应地，前序、中序和后序遍历都属于<u>深度优先遍历（depth-first traversal）</u>，也称<u>深度优先搜索（depth-first search, DFS）</u>，它体现了一种“先走到尽头，再回溯继续”的遍历方式。</p>
 <p>图 7-10 展示了对二叉树进行深度优先遍历的工作原理。<strong>深度优先遍历就像是绕着整棵二叉树的外围“走”一圈</strong>，在每个节点都会遇到三个位置，分别对应前序遍历、中序遍历和后序遍历。</p>
 <p><a class="glightbox" href="../binary_tree_traversal.assets/binary_tree_dfs.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二叉搜索树的前序、中序、后序遍历" class="animation-figure" src="../binary_tree_traversal.assets/binary_tree_dfs.png" /></a></p>
 <p align="center"> 图 7-10 &nbsp; 二叉搜索树的前序、中序、后序遍历 </p>
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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+              
+  
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27l-2.77 6.1M5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83-.78.78-2.05.78-2.83 0m-5.66 7.07-1.41-1.41 1.41 1.41M6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22h1.41M2 22h1.41l4.77-4.76-1.42-1.41L2 20.59V22m0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76v1.41Z"/></svg>
+  
+  <span class="md-ellipsis">
+    序
+  </span>
+  
+
+            </a>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
         
       
         
@@ -349,7 +406,7 @@
         
         
         
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1466,7 +1523,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" checked>
         
           
           
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             </a>
             
               
-              <label class="md-nav__link md-nav__link--active" for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link md-nav__link--active" for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1662,7 +1719,7 @@
         
         
         
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1816,7 +1873,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -1974,7 +2031,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2184,7 +2241,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2487,7 +2544,7 @@
         
         
         
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
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-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
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-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
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-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
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             第 15 章 &nbsp; 贪心
           </label>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
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             第 16 章 &nbsp; 附录
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             参考文献
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     纸质书
@@ -3446,8 +3503,8 @@
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_19">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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             纸质书
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@@ -3736,7 +3793,7 @@ aria-label="页脚"
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-      Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
     </div>
   
   
diff --git a/chapter_tree/summary/index.html b/chapter_tree/summary/index.html
index 8bba3d42c..43d67f07c 100644
--- a/chapter_tree/summary/index.html
+++ b/chapter_tree/summary/index.html
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+    
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+      
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+        
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        
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+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_hello_algo/" class="md-nav__link ">
+              
+  
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+  
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+  
+
+            </a>
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+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_1">
+            <span class="md-nav__icon md-icon"></span>
+            序
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
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+    
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@@ -349,7 +406,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
         
           
           
@@ -367,14 +424,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_1" id="__nav_1_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
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               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_1">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_2">
             <span class="md-nav__icon md-icon"></span>
             第 0 章 &nbsp; 前言
           </label>
@@ -480,7 +537,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
         
           
           
@@ -498,14 +555,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_2" id="__nav_2_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_2">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3">
             <span class="md-nav__icon md-icon"></span>
             第 1 章 &nbsp; 初识算法
           </label>
@@ -615,7 +672,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
         
           
           
@@ -633,14 +690,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_3">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_4">
             <span class="md-nav__icon md-icon"></span>
             第 2 章 &nbsp; 复杂度分析
           </label>
@@ -792,7 +849,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
         
           
           
@@ -810,14 +867,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_4" id="__nav_4_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_4">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_5">
             <span class="md-nav__icon md-icon"></span>
             第 3 章 &nbsp; 数据结构
           </label>
@@ -969,7 +1026,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_5" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
         
           
           
@@ -987,14 +1044,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_5">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_6">
             <span class="md-nav__icon md-icon"></span>
             第 4 章 &nbsp; 数组与链表
           </label>
@@ -1152,7 +1209,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
         
           
           
@@ -1170,14 +1227,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_6">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
             <span class="md-nav__icon md-icon"></span>
             第 5 章 &nbsp; 栈与队列
           </label>
@@ -1306,7 +1363,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
         
           
           
@@ -1324,14 +1381,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_7">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
             <span class="md-nav__icon md-icon"></span>
             第 6 章 &nbsp; 哈希表
           </label>
@@ -1466,7 +1523,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" checked>
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" checked>
         
           
           
@@ -1484,14 +1541,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="true">
-          <label class="md-nav__title" for="__nav_8">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_9">
             <span class="md-nav__icon md-icon"></span>
             第 7 章 &nbsp; 树
           </label>
@@ -1720,7 +1777,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
         
           
           
@@ -1738,14 +1795,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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               </label>
             
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_9">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
             <span class="md-nav__icon md-icon"></span>
             第 8 章 &nbsp; 堆
           </label>
@@ -1874,7 +1931,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
         
           
           
@@ -1892,14 +1949,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_10">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
             <span class="md-nav__icon md-icon"></span>
             第 9 章 &nbsp; 图
           </label>
@@ -2032,7 +2089,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
         
           
           
@@ -2050,14 +2107,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
           </div>
         
-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_11">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
             <span class="md-nav__icon md-icon"></span>
             第 10 章 &nbsp; 搜索
           </label>
@@ -2242,7 +2299,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
         
           
           
@@ -2260,14 +2317,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_12">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
             <span class="md-nav__icon md-icon"></span>
             第 11 章 &nbsp; 排序
           </label>
@@ -2545,7 +2602,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
         
           
           
@@ -2563,14 +2620,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_13">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
             <span class="md-nav__icon md-icon"></span>
             第 12 章 &nbsp; 分治
           </label>
@@ -2722,7 +2779,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
         
           
           
@@ -2740,14 +2797,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_14">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
             <span class="md-nav__icon md-icon"></span>
             第 13 章 &nbsp; 回溯
           </label>
@@ -2903,7 +2960,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
         
           
           
@@ -2921,14 +2978,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_16">
             <span class="md-nav__icon md-icon"></span>
             第 14 章 &nbsp; 动态规划
           </label>
@@ -3122,7 +3179,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_16" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
         
           
           
@@ -3140,14 +3197,14 @@
             </a>
             
               
-              <label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_17">
             <span class="md-nav__icon md-icon"></span>
             第 15 章 &nbsp; 贪心
           </label>
@@ -3295,7 +3352,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
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             </a>
             
               
-              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
+              <label class="md-nav__link " for="__nav_18" id="__nav_18_label" tabindex="0">
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_17_label" aria-expanded="false">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_18">
             <span class="md-nav__icon md-icon"></span>
             第 16 章 &nbsp; 附录
           </label>
@@ -3420,7 +3477,7 @@
         
         
         
-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
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-        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
-          <label class="md-nav__title" for="__nav_18">
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_19_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_19">
             <span class="md-nav__icon md-icon"></span>
             参考文献
           </label>
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-        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_19" >
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
         
           
           
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 <h2 id="021-writing-conventions">0.2.1 &nbsp; Writing Conventions<a class="headerlink" href="#021-writing-conventions" title="Permanent link">&para;</a></h2>
 <ul>
 <li>Chapters marked with '*' after the title are optional and contain relatively challenging content. If you are short on time, it is advisable to skip them.</li>
-<li>Key technical terms and their English equivalents are enclosed in <strong>Bold</strong> + <em>italics</em> brackets, for example, <strong><em>array</em></strong>. It's advisable to familiarize yourself with these for better comprehension of technical texts.</li>
-<li>Proprietary terms and words with specific meanings are indicated with “quotation marks” to avoid ambiguity.</li>
+<li>Technical terms will be in boldface (in the print and PDF versions) or underlined (in the web version), for instance, <u>array</u>. It's advisable to familiarize yourself with these for better comprehension of technical texts.</li>
 <li><strong>Bolded text</strong> indicates key content or summary statements, which deserve special attention.</li>
-<li>When it comes to terms that are inconsistent between programming languages, this book follows Python, for example using <span class="arithmatex">\(\text{None}\)</span> to mean "null".</li>
+<li>Words and phrases with specific meanings are indicated with “quotation marks” to avoid ambiguity.</li>
+<li>When it comes to terms that are inconsistent between programming languages, this book follows Python, for example using <code>None</code> to mean <code>null</code>.</li>
 <li>This book partially ignores the comment conventions for programming languages in exchange for a more compact layout of the content. The comments primarily consist of three types: title comments, content comments, and multi-line comments.</li>
 </ul>
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+      Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
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-{"config":{"lang":["en"],"separator":"[\\s\\-]+","pipeline":["stopWordFilter"]},"docs":[{"location":"","title":"Home","text":"Hello Algo <p>Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code</p> <p> </p> <p>        Dive In           Clone Repo           Get PDF    </p> <p>The English edition is brewing...</p> <p>Feel free to engage in Chinese-to-English translation and pull request review! For guidelines, please see #914.</p> <p>Quote</p> <p>\"An easy-to-understand book on data structures and algorithms, which guides readers to learn by minds-on and hands-on. Strongly recommended for algorithm beginners!\"</p> <p>\u2014\u2014 Junhui Deng, Professor of Computer Science, Tsinghua University</p> <p>Quote</p> <p>\"If I had 'Hello Algo' when I was learning data structures and algorithms, it would have been 10 times easier!\"</p> <p>\u2014\u2014 Mu Li, Senior Principal Scientist, Amazon</p> Animated illustrations <p>Easy to understandSmooth learning curve</p> <p>\"A picture is worth a thousand words.\"</p> Off-the-Shelf Code <p>Multi programming languagesRun with one click</p> <p>\"Talk is cheap. Show me the code.\"</p> Learning Together <p>Discussion and questions welcomeReaders progress together</p> <p>\"Learning by teaching.\"</p>  Preface  <p>Two years ago, I shared the \"Sword Offer\" series of problem solutions on LeetCode, which received much love and support from many students. During my interactions with readers, the most common question I encountered was \"How to get started with algorithms.\" Gradually, I developed a deep interest in this question.</p> <p>Blindly solving problems seems to be the most popular method, being simple, direct, and effective. However, problem-solving is like playing a \"Minesweeper\" game, where students with strong self-learning abilities can successfully clear the mines one by one, but those with insufficient foundations may end up bruised from explosions, retreating step by step in frustration. Thoroughly reading textbooks is also common, but for students aiming for job applications, the energy consumed by graduation, resume submissions, and preparing for written tests and interviews makes tackling thick books a daunting challenge.</p> <p>If you are facing similar troubles, then you are lucky to have found this book. This book is my answer to this question, not necessarily the best solution, but at least an active attempt. Although this book won't directly land you an Offer, it will guide you through the \"knowledge map\" of data structures and algorithms, help you understand the shape, size, and distribution of different \"mines,\" and equip you with various \"demining methods.\" With these skills, I believe you can more comfortably solve problems and read literature, gradually building a complete knowledge system.</p> <p>I deeply agree with Professor Feynman's saying: \"Knowledge isn't free. You have to pay attention.\" In this sense, this book is not entirely \"free.\" To not disappoint the precious \"attention\" you pay to this book, I will do my utmost, investing the greatest \"attention\" to complete the creation of this book.</p>  Author  <p>Yudong Jin(krahets), Senior Algorithm Engineer in a top tech company, Master's degree from Shanghai Jiao Tong University. The highest-read blogger across the entire LeetCode, his published \"Illustration of Algorithm Data Structures\" has been subscribed to by over 300k.</p>  Contribution  <p>This book is continuously improved with the joint efforts of many contributors from the open-source community. Thanks to each writer who invested their time and energy, listed in the order generated by GitHub:</p> <p> </p> <p>The code review work for this book was completed by codingonion, Gonglja, gvenusleo, hpstory, justin\u2010tse, krahets, night-cruise, nuomi1, and Reanon (listed in alphabetical order). Thanks to them for their time and effort, ensuring the standardization and uniformity of the code in various languages.</p> <sub>codingonion</sub><sub>Rust, Zig</sub> <sub>Gonglja</sub><sub>C, C++</sub> <sub>gvenusleo</sub><sub>Dart</sub> <sub>hpstory</sub><sub>C#</sub> <sub>justin-tse</sub><sub>JS, TS</sub> <sub>krahets</sub><sub>Java, Python</sub> <sub>night-cruise</sub><sub>Rust</sub> <sub>nuomi1</sub><sub>Swift</sub> <sub>Reanon</sub><sub>Go, C</sub>"},{"location":"chapter_array_and_linkedlist/","title":"Chapter 4. \u00a0 Arrays and Linked Lists","text":"<p>Abstract</p> <p>The world of data structures resembles a sturdy brick wall.</p> <p>In arrays, envision bricks snugly aligned, each resting seamlessly beside the next, creating a unified formation. Meanwhile, in linked lists, these bricks disperse freely, embraced by vines gracefully knitting connections between them.</p>"},{"location":"chapter_array_and_linkedlist/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>4.1 \u00a0 Array</li> <li>4.2 \u00a0 Linked List</li> <li>4.3 \u00a0 List</li> <li>4.4 \u00a0 Memory and Cache</li> <li>4.5 \u00a0 Summary</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/","title":"4.1 \u00a0 Arrays","text":"<p>An \"array\" is a linear data structure that operates as a lineup of similar items, stored together in a computer's memory in contiguous spaces. It's like a sequence that maintains organized storage. Each item in this lineup has its unique 'spot' known as an \"index\". Please refer to the Figure 4-1  to observe how arrays work and grasp these key terms.</p> <p></p> <p> Figure 4-1 \u00a0 Array Definition and Storage Method </p>"},{"location":"chapter_array_and_linkedlist/array/#411-common-operations-on-arrays","title":"4.1.1 \u00a0 Common Operations on Arrays","text":""},{"location":"chapter_array_and_linkedlist/array/#1-initializing-arrays","title":"1. \u00a0 Initializing Arrays","text":"<p>Arrays can be initialized in two ways depending on the needs: either without initial values or with specified initial values. When initial values are not specified, most programming languages will set the array elements to \\(0\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig array.py<pre><code># Initialize array\narr: list[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> array.cpp<pre><code>/* Initialize array */\n// Stored on stack\nint arr[5];\nint nums[5] = { 1, 3, 2, 5, 4 };\n// Stored on heap (manual memory release needed)\nint* arr1 = new int[5];\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n</code></pre> array.java<pre><code>/* Initialize array */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n</code></pre> array.cs<pre><code>/* Initialize array */\nint[] arr = new int[5]; // [ 0, 0, 0, 0, 0 ]\nint[] nums = [1, 3, 2, 5, 4];\n</code></pre> array.go<pre><code>/* Initialize array */\nvar arr [5]int\n// In Go, specifying the length ([5]int) denotes an array, while not specifying it ([]int) denotes a slice.\n// Since Go's arrays are designed to have compile-time fixed length, only constants can be used to specify the length.\n// For convenience in implementing the extend() method, the Slice will be considered as an Array here.\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> array.swift<pre><code>/* Initialize array */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n</code></pre> array.js<pre><code>/* Initialize array */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n</code></pre> array.ts<pre><code>/* Initialize array */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> array.dart<pre><code>/* Initialize array */\nList&lt;int&gt; arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> array.rs<pre><code>/* Initialize array */\nlet arr: Vec&lt;i32&gt; = vec![0; 5]; // [0, 0, 0, 0, 0]\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> array.c<pre><code>/* Initialize array */\nint arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n</code></pre> array.kt<pre><code>\n</code></pre> array.zig<pre><code>// Initialize array\nvar arr = [_]i32{0} ** 5; // { 0, 0, 0, 0, 0 }\nvar nums = [_]i32{ 1, 3, 2, 5, 4 };\n</code></pre>"},{"location":"chapter_array_and_linkedlist/array/#2-accessing-elements","title":"2. \u00a0 Accessing Elements","text":"<p>Elements in an array are stored in contiguous memory spaces, making it simpler to compute each element's memory address. The formula shown in the Figure below aids in determining an element's memory address, utilizing the array's memory address (specifically, the first element's address) and the element's index. This computation streamlines direct access to the desired element.</p> <p></p> <p> Figure 4-2 \u00a0 Memory Address Calculation for Array Elements </p> <p>As observed in the above illustration, array indexing conventionally begins at \\(0\\). While this might appear counterintuitive, considering counting usually starts at \\(1\\), within the address calculation formula, an index is essentially an offset from the memory address. For the first element's address, this offset is \\(0\\), validating its index as \\(0\\).</p> <p>Accessing elements in an array is highly efficient, allowing us to randomly access any element in \\(O(1)\\) time.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def random_access(nums: list[int]) -&gt; int:\n    \"\"\"\u968f\u673a\u8bbf\u95ee\u5143\u7d20\"\"\"\n    # \u5728\u533a\u95f4 [0, len(nums)-1] \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    random_index = random.randint(0, len(nums) - 1)\n    # \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    random_num = nums[random_index]\n    return random_num\n</code></pre> array.cpp<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n    // \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = rand() % size;\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.java<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int[] nums) {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.cs<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint RandomAccess(int[] nums) {\n    Random random = new();\n    // \u5728\u533a\u95f4 [0, nums.Length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = random.Next(nums.Length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.go<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums []int) (randomNum int) {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    randomIndex := rand.Intn(len(nums))\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    randomNum = nums[randomIndex]\n    return\n}\n</code></pre> array.swift<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums: [Int]) -&gt; Int {\n    // \u5728\u533a\u95f4 [0, nums.count) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    let randomIndex = nums.indices.randomElement()!\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    let randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.js<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums) {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    const random_index = Math.floor(Math.random() * nums.length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.ts<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums: number[]): number {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    const random_index = Math.floor(Math.random() * nums.length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.dart<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(List&lt;int&gt; nums) {\n  // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n  int randomIndex = Random().nextInt(nums.length);\n  // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n  int randomNum = nums[randomIndex];\n  return randomNum;\n}\n</code></pre> array.rs<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfn random_access(nums: &amp;[i32]) -&gt; i32 {\n    // \u5728\u533a\u95f4 [0, nums.len()) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    let random_index = rand::thread_rng().gen_range(0..nums.len());\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    let random_num = nums[random_index];\n    random_num\n}\n</code></pre> array.c<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n    // \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = rand() % size;\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.kt<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfun randomAccess(nums: IntArray): Int {\n    // \u5728\u533a\u95f4 [0, nums.size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    val randomIndex = ThreadLocalRandom.current().nextInt(0, nums.size)\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    val randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.rb<pre><code>### \u968f\u673a\u8bbf\u95ee\u5143\u7d20 ###\ndef random_access(nums)\n  # \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n  random_index = Random.rand 0...(nums.length - 1)\n\n  # \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n  nums[random_index]\nend\n</code></pre> array.zig<pre><code>// \u968f\u673a\u8bbf\u95ee\u5143\u7d20\nfn randomAccess(nums: []i32) i32 {\n    // \u5728\u533a\u95f4 [0, nums.len) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6574\u6570\n    var randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    var randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#3-inserting-elements","title":"3. \u00a0 Inserting Elements","text":"<p>Array elements are tightly packed in memory, with no space available to accommodate additional data between them. Illustrated in Figure below, inserting an element in the middle of an array requires shifting all subsequent elements back by one position to create room for the new element.</p> <p></p> <p> Figure 4-3 \u00a0 Array Element Insertion Example </p> <p>It's important to note that due to the fixed length of an array, inserting an element will unavoidably result in the loss of the last element in the array. Solutions to address this issue will be explored in the \"List\" chapter.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def insert(nums: list[int], num: int, index: int):\n    \"\"\"\u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\"\"\"\n    # \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i in range(len(nums) - 1, index, -1):\n        nums[i] = nums[i - 1]\n    # \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n</code></pre> array.cpp<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.java<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid Insert(int[] nums, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = nums.Length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.go<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums []int, num int, index int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i := len(nums) - 1; i &gt; index; i-- {\n        nums[i] = nums[i-1]\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.swift<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i in nums.indices.dropFirst(index).reversed() {\n        nums[i] = nums[i - 1]\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.js<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums, num, index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums: number[], num: number, index: number): void {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 _num */\nvoid insert(List&lt;int&gt; nums, int _num, int index) {\n  // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n  for (var i = nums.length - 1; i &gt; index; i--) {\n    nums[i] = nums[i - 1];\n  }\n  // \u5c06 _num \u8d4b\u7ed9 index \u5904\u5143\u7d20\n  nums[index] = _num;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfn insert(nums: &amp;mut Vec&lt;i32&gt;, num: i32, index: usize) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i in (index + 1..nums.len()).rev() {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.c<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.kt<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfun insert(nums: IntArray, num: Int, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (i in nums.size - 1 downTo index + 1) {\n        nums[i] = nums[i - 1]\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.rb<pre><code>### \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num ###\ndef insert(nums, num, index)\n  # \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n  for i in (nums.length - 1).downto(index + 1)\n    nums[i] = nums[i - 1]\n  end\n\n  # \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n  nums[index] = num\nend\n</code></pre> array.zig<pre><code>// \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\nfn insert(nums: []i32, num: i32, index: usize) void {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    var i = nums.len - 1;\n    while (i &gt; index) : (i -= 1) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#4-deleting-elements","title":"4. \u00a0 Deleting Elements","text":"<p>Similarly, as depicted in the Figure 4-4 , to delete an element at index \\(i\\), all elements following index \\(i\\) must be moved forward by one position.</p> <p></p> <p> Figure 4-4 \u00a0 Array Element Deletion Example </p> <p>Please note that after deletion, the former last element becomes \"meaningless,\" hence requiring no specific modification.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def remove(nums: list[int], index: int):\n    \"\"\"\u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\"\"\"\n    # \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i in range(index, len(nums) - 1):\n        nums[i] = nums[i + 1]\n</code></pre> array.cpp<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int *nums, int size, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.java<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.cs<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid Remove(int[] nums, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; nums.Length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.go<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums []int, index int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i := index; i &lt; len(nums)-1; i++ {\n        nums[i] = nums[i+1]\n    }\n}\n</code></pre> array.swift<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums: inout [Int], index: Int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i in nums.indices.dropFirst(index).dropLast() {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.js<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums, index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.ts<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums: number[], index: number): void {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.dart<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(List&lt;int&gt; nums, int index) {\n  // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n  for (var i = index; i &lt; nums.length - 1; i++) {\n    nums[i] = nums[i + 1];\n  }\n}\n</code></pre> array.rs<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfn remove(nums: &amp;mut Vec&lt;i32&gt;, index: usize) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i in index..nums.len() - 1 {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.c<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(int *nums, int size, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.kt<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfun remove(nums: IntArray, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (i in index..&lt;nums.size - 1) {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.rb<pre><code>### \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 ###\ndef remove(nums, index)\n  # \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n  for i in index...nums.length\n    nums[i] = nums[i + 1] || 0\n  end\nend\n</code></pre> array.zig<pre><code>// \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\nfn remove(nums: []i32, index: usize) void {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    var i = index;\n    while (i &lt; nums.len - 1) : (i += 1) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>In summary, the insertion and deletion operations in arrays present the following disadvantages:</p> <ul> <li>High Time Complexity: Both insertion and deletion in an array have an average time complexity of \\(O(n)\\), where \\(n\\) is the length of the array.</li> <li>Loss of Elements: Due to the fixed length of arrays, elements that exceed the array's capacity are lost during insertion.</li> <li>Waste of Memory: Initializing a longer array and utilizing only the front part results in \"meaningless\" end elements during insertion, leading to some wasted memory space.</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#5-traversing-arrays","title":"5. \u00a0 Traversing Arrays","text":"<p>In most programming languages, we can traverse an array either by using indices or by directly iterating over each element:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def traverse(nums: list[int]):\n    \"\"\"\u904d\u5386\u6570\u7ec4\"\"\"\n    count = 0\n    # \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i in range(len(nums)):\n        count += nums[i]\n    # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for num in nums:\n        count += num\n    # \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n    for i, num in enumerate(nums):\n        count += nums[i]\n        count += num\n</code></pre> array.cpp<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.java<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (int num : nums) {\n        count += num;\n    }\n}\n</code></pre> array.cs<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid Traverse(int[] nums) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; nums.Length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    foreach (int num in nums) {\n        count += num;\n    }\n}\n</code></pre> array.go<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums []int) {\n    count := 0\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i := 0; i &lt; len(nums); i++ {\n        count += nums[i]\n    }\n    count = 0\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for _, num := range nums {\n        count += num\n    }\n    // \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n    for i, num := range nums {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.swift<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums: [Int]) {\n    var count = 0\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i in nums.indices {\n        count += nums[i]\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for num in nums {\n        count += num\n    }\n    // \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n    for (i, num) in nums.enumerated() {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.js<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums) {\n    let count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.ts<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums: number[]): void {\n    let count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.dart<pre><code>/* \u904d\u5386\u6570\u7ec4\u5143\u7d20 */\nvoid traverse(List&lt;int&gt; nums) {\n  int count = 0;\n  // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n  for (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n  }\n  // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n  for (int _num in nums) {\n    count += _num;\n  }\n  // \u901a\u8fc7 forEach \u65b9\u6cd5\u904d\u5386\u6570\u7ec4\n  nums.forEach((_num) {\n    count += _num;\n  });\n}\n</code></pre> array.rs<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfn traverse(nums: &amp;[i32]) {\n    let mut _count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i in 0..nums.len() {\n        _count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for num in nums {\n        _count += num;\n    }\n}\n</code></pre> array.c<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.kt<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfun traverse(nums: IntArray) {\n    var count = 0\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (i in nums.indices) {\n        count += nums[i]\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (j: Int in nums) {\n        count += j\n    }\n}\n</code></pre> array.rb<pre><code>### \u904d\u5386\u6570\u7ec4 ###\ndef traverse(nums)\n  count = 0\n\n  # \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n  for i in 0...nums.length\n    count += nums[i]\n  end\n\n  # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n  for num in nums\n    count += num\n  end\nend\n</code></pre> array.zig<pre><code>// \u904d\u5386\u6570\u7ec4\nfn traverse(nums: []i32) void {\n    var count: i32 = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    var i: i32 = 0;\n    while (i &lt; nums.len) : (i += 1) {\n        count += nums[i];\n    }\n    count = 0;\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (nums) |num| {\n        count += num;\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#6-finding-elements","title":"6. \u00a0 Finding Elements","text":"<p>Locating a specific element within an array involves iterating through the array, checking each element to determine if it matches the desired value.</p> <p>Because arrays are linear data structures, this operation is commonly referred to as \"linear search.\"</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def find(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\"\"\"\n    for i in range(len(nums)):\n        if nums[i] == target:\n            return i\n    return -1\n</code></pre> array.cpp<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.java<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint Find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.go<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums []int, target int) (index int) {\n    index = -1\n    for i := 0; i &lt; len(nums); i++ {\n        if nums[i] == target {\n            index = i\n            break\n        }\n    }\n    return\n}\n</code></pre> array.swift<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums: [Int], target: Int) -&gt; Int {\n    for i in nums.indices {\n        if nums[i] == target {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> array.js<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums, target) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) return i;\n    }\n    return -1;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums: number[], target: number): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(List&lt;int&gt; nums, int target) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    if (nums[i] == target) return i;\n  }\n  return -1;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfn find(nums: &amp;[i32], target: i32) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        if nums[i] == target {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> array.c<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.kt<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfun find(nums: IntArray, target: Int): Int {\n    for (i in nums.indices) {\n        if (nums[i] == target) return i\n    }\n    return -1\n}\n</code></pre> array.rb<pre><code>### \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 ###\ndef find(nums, target)\n  for i in 0...nums.length\n    return i if nums[i] == target\n  end\n\n  -1\nend\n</code></pre> array.zig<pre><code>// \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\nfn find(nums: []i32, target: i32) i32 {\n    for (nums, 0..) |num, i| {\n        if (num == target) return @intCast(i);\n    }\n    return -1;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#7-expanding-arrays","title":"7. \u00a0 Expanding Arrays","text":"<p>In complex system environments, ensuring the availability of memory space after an array for safe capacity extension becomes challenging. Consequently, in most programming languages, the length of an array is immutable.</p> <p>To expand an array,  it's necessary to create a larger array and then copy the elements from the original array. This operation has a time complexity of \\(O(n)\\) and can be time-consuming for large arrays. The code are as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def extend(nums: list[int], enlarge: int) -&gt; list[int]:\n    \"\"\"\u6269\u5c55\u6570\u7ec4\u957f\u5ea6\"\"\"\n    # \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    res = [0] * (len(nums) + enlarge)\n    # \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for i in range(len(nums)):\n        res[i] = nums[i]\n    # \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n</code></pre> array.cpp<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int *res = new int[size + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // \u91ca\u653e\u5185\u5b58\n    delete[] nums;\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.java<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int[] res = new int[nums.length + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.cs<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] Extend(int[] nums, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int[] res = new int[nums.Length + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; nums.Length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.go<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums []int, enlarge int) []int {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    res := make([]int, len(nums)+enlarge)\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for i, num := range nums {\n        res[i] = num\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n}\n</code></pre> array.swift<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums: [Int], enlarge: Int) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    var res = Array(repeating: 0, count: nums.count + enlarge)\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for i in nums.indices {\n        res[i] = nums[i]\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n}\n</code></pre> array.js<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cJavaScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums, enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    const res = new Array(nums.length + enlarge).fill(0);\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.ts<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cTypeScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums: number[], enlarge: number): number[] {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    const res = new Array(nums.length + enlarge).fill(0);\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.dart<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nList&lt;int&gt; extend(List&lt;int&gt; nums, int enlarge) {\n  // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n  List&lt;int&gt; res = List.filled(nums.length + enlarge, 0);\n  // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n  for (var i = 0; i &lt; nums.length; i++) {\n    res[i] = nums[i];\n  }\n  // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n  return res;\n}\n</code></pre> array.rs<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfn extend(nums: Vec&lt;i32&gt;, enlarge: usize) -&gt; Vec&lt;i32&gt; {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    let mut res: Vec&lt;i32&gt; = vec![0; nums.len() + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\n    for i in 0..nums.len() {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    res\n}\n</code></pre> array.c<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int *res = (int *)malloc(sizeof(int) * (size + enlarge));\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // \u521d\u59cb\u5316\u6269\u5c55\u540e\u7684\u7a7a\u95f4\n    for (int i = size; i &lt; size + enlarge; i++) {\n        res[i] = 0;\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.kt<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfun extend(nums: IntArray, enlarge: Int): IntArray {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    val res = IntArray(nums.size + enlarge)\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (i in nums.indices) {\n        res[i] = nums[i]\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n}\n</code></pre> array.rb<pre><code>### \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 ###\n# \u8bf7\u6ce8\u610f\uff0cRuby \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n# \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\ndef extend(nums, enlarge)\n  # \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n  res = Array.new(nums.length + enlarge, 0)\n\n  # \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n  for i in 0...nums.length\n    res[i] = nums[i]\n  end\n\n  # \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n  res\nend\n</code></pre> array.zig<pre><code>// \u6269\u5c55\u6570\u7ec4\u957f\u5ea6\nfn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    var res = try mem_allocator.alloc(i32, nums.len + enlarge);\n    @memset(res, 0);\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    std.mem.copy(i32, res, nums);\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#412-advantages-and-limitations-of-arrays","title":"4.1.2 \u00a0 Advantages and Limitations of Arrays","text":"<p>Arrays are stored in contiguous memory spaces and consist of elements of the same type. This approach provides substantial prior information that systems can leverage to optimize the efficiency of data structure operations.</p> <ul> <li>High Space Efficiency: Arrays allocate a contiguous block of memory for data, eliminating the need for additional structural overhead.</li> <li>Support for Random Access: Arrays allow \\(O(1)\\) time access to any element.</li> <li>Cache Locality: When accessing array elements, the computer not only loads them but also caches the surrounding data, utilizing high-speed cache to enchance subsequent operation speeds.</li> </ul> <p>However, continuous space storage is a double-edged sword, with the following limitations:</p> <ul> <li>Low Efficiency in Insertion and Deletion: As arrays accumulate many elements, inserting or deleting elements requires shifting a large number of elements.</li> <li>Fixed Length: The length of an array is fixed after initialization. Expanding an array requires copying all data to a new array, incurring significant costs.</li> <li>Space Wastage: If the allocated array size exceeds the what is necessary, the extra space is wasted.</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#413-typical-applications-of-arrays","title":"4.1.3 \u00a0 Typical Applications of Arrays","text":"<p>Arrays are fundamental and widely used data structures. They find frequent application in various algorithms and serve in the implementation of complex data structures.</p> <ul> <li>Random Access: Arrays are ideal for storing data when random sampling is required. By generating a random sequence based on indices, we can achieve random sampling efficiently.</li> <li>Sorting and Searching: Arrays are the most commonly used data structure for sorting and searching algorithms.  Techniques like quick sort, merge sort, binary search, etc., are primarily operate on arrays.</li> <li>Lookup Tables: Arrays serve as efficient lookup tables for quick element or relationship retrieval. For instance, mapping characters to ASCII codes becomes seamless by using the ASCII code values as indices and storing corresponding elements in the array.</li> <li>Machine Learning: Within the domain of neural networks, arrays play a pivotal role in executing crucial linear algebra operations involving vectors, matrices, and tensors. Arrays serve as the primary and most extensively used data structure in neural network programming.</li> <li>Data Structure Implementation:  Arrays serve as the building blocks for implementing various data structures like stacks, queues, hash tables, heaps, graphs, etc. For instance, the adjacency matrix representation of a graph is essentially a two-dimensional array.</li> </ul>"},{"location":"chapter_array_and_linkedlist/linked_list/","title":"4.2 \u00a0 Linked Lists","text":"<p>Memory space is a shared resource among all programs. In a complex system environment, available memory can be dispersed throughout the memory space. We understand that the memory allocated for an array must be continuous. However, for very large arrays, finding a sufficiently large contiguous memory space might be challenging. This is where the flexible advantage of linked lists becomes evident.</p> <p>A \"linked list\" is a linear data structure in which each element is a node object, and the nodes are interconnected through \"references\". These references hold the memory addresses of subsequent nodes, enabling navigation from one node to the next.</p> <p>The design of linked lists allows for their nodes to be distributed across memory locations without requiring contiguous memory addresses.</p> <p></p> <p> Figure 4-5 \u00a0 Linked List Definition and Storage Method </p> <p>As shown in the figure, we see that the basic building block of a linked list is the \"node\" object. Each node comprises two key components: the node's \"value\" and a \"reference\" to the next node.</p> <ul> <li>The first node in a linked list is the \"head node\", and the final one is the \"tail node\".</li> <li>The tail node points to \"null\", designated as <code>null</code> in Java, <code>nullptr</code> in C++, and <code>None</code> in Python.</li> <li>In languages that support pointers, like C, C++, Go, and Rust, this \"reference\" is typically implemented as a \"pointer\".</li> </ul> <p>As the code below illustrates, a <code>ListNode</code> in a linked list, besides holding a value, must also maintain an additional reference (or pointer). Therefore, a linked list occupies more memory space than an array when storing the same quantity of data..</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>class ListNode:\n    \"\"\"Linked List Node Class\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val               # Node value\n        self.next: ListNode | None = None # Reference to the next node\n</code></pre> <pre><code>/* Linked List Node Structure */\nstruct ListNode {\n    int val;         // Node value\n    ListNode *next;  // Pointer to the next node\n    ListNode(int x) : val(x), next(nullptr) {}  // Constructor\n};\n</code></pre> <pre><code>/* Linked List Node Class */\nclass ListNode {\n    int val;        // Node value\n    ListNode next;  // Reference to the next node\n    ListNode(int x) { val = x; }  // Constructor\n}\n</code></pre> <pre><code>/* Linked List Node Class */\nclass ListNode(int x) {  // Constructor\n    int val = x;         // Node value\n    ListNode? next;      // Reference to the next node\n}\n</code></pre> <pre><code>/* Linked List Node Structure */\ntype ListNode struct {\n    Val  int       // Node value\n    Next *ListNode // Pointer to the next node\n}\n\n// NewListNode Constructor, creates a new linked list\nfunc NewListNode(val int) *ListNode {\n    return &amp;ListNode{\n        Val:  val,\n        Next: nil,\n    }\n}\n</code></pre> <pre><code>/* Linked List Node Class */\nclass ListNode {\n    var val: Int // Node value\n    var next: ListNode? // Reference to the next node\n\n    init(x: Int) { // Constructor\n        val = x\n    }\n}\n</code></pre> <pre><code>/* Linked List Node Class */\nclass ListNode {\n    constructor(val, next) {\n        this.val = (val === undefined ? 0 : val);       // Node value\n        this.next = (next === undefined ? null : next); // Reference to the next node\n    }\n}\n</code></pre> <pre><code>/* Linked List Node Class */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    constructor(val?: number, next?: ListNode | null) {\n        this.val = val === undefined ? 0 : val;        // Node value\n        this.next = next === undefined ? null : next;  // Reference to the next node\n    }\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n  int val; // Node value\n  ListNode? next; // Reference to the next node\n  ListNode(this.val, [this.next]); // Constructor\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* Linked List Node Class */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // Node value\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // Pointer to the next node\n}\n</code></pre> <pre><code>/* Linked List Node Structure */\ntypedef struct ListNode {\n    int val;               // Node value\n    struct ListNode *next; // Pointer to the next node\n} ListNode;\n\n/* Constructor */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    return node;\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>// Linked List Node Class\npub fn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = 0, // Node value\n        next: ?*Self = null, // Pointer to the next node\n\n        // Constructor\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#421-common-operations-on-linked-lists","title":"4.2.1 \u00a0 Common Operations on Linked Lists","text":""},{"location":"chapter_array_and_linkedlist/linked_list/#1-initializing-a-linked-list","title":"1. \u00a0 Initializing a Linked List","text":"<p>Constructing a linked list is a two-step process: first, initializing each node object, and second, forming the reference links between the nodes. After initialization, we can traverse all nodes sequentially from the head node by following the <code>next</code> reference.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig linked_list.py<pre><code># Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# Initialize each node\nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# Build references between nodes\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.cpp<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// Build references between nodes\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.java<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.cs<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nListNode n0 = new(1);\nListNode n1 = new(3);\nListNode n2 = new(2);\nListNode n3 = new(5);\nListNode n4 = new(4);\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.go<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// Build references between nodes\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n</code></pre> linked_list.swift<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// Build references between nodes\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.js<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.ts<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.dart<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rs<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nlet n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None }));\nlet n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None }));\nlet n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None }));\nlet n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None }));\nlet n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None }));\n\n// Build references between nodes\nn0.borrow_mut().next = Some(n1.clone());\nn1.borrow_mut().next = Some(n2.clone());\nn2.borrow_mut().next = Some(n3.clone());\nn3.borrow_mut().next = Some(n4.clone());\n</code></pre> linked_list.c<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// Build references between nodes\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.kt<pre><code>\n</code></pre> linked_list.zig<pre><code>// Initialize linked list\n// Initialize each node\nvar n0 = inc.ListNode(i32){.val = 1};\nvar n1 = inc.ListNode(i32){.val = 3};\nvar n2 = inc.ListNode(i32){.val = 2};\nvar n3 = inc.ListNode(i32){.val = 5};\nvar n4 = inc.ListNode(i32){.val = 4};\n// Build references between nodes\nn0.next = &amp;n1;\nn1.next = &amp;n2;\nn2.next = &amp;n3;\nn3.next = &amp;n4;\n</code></pre> <p>The array as a whole is a variable, for instance, the array <code>nums</code> includes elements like <code>nums[0]</code>, <code>nums[1]</code>, and so on, whereas a linked list is made up of several distinct node objects. We typically refer to a linked list by its head node, for example, the linked list in the previous code snippet is referred to as <code>n0</code>.</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#2-inserting-a-node","title":"2. \u00a0 Inserting a Node","text":"<p>Inserting a node into a linked list is very easy. As shown in the figure, let's assume we aim to insert a new node <code>P</code> between two adjacent nodes <code>n0</code> and <code>n1</code>. This can be achieved by simply modifying two node references (pointers), with a time complexity of \\(O(1)\\).</p> <p>By comparison, inserting an element into an array has a time complexity of \\(O(n)\\), which becomes less efficient when dealing with large data volumes.</p> <p></p> <p> Figure 4-6 \u00a0 Linked List Node Insertion Example </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def insert(n0: ListNode, P: ListNode):\n    \"\"\"\u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\"\"\"\n    n1 = n0.next\n    P.next = n1\n    n0.next = P\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\n    ListNode n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid Insert(ListNode n0, ListNode P) {\n    ListNode? n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insertNode(n0 *ListNode, P *ListNode) {\n    n1 := n0.Next\n    P.Next = n1\n    n0.Next = P\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insert(n0: ListNode, P: ListNode) {\n    let n1 = n0.next\n    P.next = n1\n    n0.next = P\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0, P) {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0: ListNode, P: ListNode): void {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\n  ListNode? n1 = n0.next;\n  P.next = n1;\n  n0.next = P;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\n#[allow(non_snake_case)]\npub fn insert&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, P: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    let n1 = n0.borrow_mut().next.take();\n    P.borrow_mut().next = n1;\n    n0.borrow_mut().next = Some(P);\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.kt<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9p */\nfun insert(n0: ListNode?, p: ListNode?) {\n    val n1 = n0?.next\n    p?.next = n1\n    n0?.next = p\n}\n</code></pre> linked_list.rb<pre><code>### \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 _p ###\n# Ruby \u7684 `p` \u662f\u4e00\u4e2a\u5185\u7f6e\u51fd\u6570\uff0c `P` \u662f\u4e00\u4e2a\u5e38\u91cf\uff0c\u6240\u4ee5\u53ef\u4ee5\u4f7f\u7528 `_p` \u4ee3\u66ff\ndef insert(n0, _p)\n  n1 = n0.next\n  _p.next = n1\n  n0.next = _p\nend\n</code></pre> linked_list.zig<pre><code>// \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\nfn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {\n    var n1 = n0.?.next;\n    P.?.next = n1;\n    n0.?.next = P;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#3-deleting-a-node","title":"3. \u00a0 Deleting a Node","text":"<p>As shown in the figure, deleting a node from a linked list is also very easy, involving only the modification of a single node's reference (pointer).</p> <p>It's important to note that even though node <code>P</code> continues to point to <code>n1</code> after being deleted, it becomes inaccessible during linked list traversal. This effectively means that <code>P</code> is no longer a part of the linked list.</p> <p></p> <p> Figure 4-7 \u00a0 Linked List Node Deletion </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def remove(n0: ListNode):\n    \"\"\"\u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\"\"\"\n    if not n0.next:\n        return\n    # n0 -&gt; P -&gt; n1\n    P = n0.next\n    n1 = P.next\n    n0.next = n1\n</code></pre> linked_list.cpp<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode *n0) {\n    if (n0-&gt;next == nullptr)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // \u91ca\u653e\u5185\u5b58\n    delete P;\n}\n</code></pre> linked_list.java<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid Remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode? n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.go<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc removeItem(n0 *ListNode) {\n    if n0.Next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    P := n0.Next\n    n1 := P.Next\n    n0.Next = n1\n}\n</code></pre> linked_list.swift<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc remove(n0: ListNode) {\n    if n0.next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    let P = n0.next\n    let n1 = P?.next\n    n0.next = n1\n}\n</code></pre> linked_list.js<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0) {\n    if (!n0.next) return;\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0: ListNode): void {\n    if (!n0.next) {\n        return;\n    }\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\n  if (n0.next == null) return;\n  // n0 -&gt; P -&gt; n1\n  ListNode P = n0.next!;\n  ListNode? n1 = P.next;\n  n0.next = n1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n#[allow(non_snake_case)]\npub fn remove&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    if n0.borrow().next.is_none() {\n        return;\n    };\n    // n0 -&gt; P -&gt; n1\n    let P = n0.borrow_mut().next.take();\n    if let Some(node) = P {\n        let n1 = node.borrow_mut().next.take();\n        n0.borrow_mut().next = n1;\n    }\n}\n</code></pre> linked_list.c<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(ListNode *n0) {\n    if (!n0-&gt;next)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // \u91ca\u653e\u5185\u5b58\n    free(P);\n}\n</code></pre> linked_list.kt<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfun remove(n0: ListNode?) {\n    val p = n0?.next\n    val n1 = p?.next\n    n0?.next = n1\n}\n</code></pre> linked_list.rb<pre><code>### \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 ###\ndef remove(n0)\n  return if n0.next.nil?\n\n  # n0 -&gt; remove_node -&gt; n1\n  remove_node = n0.next\n  n1 = remove_node.next\n  n0.next = n1\nend\n</code></pre> linked_list.zig<pre><code>// \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\nfn remove(n0: ?*inc.ListNode(i32)) void {\n    if (n0.?.next == null) return;\n    // n0 -&gt; P -&gt; n1\n    var P = n0.?.next;\n    var n1 = P.?.next;\n    n0.?.next = n1;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#4-accessing-nodes","title":"4. \u00a0 Accessing Nodes","text":"<p>Accessing nodes in a linked list is less efficient. As previously mentioned, any element in an array can be accessed in \\(O(1)\\) time. In contrast, with a linked list, the program involves starting from the head node and sequentially traversing through the nodes until the desired node is found. In other words, to access the \\(i\\)-th node in a linked list, the program must iterate through \\(i - 1\\) nodes, resulting in a time complexity of \\(O(n)\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def access(head: ListNode, index: int) -&gt; ListNode | None:\n    \"\"\"\u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\"\"\"\n    for _ in range(index):\n        if not head:\n            return None\n        head = head.next\n    return head\n</code></pre> linked_list.cpp<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == nullptr)\n            return nullptr;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.java<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode access(ListNode head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.cs<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? Access(ListNode? head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.go<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head *ListNode, index int) *ListNode {\n    for i := 0; i &lt; index; i++ {\n        if head == nil {\n            return nil\n        }\n        head = head.Next\n    }\n    return head\n}\n</code></pre> linked_list.swift<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head: ListNode, index: Int) -&gt; ListNode? {\n    var head: ListNode? = head\n    for _ in 0 ..&lt; index {\n        if head == nil {\n            return nil\n        }\n        head = head?.next\n    }\n    return head\n}\n</code></pre> linked_list.js<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head, index) {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.ts<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head: ListNode | null, index: number): ListNode | null {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.dart<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? access(ListNode? head, int index) {\n  for (var i = 0; i &lt; index; i++) {\n    if (head == null) return null;\n    head = head.next;\n  }\n  return head;\n}\n</code></pre> linked_list.rs<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\npub fn access&lt;T&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, index: i32) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n    if index &lt;= 0 {\n        return head;\n    };\n    if let Some(node) = &amp;head.borrow().next {\n        return access(node.clone(), index - 1);\n    }\n\n    return head;\n}\n</code></pre> linked_list.c<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == NULL)\n            return NULL;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.kt<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfun access(head: ListNode?, index: Int): ListNode? {\n    var h = head\n    for (i in 0..&lt;index) {\n        h = h?.next\n    }\n    return h\n}\n</code></pre> linked_list.rb<pre><code>### \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 ###\ndef access(head, index)\n  for i in 0...index\n    return nil if head.nil?\n    head = head.next\n  end\n\n  head\nend\n</code></pre> linked_list.zig<pre><code>// \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\nfn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {\n    var head = node;\n    var i: i32 = 0;\n    while (i &lt; index) : (i += 1) {\n        head = head.?.next;\n        if (head == null) return null;\n    }\n    return head;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#5-finding-nodes","title":"5. \u00a0 Finding Nodes","text":"<p>Traverse the linked list to locate a node whose value matches <code>target</code>, and then output the index of that node within the linked list. This procedure is also an example of linear search. The corresponding code is provided below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def find(head: ListNode, target: int) -&gt; int:\n    \"\"\"\u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\"\"\"\n    index = 0\n    while head:\n        if head.val == target:\n            return index\n        head = head.next\n        index += 1\n    return -1\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head != nullptr) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint Find(ListNode? head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc findNode(head *ListNode, target int) int {\n    index := 0\n    for head != nil {\n        if head.Val == target {\n            return index\n        }\n        head = head.Next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc find(head: ListNode, target: Int) -&gt; Int {\n    var head: ListNode? = head\n    var index = 0\n    while head != nil {\n        if head?.val == target {\n            return index\n        }\n        head = head?.next\n        index += 1\n    }\n    return -1\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head, target) {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head: ListNode | null, target: number): number {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode? head, int target) {\n  int index = 0;\n  while (head != null) {\n    if (head.val == target) {\n      return index;\n    }\n    head = head.next;\n    index++;\n  }\n  return -1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\npub fn find&lt;T: PartialEq&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, target: T, index: i32) -&gt; i32 {\n    if head.borrow().val == target {\n        return index;\n    };\n    if let Some(node) = &amp;head.borrow_mut().next {\n        return find(node.clone(), target, index + 1);\n    }\n    return -1;\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.kt<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfun find(head: ListNode?, target: Int): Int {\n    var index = 0\n    var h = head\n    while (h != null) {\n        if (h.value == target) return index\n        h = h.next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.rb<pre><code>### \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 ###\ndef find(head, target)\n  index = 0\n  while head\n    return index if head.val == target\n    head = head.next\n    index += 1\n  end\n\n  -1\nend\n</code></pre> linked_list.zig<pre><code>// \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\nfn find(node: ?*inc.ListNode(i32), target: i32) i32 {\n    var head = node;\n    var index: i32 = 0;\n    while (head != null) {\n        if (head.?.val == target) return index;\n        head = head.?.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#422-arrays-vs-linked-lists","title":"4.2.2 \u00a0 Arrays vs. Linked Lists","text":"<p>The Table 4-1  summarizes the characteristics of arrays and linked lists, and it also compares their efficiencies in various operations. Because they utilize opposing storage strategies, their respective properties and operational efficiencies exhibit distinct contrasts.</p> <p> Table 4-1 \u00a0 Efficiency Comparison of Arrays and Linked Lists </p> Arrays Linked Lists Storage Contiguous Memory Space Dispersed Memory Space Capacity Expansion Fixed Length Flexible Expansion Memory Efficiency Less Memory per Element, Potential Space Wastage More Memory per Element Accessing Elements \\(O(1)\\) \\(O(n)\\) Adding Elements \\(O(n)\\) \\(O(1)\\) Deleting Elements \\(O(n)\\) \\(O(1)\\)"},{"location":"chapter_array_and_linkedlist/linked_list/#423-common-types-of-linked-lists","title":"4.2.3 \u00a0 Common Types of Linked Lists","text":"<p>As shown in the figure, there are three common types of linked lists.</p> <ul> <li>Singly Linked List: This is the standard linked list described earlier. Nodes in a singly linked list include a value and a reference to the next node. The first node is known as the head node, and the last node, which points to null (<code>None</code>), is the tail node.</li> <li>Circular Linked List: This is formed when the tail node of a singly linked list points back to the head node, creating a loop. In a circular linked list, any node can function as the head node.</li> <li>Doubly Linked List: In contrast to a singly linked list, a doubly linked list maintains references in two directions. Each node contains references (pointer) to both its successor (the next node) and predecessor (the previous node). Although doubly linked lists offer more flexibility for traversing in either direction, they also consume more memory space.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>class ListNode:\n    \"\"\"Bidirectional linked list node class\"\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # Node value\n        self.next: ListNode | None = None  # Reference to the successor node\n        self.prev: ListNode | None = None  # Reference to a predecessor node\n</code></pre> <pre><code>/* Bidirectional linked list node structure */\nstruct ListNode {\n    int val;         // Node value\n    ListNode *next;  // Pointer to the successor node\n    ListNode *prev;  // Pointer to the predecessor node\n    ListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // Constructor\n};\n</code></pre> <pre><code>/* Bidirectional linked list node class */\nclass ListNode {\n    int val;        // Node value\n    ListNode next;  // Reference to the next node\n    ListNode prev;  // Reference to the predecessor node\n    ListNode(int x) { val = x; }  // Constructor\n}\n</code></pre> <pre><code>/* Bidirectional linked list node class */\nclass ListNode(int x) {  // Constructor\n    int val = x;    // Node value\n    ListNode next;  // Reference to the next node\n    ListNode prev;  // Reference to the predecessor node\n}\n</code></pre> <pre><code>/* Bidirectional linked list node structure */\ntype DoublyListNode struct {\n    Val  int             // Node value\n    Next *DoublyListNode // Pointer to the successor node\n    Prev *DoublyListNode // Pointer to the predecessor node\n}\n\n// NewDoublyListNode initialization\nfunc NewDoublyListNode(val int) *DoublyListNode {\n    return &amp;DoublyListNode{\n        Val:  val,\n        Next: nil,\n        Prev: nil,\n    }\n}\n</code></pre> <pre><code>/* Bidirectional linked list node class */\nclass ListNode {\n    var val: Int // Node value\n    var next: ListNode? // Reference to the next node\n    var prev: ListNode? // Reference to the predecessor node\n\n    init(x: Int) { // Constructor\n        val = x\n    }\n}\n</code></pre> <pre><code>/* Bidirectional linked list node class */\nclass ListNode {\n    constructor(val, next, prev) {\n        this.val = val  ===  undefined ? 0 : val;        // Node value\n        this.next = next  ===  undefined ? null : next;  // Reference to the successor node\n        this.prev = prev  ===  undefined ? null : prev;  // Reference to the predecessor node\n    }\n}\n</code></pre> <pre><code>/* Bidirectional linked list node class */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    prev: ListNode | null;\n    constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\n        this.val = val  ===  undefined ? 0 : val;        // Node value\n        this.next = next  ===  undefined ? null : next;  // Reference to the successor node\n        this.prev = prev  ===  undefined ? null : prev;  // Reference to the predecessor node\n    }\n}\n</code></pre> <pre><code>/* Bidirectional linked list node class */\nclass ListNode {\n    int val;        // Node value\n    ListNode next;  // Reference to the next node\n    ListNode prev;  // Reference to the predecessor node\n    ListNode(this.val, [this.next, this.prev]);  // Constructor\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* Bidirectional linked list node type */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // Node value\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // Pointer to successor node\n    prev: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // Pointer to predecessor node\n}\n\n/* Constructors */\nimpl ListNode {\n    fn new(val: i32) -&gt; Self {\n        ListNode {\n            val,\n            next: None,\n            prev: None,\n        }\n    }\n}\n</code></pre> <pre><code>/* Bidirectional linked list node structure */\ntypedef struct ListNode {\n    int val;               // Node value\n    struct ListNode *next; // Pointer to the successor node\n    struct ListNode *prev; // Pointer to the predecessor node\n} ListNode;\n\n/* Constructors */\nListNode *newListNode(int val) {\n    ListNode *node, *next;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    node-&gt;prev = NULL;\n    return node;\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>// Bidirectional linked list node class\npub fn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = 0, // Node value\n        next: ?*Self = null, // Pointer to the successor node\n        prev: ?*Self = null, // Pointer to the predecessor node\n\n        // Constructor\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n            self.prev = null;\n        }\n    };\n}\n</code></pre> <p></p> <p> Figure 4-8 \u00a0 Common Types of Linked Lists </p>"},{"location":"chapter_array_and_linkedlist/linked_list/#424-typical-applications-of-linked-lists","title":"4.2.4 \u00a0 Typical Applications of Linked Lists","text":"<p>Singly linked lists are frequently utilized in implementing stacks, queues, hash tables, and graphs.</p> <ul> <li>Stacks and Queues: In singly linked lists, if insertions and deletions occur at the same end, it behaves like a stack (last-in-first-out). Conversely, if insertions are at one end and deletions at the other, it functions like a queue (first-in-first-out).</li> <li>Hash Tables: Linked lists are used in chaining, a popular method for resolving hash collisions. Here, all collided elements are grouped into a linked list.</li> <li>Graphs: Adjacency lists, a standard method for graph representation, associate each graph vertex with a linked list. This list contains elements that represent vertices connected to the corresponding vertex.</li> </ul> <p>Doubly linked lists are ideal for scenarios requiring rapid access to preceding and succeeding elements.</p> <ul> <li>Advanced Data Structures: In structures like red-black trees and B-trees, accessing a node's parent is essential. This is achieved by incorporating a reference to the parent node in each node, akin to a doubly linked list.</li> <li>Browser History: In web browsers, doubly linked lists facilitate navigating the history of visited pages when users click forward or back.</li> <li>LRU Algorithm: Doubly linked lists are apt for Least Recently Used (LRU) cache eviction algorithms, enabling swift identification of the least recently used data and facilitating fast node addition and removal.</li> </ul> <p>Circular linked lists are ideal for applications that require periodic operations, such as resource scheduling in operating systems.</p> <ul> <li>Round-Robin Scheduling Algorithm: In operating systems, the round-robin scheduling algorithm is a common CPU scheduling method, requiring cycling through a group of processes. Each process is assigned a time slice, and upon expiration, the CPU rotates to the next process. This cyclical operation can be efficiently realized using a circular linked list, allowing for a fair and time-shared system among all processes.</li> <li>Data Buffers: Circular linked lists are also used in data buffers, like in audio and video players, where the data stream is divided into multiple buffer blocks arranged in a circular fashion for seamless playback.</li> </ul>"},{"location":"chapter_array_and_linkedlist/list/","title":"4.3 \u00a0 List","text":"<p>A \"list\" is an abstract data structure concept that represents an ordered collection of elements, supporting operations such as element access, modification, addition, deletion, and traversal, without requiring users to consider capacity limitations. Lists can be implemented based on linked lists or arrays.</p> <ul> <li>A linked list inherently serves as a list, supporting operations for adding, deleting, searching, and modifying elements, with the flexibility to dynamically adjust its size.</li> <li>Arrays also support these operations, but due to their immutable length, they can be considered as a list with a length limit.</li> </ul> <p>When implementing lists using arrays, the immutability of length reduces the practicality of the list. This is because predicting the amount of data to be stored in advance is often challenging, making it difficult to choose an appropriate list length. If the length is too small, it may not meet the requirements; if too large, it may waste memory space.</p> <p>To solve this problem, we can implement lists using a \"dynamic array.\" It inherits the advantages of arrays and can dynamically expand during program execution.</p> <p>In fact, many programming languages' standard libraries implement lists using dynamic arrays, such as Python's <code>list</code>, Java's <code>ArrayList</code>, C++'s <code>vector</code>, and C#'s <code>List</code>. In the following discussion, we will consider \"list\" and \"dynamic array\" as synonymous concepts.</p>"},{"location":"chapter_array_and_linkedlist/list/#431-common-list-operations","title":"4.3.1 \u00a0 Common List Operations","text":""},{"location":"chapter_array_and_linkedlist/list/#1-initializing-a-list","title":"1. \u00a0 Initializing a List","text":"<p>We typically use two initialization methods: \"without initial values\" and \"with initial values\".</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig list.py<pre><code># Initialize list\n# Without initial values\nnums1: list[int] = []\n# With initial values\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> list.cpp<pre><code>/* Initialize list */\n// Note, in C++ the vector is the equivalent of nums described here\n// Without initial values\nvector&lt;int&gt; nums1;\n// With initial values\nvector&lt;int&gt; nums = { 1, 3, 2, 5, 4 };\n</code></pre> list.java<pre><code>/* Initialize list */\n// Without initial values\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;();\n// With initial values (note the element type should be the wrapper class Integer[] for int[])\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList&lt;Integer&gt; nums = new ArrayList&lt;&gt;(Arrays.asList(numbers));\n</code></pre> list.cs<pre><code>/* Initialize list */\n// Without initial values\nList&lt;int&gt; nums1 = [];\n// With initial values\nint[] numbers = [1, 3, 2, 5, 4];\nList&lt;int&gt; nums = [.. numbers];\n</code></pre> list_test.go<pre><code>/* Initialize list */\n// Without initial values\nnums1 := []int{}\n// With initial values\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> list.swift<pre><code>/* Initialize list */\n// Without initial values\nlet nums1: [Int] = []\n// With initial values\nvar nums = [1, 3, 2, 5, 4]\n</code></pre> list.js<pre><code>/* Initialize list */\n// Without initial values\nconst nums1 = [];\n// With initial values\nconst nums = [1, 3, 2, 5, 4];\n</code></pre> list.ts<pre><code>/* Initialize list */\n// Without initial values\nconst nums1: number[] = [];\n// With initial values\nconst nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> list.dart<pre><code>/* Initialize list */\n// Without initial values\nList&lt;int&gt; nums1 = [];\n// With initial values\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> list.rs<pre><code>/* Initialize list */\n// Without initial values\nlet nums1: Vec&lt;i32&gt; = Vec::new();\n// With initial values\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>\n</code></pre> list.zig<pre><code>// Initialize list\nvar nums = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums.deinit();\ntry nums.appendSlice(&amp;[_]i32{ 1, 3, 2, 5, 4 });\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#2-accessing-elements","title":"2. \u00a0 Accessing Elements","text":"<p>Lists are essentially arrays, thus they can access and update elements in \\(O(1)\\) time, which is very efficient.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig list.py<pre><code># Access elements\nnum: int = nums[1]  # Access the element at index 1\n\n# Update elements\nnums[1] = 0    # Update the element at index 1 to 0\n</code></pre> list.cpp<pre><code>/* Access elements */\nint num = nums[1];  // Access the element at index 1\n\n/* Update elements */\nnums[1] = 0;  // Update the element at index 1 to 0\n</code></pre> list.java<pre><code>/* Access elements */\nint num = nums.get(1);  // Access the element at index 1\n\n/* Update elements */\nnums.set(1, 0);  // Update the element at index 1 to 0\n</code></pre> list.cs<pre><code>/* Access elements */\nint num = nums[1];  // Access the element at index 1\n\n/* Update elements */\nnums[1] = 0;  // Update the element at index 1 to 0\n</code></pre> list_test.go<pre><code>/* Access elements */\nnum := nums[1]  // Access the element at index 1\n\n/* Update elements */\nnums[1] = 0     // Update the element at index 1 to 0\n</code></pre> list.swift<pre><code>/* Access elements */\nlet num = nums[1] // Access the element at index 1\n\n/* Update elements */\nnums[1] = 0 // Update the element at index 1 to 0\n</code></pre> list.js<pre><code>/* Access elements */\nconst num = nums[1];  // Access the element at index 1\n\n/* Update elements */\nnums[1] = 0;  // Update the element at index 1 to 0\n</code></pre> list.ts<pre><code>/* Access elements */\nconst num: number = nums[1];  // Access the element at index 1\n\n/* Update elements */\nnums[1] = 0;  // Update the element at index 1 to 0\n</code></pre> list.dart<pre><code>/* Access elements */\nint num = nums[1];  // Access the element at index 1\n\n/* Update elements */\nnums[1] = 0;  // Update the element at index 1 to 0\n</code></pre> list.rs<pre><code>/* Access elements */\nlet num: i32 = nums[1];  // Access the element at index 1\n/* Update elements */\nnums[1] = 0;             // Update the element at index 1 to 0\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>\n</code></pre> list.zig<pre><code>// Access elements\nvar num = nums.items[1]; // Access the element at index 1\n\n// Update elements\nnums.items[1] = 0; // Update the element at index 1 to 0  \n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#3-inserting-and-removing-elements","title":"3. \u00a0 Inserting and Removing Elements","text":"<p>Compared to arrays, lists offer more flexibility in adding and removing elements. While adding elements to the end of a list is an \\(O(1)\\) operation, the efficiency of inserting and removing elements elsewhere in the list remains the same as in arrays, with a time complexity of \\(O(n)\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig list.py<pre><code># Clear list\nnums.clear()\n\n# Append elements at the end\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n# Insert element in the middle\nnums.insert(3, 6)  # Insert number 6 at index 3\n\n# Remove elements\nnums.pop(3)        # Remove the element at index 3\n</code></pre> list.cpp<pre><code>/* Clear list */\nnums.clear();\n\n/* Append elements at the end */\nnums.push_back(1);\nnums.push_back(3);\nnums.push_back(2);\nnums.push_back(5);\nnums.push_back(4);\n\n/* Insert element in the middle */\nnums.insert(nums.begin() + 3, 6);  // Insert number 6 at index 3\n\n/* Remove elements */\nnums.erase(nums.begin() + 3);      // Remove the element at index 3\n</code></pre> list.java<pre><code>/* Clear list */\nnums.clear();\n\n/* Append elements at the end */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* Insert element in the middle */\nnums.add(3, 6);  // Insert number 6 at index 3\n\n/* Remove elements */\nnums.remove(3);  // Remove the element at index 3\n</code></pre> list.cs<pre><code>/* Clear list */\nnums.Clear();\n\n/* Append elements at the end */\nnums.Add(1);\nnums.Add(3);\nnums.Add(2);\nnums.Add(5);\nnums.Add(4);\n\n/* Insert element in the middle */\nnums.Insert(3, 6);\n\n/* Remove elements */\nnums.RemoveAt(3);\n</code></pre> list_test.go<pre><code>/* Clear list */\nnums = nil\n\n/* Append elements at the end */\nnums = append(nums, 1)\nnums = append(nums, 3)\nnums = append(nums, 2)\nnums = append(nums, 5)\nnums = append(nums, 4)\n\n/* Insert element in the middle */\nnums = append(nums[:3], append([]int{6}, nums[3:]...)...) // Insert number 6 at index 3\n\n/* Remove elements */\nnums = append(nums[:3], nums[4:]...) // Remove the element at index 3\n</code></pre> list.swift<pre><code>/* Clear list */\nnums.removeAll()\n\n/* Append elements at the end */\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n/* Insert element in the middle */\nnums.insert(6, at: 3) // Insert number 6 at index 3\n\n/* Remove elements */\nnums.remove(at: 3) // Remove the element at index 3\n</code></pre> list.js<pre><code>/* Clear list */\nnums.length = 0;\n\n/* Append elements at the end */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* Insert element in the middle */\nnums.splice(3, 0, 6);\n\n/* Remove elements */\nnums.splice(3, 1);\n</code></pre> list.ts<pre><code>/* Clear list */\nnums.length = 0;\n\n/* Append elements at the end */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* Insert element in the middle */\nnums.splice(3, 0, 6);\n\n/* Remove elements */\nnums.splice(3, 1);\n</code></pre> list.dart<pre><code>/* Clear list */\nnums.clear();\n\n/* Append elements at the end */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* Insert element in the middle */\nnums.insert(3, 6); // Insert number 6 at index 3\n\n/* Remove elements */\nnums.removeAt(3); // Remove the element at index 3\n</code></pre> list.rs<pre><code>/* Clear list */\nnums.clear();\n\n/* Append elements at the end */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* Insert element in the middle */\nnums.insert(3, 6);  // Insert number 6 at index 3\n\n/* Remove elements */\nnums.remove(3);    // Remove the element at index 3\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>\n</code></pre> list.zig<pre><code>// Clear list\nnums.clearRetainingCapacity();\n\n// Append elements at the end\ntry nums.append(1);\ntry nums.append(3);\ntry nums.append(2);\ntry nums.append(5);\ntry nums.append(4);\n\n// Insert element in the middle\ntry nums.insert(3, 6); // Insert number 6 at index 3\n\n// Remove elements\n_ = nums.orderedRemove(3); // Remove the element at index 3\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#4-iterating-the-list","title":"4. \u00a0 Iterating the List","text":"<p>Similar to arrays, lists can be iterated either by using indices or by directly iterating through each element.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig list.py<pre><code># Iterate through the list by index\ncount = 0\nfor i in range(len(nums)):\n    count += nums[i]\n\n# Iterate directly through list elements\nfor num in nums:\n    count += num\n</code></pre> list.cpp<pre><code>/* Iterate through the list by index */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums[i];\n}\n\n/* Iterate directly through list elements */\ncount = 0;\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.java<pre><code>/* Iterate through the list by index */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums.get(i);\n}\n\n/* Iterate directly through list elements */\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.cs<pre><code>/* Iterate through the list by index */\nint count = 0;\nfor (int i = 0; i &lt; nums.Count; i++) {\n    count += nums[i];\n}\n\n/* Iterate directly through list elements */\ncount = 0;\nforeach (int num in nums) {\n    count += num;\n}\n</code></pre> list_test.go<pre><code>/* Iterate through the list by index */\ncount := 0\nfor i := 0; i &lt; len(nums); i++ {\n    count += nums[i]\n}\n\n/* Iterate directly through list elements */\ncount = 0\nfor _, num := range nums {\n    count += num\n}\n</code></pre> list.swift<pre><code>/* Iterate through the list by index */\nvar count = 0\nfor i in nums.indices {\n    count += nums[i]\n}\n\n/* Iterate directly through list elements */\ncount = 0\nfor num in nums {\n    count += num\n}\n</code></pre> list.js<pre><code>/* Iterate through the list by index */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* Iterate directly through list elements */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.ts<pre><code>/* Iterate through the list by index */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* Iterate directly through list elements */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.dart<pre><code>/* Iterate through the list by index */\nint count = 0;\nfor (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* Iterate directly through list elements */\ncount = 0;\nfor (var num in nums) {\n    count += num;\n}\n</code></pre> list.rs<pre><code>// Iterate through the list by index\nlet mut _count = 0;\nfor i in 0..nums.len() {\n    _count += nums[i];\n}\n\n// Iterate directly through list elements\n_count = 0;\nfor num in &amp;nums {\n    _count += num;\n}\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>\n</code></pre> list.zig<pre><code>// Iterate through the list by index\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i &lt; nums.items.len) : (i += 1) {\n    count += nums[i];\n}\n\n// Iterate directly through list elements\ncount = 0;\nfor (nums.items) |num| {\n    count += num;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#5-concatenating-lists","title":"5. \u00a0 Concatenating Lists","text":"<p>Given a new list <code>nums1</code>, we can append it to the end of the original list.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig list.py<pre><code># Concatenate two lists\nnums1: list[int] = [6, 8, 7, 10, 9]\nnums += nums1  # Concatenate nums1 to the end of nums\n</code></pre> list.cpp<pre><code>/* Concatenate two lists */\nvector&lt;int&gt; nums1 = { 6, 8, 7, 10, 9 };\n// Concatenate nums1 to the end of nums\nnums.insert(nums.end(), nums1.begin(), nums1.end());\n</code></pre> list.java<pre><code>/* Concatenate two lists */\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nnums.addAll(nums1);  // Concatenate nums1 to the end of nums\n</code></pre> list.cs<pre><code>/* Concatenate two lists */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.AddRange(nums1);  // Concatenate nums1 to the end of nums\n</code></pre> list_test.go<pre><code>/* Concatenate two lists */\nnums1 := []int{6, 8, 7, 10, 9}\nnums = append(nums, nums1...)  // Concatenate nums1 to the end of nums\n</code></pre> list.swift<pre><code>/* Concatenate two lists */\nlet nums1 = [6, 8, 7, 10, 9]\nnums.append(contentsOf: nums1) // Concatenate nums1 to the end of nums\n</code></pre> list.js<pre><code>/* Concatenate two lists */\nconst nums1 = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // Concatenate nums1 to the end of nums\n</code></pre> list.ts<pre><code>/* Concatenate two lists */\nconst nums1: number[] = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // Concatenate nums1 to the end of nums\n</code></pre> list.dart<pre><code>/* Concatenate two lists */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.addAll(nums1);  // Concatenate nums1 to the end of nums\n</code></pre> list.rs<pre><code>/* Concatenate two lists */\nlet nums1: Vec&lt;i32&gt; = vec![6, 8, 7, 10, 9];\nnums.extend(nums1);\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>\n</code></pre> list.zig<pre><code>// Concatenate two lists\nvar nums1 = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums1.deinit();\ntry nums1.appendSlice(&amp;[_]i32{ 6, 8, 7, 10, 9 });\ntry nums.insertSlice(nums.items.len, nums1.items); // Concatenate nums1 to the end of nums\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#6-sorting-the-list","title":"6. \u00a0 Sorting the List","text":"<p>Once the list is sorted, we can employ algorithms commonly used in array-related algorithm problems, such as \"binary search\" and \"two-pointer\" algorithms.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig list.py<pre><code># Sort the list\nnums.sort()  # After sorting, the list elements are in ascending order\n</code></pre> list.cpp<pre><code>/* Sort the list */\nsort(nums.begin(), nums.end());  // After sorting, the list elements are in ascending order\n</code></pre> list.java<pre><code>/* Sort the list */\nCollections.sort(nums);  // After sorting, the list elements are in ascending order\n</code></pre> list.cs<pre><code>/* Sort the list */\nnums.Sort(); // After sorting, the list elements are in ascending order\n</code></pre> list_test.go<pre><code>/* Sort the list */\nsort.Ints(nums)  // After sorting, the list elements are in ascending order\n</code></pre> list.swift<pre><code>/* Sort the list */\nnums.sort() // After sorting, the list elements are in ascending order\n</code></pre> list.js<pre><code>/* Sort the list */  \nnums.sort((a, b) =&gt; a - b);  // After sorting, the list elements are in ascending order\n</code></pre> list.ts<pre><code>/* Sort the list */\nnums.sort((a, b) =&gt; a - b);  // After sorting, the list elements are in ascending order\n</code></pre> list.dart<pre><code>/* Sort the list */\nnums.sort(); // After sorting, the list elements are in ascending order\n</code></pre> list.rs<pre><code>/* Sort the list */\nnums.sort(); // After sorting, the list elements are in ascending order\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>\n</code></pre> list.zig<pre><code>// Sort the list\nstd.sort.sort(i32, nums.items, {}, comptime std.sort.asc(i32));\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#432-list-implementation","title":"4.3.2 \u00a0 List Implementation","text":"<p>Many programming languages come with built-in lists, including Java, C++, Python, etc. Their implementations tend to be intricate, featuring carefully considered settings for various parameters, like initial capacity and expansion factors. Readers who are curious can delve into the source code for further learning.</p> <p>To enhance our understanding of how lists work, we will attempt to implement a simplified version of a list, focusing on three crucial design aspects:</p> <ul> <li>Initial Capacity: Choose a reasonable initial capacity for the array. In this example, we choose 10 as the initial capacity.</li> <li>Size Recording: Declare a variable <code>size</code> to record the current number of elements in the list, updating in real-time with element insertion and deletion. With this variable, we can locate the end of the list and determine whether expansion is needed.</li> <li>Expansion Mechanism: If the list reaches full capacity upon an element insertion, an expansion process is required. This involves creating a larger array based on the expansion factor, and then transferring all elements from the current array to the new one. In this example, we stipulate that the array size should double with each expansion.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_list.py<pre><code>class MyList:\n    \"\"\"\u5217\u8868\u7c7b\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._capacity: int = 10  # \u5217\u8868\u5bb9\u91cf\n        self._arr: list[int] = [0] * self._capacity  # \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n        self._size: int = 0  # \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        self._extend_ratio: int = 2  # \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\"\"\"\n        return self._size\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u5217\u8868\u5bb9\u91cf\"\"\"\n        return self._capacity\n\n    def get(self, index: int) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u5143\u7d20\"\"\"\n        # \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        return self._arr[index]\n\n    def set(self, num: int, index: int):\n        \"\"\"\u66f4\u65b0\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        self._arr[index] = num\n\n    def add(self, num: int):\n        \"\"\"\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\"\"\"\n        # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self.size() == self.capacity():\n            self.extend_capacity()\n        self._arr[self._size] = num\n        self._size += 1\n\n    def insert(self, num: int, index: int):\n        \"\"\"\u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self._size == self.capacity():\n            self.extend_capacity()\n        # \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for j in range(self._size - 1, index - 1, -1):\n            self._arr[j + 1] = self._arr[j]\n        self._arr[index] = num\n        # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self._size += 1\n\n    def remove(self, index: int) -&gt; int:\n        \"\"\"\u5220\u9664\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        num = self._arr[index]\n        # \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for j in range(index, self._size - 1):\n            self._arr[j] = self._arr[j + 1]\n        # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self._size -= 1\n        # \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num\n\n    def extend_capacity(self):\n        \"\"\"\u5217\u8868\u6269\u5bb9\"\"\"\n        # \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 _extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        self._arr = self._arr + [0] * self.capacity() * (self._extend_ratio - 1)\n        # \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        self._capacity = len(self._arr)\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868\"\"\"\n        return self._arr[: self._size]\n</code></pre> my_list.cpp<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n  private:\n    int *arr;             // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    int arrCapacity = 10; // \u5217\u8868\u5bb9\u91cf\n    int arrSize = 0;      // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    int extendRatio = 2;   // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~MyList() {\n        delete[] arr;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    int size() {\n        return arrSize;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    int capacity() {\n        return arrCapacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    int get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    void add(int num) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size() == capacity())\n            extendCapacity();\n        arr[size()] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size() == capacity())\n            extendCapacity();\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (int j = size() - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    int remove(int index) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (int j = index; j &lt; size() - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    void extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\n        int newCapacity = capacity() * extendRatio;\n        int *tmp = arr;\n        arr = new int[newCapacity];\n        // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        for (int i = 0; i &lt; size(); i++) {\n            arr[i] = tmp[i];\n        }\n        // \u91ca\u653e\u5185\u5b58\n        delete[] tmp;\n        arrCapacity = newCapacity;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Vector \u7528\u4e8e\u6253\u5370 */\n    vector&lt;int&gt; toVector() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        vector&lt;int&gt; vec(size());\n        for (int i = 0; i &lt; size(); i++) {\n            vec[i] = arr[i];\n        }\n        return vec;\n    }\n};\n</code></pre> my_list.java<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private int[] arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private int capacity = 10; // \u5217\u8868\u5bb9\u91cf\n    private int size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private int extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public MyList() {\n        arr = new int[capacity];\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\n    public int size() {\n        return size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    public int capacity() {\n        return capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    public int get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    public void add(int num) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity();\n        arr[size] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    public void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity();\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (int j = size - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    public int remove(int index) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (int j = index; j &lt; size - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    public void extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        arr = Arrays.copyOf(arr, capacity() * extendRatio);\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        capacity = arr.length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    public int[] toArray() {\n        int size = size();\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] arr = new int[size];\n        for (int i = 0; i &lt; size; i++) {\n            arr[i] = get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.cs<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private int[] arr;           // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private int arrCapacity = 10;    // \u5217\u8868\u5bb9\u91cf\n    private int arrSize = 0;         // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private readonly int extendRatio = 2;  // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    public int Size() {\n        return arrSize;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    public int Capacity() {\n        return arrCapacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    public int Get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public void Set(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    public void Add(int num) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        arr[arrSize] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    public void Insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (int j = arrSize - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    public int Remove(int index) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (int j = index; j &lt; arrSize - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    public void ExtendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a arrCapacity * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        Array.Resize(ref arr, arrCapacity * extendRatio);\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        arrCapacity = arr.Length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    public int[] ToArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] arr = new int[arrSize];\n        for (int i = 0; i &lt; arrSize; i++) {\n            arr[i] = Get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.go<pre><code>/* \u5217\u8868\u7c7b */\ntype myList struct {\n    arrCapacity int\n    arr         []int\n    arrSize     int\n    extendRatio int\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newMyList() *myList {\n    return &amp;myList{\n        arrCapacity: 10,              // \u5217\u8868\u5bb9\u91cf\n        arr:         make([]int, 10), // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n        arrSize:     0,               // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        extendRatio: 2,               // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n    }\n}\n\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\nfunc (l *myList) size() int {\n    return l.arrSize\n}\n\n/*  \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc (l *myList) capacity() int {\n    return l.arrCapacity\n}\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc (l *myList) get(index int) int {\n    // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    return l.arr[index]\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc (l *myList) set(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    l.arr[index] = num\n}\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc (l *myList) add(num int) {\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    l.arr[l.arrSize] = num\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    l.arrSize++\n}\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc (l *myList) insert(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for j := l.arrSize - 1; j &gt;= index; j-- {\n        l.arr[j+1] = l.arr[j]\n    }\n    l.arr[index] = num\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    l.arrSize++\n}\n\n/* \u5220\u9664\u5143\u7d20 */\nfunc (l *myList) remove(index int) int {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    num := l.arr[index]\n    // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for j := index; j &lt; l.arrSize-1; j++ {\n        l.arr[j] = l.arr[j+1]\n    }\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    l.arrSize--\n    // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n    return num\n}\n\n/* \u5217\u8868\u6269\u5bb9 */\nfunc (l *myList) extendCapacity() {\n    // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    l.arr = append(l.arr, make([]int, l.arrCapacity*(l.extendRatio-1))...)\n    // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n    l.arrCapacity = len(l.arr)\n}\n\n/* \u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868 */\nfunc (l *myList) toArray() []int {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    return l.arr[:l.arrSize]\n}\n</code></pre> my_list.swift<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private var arr: [Int] // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private var _capacity: Int // \u5217\u8868\u5bb9\u91cf\n    private var _size: Int // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private let extendRatio: Int // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init() {\n        _capacity = 10\n        _size = 0\n        extendRatio = 2\n        arr = Array(repeating: 0, count: _capacity)\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        _capacity\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    func get(index: Int) -&gt; Int {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u9519\u8bef\uff0c\u4e0b\u540c\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        return arr[index]\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    func set(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        arr[index] = num\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    func add(num: Int) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if size() == capacity() {\n            extendCapacity()\n        }\n        arr[size()] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        _size += 1\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    func insert(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if size() == capacity() {\n            extendCapacity()\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for j in (index ..&lt; size()).reversed() {\n            arr[j + 1] = arr[j]\n        }\n        arr[index] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        _size += 1\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    @discardableResult\n    func remove(index: Int) -&gt; Int {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        let num = arr[index]\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for j in index ..&lt; (size() - 1) {\n            arr[j] = arr[j + 1]\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        _size -= 1\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    func extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        arr = arr + Array(repeating: 0, count: capacity() * (extendRatio - 1))\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        _capacity = arr.count\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    func toArray() -&gt; [Int] {\n        Array(arr.prefix(size()))\n    }\n}\n</code></pre> my_list.js<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    #arr = new Array(); // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    #capacity = 10; // \u5217\u8868\u5bb9\u91cf\n    #size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    #extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#arr = new Array(this.#capacity);\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    size() {\n        return this.#size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    capacity() {\n        return this.#capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    get(index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        return this.#arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    set(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        this.#arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    add(num) {\n        // \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\n        this.#arr[this.#size] = num;\n        this.#size++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    insert(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (let j = this.#size - 1; j &gt;= index; j--) {\n            this.#arr[j + 1] = this.#arr[j];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this.#arr[index] = num;\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    remove(index) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        let num = this.#arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (let j = index; j &lt; this.#size - 1; j++) {\n            this.#arr[j] = this.#arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this.#size--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        this.#arr = this.#arr.concat(\n            new Array(this.capacity() * (this.#extendRatio - 1))\n        );\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        this.#capacity = this.#arr.length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    toArray() {\n        let size = this.size();\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.ts<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private arr: Array&lt;number&gt;; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private _capacity: number = 10; // \u5217\u8868\u5bb9\u91cf\n    private _size: number = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private extendRatio: number = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.arr = new Array(this._capacity);\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    public size(): number {\n        return this._size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    public capacity(): number {\n        return this._capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    public get(index: number): number {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        return this.arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public set(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        this.arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    public add(num: number): void {\n        // \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\n        if (this._size === this._capacity) this.extendCapacity();\n        // \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\n        this.arr[this._size] = num;\n        this._size++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    public insert(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (this._size === this._capacity) {\n            this.extendCapacity();\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (let j = this._size - 1; j &gt;= index; j--) {\n            this.arr[j + 1] = this.arr[j];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this.arr[index] = num;\n        this._size++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    public remove(index: number): number {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        let num = this.arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (let j = index; j &lt; this._size - 1; j++) {\n            this.arr[j] = this.arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this._size--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    public extendCapacity(): void {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        this.arr = this.arr.concat(\n            new Array(this.capacity() * (this.extendRatio - 1))\n        );\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        this._capacity = this.arr.length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    public toArray(): number[] {\n        let size = this.size();\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.dart<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n  late List&lt;int&gt; _arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n  int _capacity = 10; // \u5217\u8868\u5bb9\u91cf\n  int _size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n  int _extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  MyList() {\n    _arr = List.filled(_capacity, 0);\n  }\n\n  /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n  int size() =&gt; _size;\n\n  /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n  int capacity() =&gt; _capacity;\n\n  /* \u8bbf\u95ee\u5143\u7d20 */\n  int get(int index) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    return _arr[index];\n  }\n\n  /* \u66f4\u65b0\u5143\u7d20 */\n  void set(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    _arr[index] = _num;\n  }\n\n  /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n  void add(int _num) {\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if (_size == _capacity) extendCapacity();\n    _arr[_size] = _num;\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    _size++;\n  }\n\n  /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n  void insert(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if (_size == _capacity) extendCapacity();\n    // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (var j = _size - 1; j &gt;= index; j--) {\n      _arr[j + 1] = _arr[j];\n    }\n    _arr[index] = _num;\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    _size++;\n  }\n\n  /* \u5220\u9664\u5143\u7d20 */\n  int remove(int index) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    int _num = _arr[index];\n    // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (var j = index; j &lt; _size - 1; j++) {\n      _arr[j] = _arr[j + 1];\n    }\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    _size--;\n    // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n    return _num;\n  }\n\n  /* \u5217\u8868\u6269\u5bb9 */\n  void extendCapacity() {\n    // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 _extendRatio \u500d\u7684\u65b0\u6570\u7ec4\n    final _newNums = List.filled(_capacity * _extendRatio, 0);\n    // \u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    List.copyRange(_newNums, 0, _arr);\n    // \u66f4\u65b0 _arr \u7684\u5f15\u7528\n    _arr = _newNums;\n    // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n    _capacity = _arr.length;\n  }\n\n  /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n  List&lt;int&gt; toArray() {\n    List&lt;int&gt; arr = [];\n    for (var i = 0; i &lt; _size; i++) {\n      arr.add(get(i));\n    }\n    return arr;\n  }\n}\n</code></pre> my_list.rs<pre><code>/* \u5217\u8868\u7c7b */\n#[allow(dead_code)]\nstruct MyList {\n    arr: Vec&lt;i32&gt;,       // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    capacity: usize,     // \u5217\u8868\u5bb9\u91cf\n    size: usize,         // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    extend_ratio: usize, // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n}\n\n#[allow(unused, unused_comparisons)]\nimpl MyList {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(capacity: usize) -&gt; Self {\n        let mut vec = Vec::new();\n        vec.resize(capacity, 0);\n        Self {\n            arr: vec,\n            capacity,\n            size: 0,\n            extend_ratio: 2,\n        }\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    pub fn size(&amp;self) -&gt; usize {\n        return self.size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    pub fn capacity(&amp;self) -&gt; usize {\n        return self.capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    pub fn get(&amp;self, index: usize) -&gt; i32 {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if index &gt;= self.size {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        return self.arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    pub fn set(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        self.arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    pub fn add(&amp;mut self, num: i32) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        self.arr[self.size] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self.size += 1;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    pub fn insert(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size() {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for j in (index..self.size).rev() {\n            self.arr[j + 1] = self.arr[j];\n        }\n        self.arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self.size += 1;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    pub fn remove(&amp;mut self, index: usize) -&gt; i32 {\n        if index &gt;= self.size() {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        let num = self.arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for j in (index..self.size - 1) {\n            self.arr[j] = self.arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self.size -= 1;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    pub fn extend_capacity(&amp;mut self) {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        let new_capacity = self.capacity * self.extend_ratio;\n        self.arr.resize(new_capacity, 0);\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        self.capacity = new_capacity;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    pub fn to_array(&amp;mut self) -&gt; Vec&lt;i32&gt; {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        let mut arr = Vec::new();\n        for i in 0..self.size {\n            arr.push(self.get(i));\n        }\n        arr\n    }\n}\n</code></pre> my_list.c<pre><code>/* \u5217\u8868\u7c7b */\ntypedef struct {\n    int *arr;        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    int capacity;    // \u5217\u8868\u5bb9\u91cf\n    int size;        // \u5217\u8868\u5927\u5c0f\n    int extendRatio; // \u5217\u8868\u6bcf\u6b21\u6269\u5bb9\u7684\u500d\u6570\n} MyList;\n\n/* \u6784\u9020\u51fd\u6570 */\nMyList *newMyList() {\n    MyList *nums = malloc(sizeof(MyList));\n    nums-&gt;capacity = 10;\n    nums-&gt;arr = malloc(sizeof(int) * nums-&gt;capacity);\n    nums-&gt;size = 0;\n    nums-&gt;extendRatio = 2;\n    return nums;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delMyList(MyList *nums) {\n    free(nums-&gt;arr);\n    free(nums);\n}\n\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6 */\nint size(MyList *nums) {\n    return nums-&gt;size;\n}\n\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity(MyList *nums) {\n    return nums-&gt;capacity;\n}\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    return nums-&gt;arr[index];\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    nums-&gt;arr[index] = num;\n}\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(MyList *nums, int num) {\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // \u6269\u5bb9\n    }\n    nums-&gt;arr[size(nums)] = num;\n    nums-&gt;size++;\n}\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // \u6269\u5bb9\n    }\n    for (int i = size(nums); i &gt; index; --i) {\n        nums-&gt;arr[i] = nums-&gt;arr[i - 1];\n    }\n    nums-&gt;arr[index] = num;\n    nums-&gt;size++;\n}\n\n/* \u5220\u9664\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nint removeItem(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    int num = nums-&gt;arr[index];\n    for (int i = index; i &lt; size(nums) - 1; i++) {\n        nums-&gt;arr[i] = nums-&gt;arr[i + 1];\n    }\n    nums-&gt;size--;\n    return num;\n}\n\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity(MyList *nums) {\n    // \u5148\u5206\u914d\u7a7a\u95f4\n    int newCapacity = capacity(nums) * nums-&gt;extendRatio;\n    int *extend = (int *)malloc(sizeof(int) * newCapacity);\n    int *temp = nums-&gt;arr;\n\n    // \u62f7\u8d1d\u65e7\u6570\u636e\u5230\u65b0\u6570\u636e\n    for (int i = 0; i &lt; size(nums); i++)\n        extend[i] = nums-&gt;arr[i];\n\n    // \u91ca\u653e\u65e7\u6570\u636e\n    free(temp);\n\n    // \u66f4\u65b0\u65b0\u6570\u636e\n    nums-&gt;arr = extend;\n    nums-&gt;capacity = newCapacity;\n}\n\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Array \u7528\u4e8e\u6253\u5370 */\nint *toArray(MyList *nums) {\n    return nums-&gt;arr;\n}\n</code></pre> my_list.kt<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private var arr: IntArray = intArrayOf() // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private var capacity = 10 // \u5217\u8868\u5bb9\u91cf\n    private var size = 0 // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private var extendRatio = 2 // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u51fd\u6570 */\n    init {\n        arr = IntArray(capacity)\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\n    fun size(): Int {\n        return size\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    fun capacity(): Int {\n        return capacity\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    fun get(index: Int): Int {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException()\n        return arr[index]\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    fun set(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        arr[index] = num\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    fun add(num: Int) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity()\n        arr[size] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    fun insert(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity()\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (j in size - 1 downTo index)\n            arr[j + 1] = arr[j]\n        arr[index] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    fun remove(index: Int): Int {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        val num: Int = arr[index]\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (j in index..&lt;size - 1)\n            arr[j] = arr[j + 1]\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size--\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    fun extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        arr = arr.copyOf(capacity() * extendRatio)\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        capacity = arr.size\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    fun toArray(): IntArray {\n        val size = size()\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        val arr = IntArray(size)\n        for (i in 0..&lt;size) {\n            arr[i] = get(i)\n        }\n        return arr\n    }\n}\n</code></pre> my_list.rb<pre><code>### \u5217\u8868\u7c7b ###\nclass MyList\n  attr_reader :size       # \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n  attr_reader :capacity   # \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\n\n  ### \u6784\u9020\u65b9\u6cd5 ###\n  def initialize\n    @capacity = 10\n    @size = 0\n    @extend_ratio = 2\n    @arr = Array.new capacity\n  end\n\n  ### \u8bbf\u95ee\u5143\u7d20 ###\n  def get(index)\n    # \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    @arr[index]\n  end\n\n  ### \u8bbf\u95ee\u5143\u7d20 ###\n  def set(index, num)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    @arr[index] = num\n  end\n\n  ### \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 ###\n  def add(num)\n    # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    extend_capacity if size == capacity\n    @arr[size] = num\n\n    # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    @size += 1\n  end\n\n  ### \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 ###\n  def insert(index, num)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n\n    # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    extend_capacity if size == capacity\n\n    # \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for j in (size - 1).downto(index)\n      @arr[j + 1] = @arr[j] \n    end\n    @arr[index] = num\n\n    # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    @size += 1\n  end\n\n  ### \u5220\u9664\u5143\u7d20 ###\n  def remove(index)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    num = @arr[index]\n\n    # \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for j in index...size\n      @arr[j] = @arr[j + 1]\n    end\n\n    # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    @size -= 1\n\n    # \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n    num\n  end\n\n  ### \u5217\u8868\u6269\u5bb9 ###\n  def extend_capacity\n    # \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    arr = @arr.dup + Array.new(capacity * (@extend_ratio - 1))\n    # \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n    @capacity = arr.length\n  end\n\n  ### \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 ###\n  def to_array\n    sz = size\n    # \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    arr = Array.new sz\n    for i in 0...sz\n      arr[i] = get i\n    end\n    arr\n  end\nend\n</code></pre> my_list.zig<pre><code>// \u5217\u8868\u7c7b\nfn MyList(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        arr: []T = undefined,                        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n        arrCapacity: usize = 10,                     // \u5217\u8868\u5bb9\u91cf\n        numSize: usize = 0,                           // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        extendRatio: usize = 2,                       // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u5217\u8868\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.arr = try self.mem_allocator.alloc(T, self.arrCapacity);\n            @memset(self.arr, @as(T, 0));\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        pub fn size(self: *Self) usize {\n            return self.numSize;\n        }\n\n        // \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\n        pub fn capacity(self: *Self) usize {\n            return self.arrCapacity;\n        }\n\n        // \u8bbf\u95ee\u5143\u7d20\n        pub fn get(self: *Self, index: usize) T {\n            // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            return self.arr[index];\n        }  \n\n        // \u66f4\u65b0\u5143\u7d20\n        pub fn set(self: *Self, index: usize, num: T) void {\n            // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            self.arr[index] = num;\n        }  \n\n        // \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\n        pub fn add(self: *Self, num: T) !void {\n            // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n            if (self.size() == self.capacity()) try self.extendCapacity();\n            self.arr[self.size()] = num;\n            // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n            self.numSize += 1;\n        }  \n\n        // \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\n        pub fn insert(self: *Self, index: usize, num: T) !void {\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n            if (self.size() == self.capacity()) try self.extendCapacity();\n            // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n            var j = self.size() - 1;\n            while (j &gt;= index) : (j -= 1) {\n                self.arr[j + 1] = self.arr[j];\n            }\n            self.arr[index] = num;\n            // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n            self.numSize += 1;\n        }\n\n        // \u5220\u9664\u5143\u7d20\n        pub fn remove(self: *Self, index: usize) T {\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            var num = self.arr[index];\n            // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n            var j = index;\n            while (j &lt; self.size() - 1) : (j += 1) {\n                self.arr[j] = self.arr[j + 1];\n            }\n            // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n            self.numSize -= 1;\n            // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n            return num;\n        }\n\n        // \u5217\u8868\u6269\u5bb9\n        pub fn extendCapacity(self: *Self) !void {\n            // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n            var newCapacity = self.capacity() * self.extendRatio;\n            var extend = try self.mem_allocator.alloc(T, newCapacity);\n            @memset(extend, @as(T, 0));\n            // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n            std.mem.copy(T, extend, self.arr);\n            self.arr = extend;\n            // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n            self.arrCapacity = newCapacity;\n        }\n\n        // \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n            var arr = try self.mem_allocator.alloc(T, self.size());\n           @memset(arr, @as(T, 0));\n            for (arr, 0..) |*num, i| {\n                num.* = self.get(i);\n            }\n            return arr;\n        }\n    };\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/","title":"4.4 \u00a0 Memory and Cache *","text":"<p>In the first two sections of this chapter, we explored arrays and linked lists, two fundamental and important data structures, representing \"continuous storage\" and \"dispersed storage\" respectively.</p> <p>In fact, the physical structure largely determines the efficiency of a program's use of memory and cache, which in turn affects the overall performance of the algorithm.</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#441-computer-storage-devices","title":"4.4.1 \u00a0 Computer Storage Devices","text":"<p>There are three types of storage devices in computers: \"hard disk,\" \"random-access memory (RAM),\" and \"cache memory.\" The following table shows their different roles and performance characteristics in computer systems.</p> <p> Table 4-2 \u00a0 Computer Storage Devices </p> Hard Disk Memory Cache Usage Long-term storage of data, including OS, programs, files, etc. Temporary storage of currently running programs and data being processed Stores frequently accessed data and instructions, reducing the number of CPU accesses to memory Volatility Data is not lost after power off Data is lost after power off Data is lost after power off Capacity Larger, TB level Smaller, GB level Very small, MB level Speed Slower, several hundred to thousands MB/s Faster, several tens of GB/s Very fast, several tens to hundreds of GB/s Price Cheaper, several cents to yuan / GB More expensive, tens to hundreds of yuan / GB Very expensive, priced with CPU <p>We can imagine the computer storage system as a pyramid structure shown in the Figure 4-9 . The storage devices closer to the top of the pyramid are faster, have smaller capacity, and are more costly. This multi-level design is not accidental, but the result of careful consideration by computer scientists and engineers.</p> <ul> <li>Hard disks are difficult to replace with memory. Firstly, data in memory is lost after power off, making it unsuitable for long-term data storage; secondly, the cost of memory is dozens of times that of hard disks, making it difficult to popularize in the consumer market.</li> <li>It is difficult for caches to have both large capacity and high speed. As the capacity of L1, L2, L3 caches gradually increases, their physical size becomes larger, increasing the physical distance from the CPU core, leading to increased data transfer time and higher element access latency. Under current technology, a multi-level cache structure is the best balance between capacity, speed, and cost.</li> </ul> <p></p> <p> Figure 4-9 \u00a0 Computer Storage System </p> <p>Note</p> <p>The storage hierarchy of computers reflects a delicate balance between speed, capacity, and cost. In fact, this kind of trade-off is common in all industrial fields, requiring us to find the best balance between different advantages and limitations.</p> <p>Overall, hard disks are used for long-term storage of large amounts of data, memory is used for temporary storage of data being processed during program execution, and cache is used to store frequently accessed data and instructions to improve program execution efficiency. Together, they ensure the efficient operation of computer systems.</p> <p>As shown in the Figure 4-10 , during program execution, data is read from the hard disk into memory for CPU computation. The cache can be considered a part of the CPU, smartly loading data from memory to provide fast data access to the CPU, significantly enhancing program execution efficiency and reducing reliance on slower memory.</p> <p></p> <p> Figure 4-10 \u00a0 Data Flow Between Hard Disk, Memory, and Cache </p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#442-memory-efficiency-of-data-structures","title":"4.4.2 \u00a0 Memory Efficiency of Data Structures","text":"<p>In terms of memory space utilization, arrays and linked lists have their advantages and limitations.</p> <p>On one hand, memory is limited and cannot be shared by multiple programs, so we hope that data structures can use space as efficiently as possible. The elements of an array are tightly packed without extra space for storing references (pointers) between linked list nodes, making them more space-efficient. However, arrays require allocating sufficient continuous memory space at once, which may lead to memory waste, and array expansion also requires additional time and space costs. In contrast, linked lists allocate and reclaim memory dynamically on a per-node basis, providing greater flexibility.</p> <p>On the other hand, during program execution, as memory is repeatedly allocated and released, the degree of fragmentation of free memory becomes higher, leading to reduced memory utilization efficiency. Arrays, due to their continuous storage method, are relatively less likely to cause memory fragmentation. In contrast, the elements of a linked list are dispersedly stored, and frequent insertion and deletion operations make memory fragmentation more likely.</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#443-cache-efficiency-of-data-structures","title":"4.4.3 \u00a0 Cache Efficiency of Data Structures","text":"<p>Although caches are much smaller in space capacity than memory, they are much faster and play a crucial role in program execution speed. Since the cache's capacity is limited and can only store a small part of frequently accessed data, when the CPU tries to access data not in the cache, a \"cache miss\" occurs, forcing the CPU to load the needed data from slower memory.</p> <p>Clearly, the fewer the cache misses, the higher the CPU's data read-write efficiency, and the better the program performance. The proportion of successful data retrieval from the cache by the CPU is called the \"cache hit rate,\" a metric often used to measure cache efficiency.</p> <p>To achieve higher efficiency, caches adopt the following data loading mechanisms.</p> <ul> <li>Cache Lines: Caches don't store and load data byte by byte but in units of cache lines. Compared to byte-by-byte transfer, the transmission of cache lines is more efficient.</li> <li>Prefetch Mechanism: Processors try to predict data access patterns (such as sequential access, fixed stride jumping access, etc.) and load data into the cache according to specific patterns to improve the hit rate.</li> <li>Spatial Locality: If data is accessed, data nearby is likely to be accessed in the near future. Therefore, when loading certain data, the cache also loads nearby data to improve the hit rate.</li> <li>Temporal Locality: If data is accessed, it's likely to be accessed again in the near future. Caches use this principle to retain recently accessed data to improve the hit rate.</li> </ul> <p>In fact, arrays and linked lists have different cache utilization efficiencies, mainly reflected in the following aspects.</p> <ul> <li>Occupied Space: Linked list elements occupy more space than array elements, resulting in less effective data volume in the cache.</li> <li>Cache Lines: Linked list data is scattered throughout memory, and since caches load \"by line,\" the proportion of loading invalid data is higher.</li> <li>Prefetch Mechanism: The data access pattern of arrays is more \"predictable\" than that of linked lists, meaning the system is more likely to guess which data will be loaded next.</li> <li>Spatial Locality: Arrays are stored in concentrated memory spaces, so the data near the loaded data is more likely to be accessed next.</li> </ul> <p>Overall, arrays have a higher cache hit rate and are generally more efficient in operation than linked lists. This makes data structures based on arrays more popular in solving algorithmic problems.</p> <p>It should be noted that high cache efficiency does not mean that arrays are always better than linked lists. Which data structure to choose in actual applications should be based on specific requirements. For example, both arrays and linked lists can implement the \"stack\" data structure (which will be detailed in the next chapter), but they are suitable for different scenarios.</p> <ul> <li>In algorithm problems, we tend to choose stacks based on arrays because they provide higher operational efficiency and random access capabilities, with the only cost being the need to pre-allocate a certain amount of memory space for the array.</li> <li>If the data volume is very large, highly dynamic, and the expected size of the stack is difficult to estimate, then a stack based on a linked list is more appropriate. Linked lists can disperse a large amount of data in different parts of the memory and avoid the additional overhead of array expansion.</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/","title":"4.5 \u00a0 Summary","text":""},{"location":"chapter_array_and_linkedlist/summary/#1-key-review","title":"1. \u00a0 Key Review","text":"<ul> <li>Arrays and linked lists are two basic data structures, representing two storage methods in computer memory: contiguous space storage and non-contiguous space storage. Their characteristics complement each other.</li> <li>Arrays support random access and use less memory; however, they are inefficient in inserting and deleting elements and have a fixed length after initialization.</li> <li>Linked lists implement efficient node insertion and deletion through changing references (pointers) and can flexibly adjust their length; however, they have lower node access efficiency and consume more memory.</li> <li>Common types of linked lists include singly linked lists, circular linked lists, and doubly linked lists, each with its own application scenarios.</li> <li>Lists are ordered collections of elements that support addition, deletion, and modification, typically implemented based on dynamic arrays, retaining the advantages of arrays while allowing flexible length adjustment.</li> <li>The advent of lists significantly enhanced the practicality of arrays but may lead to some memory space wastage.</li> <li>During program execution, data is mainly stored in memory. Arrays provide higher memory space efficiency, while linked lists are more flexible in memory usage.</li> <li>Caches provide fast data access to CPUs through mechanisms like cache lines, prefetching, spatial locality, and temporal locality, significantly enhancing program execution efficiency.</li> <li>Due to higher cache hit rates, arrays are generally more efficient than linked lists. When choosing a data structure, the appropriate choice should be made based on specific needs and scenarios.</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: Does storing arrays on the stack versus the heap affect time and space efficiency?</p> <p>Arrays stored on both the stack and heap are stored in contiguous memory spaces, and data operation efficiency is essentially the same. However, stacks and heaps have their own characteristics, leading to the following differences.</p> <ol> <li>Allocation and release efficiency: The stack is a smaller memory block, allocated automatically by the compiler; the heap memory is relatively larger and can be dynamically allocated in the code, more prone to fragmentation. Therefore, allocation and release operations on the heap are generally slower than on the stack.</li> <li>Size limitation: Stack memory is relatively small, while the heap size is generally limited by available memory. Therefore, the heap is more suitable for storing large arrays.</li> <li>Flexibility: The size of arrays on the stack needs to be determined at compile-time, while the size of arrays on the heap can be dynamically determined at runtime.</li> </ol> <p>Q: Why do arrays require elements of the same type, while linked lists do not emphasize same-type elements?</p> <p>Linked lists consist of nodes connected by references (pointers), and each node can store data of different types, such as int, double, string, object, etc.</p> <p>In contrast, array elements must be of the same type, allowing the calculation of offsets to access the corresponding element positions. For example, an array containing both int and long types, with single elements occupying 4 bytes and 8 bytes respectively, cannot use the following formula to calculate offsets, as the array contains elements of two different lengths.</p> <pre><code># Element memory address = Array memory address + Element length * Element index\n</code></pre> <p>Q: After deleting a node, is it necessary to set <code>P.next</code> to <code>None</code>?</p> <p>Not modifying <code>P.next</code> is also acceptable. From the perspective of the linked list, traversing from the head node to the tail node will no longer encounter <code>P</code>. This means that node <code>P</code> has been effectively removed from the list, and where <code>P</code> points no longer affects the list.</p> <p>From a garbage collection perspective, for languages with automatic garbage collection mechanisms like Java, Python, and Go, whether node <code>P</code> is collected depends on whether there are still references pointing to it, not on the value of <code>P.next</code>. In languages like C and C++, we need to manually free the node's memory.</p> <p>Q: In linked lists, the time complexity for insertion and deletion operations is <code>O(1)</code>. But searching for the element before insertion or deletion takes <code>O(n)</code> time, so why isn't the time complexity <code>O(n)</code>?</p> <p>If an element is searched first and then deleted, the time complexity is indeed <code>O(n)</code>. However, the <code>O(1)</code> advantage of linked lists in insertion and deletion can be realized in other applications. For example, in the implementation of double-ended queues using linked lists, we maintain pointers always pointing to the head and tail nodes, making each insertion and deletion operation <code>O(1)</code>.</p> <p>Q: In the image \"Linked List Definition and Storage Method\", do the light blue storage nodes occupy a single memory address, or do they share half with the node value?</p> <p>The diagram is just a qualitative representation; quantitative analysis depends on specific situations.</p> <ul> <li>Different types of node values occupy different amounts of space, such as int, long, double, and object instances.</li> <li>The memory space occupied by pointer variables depends on the operating system and compilation environment used, usually 8 bytes or 4 bytes.</li> </ul> <p>Q: Is adding elements to the end of a list always <code>O(1)</code>?</p> <p>If adding an element exceeds the list length, the list needs to be expanded first. The system will request a new memory block and move all elements of the original list over, in which case the time complexity becomes <code>O(n)</code>.</p> <p>Q: The statement \"The emergence of lists greatly improves the practicality of arrays, but may lead to some memory space wastage\" - does this refer to the memory occupied by additional variables like capacity, length, and expansion multiplier?</p> <p>The space wastage here mainly refers to two aspects: on the one hand, lists are set with an initial length, which we may not always need; on the other hand, to prevent frequent expansion, expansion usually multiplies by a coefficient, such as \\(\\times 1.5\\). This results in many empty slots, which we typically cannot fully fill.</p> <p>Q: In Python, after initializing <code>n = [1, 2, 3]</code>, the addresses of these 3 elements are contiguous, but initializing <code>m = [2, 1, 3]</code> shows that each element's <code>id</code> is not consecutive but identical to those in <code>n</code>. If the addresses of these elements are not contiguous, is <code>m</code> still an array?</p> <p>If we replace list elements with linked list nodes <code>n = [n1, n2, n3, n4, n5]</code>, these 5 node objects are also typically dispersed throughout memory. However, given a list index, we can still access the node's memory address in <code>O(1)</code> time, thereby accessing the corresponding node. This is because the array stores references to the nodes, not the nodes themselves.</p> <p>Unlike many languages, in Python, numbers are also wrapped as objects, and lists store references to these numbers, not the numbers themselves. Therefore, we find that the same number in two arrays has the same <code>id</code>, and these numbers' memory addresses need not be contiguous.</p> <p>Q: The <code>std::list</code> in C++ STL has already implemented a doubly linked list, but it seems that some algorithm books don't directly use it. Is there any limitation?</p> <p>On the one hand, we often prefer to use arrays to implement algorithms, only using linked lists when necessary, mainly for two reasons.</p> <ul> <li>Space overhead: Since each element requires two additional pointers (one for the previous element and one for the next), <code>std::list</code> usually occupies more space than <code>std::vector</code>.</li> <li>Cache unfriendly: As the data is not stored continuously, <code>std::list</code> has a lower cache utilization rate. Generally, <code>std::vector</code> performs better.</li> </ul> <p>On the other hand, linked lists are primarily necessary for binary trees and graphs. Stacks and queues are often implemented using the programming language's <code>stack</code> and <code>queue</code> classes, rather than linked lists.</p> <p>Q: Does initializing a list <code>res = [0] * self.size()</code> result in each element of <code>res</code> referencing the same address?</p> <p>No. However, this issue arises with two-dimensional arrays, for example, initializing a two-dimensional list <code>res = [[0] * self.size()]</code> would reference the same list <code>[0]</code> multiple times.</p> <p>Q: In deleting a node, is it necessary to break the reference to its successor node?</p> <p>From the perspective of data structures and algorithms (problem-solving), it's okay not to break the link, as long as the program's logic is correct. From the perspective of standard libraries, breaking the link is safer and more logically clear. If the link is not broken, and the deleted node is not properly recycled, it could affect the recycling of the successor node's memory.</p>"},{"location":"chapter_computational_complexity/","title":"Chapter 2. \u00a0 Complexity Analysis","text":"<p>Abstract</p> <p>Complexity analysis is like a space-time navigator in the vast universe of algorithms.</p> <p>It guides us in exploring deeper within the the dimensions of time and space, seeking more elegant solutions.</p>"},{"location":"chapter_computational_complexity/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>2.1 \u00a0 Algorithm Efficiency Assessment</li> <li>2.2 \u00a0 Iteration and Recursion</li> <li>2.3 \u00a0 Time Complexity</li> <li>2.4 \u00a0 Space Complexity</li> <li>2.5 \u00a0 Summary</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/","title":"2.2 \u00a0 Iteration and Recursion","text":"<p>In algorithms, the repeated execution of a task is quite common and is closely related to the analysis of complexity. Therefore, before delving into the concepts of time complexity and space complexity, let's first explore how to implement repetitive tasks in programming. This involves understanding two fundamental programming control structures: iteration and recursion.</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#221-iteration","title":"2.2.1 \u00a0 Iteration","text":"<p>\"Iteration\" is a control structure for repeatedly performing a task. In iteration, a program repeats a block of code as long as a certain condition is met until this condition is no longer satisfied.</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for-loops","title":"1. \u00a0 For Loops","text":"<p>The <code>for</code> loop is one of the most common forms of iteration, and it's particularly suitable when the number of iterations is known in advance.</p> <p>The following function uses a <code>for</code> loop to perform a summation of \\(1 + 2 + \\dots + n\\), with the sum being stored in the variable <code>res</code>. It's important to note that in Python, <code>range(a, b)</code> creates an interval that is inclusive of <code>a</code> but exclusive of <code>b</code>, meaning it iterates over the range from \\(a\\) up to \\(b\u22121\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def for_loop(n: int) -&gt; int:\n    \"\"\"for \u5faa\u73af\"\"\"\n    res = 0\n    # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        res += i\n    return res\n</code></pre> iteration.cpp<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* for \u5faa\u73af */\nint ForLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* for \u5faa\u73af */\nfunc forLoop(n int) int {\n    res := 0\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* for \u5faa\u73af */\nfunc forLoop(n: Int) -&gt; Int {\n    var res = 0\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* for \u5faa\u73af */\nfunction forLoop(n) {\n    let res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* for \u5faa\u73af */\nfunction forLoop(n: number): number {\n    let res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n  int res = 0;\n  // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    res += i;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* for \u5faa\u73af */\nfn for_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i in 1..=n {\n        res += i;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* for \u5faa\u73af */\nfun forLoop(n: Int): Int {\n    var res = 0\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (i in 1..n) {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{for_loop}\n</code></pre> iteration.zig<pre><code>// for \u5faa\u73af\nfn forLoop(n: usize) i32 {\n    var res: i32 = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (1..n+1) |i| {\n        res = res + @as(i32, @intCast(i));\n    }\n    return res;\n} \n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The flowchart below represents this sum function.</p> <p></p> <p> Figure 2-1 \u00a0 Flowchart of the Sum Function </p> <p>The number of operations in this summation function is proportional to the size of the input data \\(n\\), or in other words, it has a \"linear relationship.\" This \"linear relationship\" is what time complexity describes. This topic will be discussed in more detail in the next section.</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while-loops","title":"2. \u00a0 While Loops","text":"<p>Similar to <code>for</code> loops, <code>while</code> loops are another approach for implementing iteration. In a <code>while</code> loop, the program checks a condition at the beginning of each iteration; if the condition is true, the execution continues, otherwise, the loop ends.</p> <p>Below we use a <code>while</code> loop to implement the sum \\(1 + 2 + \\dots + n\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def while_loop(n: int) -&gt; int:\n    \"\"\"while \u5faa\u73af\"\"\"\n    res = 0\n    i = 1  # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n:\n        res += i\n        i += 1  # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    return res\n</code></pre> iteration.cpp<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u5faa\u73af */\nint WhileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while \u5faa\u73af */\nfunc whileLoop(n int) int {\n    res := 0\n    // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    i := 1\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while \u5faa\u73af */\nfunc whileLoop(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i\n        i += 1 // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while \u5faa\u73af */\nfunction whileLoop(n) {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u5faa\u73af */\nfunction whileLoop(n: number): number {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n  int res = 0;\n  int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n  // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n  while (i &lt;= n) {\n    res += i;\n    i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u5faa\u73af */\nfn while_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i;\n        i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while \u5faa\u73af */\nfun whileLoop(n: Int): Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i\n        i++ // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{while_loop}\n</code></pre> iteration.zig<pre><code>// while \u5faa\u73af\nfn whileLoop(n: i32) i32 {\n    var res: i32 = 0;\n    var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += @intCast(i);\n        i += 1;\n    }\n    return res;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p><code>While</code> loops provide more flexibility than <code>for</code> loops, especially since they allow for custom initialization and modification of the condition variable at each step.</p> <p>For example, in the following code, the condition variable \\(i\\) is updated twice each round, which would be inconvenient to implement with a <code>for</code> loop.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def while_loop_ii(n: int) -&gt; int:\n    \"\"\"while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\"\"\"\n    res = 0\n    i = 1  # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    # \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n:\n        res += i\n        # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1\n        i *= 2\n    return res\n</code></pre> iteration.cpp<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint WhileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1; \n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n int) int {\n    res := 0\n    // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    i := 1\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    for i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n) {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n: number): number {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n  int res = 0;\n  int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n  // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n  while (i &lt;= n) {\n    res += i;\n    // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    i++;\n    i *= 2;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfn while_loop_ii(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1;\n        i *= 2;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfun whileLoopII(n: Int): Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{while_loop_ii}\n</code></pre> iteration.zig<pre><code>//  while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\nfn whileLoopII(n: i32) i32 {\n    var res: i32 = 0;\n    var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += @intCast(i);\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Overall, <code>for</code> loops are more concise, while <code>while</code> loops are more flexible. Both can implement iterative structures. Which one to use should be determined based on the specific requirements of the problem.</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3-nested-loops","title":"3. \u00a0 Nested Loops","text":"<p>We can nest one loop structure within another. Below is an example using <code>for</code> loops:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def nested_for_loop(n: int) -&gt; str:\n    \"\"\"\u53cc\u5c42 for \u5faa\u73af\"\"\"\n    res = \"\"\n    # \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        # \u5faa\u73af j = 1, 2, ..., n-1, n\n        for j in range(1, n + 1):\n            res += f\"({i}, {j}), \"\n    return res\n</code></pre> iteration.cpp<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nstring nestedForLoop(int n) {\n    ostringstream res;\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; ++j) {\n            res &lt;&lt; \"(\" &lt;&lt; i &lt;&lt; \", \" &lt;&lt; j &lt;&lt; \"), \";\n        }\n    }\n    return res.str();\n}\n</code></pre> iteration.java<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n    StringBuilder res = new StringBuilder();\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.append(\"(\" + i + \", \" + j + \"), \");\n        }\n    }\n    return res.toString();\n}\n</code></pre> iteration.cs<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nstring NestedForLoop(int n) {\n    StringBuilder res = new();\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.Append($\"({i}, {j}), \");\n        }\n    }\n    return res.ToString();\n}\n</code></pre> iteration.go<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n int) string {\n    res := \"\"\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= n; j++ {\n            // \u5faa\u73af j = 1, 2, ..., n-1, n\n            res += fmt.Sprintf(\"(%d, %d), \", i, j)\n        }\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n: Int) -&gt; String {\n    var res = \"\"\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for j in 1 ... n {\n            res.append(\"(\\(i), \\(j)), \")\n        }\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n) {\n    let res = '';\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n: number): string {\n    let res = '';\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n  String res = \"\";\n  // \u5faa\u73af i = 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    // \u5faa\u73af j = 1, 2, ..., n-1, n\n    for (int j = 1; j &lt;= n; j++) {\n      res += \"($i, $j), \";\n    }\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfn nested_for_loop(n: i32) -&gt; String {\n    let mut res = vec![];\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i in 1..=n {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for j in 1..=n {\n            res.push(format!(\"({}, {}), \", i, j));\n        }\n    }\n    res.join(\"\")\n}\n</code></pre> iteration.c<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nchar *nestedForLoop(int n) {\n    // n * n \u4e3a\u5bf9\u5e94\u70b9\u6570\u91cf\uff0c\"(i, j), \" \u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u6700\u5927\u4e3a 6+10*2\uff0c\u52a0\u4e0a\u6700\u540e\u4e00\u4e2a\u7a7a\u5b57\u7b26 \\0 \u7684\u989d\u5916\u7a7a\u95f4\n    int size = n * n * 26 + 1;\n    char *res = malloc(size * sizeof(char));\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            char tmp[26];\n            snprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\n            strncat(res, tmp, size - strlen(res) - 1);\n        }\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfun nestedForLoop(n: Int): String {\n    val res = StringBuilder()\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (i in 1..n) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (j in 1..n) {\n            res.append(\" ($i, $j), \")\n        }\n    }\n    return res.toString()\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{nested_for_loop}\n</code></pre> iteration.zig<pre><code>// \u53cc\u5c42 for \u5faa\u73af\nfn nestedForLoop(allocator: Allocator, n: usize) ![]const u8 {\n    var res = std.ArrayList(u8).init(allocator);\n    defer res.deinit();\n    var buffer: [20]u8 = undefined;\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (1..n+1) |i| {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (1..n+1) |j| {\n            var _str = try std.fmt.bufPrint(&amp;buffer, \"({d}, {d}), \", .{i, j});\n            try res.appendSlice(_str);\n        }\n    }\n    return res.toOwnedSlice();\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The flowchart below represents this nested loop.</p> <p></p> <p> Figure 2-2 \u00a0 Flowchart of the Nested Loop </p> <p>In such cases, the number of operations of the function is proportional to \\(n^2\\), meaning the algorithm's runtime and the size of the input data \\(n\\) has a 'quadratic relationship.'</p> <p>We can further increase the complexity by adding more nested loops, each level of nesting effectively \"increasing the dimension,\" which raises the time complexity to \"cubic,\" \"quartic,\" and so on.</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#222-recursion","title":"2.2.2 \u00a0 Recursion","text":"<p>\"Recursion\" is an algorithmic strategy where a function solves a problem by calling itself. It primarily involves two phases:</p> <ol> <li>Calling: This is where the program repeatedly calls itself, often with progressively smaller or simpler arguments, moving towards the \"termination condition.\"</li> <li>Returning: Upon triggering the \"termination condition,\" the program begins to return from the deepest recursive function, aggregating the results of each layer.</li> </ol> <p>From an implementation perspective, recursive code mainly includes three elements.</p> <ol> <li>Termination Condition: Determines when to switch from \"calling\" to \"returning.\"</li> <li>Recursive Call: Corresponds to \"calling,\" where the function calls itself, usually with smaller or more simplified parameters.</li> <li>Return Result: Corresponds to \"returning,\" where the result of the current recursion level is returned to the previous layer.</li> </ol> <p>Observe the following code, where simply calling the function <code>recur(n)</code> can compute the sum of \\(1 + 2 + \\dots + n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def recur(n: int) -&gt; int:\n    \"\"\"\u9012\u5f52\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6\n    if n == 1:\n        return 1\n    # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    res = recur(n - 1)\n    # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res\n</code></pre> recursion.cpp<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.java<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.cs<pre><code>/* \u9012\u5f52 */\nint Recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = Recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.go<pre><code>/* \u9012\u5f52 */\nfunc recur(n int) int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 1 {\n        return 1\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    res := recur(n - 1)\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res\n}\n</code></pre> recursion.swift<pre><code>/* \u9012\u5f52 */\nfunc recur(n: Int) -&gt; Int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 1 {\n        return 1\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    let res = recur(n: n - 1)\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res\n}\n</code></pre> recursion.js<pre><code>/* \u9012\u5f52 */\nfunction recur(n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 1) return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    const res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.ts<pre><code>/* \u9012\u5f52 */\nfunction recur(n: number): number {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 1) return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    const res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.dart<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n  // \u7ec8\u6b62\u6761\u4ef6\n  if (n == 1) return 1;\n  // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n  int res = recur(n - 1);\n  // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n  return n + res;\n}\n</code></pre> recursion.rs<pre><code>/* \u9012\u5f52 */\nfn recur(n: i32) -&gt; i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 1 {\n        return 1;\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    let res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    n + res\n}\n</code></pre> recursion.c<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.kt<pre><code>/* \u9012\u5f52 */\nfun recur(n: Int): Int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1\n    // \u9012: \u9012\u5f52\u8c03\u7528\n    val res = recur(n - 1)\n    // \u5f52: \u8fd4\u56de\u7ed3\u679c\n    return n + res\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{recur}\n</code></pre> recursion.zig<pre><code>// \u9012\u5f52\u51fd\u6570\nfn recur(n: i32) i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1) {\n        return 1;\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    var res: i32 = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The Figure 2-3  shows the recursive process of this function.</p> <p></p> <p> Figure 2-3 \u00a0 Recursive Process of the Sum Function </p> <p>Although iteration and recursion can achieve the same results from a computational standpoint, they represent two entirely different paradigms of thinking and problem-solving.</p> <ul> <li>Iteration: Solves problems \"from the bottom up.\" It starts with the most basic steps, and then repeatedly adds or accumulates these steps until the task is complete.</li> <li>Recursion: Solves problems \"from the top down.\" It breaks down the original problem into smaller sub-problems, each of which has the same form as the original problem. These sub-problems are then further decomposed into even smaller sub-problems, stopping at the base case whose solution is known.</li> </ul> <p>Let's take the earlier example of the summation function, defined as \\(f(n) = 1 + 2 + \\dots + n\\).</p> <ul> <li>Iteration: In this approach, we simulate the summation process within a loop. Starting from \\(1\\) and traversing to \\(n\\), we perform the summation operation in each iteration to eventually compute \\(f(n)\\).</li> <li>Recursion: Here, the problem is broken down into a sub-problem: \\(f(n) = n + f(n-1)\\). This decomposition continues recursively until reaching the base case, \\(f(1) = 1\\), at which point the recursion terminates.</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-call-stack","title":"1. \u00a0 Call Stack","text":"<p>Every time a recursive function calls itself, the system allocates memory for the newly initiated function to store local variables, the return address, and other relevant information. This leads to two primary outcomes.</p> <ul> <li>The function's context data is stored in a memory area called \"stack frame space\" and is only released after the function returns. Therefore, recursion generally consumes more memory space than iteration.</li> <li>Recursive calls introduce additional overhead. Hence, recursion is usually less time-efficient than loops.</li> </ul> <p>As shown in the Figure 2-4 , there are \\(n\\) unreturned recursive functions before triggering the termination condition, indicating a recursion depth of \\(n\\).</p> <p></p> <p> Figure 2-4 \u00a0 Recursion Call Depth </p> <p>In practice, the depth of recursion allowed by programming languages is usually limited, and excessively deep recursion can lead to stack overflow errors.</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-tail-recursion","title":"2. \u00a0 Tail Recursion","text":"<p>Interestingly, if a function performs its recursive call as the very last step before returning, it can be optimized by the compiler or interpreter to be as space-efficient as iteration. This scenario is known as \"tail recursion.\"</p> <ul> <li>Regular Recursion: In standard recursion, when the function returns to the previous level, it continues to execute more code, requiring the system to save the context of the previous call.</li> <li>Tail Recursion: Here, the recursive call is the final operation before the function returns. This means that upon returning to the previous level, no further actions are needed, so the system does not need to save the context of the previous level.</li> </ul> <p>For example, in calculating \\(1 + 2 + \\dots + n\\), we can make the result variable <code>res</code> a parameter of the function, thereby achieving tail recursion:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def tail_recur(n, res):\n    \"\"\"\u5c3e\u9012\u5f52\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6\n    if n == 0:\n        return res\n    # \u5c3e\u9012\u5f52\u8c03\u7528\n    return tail_recur(n - 1, res + n)\n</code></pre> recursion.cpp<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.java<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.cs<pre><code>/* \u5c3e\u9012\u5f52 */\nint TailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return TailRecur(n - 1, res + n);\n}\n</code></pre> recursion.go<pre><code>/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n int, res int) int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 0 {\n        return res\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n-1, res+n)\n}\n</code></pre> recursion.swift<pre><code>/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n: Int, res: Int) -&gt; Int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 0 {\n        return res\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n: n - 1, res: res + n)\n}\n</code></pre> recursion.js<pre><code>/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n, res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 0) return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.ts<pre><code>/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n: number, res: number): number {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 0) return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.dart<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n  // \u7ec8\u6b62\u6761\u4ef6\n  if (n == 0) return res;\n  // \u5c3e\u9012\u5f52\u8c03\u7528\n  return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.rs<pre><code>/* \u5c3e\u9012\u5f52 */\nfn tail_recur(n: i32, res: i32) -&gt; i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 0 {\n        return res;\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    tail_recur(n - 1, res + n)\n}\n</code></pre> recursion.c<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.kt<pre><code>/* \u5c3e\u9012\u5f52 */\ntailrec fun tailRecur(n: Int, res: Int): Int {\n    // \u6dfb\u52a0 tailrec \u5173\u952e\u8bcd\uff0c\u4ee5\u5f00\u542f\u5c3e\u9012\u5f52\u4f18\u5316\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n)\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{tail_recur}\n</code></pre> recursion.zig<pre><code>// \u5c3e\u9012\u5f52\u51fd\u6570\nfn tailRecur(n: i32, res: i32) i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0) {\n        return res;\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The execution process of tail recursion is shown in the following figure. Comparing regular recursion and tail recursion, the point of the summation operation is different.</p> <ul> <li>Regular Recursion: The summation operation occurs during the \"returning\" phase, requiring another summation after each layer returns.</li> <li>Tail Recursion: The summation operation occurs during the \"calling\" phase, and the \"returning\" phase only involves returning through each layer.</li> </ul> <p></p> <p> Figure 2-5 \u00a0 Tail Recursion Process </p> <p>Tip</p> <p>Note that many compilers or interpreters do not support tail recursion optimization. For example, Python does not support tail recursion optimization by default, so even if the function is in the form of tail recursion, it may still encounter stack overflow issues.</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3-recursion-tree","title":"3. \u00a0 Recursion Tree","text":"<p>When dealing with algorithms related to \"divide and conquer\", recursion often offers a more intuitive approach and more readable code than iteration. Take the \"Fibonacci sequence\" as an example.</p> <p>Question</p> <p>Given a Fibonacci sequence \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\), find the \\(n\\)th number in the sequence.</p> <p>Let the \\(n\\)th number of the Fibonacci sequence be \\(f(n)\\), it's easy to deduce two conclusions:</p> <ul> <li>The first two numbers of the sequence are \\(f(1) = 0\\) and \\(f(2) = 1\\).</li> <li>Each number in the sequence is the sum of the two preceding ones, that is, \\(f(n) = f(n - 1) + f(n - 2)\\).</li> </ul> <p>Using the recursive relation, and considering the first two numbers as termination conditions, we can write the recursive code. Calling <code>fib(n)</code> will yield the \\(n\\)th number of the Fibonacci sequence:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def fib(n: int) -&gt; int:\n    \"\"\"\u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 or n == 2:\n        return n - 1\n    # \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    res = fib(n - 1) + fib(n - 2)\n    # \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n</code></pre> recursion.cpp<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint Fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = Fib(n - 1) + Fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n int) int {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    res := fib(n-1) + fib(n-2)\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n: Int) -&gt; Int {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    let res = fib(n: n - 1) + fib(n: n - 2)\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n}\n</code></pre> recursion.js<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n: number): number {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n  // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n  if (n == 1 || n == 2) return n - 1;\n  // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n  int res = fib(n - 1) + fib(n - 2);\n  // \u8fd4\u56de\u7ed3\u679c f(n)\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfn fib(n: i32) -&gt; i32 {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1;\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    let res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c\n    res\n}\n</code></pre> recursion.c<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfun fib(n: Int): Int {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    val res = fib(n - 1) + fib(n - 2)\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{fib}\n</code></pre> recursion.zig<pre><code>// \u6590\u6ce2\u90a3\u5951\u6570\u5217\nfn fib(n: i32) i32 {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 or n == 2) {\n        return n - 1;\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    var res: i32 = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Observing the above code, we see that it recursively calls two functions within itself, meaning that one call generates two branching calls. As illustrated below, this continuous recursive calling eventually creates a \"recursion tree\" with a depth of \\(n\\).</p> <p></p> <p> Figure 2-6 \u00a0 Fibonacci Sequence Recursion Tree </p> <p>Fundamentally, recursion embodies the paradigm of \"breaking down a problem into smaller sub-problems.\" This divide-and-conquer strategy is crucial.</p> <ul> <li>From an algorithmic perspective, many important strategies like searching, sorting, backtracking, divide-and-conquer, and dynamic programming directly or indirectly use this way of thinking.</li> <li>From a data structure perspective, recursion is naturally suited for dealing with linked lists, trees, and graphs, as they are well suited for analysis using the divide-and-conquer approach.</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#223-comparison","title":"2.2.3 \u00a0 Comparison","text":"<p>Summarizing the above content, the following table shows the differences between iteration and recursion in terms of implementation, performance, and applicability.</p> <p> Table: Comparison of Iteration and Recursion Characteristics </p> Iteration Recursion Approach Loop structure Function calls itself Time Efficiency Generally higher efficiency, no function call overhead Each function call generates overhead Memory Usage Typically uses a fixed size of memory space Accumulative function calls can use a substantial amount of stack frame space Suitable Problems Suitable for simple loop tasks, intuitive and readable code Suitable for problem decomposition, like trees, graphs, divide-and-conquer, backtracking, etc., concise and clear code structure <p>Tip</p> <p>If you find the following content difficult to understand, consider revisiting it after reading the \"Stack\" chapter.</p> <p>So, what is the intrinsic connection between iteration and recursion? Taking the above recursive function as an example, the summation operation occurs during the recursion's \"return\" phase. This means that the initially called function is the last to complete its summation operation, mirroring the \"last in, first out\" principle of a stack.</p> <p>Recursive terms like \"call stack\" and \"stack frame space\" hint at the close relationship between recursion and stacks.</p> <ol> <li>Calling: When a function is called, the system allocates a new stack frame on the \"call stack\" for that function, storing local variables, parameters, return addresses, and other data.</li> <li>Returning: When a function completes execution and returns, the corresponding stack frame is removed from the \"call stack,\" restoring the execution environment of the previous function.</li> </ol> <p>Therefore, we can use an explicit stack to simulate the behavior of the call stack, thus transforming recursion into an iterative form:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def for_loop_recur(n: int) -&gt; int:\n    \"\"\"\u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\"\"\"\n    # \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    stack = []\n    res = 0\n    # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i in range(n, 0, -1):\n        # \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.append(i)\n    # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while stack:\n        # \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop()\n    # res = 1+2+3+...+n\n    return res\n</code></pre> recursion.cpp<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    stack&lt;int&gt; stack;\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (!stack.empty()) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.top();\n        stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    Stack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (!stack.isEmpty()) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint ForLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    Stack&lt;int&gt; stack = new();\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.Push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (stack.Count &gt; 0) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.Pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n int) int {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    stack := list.New()\n    res := 0\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i := n; i &gt; 0; i-- {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.PushBack(i)\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    for stack.Len() != 0 {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.Back().Value.(int)\n        stack.Remove(stack.Back())\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n: Int) -&gt; Int {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    var stack: [Int] = []\n    var res = 0\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i in (1 ... n).reversed() {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.append(i)\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while !stack.isEmpty {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.removeLast()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.js<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    const stack = [];\n    let res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (let i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (stack.length) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n: number): number {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808 \n    const stack: number[] = [];\n    let res: number = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (let i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (stack.length) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n  // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n  List&lt;int&gt; stack = [];\n  int res = 0;\n  // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n  for (int i = n; i &gt; 0; i--) {\n    // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n    stack.add(i);\n  }\n  // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n  while (!stack.isEmpty) {\n    // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n    res += stack.removeLast();\n  }\n  // res = 1+2+3+...+n\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfn for_loop_recur(n: i32) -&gt; i32 {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    let mut stack = Vec::new();\n    let mut res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i in (1..=n).rev() {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while !stack.is_empty() {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop().unwrap();\n    }\n    // res = 1+2+3+...+n\n    res\n}\n</code></pre> recursion.c<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n    int stack[1000]; // \u501f\u52a9\u4e00\u4e2a\u5927\u6570\u7ec4\u6765\u6a21\u62df\u6808\n    int top = -1;    // \u6808\u9876\u7d22\u5f15\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack[1 + top++] = i;\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (top &gt;= 0) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack[top--];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfun forLoopRecur(n: Int): Int {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    val stack = Stack&lt;Int&gt;()\n    var res = 0\n    // \u9012: \u9012\u5f52\u8c03\u7528\n    for (i in n downTo 0) {\n        stack.push(i)\n    }\n    // \u5f52: \u8fd4\u56de\u7ed3\u679c\n    while (stack.isNotEmpty()) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{for_loop_recur}\n</code></pre> recursion.zig<pre><code>// \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\nfn forLoopRecur(comptime n: i32) i32 {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    var stack: [n]i32 = undefined;\n    var res: i32 = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    var i: usize = n;\n    while (i &gt; 0) {\n        stack[i - 1] = @intCast(i);\n        i -= 1;\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    var index: usize = n;\n    while (index &gt; 0) {\n        index -= 1;\n        res += stack[index];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Observing the above code, when recursion is transformed into iteration, the code becomes more complex. Although iteration and recursion can often be transformed into each other, it's not always advisable to do so for two reasons:</p> <ul> <li>The transformed code may become more challenging to understand and less readable.</li> <li>For some complex problems, simulating the behavior of the system's call stack can be quite challenging.</li> </ul> <p>In conclusion, whether to choose iteration or recursion depends on the specific nature of the problem. In programming practice, it's crucial to weigh the pros and cons of both and choose the most suitable approach for the situation at hand.</p>"},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1 \u00a0 Algorithm Efficiency Assessment","text":"<p>In algorithm design, we pursue the following two objectives in sequence.</p> <ol> <li>Finding a Solution to the Problem: The algorithm should reliably find the correct solution within the stipulated range of inputs.</li> <li>Seeking the Optimal Solution: For the same problem, multiple solutions might exist, and we aim to find the most efficient algorithm possible.</li> </ol> <p>In other words, under the premise of being able to solve the problem, algorithm efficiency has become the main criterion for evaluating the merits of an algorithm, which includes the following two dimensions.</p> <ul> <li>Time Efficiency: The speed at which an algorithm runs.</li> <li>Space Efficiency: The size of the memory space occupied by an algorithm.</li> </ul> <p>In short, our goal is to design data structures and algorithms that are both fast and memory-efficient. Effectively assessing algorithm efficiency is crucial because only then can we compare various algorithms and guide the process of algorithm design and optimization.</p> <p>There are mainly two methods of efficiency assessment: actual testing and theoretical estimation.</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#211-actual-testing","title":"2.1.1 \u00a0 Actual Testing","text":"<p>Suppose we have algorithms <code>A</code> and <code>B</code>, both capable of solving the same problem, and we need to compare their efficiencies. The most direct method is to use a computer to run these two algorithms and monitor and record their runtime and memory usage. This assessment method reflects the actual situation but has significant limitations.</p> <p>On one hand, it's difficult to eliminate interference from the testing environment. Hardware configurations can affect algorithm performance. For example, algorithm <code>A</code> might run faster than <code>B</code> on one computer, but the opposite result may occur on another computer with different configurations. This means we would need to test on a variety of machines to calculate average efficiency, which is impractical.</p> <p>On the other hand, conducting a full test is very resource-intensive. As the volume of input data changes, the efficiency of the algorithms may vary. For example, with smaller data volumes, algorithm <code>A</code> might run faster than <code>B</code>, but the opposite might be true with larger data volumes. Therefore, to draw convincing conclusions, we need to test a wide range of input data sizes, which requires significant computational resources.</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#212-theoretical-estimation","title":"2.1.2 \u00a0 Theoretical Estimation","text":"<p>Due to the significant limitations of actual testing, we can consider evaluating algorithm efficiency solely through calculations. This estimation method is known as \"asymptotic complexity analysis,\" or simply \"complexity analysis.\"</p> <p>Complexity analysis reflects the relationship between the time and space resources required for algorithm execution and the size of the input data. It describes the trend of growth in the time and space required by the algorithm as the size of the input data increases. This definition might sound complex, but we can break it down into three key points to understand it better.</p> <ul> <li>\"Time and space resources\" correspond to \"time complexity\" and \"space complexity,\" respectively.</li> <li>\"As the size of input data increases\" means that complexity reflects the relationship between algorithm efficiency and the volume of input data.</li> <li>\"The trend of growth in time and space\" indicates that complexity analysis focuses not on the specific values of runtime or space occupied but on the \"rate\" at which time or space grows.</li> </ul> <p>Complexity analysis overcomes the disadvantages of actual testing methods, reflected in the following aspects:</p> <ul> <li>It is independent of the testing environment and applicable to all operating platforms.</li> <li>It can reflect algorithm efficiency under different data volumes, especially in the performance of algorithms with large data volumes.</li> </ul> <p>Tip</p> <p>If you're still confused about the concept of complexity, don't worry. We will introduce it in detail in subsequent chapters.</p> <p>Complexity analysis provides us with a \"ruler\" to measure the time and space resources needed to execute an algorithm and compare the efficiency between different algorithms.</p> <p>Complexity is a mathematical concept and may be abstract and challenging for beginners. From this perspective, complexity analysis might not be the best content to introduce first. However, when discussing the characteristics of a particular data structure or algorithm, it's hard to avoid analyzing its speed and space usage.</p> <p>In summary, it's recommended that you establish a preliminary understanding of complexity analysis before diving deep into data structures and algorithms, so that you can carry out simple complexity analyses of algorithms.</p>"},{"location":"chapter_computational_complexity/space_complexity/","title":"2.4 \u00a0 Space Complexity","text":"<p>\"Space complexity\" is used to measure the growth trend of the memory space occupied by an algorithm as the amount of data increases. This concept is very similar to time complexity, except that \"running time\" is replaced with \"occupied memory space\".</p>"},{"location":"chapter_computational_complexity/space_complexity/#241-space-related-to-algorithms","title":"2.4.1 \u00a0 Space Related to Algorithms","text":"<p>The memory space used by an algorithm during its execution mainly includes the following types.</p> <ul> <li>Input Space: Used to store the input data of the algorithm.</li> <li>Temporary Space: Used to store variables, objects, function contexts, and other data during the algorithm's execution.</li> <li>Output Space: Used to store the output data of the algorithm.</li> </ul> <p>Generally, the scope of space complexity statistics includes both \"Temporary Space\" and \"Output Space\".</p> <p>Temporary space can be further divided into three parts.</p> <ul> <li>Temporary Data: Used to save various constants, variables, objects, etc., during the algorithm's execution.</li> <li>Stack Frame Space: Used to save the context data of the called function. The system creates a stack frame at the top of the stack each time a function is called, and the stack frame space is released after the function returns.</li> <li>Instruction Space: Used to store compiled program instructions, which are usually negligible in actual statistics.</li> </ul> <p>When analyzing the space complexity of a program, we typically count the Temporary Data, Stack Frame Space, and Output Data, as shown in the Figure 2-15 .</p> <p></p> <p> Figure 2-15 \u00a0 Space Types Used in Algorithms </p> <p>The relevant code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>class Node:\n    \"\"\"Classes\"\"\"\"\n    def __init__(self, x: int):\n        self.val: int = x               # node value\n        self.next: Node | None = None   # reference to the next node\n\ndef function() -&gt; int:\n    \"\"\"\"Functions\"\"\"\"\"\n    # Perform certain operations...\n    return 0\n\ndef algorithm(n) -&gt; int:    # input data\n    A = 0                   # temporary data (constant, usually in uppercase)\n    b = 0                   # temporary data (variable)\n    node = Node(0)          # temporary data (object)\n    c = function()          # Stack frame space (call function)\n    return A + b + c        # output data\n</code></pre> <pre><code>/* Structures */\nstruct Node {\n    int val;\n    Node *next;\n    Node(int x) : val(x), next(nullptr) {}\n};\n\n/* Functions */\nint func() {\n    // Perform certain operations...\n    return 0;\n}\n\nint algorithm(int n) {          // input data\n    const int a = 0;            // temporary data (constant)\n    int b = 0;                  // temporary data (variable)\n    Node* node = new Node(0);   // temporary data (object)\n    int c = func();             // stack frame space (call function)\n    return a + b + c;           // output data\n}\n</code></pre> <pre><code>/* Classes */\nclass Node {\n    int val;\n    Node next;\n    Node(int x) { val = x; }\n}\n\n/* Functions */\nint function() {\n    // Perform certain operations...\n    return 0;\n}\n\nint algorithm(int n) {          // input data\n    final int a = 0;            // temporary data (constant)\n    int b = 0;                  // temporary data (variable)\n    Node node = new Node(0);    // temporary data (object)\n    int c = function();         // stack frame space (call function)\n    return a + b + c;           // output data\n}\n</code></pre> <pre><code>/* Classes */\nclass Node {\n    int val;\n    Node next;\n    Node(int x) { val = x; }\n}\n\n/* Functions */\nint Function() {\n    // Perform certain operations...\n    return 0;\n}\n\nint Algorithm(int n) {  // input data\n    const int a = 0;    // temporary data (constant)\n    int b = 0;          // temporary data (variable)\n    Node node = new(0); // temporary data (object)\n    int c = Function(); // stack frame space (call function)\n    return a + b + c;   // output data\n}\n</code></pre> <pre><code>/* Structures */\ntype node struct {\n    val  int\n    next *node\n}\n\n/* Create node structure */\nfunc newNode(val int) *node {\n    return &amp;node{val: val}\n}\n\n/* Functions */\nfunc function() int {\n    // Perform certain operations...\n    return 0\n}\n\nfunc algorithm(n int) int { // input data\n    const a = 0             // temporary data (constant)\n    b := 0                  // temporary storage of data (variable)\n    newNode(0)              // temporary data (object)\n    c := function()         // stack frame space (call function)\n    return a + b + c        // output data\n}\n</code></pre> <pre><code>/* Classes */\nclass Node {\n    var val: Int\n    var next: Node?\n\n    init(x: Int) {\n        val = x\n    }\n}\n\n/* Functions */\nfunc function() -&gt; Int {\n    // Perform certain operations...\n    return 0\n}\n\nfunc algorithm(n: Int) -&gt; Int { // input data\n    let a = 0                   // temporary data (constant)\n    var b = 0                   // temporary data (variable)\n    let node = Node(x: 0)       // temporary data (object)\n    let c = function()          // stack frame space (call function)\n    return a + b + c            // output data\n}\n</code></pre> <pre><code>/* Classes */\nclass Node {\n    val;\n    next;\n    constructor(val) {\n        this.val = val === undefined ? 0 : val; // node value\n        this.next = null;                       // reference to the next node\n    }\n}\n\n/* Functions */\nfunction constFunc() {\n    // Perform certain operations\n    return 0;\n}\n\nfunction algorithm(n) {         // input data\n    const a = 0;                // temporary data (constant)\n    let b = 0;                  // temporary data (variable)\n    const node = new Node(0);   // temporary data (object)\n    const c = constFunc();      // Stack frame space (calling function)\n    return a + b + c;           // output data\n}\n</code></pre> <pre><code>/* Classes */\nclass Node {\n    val: number;\n    next: Node | null;\n    constructor(val?: number) {\n        this.val = val === undefined ? 0 : val; // node value\n        this.next = null;                       // reference to the next node\n    }\n}\n\n/* Functions */\nfunction constFunc(): number {\n    // Perform certain operations\n    return 0;\n}\n\nfunction algorithm(n: number): number { // input data\n    const a = 0;                        // temporary data (constant)\n    let b = 0;                          // temporary data (variable)\n    const node = new Node(0);           // temporary data (object)\n    const c = constFunc();              // Stack frame space (calling function)\n    return a + b + c;                   // output data\n}\n</code></pre> <pre><code>/* Classes */\nclass Node {\n  int val;\n  Node next;\n  Node(this.val, [this.next]);\n}\n\n/* Functions */\nint function() {\n  // Perform certain operations...\n  return 0;\n}\n\nint algorithm(int n) {  // input data\n  const int a = 0;      // temporary data (constant)\n  int b = 0;            // temporary data (variable)\n  Node node = Node(0);  // temporary data (object)\n  int c = function();   // stack frame space (call function)\n  return a + b + c;     // output data\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* Structures */\nstruct Node {\n    val: i32,\n    next: Option&lt;Rc&lt;RefCell&lt;Node&gt;&gt;&gt;,\n}\n\n/* Creating a Node structure */\nimpl Node {\n    fn new(val: i32) -&gt; Self {\n        Self { val: val, next: None }\n    }\n}\n\n/* Functions */\nfn function() -&gt; i32 {     \n    // Perform certain operations...\n    return 0;\n}\n\nfn algorithm(n: i32) -&gt; i32 {   // input data\n    const a: i32 = 0;           // temporary data (constant)\n    let mut b = 0;              // temporary data (variable)\n    let node = Node::new(0);    // temporary data (object)\n    let c = function();         // stack frame space (call function)\n    return a + b + c;           // output data\n}\n</code></pre> <pre><code>/* Functions */\nint func() {\n    // Perform certain operations...\n    return 0;\n}\n\nint algorithm(int n) {  // input data\n    const int a = 0;    // temporary data (constant)\n    int b = 0;          // temporary data (variable)\n    int c = func();     // stack frame space (call function)\n    return a + b + c;   // output data\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre>"},{"location":"chapter_computational_complexity/space_complexity/#242-calculation-method","title":"2.4.2 \u00a0 Calculation Method","text":"<p>The method for calculating space complexity is roughly similar to that of time complexity, with the only change being the shift of the statistical object from \"number of operations\" to \"size of used space\".</p> <p>However, unlike time complexity, we usually only focus on the worst-case space complexity. This is because memory space is a hard requirement, and we must ensure that there is enough memory space reserved under all input data.</p> <p>Consider the following code, the term \"worst-case\" in worst-case space complexity has two meanings.</p> <ol> <li>Based on the worst input data: When \\(n &lt; 10\\), the space complexity is \\(O(1)\\); but when \\(n &gt; 10\\), the initialized array <code>nums</code> occupies \\(O(n)\\) space, thus the worst-case space complexity is \\(O(n)\\).</li> <li>Based on the peak memory used during the algorithm's execution: For example, before executing the last line, the program occupies \\(O(1)\\) space; when initializing the array <code>nums</code>, the program occupies \\(O(n)\\) space, hence the worst-case space complexity is \\(O(n)\\).</li> </ol> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>def algorithm(n: int):\n    a = 0               # O(1)\n    b = [0] * 10000     # O(1)\n    if n &gt; 10:\n        nums = [0] * n  # O(n)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    vector&lt;int&gt; b(10000);    // O(1)\n    if (n &gt; 10)\n        vector&lt;int&gt; nums(n); // O(n)\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10)\n        int[] nums = new int[n]; // O(n)\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10) {\n        int[] nums = new int[n]; // O(n)\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 0                      // O(1)\n    b := make([]int, 10000)     // O(1)\n    var nums []int\n    if n &gt; 10 {\n        nums := make([]int, n)  // O(n)\n    }\n    fmt.Println(a, b, nums)\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    let a = 0 // O(1)\n    let b = Array(repeating: 0, count: 10000) // O(1)\n    if n &gt; 10 {\n        let nums = Array(repeating: 0, count: n) // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 0;                            // O(1)\n  List&lt;int&gt; b = List.filled(10000, 0);  // O(1)\n  if (n &gt; 10) {\n    List&lt;int&gt; nums = List.filled(n, 0); // O(n)\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let a = 0;                              // O(1)\n    let b = [0; 10000];                     // O(1)\n    if n &gt; 10 {\n        let nums = vec![0; n as usize];     // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    int b[10000];            // O(1)\n    if (n &gt; 10)\n        int nums[n] = {0};   // O(n)\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>In recursive functions, stack frame space must be taken into count. Consider the following code:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>def function() -&gt; int:\n    # Perform certain operations\n    return 0\n\ndef loop(n: int):\n    \"\"\"Loop O(1)\"\"\"\"\"\n    for _ in range(n):\n        function()\n\ndef recur(n: int):\n    \"\"\"Recursion O(n)\"\"\"\"\"\n    if n == 1:\n        return\n    return recur(n - 1)\n</code></pre> <pre><code>int func() {\n    // Perform certain operations\n    return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n    // Perform certain operations\n    return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int Function() {\n    // Perform certain operations\n    return 0;\n}\n/* Cycle O(1) */\nvoid Loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n/* Recursion O(n) */\nint Recur(int n) {\n    if (n == 1) return 1;\n    return Recur(n - 1);\n}\n</code></pre> <pre><code>func function() int {\n    // Perform certain operations\n    return 0\n}\n\n/* Cycle O(1) */\nfunc loop(n int) {\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n}\n\n/* Recursion O(n) */\nfunc recur(n int) {\n    if n == 1 {\n        return\n    }\n    recur(n - 1)\n}\n</code></pre> <pre><code>@discardableResult\nfunc function() -&gt; Int {\n    // Perform certain operations\n    return 0\n}\n\n/* Cycle O(1) */\nfunc loop(n: Int) {\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n\n/* Recursion O(n) */\nfunc recur(n: Int) {\n    if n == 1 {\n        return\n    }\n    recur(n: n - 1)\n}\n</code></pre> <pre><code>function constFunc() {\n    // Perform certain operations\n    return 0;\n}\n/* Cycle O(1) */\nfunction loop(n) {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* Recursion O(n) */\nfunction recur(n) {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>function constFunc(): number {\n    // Perform certain operations\n    return 0;\n}\n/* Cycle O(1) */\nfunction loop(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* Recursion O(n) */\nfunction recur(n: number): void {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n  // Perform certain operations\n  return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n  if (n == 1) return;\n  return recur(n - 1);\n}\n</code></pre> <pre><code>fn function() -&gt; i32 {\n    // Perform certain operations\n    return 0;\n}\n/* Cycle O(1) */\nfn loop(n: i32) {\n    for i in 0..n {\n        function();\n    }\n}\n/* Recursion O(n) */\nvoid recur(n: i32) {\n    if n == 1 {\n        return;\n    }\n    recur(n - 1);\n}\n</code></pre> <pre><code>int func() {\n    // Perform certain operations\n    return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>The time complexity of both <code>loop()</code> and <code>recur()</code> functions is \\(O(n)\\), but their space complexities differ.</p> <ul> <li>The <code>loop()</code> function calls <code>function()</code> \\(n\\) times in a loop, where each iteration's <code>function()</code> returns and releases its stack frame space, so the space complexity remains \\(O(1)\\).</li> <li>The recursive function <code>recur()</code> will have \\(n\\) instances of unreturned <code>recur()</code> existing simultaneously during its execution, thus occupying \\(O(n)\\) stack frame space.</li> </ul>"},{"location":"chapter_computational_complexity/space_complexity/#243-common-types","title":"2.4.3 \u00a0 Common Types","text":"<p>Let the size of the input data be \\(n\\), the following chart displays common types of space complexities (arranged from low to high).</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \\newline \\text{Constant Order} &lt; \\text{Logarithmic Order} &lt; \\text{Linear Order} &lt; \\text{Quadratic Order} &lt; \\text{Exponential Order} \\end{aligned} \\] <p></p> <p> Figure 2-16 \u00a0 Common Types of Space Complexity </p>"},{"location":"chapter_computational_complexity/space_complexity/#1-constant-order-o1","title":"1. \u00a0 Constant Order \\(O(1)\\)","text":"<p>Constant order is common in constants, variables, objects that are independent of the size of input data \\(n\\).</p> <p>Note that memory occupied by initializing variables or calling functions in a loop, which is released upon entering the next cycle, does not accumulate over space, thus the space complexity remains \\(O(1)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def function() -&gt; int:\n    \"\"\"\u51fd\u6570\"\"\"\n    # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n\ndef constant(n: int):\n    \"\"\"\u5e38\u6570\u9636\"\"\"\n    # \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    a = 0\n    nums = [0] * 10000\n    node = ListNode(0)\n    # \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in range(n):\n        c = 0\n    # \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in range(n):\n        function()\n</code></pre> space_complexity.cpp<pre><code>/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const int a = 0;\n    int b = 0;\n    vector&lt;int&gt; nums(10000);\n    ListNode node(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u51fd\u6570 */\nint function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    final int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new ListNode(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u51fd\u6570 */\nint Function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid Constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u51fd\u6570 */\nfunc function() int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc spaceConstant(n int) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a = 0\n    b := 0\n    nums := make([]int, 10000)\n    node := newNode(0)\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    var c int\n    for i := 0; i &lt; n; i++ {\n        c = 0\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n    b += 0\n    c += 0\n    nums[0] = 0\n    node.val = 0\n}\n</code></pre> space_complexity.swift<pre><code>/* \u51fd\u6570 */\n@discardableResult\nfunc function() -&gt; Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    let a = 0\n    var b = 0\n    let nums = Array(repeating: 0, count: 10000)\n    let node = ListNode(x: 0)\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in 0 ..&lt; n {\n        let c = 0\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n</code></pre> space_complexity.js<pre><code>/* \u51fd\u6570 */\nfunction constFunc() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u51fd\u6570 */\nfunction constFunc(): number {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n: number): void {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u51fd\u6570 */\nint function() {\n  // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n  return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n  // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n  final int a = 0;\n  int b = 0;\n  List&lt;int&gt; nums = List.filled(10000, 0);\n  ListNode node = ListNode(0);\n  // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n  for (var i = 0; i &lt; n; i++) {\n    int c = 0;\n  }\n  // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n  for (var i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u51fd\u6570 */\nfn function() -&gt; i32 {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\n#[allow(unused)]\nfn constant(n: i32) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const A: i32 = 0;\n    let b = 0;\n    let nums = vec![0; 10000];\n    let node = ListNode::new(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for i in 0..n {\n        let c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for i in 0..n {\n        function();\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const int a = 0;\n    int b = 0;\n    int nums[1000];\n    ListNode *node = newListNode(0);\n    free(node);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* \u51fd\u6570 */\nfun function(): Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfun constant(n: Int) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    val a = 0\n    var b = 0\n    val nums = Array(10000) { 0 }\n    val node = ListNode(0)\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (i in 0..&lt;n) {\n        val c = 0\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{function}\n\n[class]{}-[func]{constant}\n</code></pre> space_complexity.zig<pre><code>// \u51fd\u6570\nfn function() i32 {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n// \u5e38\u6570\u9636\nfn constant(n: i32) void {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a: i32 = 0;\n    var b: i32 = 0;\n    var nums = [_]i32{0}**10000;\n    var node = inc.ListNode(i32){.val = 0};\n    var i: i32 = 0;\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    while (i &lt; n) : (i += 1) {\n        var c: i32 = 0;\n        _ = c;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    i = 0;\n    while (i &lt; n) : (i += 1) {\n        _ = function();\n    }\n    _ = a;\n    _ = b;\n    _ = nums;\n    _ = node;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_computational_complexity/space_complexity/#2-linear-order-on","title":"2. \u00a0 Linear Order \\(O(n)\\)","text":"<p>Linear order is common in arrays, linked lists, stacks, queues, etc., where the number of elements is proportional to \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def linear(n: int):\n    \"\"\"\u7ebf\u6027\u9636\"\"\"\n    # \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    nums = [0] * n\n    # \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    hmap = dict[int, str]()\n    for i in range(n):\n        hmap[i] = str(i)\n</code></pre> space_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    vector&lt;int&gt; nums(n);\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    vector&lt;ListNode&gt; nodes;\n    for (int i = 0; i &lt; n; i++) {\n        nodes.push_back(ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    unordered_map&lt;int, string&gt; map;\n    for (int i = 0; i &lt; n; i++) {\n        map[i] = to_string(i);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    int[] nums = new int[n];\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    List&lt;ListNode&gt; nodes = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        nodes.add(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    Map&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        map.put(i, String.valueOf(i));\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7ebf\u6027\u9636 */\nvoid Linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    int[] nums = new int[n];\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    List&lt;ListNode&gt; nodes = [];\n    for (int i = 0; i &lt; n; i++) {\n        nodes.Add(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    Dictionary&lt;int, string&gt; map = [];\n    for (int i = 0; i &lt; n; i++) {\n        map.Add(i, i.ToString());\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u7ebf\u6027\u9636 */\nfunc spaceLinear(n int) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    _ = make([]int, n)\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    var nodes []*node\n    for i := 0; i &lt; n; i++ {\n        nodes = append(nodes, newNode(i))\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    m := make(map[int]string, n)\n    for i := 0; i &lt; n; i++ {\n        m[i] = strconv.Itoa(i)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    let nums = Array(repeating: 0, count: n)\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let nodes = (0 ..&lt; n).map { ListNode(x: $0) }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let map = Dictionary(uniqueKeysWithValues: (0 ..&lt; n).map { ($0, \"\\($0)\") })\n}\n</code></pre> space_complexity.js<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    const nums = new Array(n);\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const nodes = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): void {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    const nums = new Array(n);\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const nodes: ListNode[] = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n  // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n  List&lt;int&gt; nums = List.filled(n, 0);\n  // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n  List&lt;ListNode&gt; nodes = [];\n  for (var i = 0; i &lt; n; i++) {\n    nodes.add(ListNode(i));\n  }\n  // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n  Map&lt;int, String&gt; map = HashMap();\n  for (var i = 0; i &lt; n; i++) {\n    map.putIfAbsent(i, () =&gt; i.toString());\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7ebf\u6027\u9636 */\n#[allow(unused)]\nfn linear(n: i32) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    let mut nums = vec![0; n as usize];\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let mut nodes = Vec::new();\n    for i in 0..n {\n        nodes.push(ListNode::new(i))\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let mut map = HashMap::new();\n    for i in 0..n {\n        map.insert(i, i.to_string());\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u54c8\u5e0c\u8868 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n\n/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    int *nums = malloc(sizeof(int) * n);\n    free(nums);\n\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    ListNode **nodes = malloc(sizeof(ListNode *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        nodes[i] = newListNode(i);\n    }\n    // \u5185\u5b58\u91ca\u653e\n    for (int i = 0; i &lt; n; i++) {\n        free(nodes[i]);\n    }\n    free(nodes);\n\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    HashTable *h = NULL;\n    for (int i = 0; i &lt; n; i++) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = i;\n        tmp-&gt;val = i;\n        HASH_ADD_INT(h, key, tmp);\n    }\n\n    // \u5185\u5b58\u91ca\u653e\n    HashTable *curr, *tmp;\n    HASH_ITER(hh, h, curr, tmp) {\n        HASH_DEL(h, curr);\n        free(curr);\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* \u7ebf\u6027\u9636 */\nfun linear(n: Int) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    val nums = Array(n) { 0 }\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    val nodes = mutableListOf&lt;ListNode&gt;()\n    for (i in 0..&lt;n) {\n        nodes.add(ListNode(i))\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    val map = mutableMapOf&lt;Int, String&gt;()\n    for (i in 0..&lt;n) {\n        map[i] = i.toString()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{linear}\n</code></pre> space_complexity.zig<pre><code>// \u7ebf\u6027\u9636\nfn linear(comptime n: i32) !void {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    var nums = [_]i32{0}**n;\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    var nodes = std.ArrayList(i32).init(std.heap.page_allocator);\n    defer nodes.deinit();\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        try nodes.append(i);\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    var map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\n    defer map.deinit();\n    var j: i32 = 0;\n    while (j &lt; n) : (j += 1) {\n        const string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\n        defer std.heap.page_allocator.free(string);\n        try map.put(i, string);\n    }\n    _ = nums;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>As shown below, this function's recursive depth is \\(n\\), meaning there are \\(n\\) instances of unreturned <code>linear_recur()</code> function, using \\(O(n)\\) size of stack frame space:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def linear_recur(n: int):\n    \"\"\"\u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    print(\"\u9012\u5f52 n =\", n)\n    if n == 1:\n        return\n    linear_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n    cout &lt;&lt; \"\u9012\u5f52 n = \" &lt;&lt; n &lt;&lt; endl;\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n    System.out.println(\"\u9012\u5f52 n = \" + n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid LinearRecur(int n) {\n    Console.WriteLine(\"\u9012\u5f52 n = \" + n);\n    if (n == 1) return;\n    LinearRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceLinearRecur(n int) {\n    fmt.Println(\"\u9012\u5f52 n =\", n)\n    if n == 1 {\n        return\n    }\n    spaceLinearRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc linearRecur(n: Int) {\n    print(\"\u9012\u5f52 n = \\(n)\")\n    if n == 1 {\n        return\n    }\n    linearRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n) {\n    console.log(`\u9012\u5f52 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n: number): void {\n    console.log(`\u9012\u5f52 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n  print('\u9012\u5f52 n = $n');\n  if (n == 1) return;\n  linearRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn linear_recur(n: i32) {\n    println!(\"\u9012\u5f52 n = {}\", n);\n    if n == 1 {\n        return;\n    };\n    linear_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n    printf(\"\u9012\u5f52 n = %d\\r\\n\", n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.kt<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun linearRecur(n: Int) {\n    println(\"\u9012\u5f52 n = $n\")\n    if (n == 1)\n        return\n    linearRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{linear_recur}\n</code></pre> space_complexity.zig<pre><code>// \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn linearRecur(comptime n: i32) void {\n    std.debug.print(\"\u9012\u5f52 n = {}\\n\", .{n});\n    if (n == 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p></p> <p> Figure 2-17 \u00a0 Recursive Function Generating Linear Order Space Complexity </p>"},{"location":"chapter_computational_complexity/space_complexity/#3-quadratic-order-on2","title":"3. \u00a0 Quadratic Order \\(O(n^2)\\)","text":"<p>Quadratic order is common in matrices and graphs, where the number of elements is quadratic to \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def quadratic(n: int):\n    \"\"\"\u5e73\u65b9\u9636\"\"\"\n    # \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    num_matrix = [[0] * n for _ in range(n)]\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    vector&lt;vector&lt;int&gt;&gt; numMatrix;\n    for (int i = 0; i &lt; n; i++) {\n        vector&lt;int&gt; tmp;\n        for (int j = 0; j &lt; n; j++) {\n            tmp.push_back(0);\n        }\n        numMatrix.push_back(tmp);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    int[][] numMatrix = new int[n][n];\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    List&lt;List&lt;Integer&gt;&gt; numList = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            tmp.add(0);\n        }\n        numList.add(tmp);\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u9636 */\nvoid Quadratic(int n) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    int[,] numMatrix = new int[n, n];\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    List&lt;List&lt;int&gt;&gt; numList = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;int&gt; tmp = [];\n        for (int j = 0; j &lt; n; j++) {\n            tmp.Add(0);\n        }\n        numList.Add(tmp);\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u9636 */\nfunc spaceQuadratic(n int) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    numMatrix := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        numMatrix[i] = make([]int, n)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    let numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): void {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n  // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n  List&lt;List&lt;int&gt;&gt; numMatrix = List.generate(n, (_) =&gt; List.filled(n, 0));\n  // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n  List&lt;List&lt;int&gt;&gt; numList = [];\n  for (var i = 0; i &lt; n; i++) {\n    List&lt;int&gt; tmp = [];\n    for (int j = 0; j &lt; n; j++) {\n      tmp.add(0);\n    }\n    numList.add(tmp);\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u9636 */\n#[allow(unused)]\nfn quadratic(n: i32) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    let num_matrix = vec![vec![0; n as usize]; n as usize];\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    let mut num_list = Vec::new();\n    for i in 0..n {\n        let mut tmp = Vec::new();\n        for j in 0..n {\n            tmp.push(0);\n        }\n        num_list.push(tmp);\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    int **numMatrix = malloc(sizeof(int *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        int *tmp = malloc(sizeof(int) * n);\n        for (int j = 0; j &lt; n; j++) {\n            tmp[j] = 0;\n        }\n        numMatrix[i] = tmp;\n    }\n\n    // \u5185\u5b58\u91ca\u653e\n    for (int i = 0; i &lt; n; i++) {\n        free(numMatrix[i]);\n    }\n    free(numMatrix);\n}\n</code></pre> space_complexity.kt<pre><code>/* \u5e73\u65b9\u9636 */\nfun quadratic(n: Int) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    val numMatrix: Array&lt;Array&lt;Int&gt;?&gt; = arrayOfNulls(n)\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    val numList: MutableList&lt;MutableList&lt;Int&gt;&gt; = arrayListOf()\n    for (i in 0..&lt;n) {\n        val tmp = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            tmp.add(0)\n        }\n        numList.add(tmp)\n    }\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{quadratic}\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u9636\nfn quadratic(n: i32) !void {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    var nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\n    defer nodes.deinit();\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        var tmp = std.ArrayList(i32).init(std.heap.page_allocator);\n        defer tmp.deinit();\n        var j: i32 = 0;\n        while (j &lt; n) : (j += 1) {\n            try tmp.append(0);\n        }\n        try nodes.append(tmp);\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>As shown below, the recursive depth of this function is \\(n\\), and in each recursive call, an array is initialized with lengths \\(n\\), \\(n-1\\), \\(\\dots\\), \\(2\\), \\(1\\), averaging \\(n/2\\), thus overall occupying \\(O(n^2)\\) space:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def quadratic_recur(n: int) -&gt; int:\n    \"\"\"\u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n &lt;= 0:\n        return 0\n    # \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    nums = [0] * n\n    return quadratic_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    vector&lt;int&gt; nums(n);\n    cout &lt;&lt; \"\u9012\u5f52 n = \" &lt;&lt; n &lt;&lt; \" \u4e2d\u7684 nums \u957f\u5ea6 = \" &lt;&lt; nums.size() &lt;&lt; endl;\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    int[] nums = new int[n];\n    System.out.println(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.length);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint QuadraticRecur(int n) {\n    if (n &lt;= 0) return 0;\n    int[] nums = new int[n];\n    Console.WriteLine(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.Length);\n    return QuadraticRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceQuadraticRecur(n int) int {\n    if n &lt;= 0 {\n        return 0\n    }\n    nums := make([]int, n)\n    fmt.Printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d \\n\", n, len(nums))\n    return spaceQuadraticRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -&gt; Int {\n    if n &lt;= 0 {\n        return 0\n    }\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    let nums = Array(repeating: 0, count: n)\n    print(\"\u9012\u5f52 n = \\(n) \u4e2d\u7684 nums \u957f\u5ea6 = \\(nums.count)\")\n    return quadraticRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n) {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n: number): number {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n  if (n &lt;= 0) return 0;\n  List&lt;int&gt; nums = List.filled(n, 0);\n  print('\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}');\n  return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn quadratic_recur(n: i32) -&gt; i32 {\n    if n &lt;= 0 {\n        return 0;\n    };\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    let nums = vec![0; n as usize];\n    println!(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\", n, nums.len());\n    return quadratic_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    int *nums = malloc(sizeof(int) * n);\n    printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d\\r\\n\", n, n);\n    int res = quadraticRecur(n - 1);\n    free(nums);\n    return res;\n}\n</code></pre> space_complexity.kt<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\ntailrec fun quadraticRecur(n: Int): Int {\n    if (n &lt;= 0)\n        return 0\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    val nums = Array(n) { 0 }\n    println(\"\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.size}\")\n    return quadraticRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{quadratic_recur}\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn quadraticRecur(comptime n: i32) i32 {\n    if (n &lt;= 0) return 0;\n    var nums = [_]i32{0}**n;\n    std.debug.print(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\\n\", .{n, nums.len});\n    return quadraticRecur(n - 1);\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p></p> <p> Figure 2-18 \u00a0 Recursive Function Generating Quadratic Order Space Complexity </p>"},{"location":"chapter_computational_complexity/space_complexity/#4-exponential-order-o2n","title":"4. \u00a0 Exponential Order \\(O(2^n)\\)","text":"<p>Exponential order is common in binary trees. Observe the below image, a \"full binary tree\" with \\(n\\) levels has \\(2^n - 1\\) nodes, occupying \\(O(2^n)\\) space:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def build_tree(n: int) -&gt; TreeNode | None:\n    \"\"\"\u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\"\"\"\n    if n == 0:\n        return None\n    root = TreeNode(0)\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n    return root\n</code></pre> space_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return nullptr;\n    TreeNode *root = new TreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode buildTree(int n) {\n    if (n == 0)\n        return null;\n    TreeNode root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? BuildTree(int n) {\n    if (n == 0) return null;\n    TreeNode root = new(0) {\n        left = BuildTree(n - 1),\n        right = BuildTree(n - 1)\n    };\n    return root;\n}\n</code></pre> space_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n int) *TreeNode {\n    if n == 0 {\n        return nil\n    }\n    root := NewTreeNode(0)\n    root.Left = buildTree(n - 1)\n    root.Right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n: Int) -&gt; TreeNode? {\n    if n == 0 {\n        return nil\n    }\n    let root = TreeNode(x: 0)\n    root.left = buildTree(n: n - 1)\n    root.right = buildTree(n: n - 1)\n    return root\n}\n</code></pre> space_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n) {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n: number): TreeNode | null {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n) {\n  if (n == 0) return null;\n  TreeNode root = TreeNode(0);\n  root.left = buildTree(n - 1);\n  root.right = buildTree(n - 1);\n  return root;\n}\n</code></pre> space_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfn build_tree(n: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    if n == 0 {\n        return None;\n    };\n    let root = TreeNode::new(0);\n    root.borrow_mut().left = build_tree(n - 1);\n    root.borrow_mut().right = build_tree(n - 1);\n    return Some(root);\n}\n</code></pre> space_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return NULL;\n    TreeNode *root = newTreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.kt<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfun buildTree(n: Int): TreeNode? {\n    if (n == 0)\n        return null\n    val root = TreeNode(0)\n    root.left = buildTree(n - 1)\n    root.right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{build_tree}\n</code></pre> space_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\n    if (n == 0) return null;\n    const root = try mem_allocator.create(inc.TreeNode(i32));\n    root.init(0);\n    root.left = try buildTree(mem_allocator, n - 1);\n    root.right = try buildTree(mem_allocator, n - 1);\n    return root;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p></p> <p> Figure 2-19 \u00a0 Full Binary Tree Generating Exponential Order Space Complexity </p>"},{"location":"chapter_computational_complexity/space_complexity/#5-logarithmic-order-olog-n","title":"5. \u00a0 Logarithmic Order \\(O(\\log n)\\)","text":"<p>Logarithmic order is common in divide-and-conquer algorithms. For example, in merge sort, an array of length \\(n\\) is recursively divided in half each round, forming a recursion tree of height \\(\\log n\\), using \\(O(\\log n)\\) stack frame space.</p> <p>Another example is converting a number to a string. Given a positive integer \\(n\\), its number of digits is \\(\\log_{10} n + 1\\), corresponding to the length of the string, thus the space complexity is \\(O(\\log_{10} n + 1) = O(\\log n)\\).</p>"},{"location":"chapter_computational_complexity/space_complexity/#244-balancing-time-and-space","title":"2.4.4 \u00a0 Balancing Time and Space","text":"<p>Ideally, we aim for both time complexity and space complexity to be optimal. However, in practice, optimizing both simultaneously is often difficult.</p> <p>Lowering time complexity usually comes at the cost of increased space complexity, and vice versa. The approach of sacrificing memory space to improve algorithm speed is known as \"space-time tradeoff\"; the reverse is known as \"time-space tradeoff\".</p> <p>The choice depends on which aspect we value more. In most cases, time is more precious than space, so \"space-time tradeoff\" is often the more common strategy. Of course, controlling space complexity is also very important when dealing with large volumes of data.</p>"},{"location":"chapter_computational_complexity/summary/","title":"2.5 \u00a0 Summary","text":""},{"location":"chapter_computational_complexity/summary/#1-key-review","title":"1. \u00a0 Key Review","text":"<p>Algorithm Efficiency Assessment</p> <ul> <li>Time efficiency and space efficiency are the two main criteria for assessing the merits of an algorithm.</li> <li>We can assess algorithm efficiency through actual testing, but it's challenging to eliminate the influence of the test environment, and it consumes substantial computational resources.</li> <li>Complexity analysis can overcome the disadvantages of actual testing. Its results are applicable across all operating platforms and can reveal the efficiency of algorithms at different data scales.</li> </ul> <p>Time Complexity</p> <ul> <li>Time complexity measures the trend of an algorithm's running time with the increase in data volume, effectively assessing algorithm efficiency. However, it can fail in certain cases, such as with small input data volumes or when time complexities are the same, making it challenging to precisely compare the efficiency of algorithms.</li> <li>Worst-case time complexity is denoted using big O notation, representing the asymptotic upper bound, reflecting the growth level of the number of operations \\(T(n)\\) as \\(n\\) approaches infinity.</li> <li>Calculating time complexity involves two steps: first counting the number of operations, then determining the asymptotic upper bound.</li> <li>Common time complexities, arranged from low to high, include \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n \\log n)\\), \\(O(n^2)\\), \\(O(2^n)\\), and \\(O(n!)\\), among others.</li> <li>The time complexity of some algorithms is not fixed and depends on the distribution of input data. Time complexities are divided into worst, best, and average cases. The best case is rarely used because input data generally needs to meet strict conditions to achieve the best case.</li> <li>Average time complexity reflects the efficiency of an algorithm under random data inputs, closely resembling the algorithm's performance in actual applications. Calculating average time complexity requires accounting for the distribution of input data and the subsequent mathematical expectation.</li> </ul> <p>Space Complexity</p> <ul> <li>Space complexity, similar to time complexity, measures the trend of memory space occupied by an algorithm with the increase in data volume.</li> <li>The relevant memory space used during the algorithm's execution can be divided into input space, temporary space, and output space. Generally, input space is not included in space complexity calculations. Temporary space can be divided into temporary data, stack frame space, and instruction space, where stack frame space usually affects space complexity only in recursive functions.</li> <li>We usually focus only on the worst-case space complexity, which means calculating the space complexity of the algorithm under the worst input data and at the worst moment of operation.</li> <li>Common space complexities, arranged from low to high, include \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n^2)\\), and \\(O(2^n)\\), among others.</li> </ul>"},{"location":"chapter_computational_complexity/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: Is the space complexity of tail recursion \\(O(1)\\)?</p> <p>Theoretically, the space complexity of a tail-recursive function can be optimized to \\(O(1)\\). However, most programming languages (such as Java, Python, C++, Go, C#) do not support automatic optimization of tail recursion, so it's generally considered to have a space complexity of \\(O(n)\\).</p> <p>Q: What is the difference between the terms \"function\" and \"method\"?</p> <p>A \"function\" can be executed independently, with all parameters passed explicitly. A \"method\" is associated with an object and is implicitly passed to the object calling it, able to operate on the data contained within an instance of a class.</p> <p>Here are some examples from common programming languages:</p> <ul> <li>C is a procedural programming language without object-oriented concepts, so it only has functions. However, we can simulate object-oriented programming by creating structures (struct), and functions associated with these structures are equivalent to methods in other programming languages.</li> <li>Java and C# are object-oriented programming languages where code blocks (methods) are typically part of a class. Static methods behave like functions because they are bound to the class and cannot access specific instance variables.</li> <li>C++ and Python support both procedural programming (functions) and object-oriented programming (methods).</li> </ul> <p>Q: Does the \"Common Types of Space Complexity\" figure reflect the absolute size of occupied space?</p> <p>No, the figure shows space complexities, which reflect growth trends, not the absolute size of the occupied space.</p> <p>If you take \\(n = 8\\), you might find that the values of each curve don't correspond to their functions. This is because each curve includes a constant term, intended to compress the value range into a visually comfortable range.</p> <p>In practice, since we usually don't know the \"constant term\" complexity of each method, it's generally not possible to choose the best solution for \\(n = 8\\) based solely on complexity. However, for \\(n = 8^5\\), it's much easier to choose, as the growth trend becomes dominant.</p>"},{"location":"chapter_computational_complexity/time_complexity/","title":"2.3 \u00a0 Time Complexity","text":"<p>Time complexity is a concept used to measure how the run time of an algorithm increases with the size of the input data. Understanding time complexity is crucial for accurately assessing the efficiency of an algorithm.</p> <ol> <li>Determining the Running Platform: This includes hardware configuration, programming language, system environment, etc., all of which can affect the efficiency of code execution.</li> <li>Evaluating the Run Time for Various Computational Operations: For instance, an addition operation <code>+</code> might take 1 ns, a multiplication operation <code>*</code> might take 10 ns, a print operation <code>print()</code> might take 5 ns, etc.</li> <li>Counting All the Computational Operations in the Code: Summing the execution times of all these operations gives the total run time.</li> </ol> <p>For example, consider the following code with an input size of \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code># Under an operating platform\ndef algorithm(n: int):\n    a = 2      # 1 ns\n    a = a + 1  # 1 ns\n    a = a * 2  # 10 ns\n    # Cycle n times\n    for _ in range(n):  # 1 ns\n        print(0)        # 5 ns\n</code></pre> <pre><code>// Under a particular operating platform\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {  // 1 ns , every round i++ is executed\n        cout &lt;&lt; 0 &lt;&lt; endl;         // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {  // 1 ns , every round i++ is executed\n        System.out.println(0);     // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nvoid Algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {  // 1 ns , every round i++ is executed\n        Console.WriteLine(0);      // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nfunc algorithm(n int) {\n    a := 2     // 1 ns\n    a = a + 1  // 1 ns\n    a = a * 2  // 10 ns\n    // Loop n times\n    for i := 0; i &lt; n; i++ {  // 1 ns\n        fmt.Println(a)        // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nfunc algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // Loop n times\n    for _ in 0 ..&lt; n { // 1 ns\n        print(0) // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nfunction algorithm(n) {\n    var a = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // Loop n times\n    for(let i = 0; i &lt; n; i++) { // 1 ns , every round i++ is executed\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nfunction algorithm(n: number): void {\n    var a: number = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // Loop n times\n    for(let i = 0; i &lt; n; i++) { // 1 ns , every round i++ is executed\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nvoid algorithm(int n) {\n  int a = 2; // 1 ns\n  a = a + 1; // 1 ns\n  a = a * 2; // 10 ns\n  // Loop n times\n  for (int i = 0; i &lt; n; i++) { // 1 ns , every round i++ is executed\n    print(0); // 5 ns\n  }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nfn algorithm(n: i32) {\n    let mut a = 2;      // 1 ns\n    a = a + 1;          // 1 ns\n    a = a * 2;          // 10 ns\n    // Loop n times\n    for _ in 0..n {     // 1 ns for each round i++\n        println!(\"{}\", 0);  // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {   // 1 ns , every round i++ is executed\n        printf(\"%d\", 0);            // 5 ns\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>// Under a particular operating platform\nfn algorithm(n: usize) void {\n    var a: i32 = 2; // 1 ns\n    a += 1; // 1 ns\n    a *= 2; // 10 ns\n    // Loop n times\n    for (0..n) |_| { // 1 ns\n        std.debug.print(\"{}\\n\", .{0}); // 5 ns\n    }\n}\n</code></pre> <p>Using the above method, the run time of the algorithm can be calculated as \\((6n + 12)\\) ns:</p> \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] <p>However, in practice, counting the run time of an algorithm is neither practical nor reasonable. First, we don't want to tie the estimated time to the running platform, as algorithms need to run on various platforms. Second, it's challenging to know the run time for each type of operation, making the estimation process difficult.</p>"},{"location":"chapter_computational_complexity/time_complexity/#231-assessing-time-growth-trend","title":"2.3.1 \u00a0 Assessing Time Growth Trend","text":"<p>Time complexity analysis does not count the algorithm's run time, but rather the growth trend of the run time as the data volume increases.</p> <p>Let's understand this concept of \"time growth trend\" with an example. Assume the input data size is \\(n\\), and consider three algorithms <code>A</code>, <code>B</code>, and <code>C</code>:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code># Time complexity of algorithm A: constant order\ndef algorithm_A(n: int):\n    print(0)\n# Time complexity of algorithm B: linear order\ndef algorithm_B(n: int):\n    for _ in range(n):\n        print(0)\n# Time complexity of algorithm C: constant order\ndef algorithm_C(n: int):\n    for _ in range(1000000):\n        print(0)\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nvoid algorithm_A(int n) {\n    cout &lt;&lt; 0 &lt;&lt; endl;\n}\n// Time complexity of algorithm B: linear order\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nvoid algorithm_A(int n) {\n    System.out.println(0);\n}\n// Time complexity of algorithm B: linear order\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        System.out.println(0);\n    }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        System.out.println(0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nvoid AlgorithmA(int n) {\n    Console.WriteLine(0);\n}\n// Time complexity of algorithm B: linear order\nvoid AlgorithmB(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Console.WriteLine(0);\n    }\n}\n// Time complexity of algorithm C: constant order\nvoid AlgorithmC(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        Console.WriteLine(0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfunc algorithm_A(n int) {\n    fmt.Println(0)\n}\n// Time complexity of algorithm B: linear order\nfunc algorithm_B(n int) {\n    for i := 0; i &lt; n; i++ {\n        fmt.Println(0)\n    }\n}\n// Time complexity of algorithm C: constant order\nfunc algorithm_C(n int) {\n    for i := 0; i &lt; 1000000; i++ {\n        fmt.Println(0)\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfunc algorithmA(n: Int) {\n    print(0)\n}\n\n// Time complexity of algorithm B: linear order\nfunc algorithmB(n: Int) {\n    for _ in 0 ..&lt; n {\n        print(0)\n    }\n}\n\n// Time complexity of algorithm C: constant order\nfunc algorithmC(n: Int) {\n    for _ in 0 ..&lt; 1_000_000 {\n        print(0)\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfunction algorithm_A(n) {\n    console.log(0);\n}\n// Time complexity of algorithm B: linear order\nfunction algorithm_B(n) {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// Time complexity of algorithm C: constant order\nfunction algorithm_C(n) {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfunction algorithm_A(n: number): void {\n    console.log(0);\n}\n// Time complexity of algorithm B: linear order\nfunction algorithm_B(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// Time complexity of algorithm C: constant order\nfunction algorithm_C(n: number): void {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nvoid algorithmA(int n) {\n  print(0);\n}\n// Time complexity of algorithm B: linear order\nvoid algorithmB(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    print(0);\n  }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithmC(int n) {\n  for (int i = 0; i &lt; 1000000; i++) {\n    print(0);\n  }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfn algorithm_A(n: i32) {\n    println!(\"{}\", 0);\n}\n// Time complexity of algorithm B: linear order\nfn algorithm_B(n: i32) {\n    for _ in 0..n {\n        println!(\"{}\", 0);\n    }\n}\n// Time complexity of algorithm C: constant order\nfn algorithm_C(n: i32) {\n    for _ in 0..1000000 {\n        println!(\"{}\", 0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nvoid algorithm_A(int n) {\n    printf(\"%d\", 0);\n}\n// Time complexity of algorithm B: linear order\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        printf(\"%d\", 0);\n    }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        printf(\"%d\", 0);\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfn algorithm_A(n: usize) void {\n    _ = n;\n    std.debug.print(\"{}\\n\", .{0});\n}\n// Time complexity of algorithm B: linear order\nfn algorithm_B(n: i32) void {\n    for (0..n) |_| {\n        std.debug.print(\"{}\\n\", .{0});\n    }\n}\n// Time complexity of algorithm C: constant order\nfn algorithm_C(n: i32) void {\n    _ = n;\n    for (0..1000000) |_| {\n        std.debug.print(\"{}\\n\", .{0});\n    }\n}\n</code></pre> <p>The following figure shows the time complexities of these three algorithms.</p> <ul> <li>Algorithm <code>A</code> has just one print operation, and its run time does not grow with \\(n\\). Its time complexity is considered \"constant order.\"</li> <li>Algorithm <code>B</code> involves a print operation looping \\(n\\) times, and its run time grows linearly with \\(n\\). Its time complexity is \"linear order.\"</li> <li>Algorithm <code>C</code> has a print operation looping 1,000,000 times. Although it takes a long time, it is independent of the input data size \\(n\\). Therefore, the time complexity of <code>C</code> is the same as <code>A</code>, which is \"constant order.\"</li> </ul> <p></p> <p> Figure 2-7 \u00a0 Time Growth Trend of Algorithms A, B, and C </p> <p>Compared to directly counting the run time of an algorithm, what are the characteristics of time complexity analysis?</p> <ul> <li>Time complexity effectively assesses algorithm efficiency. For instance, algorithm <code>B</code> has linearly growing run time, which is slower than algorithm <code>A</code> when \\(n &gt; 1\\) and slower than <code>C</code> when \\(n &gt; 1,000,000\\). In fact, as long as the input data size \\(n\\) is sufficiently large, a \"constant order\" complexity algorithm will always be better than a \"linear order\" one, demonstrating the essence of time growth trend.</li> <li>Time complexity analysis is more straightforward. Obviously, the running platform and the types of computational operations are irrelevant to the trend of run time growth. Therefore, in time complexity analysis, we can simply treat the execution time of all computational operations as the same \"unit time,\" simplifying the \"computational operation run time count\" to a \"computational operation count.\" This significantly reduces the complexity of estimation.</li> <li>Time complexity has its limitations. For example, although algorithms <code>A</code> and <code>C</code> have the same time complexity, their actual run times can be quite different. Similarly, even though algorithm <code>B</code> has a higher time complexity than <code>C</code>, it is clearly superior when the input data size \\(n\\) is small. In these cases, it's difficult to judge the efficiency of algorithms based solely on time complexity. Nonetheless, despite these issues, complexity analysis remains the most effective and commonly used method for evaluating algorithm efficiency.</li> </ul>"},{"location":"chapter_computational_complexity/time_complexity/#232-asymptotic-upper-bound","title":"2.3.2 \u00a0 Asymptotic Upper Bound","text":"<p>Consider a function with an input size of \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>def algorithm(n: int):\n    a = 1      # +1\n    a = a + 1  # +1\n    a = a * 2  # +1\n    # Cycle n times\n    for i in range(n):  # +1\n        print(0)        # +1\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) { // +1 (execute i ++ every round)\n        cout &lt;&lt; 0 &lt;&lt; endl;    // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) { // +1 (execute i ++ every round)\n        System.out.println(0);    // +1\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {   // +1 (execute i ++ every round)\n        Console.WriteLine(0);   // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1      // +1\n    a = a + 1   // +1\n    a = a * 2   // +1\n    // Loop n times\n    for i := 0; i &lt; n; i++ {   // +1\n        fmt.Println(a)         // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // Loop n times\n    for _ in 0 ..&lt; n { // +1\n        print(0) // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    var a = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // Loop n times\n    for(let i = 0; i &lt; n; i++){ // +1 (execute i ++ every round)\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void{\n    var a: number = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // Loop n times\n    for(let i = 0; i &lt; n; i++){ // +1 (execute i ++ every round)\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +1\n  a = a + 1; // +1\n  a = a * 2; // +1\n  // Loop n times\n  for (int i = 0; i &lt; n; i++) { // +1 (execute i ++ every round)\n    print(0); // +1\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;   // +1\n    a = a + 1;      // +1\n    a = a * 2;      // +1\n\n    // Loop n times\n    for _ in 0..n { // +1 (execute i ++ every round)\n        println!(\"{}\", 0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {   // +1 (execute i ++ every round)\n        printf(\"%d\", 0);            // +1\n    }\n} \n</code></pre> <pre><code>\n</code></pre> <pre><code>fn algorithm(n: usize) void {\n    var a: i32 = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // Loop n times\n    for (0..n) |_| { // +1 (execute i ++ every round)\n        std.debug.print(\"{}\\n\", .{0}); // +1\n    }\n}\n</code></pre> <p>Given a function that represents the number of operations of an algorithm as a function of the input size \\(n\\), denoted as \\(T(n)\\), consider the following example:</p> \\[ T(n) = 3 + 2n \\] <p>Since \\(T(n)\\) is a linear function, its growth trend is linear, and therefore, its time complexity is of linear order, denoted as \\(O(n)\\). This mathematical notation, known as \"big-O notation,\" represents the \"asymptotic upper bound\" of the function \\(T(n)\\).</p> <p>In essence, time complexity analysis is about finding the asymptotic upper bound of the \"number of operations \\(T(n)\\)\". It has a precise mathematical definition.</p> <p>Asymptotic Upper Bound</p> <p>If there exist positive real numbers \\(c\\) and \\(n_0\\) such that for all \\(n &gt; n_0\\), \\(T(n) \\leq c \\cdot f(n)\\), then \\(f(n)\\) is considered an asymptotic upper bound of \\(T(n)\\), denoted as \\(T(n) = O(f(n))\\).</p> <p>As illustrated below, calculating the asymptotic upper bound involves finding a function \\(f(n)\\) such that, as \\(n\\) approaches infinity, \\(T(n)\\) and \\(f(n)\\) have the same growth order, differing only by a constant factor \\(c\\).</p> <p></p> <p> Figure 2-8 \u00a0 Asymptotic Upper Bound of a Function </p>"},{"location":"chapter_computational_complexity/time_complexity/#233-calculation-method","title":"2.3.3 \u00a0 Calculation Method","text":"<p>While the concept of asymptotic upper bound might seem mathematically dense, you don't need to fully grasp it right away. Let's first understand the method of calculation, which can be practiced and comprehended over time.</p> <p>Once \\(f(n)\\) is determined, we obtain the time complexity \\(O(f(n))\\). But how do we determine the asymptotic upper bound \\(f(n)\\)? This process generally involves two steps: counting the number of operations and determining the asymptotic upper bound.</p>"},{"location":"chapter_computational_complexity/time_complexity/#1-step-1-counting-the-number-of-operations","title":"1. \u00a0 Step 1: Counting the Number of Operations","text":"<p>This step involves going through the code line by line. However, due to the presence of the constant \\(c\\) in \\(c \\cdot f(n)\\), all coefficients and constant terms in \\(T(n)\\) can be ignored. This principle allows for simplification techniques in counting operations.</p> <ol> <li>Ignore constant terms in \\(T(n)\\), as they do not affect the time complexity being independent of \\(n\\).</li> <li>Omit all coefficients. For example, looping \\(2n\\), \\(5n + 1\\) times, etc., can be simplified to \\(n\\) times since the coefficient before \\(n\\) does not impact the time complexity.</li> <li>Use multiplication for nested loops. The total number of operations equals the product of the number of operations in each loop, applying the simplification techniques from points 1 and 2 for each loop level.</li> </ol> <p>Given a function, we can use these techniques to count operations:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>def algorithm(n: int):\n    a = 1      # +0 (trick 1)\n    a = a + n  # +0 (trick 1)\n    # +n (technique 2)\n    for i in range(5 * n + 1):\n        print(0)\n    # +n*n (technique 3)\n    for i in range(2 * n):\n        for j in range(n + 1):\n            print(0)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0 (trick 1)\n    a = a + n;  // +0 (trick 1)\n    // +n (technique 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n    // +n*n (technique 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            cout &lt;&lt; 0 &lt;&lt; endl;\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0 (trick 1)\n    a = a + n;  // +0 (trick 1)\n    // +n (technique 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        System.out.println(0);\n    }\n    // +n*n (technique 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            System.out.println(0);\n        }\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +0 (trick 1)\n    a = a + n;  // +0 (trick 1)\n    // +n (technique 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        Console.WriteLine(0);\n    }\n    // +n*n (technique 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            Console.WriteLine(0);\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1     // +0 (trick 1)\n    a = a + n  // +0 (trick 1)\n    // +n (technique 2)\n    for i := 0; i &lt; 5 * n + 1; i++ {\n        fmt.Println(0)\n    }\n    // +n*n (technique 3)\n    for i := 0; i &lt; 2 * n; i++ {\n        for j := 0; j &lt; n + 1; j++ {\n            fmt.Println(0)\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +0 (trick 1)\n    a = a + n // +0 (trick 1)\n    // +n (technique 2)\n    for _ in 0 ..&lt; (5 * n + 1) {\n        print(0)\n    }\n    // +n*n (technique 3)\n    for _ in 0 ..&lt; (2 * n) {\n        for _ in 0 ..&lt; (n + 1) {\n            print(0)\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    let a = 1;  // +0 (trick 1)\n    a = a + n;  // +0 (trick 1)\n    // +n (technique 2)\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n (technique 3)\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    let a = 1;  // +0 (trick 1)\n    a = a + n;  // +0 (trick 1)\n    // +n (technique 2)\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n (technique 3)\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +0 (trick 1)\n  a = a + n; // +0 (trick 1)\n  // +n (technique 2)\n  for (int i = 0; i &lt; 5 * n + 1; i++) {\n    print(0);\n  }\n  // +n*n (technique 3)\n  for (int i = 0; i &lt; 2 * n; i++) {\n    for (int j = 0; j &lt; n + 1; j++) {\n      print(0);\n    }\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;     // +0 (trick 1)\n    a = a + n;        // +0 (trick 1)\n\n    // +n (technique 2)\n    for i in 0..(5 * n + 1) {\n        println!(\"{}\", 0);\n    }\n\n    // +n*n (technique 3)\n    for i in 0..(2 * n) {\n        for j in 0..(n + 1) {\n            println!(\"{}\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0 (trick 1)\n    a = a + n;  // +0 (trick 1)\n    // +n (technique 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        printf(\"%d\", 0);\n    }\n    // +n*n (technique 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            printf(\"%d\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>fn algorithm(n: usize) void {\n    var a: i32 = 1;     // +0 (trick 1)\n    a = a + @as(i32, @intCast(n));        // +0 (trick 1)\n\n    // +n (technique 2)\n    for(0..(5 * n + 1)) |_| {\n        std.debug.print(\"{}\\n\", .{0});\n    }\n\n    // +n*n (technique 3)\n    for(0..(2 * n)) |_| {\n        for(0..(n + 1)) |_| {\n            std.debug.print(\"{}\\n\", .{0});\n        }\n    }\n}\n</code></pre> <p>The formula below shows the counting results before and after simplification, both leading to a time complexity of \\(O(n^2)\\):</p> \\[ \\begin{aligned} T(n) &amp; = 2n(n + 1) + (5n + 1) + 2 &amp; \\text{Complete Count (-.-|||)} \\newline &amp; = 2n^2 + 7n + 3 \\newline T(n) &amp; = n^2 + n &amp; \\text{Simplified Count (o.O)} \\end{aligned} \\]"},{"location":"chapter_computational_complexity/time_complexity/#2-step-2-determining-the-asymptotic-upper-bound","title":"2. \u00a0 Step 2: Determining the Asymptotic Upper Bound","text":"<p>The time complexity is determined by the highest order term in \\(T(n)\\). This is because, as \\(n\\) approaches infinity, the highest order term dominates, rendering the influence of other terms negligible.</p> <p>The following table illustrates examples of different operation counts and their corresponding time complexities. Some exaggerated values are used to emphasize that coefficients cannot alter the order of growth. When \\(n\\) becomes very large, these constants become insignificant.</p> <p> Table: Time Complexity for Different Operation Counts </p> Operation Count \\(T(n)\\) Time Complexity \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#234-common-types-of-time-complexity","title":"2.3.4 \u00a0 Common Types of Time Complexity","text":"<p>Let's consider the input data size as \\(n\\). The common types of time complexities are illustrated below, arranged from lowest to highest:</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n \\log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!) \\newline \\text{Constant Order} &lt; \\text{Logarithmic Order} &lt; \\text{Linear Order} &lt; \\text{Linear-Logarithmic Order} &lt; \\text{Quadratic Order} &lt; \\text{Exponential Order} &lt; \\text{Factorial Order} \\end{aligned} \\] <p></p> <p> Figure 2-9 \u00a0 Common Types of Time Complexity </p>"},{"location":"chapter_computational_complexity/time_complexity/#1-constant-order-o1","title":"1. \u00a0 Constant Order \\(O(1)\\)","text":"<p>Constant order means the number of operations is independent of the input data size \\(n\\). In the following function, although the number of operations <code>size</code> might be large, the time complexity remains \\(O(1)\\) as it's unrelated to \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def constant(n: int) -&gt; int:\n    \"\"\"\u5e38\u6570\u9636\"\"\"\n    count = 0\n    size = 100000\n    for _ in range(size):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e38\u6570\u9636 */\nint Constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e38\u6570\u9636 */\nfunc constant(n int) int {\n    count := 0\n    size := 100000\n    for i := 0; i &lt; size; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) -&gt; Int {\n    var count = 0\n    let size = 100_000\n    for _ in 0 ..&lt; size {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e38\u6570\u9636 */\nfunction constant(n: number): number {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n  int count = 0;\n  int size = 100000;\n  for (var i = 0; i &lt; size; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e38\u6570\u9636 */\nfn constant(n: i32) -&gt; i32 {\n    _ = n;\n    let mut count = 0;\n    let size = 100_000;\n    for _ in 0..size {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    int i = 0;\n    for (int i = 0; i &lt; size; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e38\u6570\u9636 */\nfun constant(n: Int): Int {\n    var count = 0\n    val size = 10_0000\n    for (i in 0..&lt;size)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{constant}\n</code></pre> time_complexity.zig<pre><code>// \u5e38\u6570\u9636\nfn constant(n: i32) i32 {\n    _ = n;\n    var count: i32 = 0;\n    const size: i32 = 100_000;\n    var i: i32 = 0;\n    while(i&lt;size) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_computational_complexity/time_complexity/#2-linear-order-on","title":"2. \u00a0 Linear Order \\(O(n)\\)","text":"<p>Linear order indicates the number of operations grows linearly with the input data size \\(n\\). Linear order commonly appears in single-loop structures:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def linear(n: int) -&gt; int:\n    \"\"\"\u7ebf\u6027\u9636\"\"\"\n    count = 0\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u9636 */\nint Linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n int) int {\n    count := 0\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) -&gt; Int {\n    var count = 0\n    for _ in 0 ..&lt; n {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): number {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n  int count = 0;\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u9636 */\nfn linear(n: i32) -&gt; i32 {\n    let mut count = 0;\n    for _ in 0..n {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7ebf\u6027\u9636 */\nfun linear(n: Int): Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (i in 0..&lt;n)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{linear}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u9636\nfn linear(n: i32) i32 {\n    var count: i32 = 0;\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Operations like array traversal and linked list traversal have a time complexity of \\(O(n)\\), where \\(n\\) is the length of the array or list:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def array_traversal(nums: list[int]) -&gt; int:\n    \"\"\"\u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\"\"\"\n    count = 0\n    # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for num in nums:\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(vector&lt;int&gt; &amp;nums) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint ArrayTraversal(int[] nums) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    foreach (int num in nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums []int) int {\n    count := 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for range nums {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums: [Int]) -&gt; Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for _ in nums {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums) {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums: number[]): number {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(List&lt;int&gt; nums) {\n  int count = 0;\n  // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n  for (var _num in nums) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfn array_traversal(nums: &amp;[i32]) -&gt; i32 {\n    let mut count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for _ in nums {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int *nums, int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfun arrayTraversal(nums: IntArray): Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (num in nums) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{array_traversal}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\nfn arrayTraversal(nums: []i32) i32 {\n    var count: i32 = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (nums) |_| {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>It's important to note that the input data size \\(n\\) should be determined based on the type of input data. For example, in the first example, \\(n\\) represents the input data size, while in the second example, the length of the array \\(n\\) is the data size.</p>"},{"location":"chapter_computational_complexity/time_complexity/#3-quadratic-order-on2","title":"3. \u00a0 Quadratic Order \\(O(n^2)\\)","text":"<p>Quadratic order means the number of operations grows quadratically with the input data size \\(n\\). Quadratic order typically appears in nested loops, where both the outer and inner loops have a time complexity of \\(O(n)\\), resulting in an overall complexity of \\(O(n^2)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def quadratic(n: int) -&gt; int:\n    \"\"\"\u5e73\u65b9\u9636\"\"\"\n    count = 0\n    # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for i in range(n):\n        for j in range(n):\n            count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u9636 */\nint Quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n int) int {\n    count := 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; n; j++ {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) -&gt; Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; n {\n            count += 1\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): number {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n  int count = 0;\n  // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n  for (int i = 0; i &lt; n; i++) {\n    for (int j = 0; j &lt; n; j++) {\n      count++;\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u9636 */\nfn quadratic(n: i32) -&gt; i32 {\n    let mut count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for _ in 0..n {\n        for _ in 0..n {\n            count += 1;\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e73\u65b9\u9636 */\nfun quadratic(n: Int): Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;n) {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{quadratic}\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u9636\nfn quadratic(n: i32) i32 {\n    var count: i32 = 0;\n    var i: i32 = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    while (i &lt; n) : (i += 1) {\n        var j: i32 = 0;\n        while (j &lt; n) : (j += 1) {\n            count += 1;\n        }\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The following image compares constant order, linear order, and quadratic order time complexities.</p> <p></p> <p> Figure 2-10 \u00a0 Constant, Linear, and Quadratic Order Time Complexities </p> <p>For instance, in bubble sort, the outer loop runs \\(n - 1\\) times, and the inner loop runs \\(n-1\\), \\(n-2\\), ..., \\(2\\), \\(1\\) times, averaging \\(n / 2\\) times, resulting in a time complexity of \\(O((n - 1) n / 2) = O(n^2)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def bubble_sort(nums: list[int]) -&gt; int:\n    \"\"\"\u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\"\"\"\n    count = 0  # \u8ba1\u6570\u5668\n    # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in range(len(nums) - 1, 0, -1):\n        # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                tmp: int = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3  # \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(vector&lt;int&gt; &amp;nums) {\n    int count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\n    int count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint BubbleSort(int[] nums) {\n    int count = 0;  // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                count += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\n    count := 0 // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                tmp := nums[j]\n                nums[j] = nums[j+1]\n                nums[j+1] = tmp\n                count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -&gt; Int {\n    var count = 0 // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\n    let count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\n    let count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(List&lt;int&gt; nums) {\n  int count = 0; // \u8ba1\u6570\u5668\n  // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n  for (var i = nums.length - 1; i &gt; 0; i--) {\n    // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n    for (var j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n      }\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfn bubble_sort(nums: &amp;mut [i32]) -&gt; i32 {\n    let mut count = 0; // \u8ba1\u6570\u5668\n\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in (1..nums.len()).rev() {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int *nums, int n) {\n    int count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = n - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfun bubbleSort(nums: IntArray): Int {\n    var count = 0\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j] = nums[j + 1].also { nums[j + 1] = nums[j] }\n                count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{bubble_sort}\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\n    var count: i32 = 0;  // \u8ba1\u6570\u5668 \n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    var i: i32 = @as(i32, @intCast(nums.len)) - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var j: usize = 0;\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_computational_complexity/time_complexity/#4-exponential-order-o2n","title":"4. \u00a0 Exponential Order \\(O(2^n)\\)","text":"<p>Biological \"cell division\" is a classic example of exponential order growth: starting with one cell, it becomes two after one division, four after two divisions, and so on, resulting in \\(2^n\\) cells after \\(n\\) divisions.</p> <p>The following image and code simulate the cell division process, with a time complexity of \\(O(2^n)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def exponential(n: int) -&gt; int:\n    \"\"\"\u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n    count = 0\n    base = 1\n    # \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in range(n):\n        for _ in range(base):\n            count += 1\n        base *= 2\n    # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Exponential(int n) {\n    int count = 0, bas = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc exponential(n int) int {\n    count, base := 0, 1\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; base; j++ {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc exponential(n: Int) -&gt; Int {\n    var count = 0\n    var base = 1\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; base {\n            count += 1\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n) {\n    let count = 0,\n        base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n: number): number {\n    let count = 0,\n        base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n  int count = 0, base = 1;\n  // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n  for (var i = 0; i &lt; n; i++) {\n    for (var j = 0; j &lt; base; j++) {\n      count++;\n    }\n    base *= 2;\n  }\n  // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn exponential(n: i32) -&gt; i32 {\n    let mut count = 0;\n    let mut base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0..n {\n        for _ in 0..base {\n            count += 1\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n    int count = 0;\n    int bas = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfun exponential(n: Int): Int {\n    var count = 0\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    var base = 1\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;base) {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{exponential}\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn exponential(n: i32) i32 {\n    var count: i32 = 0;\n    var bas: i32 = 1;\n    var i: i32 = 0;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    while (i &lt; n) : (i += 1) {\n        var j: i32 = 0;\n        while (j &lt; bas) : (j += 1) {\n            count += 1;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p></p> <p> Figure 2-11 \u00a0 Exponential Order Time Complexity </p> <p>In practice, exponential order often appears in recursive functions. For example, in the code below, it recursively splits into two halves, stopping after \\(n\\) divisions:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def exp_recur(n: int) -&gt; int:\n    \"\"\"\u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n == 1:\n        return 1\n    return exp_recur(n - 1) + exp_recur(n - 1) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint ExpRecur(int n) {\n    if (n == 1) return 1;\n    return ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc expRecur(n int) int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n-1) + expRecur(n-1) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc expRecur(n: Int) -&gt; Int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n) {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n: number): number {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n  if (n == 1) return 1;\n  return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn exp_recur(n: i32) -&gt; i32 {\n    if n == 1 {\n        return 1;\n    }\n    exp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun expRecur(n: Int): Int {\n    if (n == 1) {\n        return 1\n    }\n    return expRecur(n - 1) + expRecur(n - 1) + 1\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{exp_recur}\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn expRecur(n: i32) i32 {\n    if (n == 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Exponential order growth is extremely rapid and is commonly seen in exhaustive search methods (brute force, backtracking, etc.). For large-scale problems, exponential order is unacceptable, often requiring dynamic programming or greedy algorithms as solutions.</p>"},{"location":"chapter_computational_complexity/time_complexity/#5-logarithmic-order-olog-n","title":"5. \u00a0 Logarithmic Order \\(O(\\log n)\\)","text":"<p>In contrast to exponential order, logarithmic order reflects situations where \"the size is halved each round.\" Given an input data size \\(n\\), since the size is halved each round, the number of iterations is \\(\\log_2 n\\), the inverse function of \\(2^n\\).</p> <p>The following image and code simulate the \"halving each round\" process, with a time complexity of \\(O(\\log_2 n)\\), commonly abbreviated as \\(O(\\log n)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def logarithmic(n: int) -&gt; int:\n    \"\"\"\u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n    count = 0\n    while n &gt; 1:\n        n = n / 2\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n /= 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc logarithmic(n int) int {\n    count := 0\n    for n &gt; 1 {\n        n = n / 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc logarithmic(n: Int) -&gt; Int {\n    var count = 0\n    var n = n\n    while n &gt; 1 {\n        n = n / 2\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n) {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n: number): number {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n  int count = 0;\n  while (n &gt; 1) {\n    n = n ~/ 2;\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn logarithmic(mut n: i32) -&gt; i32 {\n    let mut count = 0;\n    while n &gt; 1 {\n        n = n / 2;\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfun logarithmic(n: Int): Int {\n    var n1 = n\n    var count = 0\n    while (n1 &gt; 1) {\n        n1 /= 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{logarithmic}\n</code></pre> time_complexity.zig<pre><code>// \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn logarithmic(n: i32) i32 {\n    var count: i32 = 0;\n    var n_var = n;\n    while (n_var &gt; 1)\n    {\n        n_var = n_var / 2;\n        count +=1;\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p></p> <p> Figure 2-12 \u00a0 Logarithmic Order Time Complexity </p> <p>Like exponential order, logarithmic order also frequently appears in recursive functions. The code below forms a recursive tree of height \\(\\log_2 n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def log_recur(n: int) -&gt; int:\n    \"\"\"\u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n &lt;= 1:\n        return 0\n    return log_recur(n / 2) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint LogRecur(int n) {\n    if (n &lt;= 1) return 0;\n    return LogRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc logRecur(n int) int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n/2) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc logRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n: n / 2) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n) {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n: number): number {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n  if (n &lt;= 1) return 0;\n  return logRecur(n ~/ 2) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 0;\n    }\n    log_recur(n / 2) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun logRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 0\n    return logRecur(n / 2) + 1\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{log_recur}\n</code></pre> time_complexity.zig<pre><code>// \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn logRecur(n: i32) i32 {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Logarithmic order is typical in algorithms based on the divide-and-conquer strategy, embodying the \"split into many\" and \"simplify complex problems\" approach. It's slow-growing and is the most ideal time complexity after constant order.</p> <p>What is the base of \\(O(\\log n)\\)?</p> <p>Technically, \"splitting into \\(m\\)\" corresponds to a time complexity of \\(O(\\log_m n)\\). Using the logarithm base change formula, we can equate different logarithmic complexities:</p> \\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\] <p>This means the base \\(m\\) can be changed without affecting the complexity. Therefore, we often omit the base \\(m\\) and simply denote logarithmic order as \\(O(\\log n)\\).</p>"},{"location":"chapter_computational_complexity/time_complexity/#6-linear-logarithmic-order-on-log-n","title":"6. \u00a0 Linear-Logarithmic Order \\(O(n \\log n)\\)","text":"<p>Linear-logarithmic order often appears in nested loops, with the complexities of the two loops being \\(O(\\log n)\\) and \\(O(n)\\) respectively. The related code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def linear_log_recur(n: int) -&gt; int:\n    \"\"\"\u7ebf\u6027\u5bf9\u6570\u9636\"\"\"\n    if n &lt;= 1:\n        return 1\n    count: int = linear_log_recur(n // 2) + linear_log_recur(n // 2)\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint LinearLogRecur(int n) {\n    if (n &lt;= 1) return 1;\n    int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n int) int {\n    if n &lt;= 1 {\n        return 1\n    }\n    count := linearLogRecur(n/2) + linearLogRecur(n/2)\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 1\n    }\n    var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\n    for _ in stride(from: 0, to: n, by: 1) {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n) {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n: number): number {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n  if (n &lt;= 1) return 1;\n  int count = linearLogRecur(n ~/ 2) + linearLogRecur(n ~/ 2);\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfn linear_log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 1;\n    }\n    let mut count = linear_log_recur(n / 2) + linear_log_recur(n / 2);\n    for _ in 0..n as i32 {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfun linearLogRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 1\n    var count = linearLogRecur(n / 2) + linearLogRecur(n / 2)\n    for (i in 0..&lt;n.toInt()) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{linear_log_recur}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u5bf9\u6570\u9636\nfn linearLogRecur(n: i32) i32 {\n    if (n &lt;= 1) return 1;\n    var count: i32 = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The image below demonstrates how linear-logarithmic order is generated. Each level of a binary tree has \\(n\\) operations, and the tree has \\(\\log_2 n + 1\\) levels, resulting in a time complexity of \\(O(n \\log n)\\).</p> <p></p> <p> Figure 2-13 \u00a0 Linear-Logarithmic Order Time Complexity </p> <p>Mainstream sorting algorithms typically have a time complexity of \\(O(n \\log n)\\), such as quicksort, mergesort, and heapsort.</p>"},{"location":"chapter_computational_complexity/time_complexity/#7-factorial-order-on","title":"7. \u00a0 Factorial Order \\(O(n!)\\)","text":"<p>Factorial order corresponds to the mathematical problem of \"full permutation.\" Given \\(n\\) distinct elements, the total number of possible permutations is:</p> \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\] <p>Factorials are typically implemented using recursion. As shown in the image and code below, the first level splits into \\(n\\) branches, the second level into \\(n - 1\\) branches, and so on, stopping after the \\(n\\)th level:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def factorial_recur(n: int) -&gt; int:\n    \"\"\"\u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n == 0:\n        return 1\n    count = 0\n    # \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for _ in range(n):\n        count += factorial_recur(n - 1)\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint FactorialRecur(int n) {\n    if (n == 0) return 1;\n    int count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (int i = 0; i &lt; n; i++) {\n        count += FactorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n int) int {\n    if n == 0 {\n        return 1\n    }\n    count := 0\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for i := 0; i &lt; n; i++ {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n: Int) -&gt; Int {\n    if n == 0 {\n        return 1\n    }\n    var count = 0\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for _ in 0 ..&lt; n {\n        count += factorialRecur(n: n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n) {\n    if (n === 0) return 1;\n    let count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n: number): number {\n    if (n === 0) return 1;\n    let count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n  if (n == 0) return 1;\n  int count = 0;\n  // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n  for (var i = 0; i &lt; n; i++) {\n    count += factorialRecur(n - 1);\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn factorial_recur(n: i32) -&gt; i32 {\n    if n == 0 {\n        return 1;\n    }\n    let mut count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for _ in 0..n {\n        count += factorial_recur(n - 1);\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun factorialRecur(n: Int): Int {\n    if (n == 0)\n        return 1\n    var count = 0\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (i in 0..&lt;n) {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{factorial_recur}\n</code></pre> time_complexity.zig<pre><code>// \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn factorialRecur(n: i32) i32 {\n    if (n == 0) return 1;\n    var count: i32 = 0;\n    var i: i32 = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    while (i &lt; n) : (i += 1) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p></p> <p> Figure 2-14 \u00a0 Factorial Order Time Complexity </p> <p>Note that factorial order grows even faster than exponential order; it's unacceptable for larger \\(n\\) values.</p>"},{"location":"chapter_computational_complexity/time_complexity/#235-worst-best-and-average-time-complexities","title":"2.3.5 \u00a0 Worst, Best, and Average Time Complexities","text":"<p>The time efficiency of an algorithm is often not fixed but depends on the distribution of the input data. Assume we have an array <code>nums</code> of length \\(n\\), consisting of numbers from \\(1\\) to \\(n\\), each appearing only once, but in a randomly shuffled order. The task is to return the index of the element \\(1\\). We can draw the following conclusions:</p> <ul> <li>When <code>nums = [?, ?, ..., 1]</code>, that is, when the last element is \\(1\\), it requires a complete traversal of the array, achieving the worst-case time complexity of \\(O(n)\\).</li> <li>When <code>nums = [1, ?, ?, ...]</code>, that is, when the first element is \\(1\\), no matter the length of the array, no further traversal is needed, achieving the best-case time complexity of \\(\\Omega(1)\\).</li> </ul> <p>The \"worst-case time complexity\" corresponds to the asymptotic upper bound, denoted by the big \\(O\\) notation. Correspondingly, the \"best-case time complexity\" corresponds to the asymptotic lower bound, denoted by \\(\\Omega\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig worst_best_time_complexity.py<pre><code>def random_numbers(n: int) -&gt; list[int]:\n    \"\"\"\u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a: 1, 2, ..., n \uff0c\u987a\u5e8f\u88ab\u6253\u4e71\"\"\"\n    # \u751f\u6210\u6570\u7ec4 nums =: 1, 2, 3, ..., n\n    nums = [i for i in range(1, n + 1)]\n    # \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    random.shuffle(nums)\n    return nums\n\ndef find_one(nums: list[int]) -&gt; int:\n    \"\"\"\u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\"\"\"\n    for i in range(len(nums)):\n        # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1:\n            return i\n    return -1\n</code></pre> worst_best_time_complexity.cpp<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nvector&lt;int&gt; randomNumbers(int n) {\n    vector&lt;int&gt; nums(n);\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u4f7f\u7528\u7cfb\u7edf\u65f6\u95f4\u751f\u6210\u968f\u673a\u79cd\u5b50\n    unsigned seed = chrono::system_clock::now().time_since_epoch().count();\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    shuffle(nums.begin(), nums.end(), default_random_engine(seed));\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector&lt;int&gt; &amp;nums) {\n    for (int i = 0; i &lt; nums.size(); i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.java<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\n    Integer[] nums = new Integer[n];\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    Collections.shuffle(Arrays.asList(nums));\n    // Integer[] -&gt; int[]\n    int[] res = new int[n];\n    for (int i = 0; i &lt; n; i++) {\n        res[i] = nums[i];\n    }\n    return res;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.cs<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] RandomNumbers(int n) {\n    int[] nums = new int[n];\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (int i = 0; i &lt; nums.Length; i++) {\n        int index = new Random().Next(i, nums.Length);\n        (nums[i], nums[index]) = (nums[index], nums[i]);\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint FindOne(int[] nums) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.go<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n int) []int {\n    nums := make([]int, n)\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for i := 0; i &lt; n; i++ {\n        nums[i] = i + 1\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    rand.Shuffle(len(nums), func(i, j int) {\n        nums[i], nums[j] = nums[j], nums[i]\n    })\n    return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\n    for i := 0; i &lt; len(nums); i++ {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.swift<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n: Int) -&gt; [Int] {\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    var nums = Array(1 ... n)\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    nums.shuffle()\n    return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -&gt; Int {\n    for i in nums.indices {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.js<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n) {\n    const nums = Array(n);\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.ts<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n: number): number[] {\n    const nums = Array(n);\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.dart<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nList&lt;int&gt; randomNumbers(int n) {\n  final nums = List.filled(n, 0);\n  // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n  for (var i = 0; i &lt; n; i++) {\n    nums[i] = i + 1;\n  }\n  // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n  nums.shuffle();\n\n  return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(List&lt;int&gt; nums) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n    // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n    if (nums[i] == 1) return i;\n  }\n\n  return -1;\n}\n</code></pre> worst_best_time_complexity.rs<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfn random_numbers(n: i32) -&gt; Vec&lt;i32&gt; {\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    let mut nums = (1..=n).collect::&lt;Vec&lt;i32&gt;&gt;();\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    nums.shuffle(&amp;mut thread_rng());\n    nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfn find_one(nums: &amp;[i32]) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1 {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> worst_best_time_complexity.c<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint *randomNumbers(int n) {\n    // \u5206\u914d\u5806\u533a\u5185\u5b58\uff08\u521b\u5efa\u4e00\u7ef4\u53ef\u53d8\u957f\u6570\u7ec4\uff1a\u6570\u7ec4\u4e2d\u5143\u7d20\u6570\u91cf\u4e3a n \uff0c\u5143\u7d20\u7c7b\u578b\u4e3a int \uff09\n    int *nums = (int *)malloc(n * sizeof(int));\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (int i = n - 1; i &gt; 0; i--) {\n        int j = rand() % (i + 1);\n        int temp = nums[i];\n        nums[i] = nums[j];\n        nums[j] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int *nums, int n) {\n    for (int i = 0; i &lt; n; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.kt<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfun randomNumbers(n: Int): Array&lt;Int?&gt; {\n    val nums = IntArray(n)\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (i in 0..&lt;n) {\n        nums[i] = i + 1\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    val mutableList = nums.toMutableList()\n    mutableList.shuffle()\n    // Integer[] -&gt; int[]\n    val res = arrayOfNulls&lt;Int&gt;(n)\n    for (i in 0..&lt;n) {\n        res[i] = mutableList[i]\n    }\n    return res\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfun findOne(nums: Array&lt;Int?&gt;): Int {\n    for (i in nums.indices) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.rb<pre><code>[class]{}-[func]{random_numbers}\n\n[class]{}-[func]{find_one}\n</code></pre> worst_best_time_complexity.zig<pre><code>// \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71\nfn randomNumbers(comptime n: usize) [n]i32 {\n    var nums: [n]i32 = undefined;\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (&amp;nums, 0..) |*num, i| {\n        num.* = @as(i32, @intCast(i)) + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    const rand = std.crypto.random;\n    rand.shuffle(i32, &amp;nums);\n    return nums;\n}\n\n// \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\nfn findOne(nums: []i32) i32 {\n    for (nums, 0..) |num, i| {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (num == 1) return @intCast(i);\n    }\n    return -1;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>It's important to note that the best-case time complexity is rarely used in practice, as it is usually only achievable under very low probabilities and might be misleading. The worst-case time complexity is more practical as it provides a safety value for efficiency, allowing us to confidently use the algorithm.</p> <p>From the above example, it's clear that both the worst-case and best-case time complexities only occur under \"special data distributions,\" which may have a small probability of occurrence and may not accurately reflect the algorithm's run efficiency. In contrast, the average time complexity can reflect the algorithm's efficiency under random input data, denoted by the \\(\\Theta\\) notation.</p> <p>For some algorithms, we can simply estimate the average case under a random data distribution. For example, in the aforementioned example, since the input array is shuffled, the probability of element \\(1\\) appearing at any index is equal. Therefore, the average number of loops for the algorithm is half the length of the array \\(n / 2\\), giving an average time complexity of \\(\\Theta(n / 2) = \\Theta(n)\\).</p> <p>However, calculating the average time complexity for more complex algorithms can be quite difficult, as it's challenging to analyze the overall mathematical expectation under the data distribution. In such cases, we usually use the worst-case time complexity as the standard for judging the efficiency of the algorithm.</p> <p>Why is the \\(\\Theta\\) symbol rarely seen?</p> <p>Possibly because the \\(O\\) notation is more commonly spoken, it is often used to represent the average time complexity. However, strictly speaking, this practice is not accurate. In this book and other materials, if you encounter statements like \"average time complexity \\(O(n)\\)\", please understand it directly as \\(\\Theta(n)\\).</p>"},{"location":"chapter_data_structure/","title":"Chapter 3. \u00a0 Data Structures","text":"<p>Abstract</p> <p>Data structures serve as a robust and diverse framework.</p> <p>They offer a blueprint for the orderly organization of data, upon which algorithms come to life.</p>"},{"location":"chapter_data_structure/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>3.1 \u00a0 Classification of Data Structures</li> <li>3.2 \u00a0 Fundamental Data Types</li> <li>3.3 \u00a0 Number Encoding *</li> <li>3.4 \u00a0 Character Encoding *</li> <li>3.5 \u00a0 Summary</li> </ul>"},{"location":"chapter_data_structure/basic_data_types/","title":"3.2 \u00a0 Basic Data Types","text":"<p>When discussing data in computers, various forms like text, images, videos, voice and 3D models comes to mind. Despite their different organizational forms, they are all composed of various basic data types.</p> <p>Basic data types are those that the CPU can directly operate on and are directly used in algorithms, mainly including the following.</p> <ul> <li>Integer types: <code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>.</li> <li>Floating-point types: <code>float</code>, <code>double</code>, used to represent decimals.</li> <li>Character type: <code>char</code>, used to represent letters, punctuation, and even emojis in various languages.</li> <li>Boolean type: <code>bool</code>, used to represent \"yes\" or \"no\" decisions.</li> </ul> <p>Basic data types are stored in computers in binary form. One binary digit is 1 bit. In most modern operating systems, 1 byte consists of 8 bits.</p> <p>The range of values for basic data types depends on the size of the space they occupy. Below, we take Java as an example.</p> <ul> <li>The integer type <code>byte</code> occupies 1 byte = 8 bits and can represent \\(2^8\\) numbers.</li> <li>The integer type <code>int</code> occupies 4 bytes = 32 bits and can represent \\(2^{32}\\) numbers.</li> </ul> <p>The following table lists the space occupied, value range, and default values of various basic data types in Java. While memorizing this table isn't necessary, having a general understanding of it and referencing it when required is recommended.</p> <p> Table 3-1 \u00a0 Space Occupied and Value Range of Basic Data Types </p> Type Symbol Space Occupied Minimum Value Maximum Value Default Value Integer <code>byte</code> 1 byte \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) 0 <code>short</code> 2 bytes \\(-2^{15}\\) \\(2^{15} - 1\\) 0 <code>int</code> 4 bytes \\(-2^{31}\\) \\(2^{31} - 1\\) 0 <code>long</code> 8 bytes \\(-2^{63}\\) \\(2^{63} - 1\\) 0 Float <code>float</code> 4 bytes \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0\\text{f}\\) <code>double</code> 8 bytes \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) 0.0 Char <code>char</code> 2 bytes 0 \\(2^{16} - 1\\) 0 Boolean <code>bool</code> 1 byte \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\) <p>Please note that the above table is specific to Java's basic data types. Every programming language has its own data type definitions, which might differ in space occupied, value ranges, and default values.</p> <ul> <li>In Python, the integer type <code>int</code> can be of any size, limited only by available memory; the floating-point <code>float</code> is double precision 64-bit; there is no <code>char</code> type, as a single character is actually a string <code>str</code> of length 1.</li> <li>C and C++ do not specify the size of basic data types, it varies with implementation and platform. The above table follows the LP64 data model, used for Unix 64-bit operating systems including Linux and macOS.</li> <li>The size of <code>char</code> in C and C++ is 1 byte, while in most programming languages, it depends on the specific character encoding method, as detailed in the \"Character Encoding\" chapter.</li> <li>Even though representing a boolean only requires 1 bit (0 or 1), it is usually stored in memory as 1 byte. This is because modern computer CPUs typically use 1 byte as the smallest addressable memory unit.</li> </ul> <p>So, what is the connection between basic data types and data structures? We know that data structures are ways to organize and store data in computers. The focus here is on \"structure\" rather than \"data\".</p> <p>If we want to represent \"a row of numbers\", we naturally think of using an array. This is because the linear structure of an array can represent the adjacency and the ordering of the numbers, but whether the stored content is an integer <code>int</code>, a decimal <code>float</code>, or a character <code>char</code>, is irrelevant to the \"data structure\".</p> <p>In other words, basic data types provide the \"content type\" of data, while data structures provide the \"way of organizing\" data. For example, in the following code, we use the same data structure (array) to store and represent different basic data types, including <code>int</code>, <code>float</code>, <code>char</code>, <code>bool</code>, etc.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code># Using various basic data types to initialize arrays\nnumbers: list[int] = [0] * 5\ndecimals: list[float] = [0.0] * 5\n# Python's characters are actually strings of length 1\ncharacters: list[str] = ['0'] * 5\nbools: list[bool] = [False] * 5\n# Python's lists can freely store various basic data types and object references\ndata = [0, 0.0, 'a', False, ListNode(0)]\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool bools[5];\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] bools = new boolean[5];\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] bools = new bool[5];\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar bools = [5]bool{}\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nlet numbers = Array(repeating: 0, count: 5)\nlet decimals = Array(repeating: 0.0, count: 5)\nlet characters: [Character] = Array(repeating: \"a\", count: 5)\nlet bools = Array(repeating: false, count: 5)\n</code></pre> <pre><code>// JavaScript's arrays can freely store various basic data types and objects\nconst array = [0, 0.0, 'a', false];\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst bools: boolean[] = [];\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nList&lt;int&gt; numbers = List.filled(5, 0);\nList&lt;double&gt; decimals = List.filled(5, 0.0);\nList&lt;String&gt; characters = List.filled(5, 'a');\nList&lt;bool&gt; bools = List.filled(5, false);\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nlet numbers: Vec&lt;i32&gt; = vec![0; 5];\nlet decimals: Vec&lt;f32&gt; = vec![0.0, 5];\nlet characters: Vec&lt;char&gt; = vec!['0'; 5];\nlet bools: Vec&lt;bool&gt; = vec![false; 5];\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool bools[10];\n</code></pre> <pre><code>\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nvar numbers: [5]i32 = undefined;\nvar decimals: [5]f32 = undefined;\nvar characters: [5]u8 = undefined;\nvar bools: [5]bool = undefined;\n</code></pre>"},{"location":"chapter_data_structure/character_encoding/","title":"3.4 \u00a0 Character Encoding *","text":"<p>In the computer system, all data is stored in binary form, and characters (represented by char) are no exception. To represent characters, we need to develop a \"character set\" that defines a one-to-one mapping between each character and binary numbers. With the character set, computers can convert binary numbers to characters by looking up the table.</p>"},{"location":"chapter_data_structure/character_encoding/#341-ascii-character-set","title":"3.4.1 \u00a0 ASCII Character Set","text":"<p>The \"ASCII code\" is one of the earliest character sets, officially known as the American Standard Code for Information Interchange. It uses 7 binary digits (the lower 7 bits of a byte) to represent a character, allowing for a maximum of 128 different characters. As shown in the Figure 3-6 , ASCII includes uppercase and lowercase English letters, numbers 0 ~ 9, various punctuation marks, and certain control characters (such as newline and tab).</p> <p></p> <p> Figure 3-6 \u00a0 ASCII Code </p> <p>However, ASCII can only represent English characters. With the globalization of computers, a character set called \"EASCII\" was developed to represent more languages. It expands from the 7-bit structure of ASCII to 8 bits, enabling the representation of 256 characters.</p> <p>Globally, various region-specific EASCII character sets have been introduced. The first 128 characters of these sets are consistent with the ASCII, while the remaining 128 characters are defined differently to accommodate the requirements of different languages.</p>"},{"location":"chapter_data_structure/character_encoding/#342-gbk-character-set","title":"3.4.2 \u00a0 GBK Character Set","text":"<p>Later, it was found that EASCII still could not meet the character requirements of many languages. For instance, there are nearly a hundred thousand Chinese characters, with several thousand used regularly. In 1980, the Standardization Administration of China released the \"GB2312\" character set, which included 6763 Chinese characters, essentially fulfilling the computer processing needs for the Chinese language.</p> <p>However, GB2312 could not handle some rare and traditional characters. The \"GBK\" character set expands GB2312 and includes 21886 Chinese characters. In the GBK encoding scheme, ASCII characters are represented with one byte, while Chinese characters use two bytes.</p>"},{"location":"chapter_data_structure/character_encoding/#343-unicode-character-set","title":"3.4.3 \u00a0 Unicode Character Set","text":"<p>With the rapid evolution of computer technology and a plethora of character sets and encoding standards, numerous problems arose. On the one hand, these character sets generally only defined characters for specific languages and could not function properly in multilingual environments. On the other hand, the existence of multiple character set standards for the same language caused garbled text when information was exchanged between computers using different encoding standards.</p> <p>Researchers of that era thought: What if a comprehensive character set encompassing all global languages and symbols was developed? Wouldn't this resolve the issues associated with cross-linguistic environments and garbled text? Inspired by this idea, the extensive character set, Unicode, was born.</p> <p>\"Unicode\" is referred to as \"\u7edf\u4e00\u7801\" (Unified Code) in Chinese, theoretically capable of accommodating over a million characters. It aims to incorporate characters from all over the world into a single set, providing a universal character set for processing and displaying various languages and reducing the issues of garbled text due to different encoding standards.</p> <p>Since its release in 1991, Unicode has continually expanded to include new languages and characters. As of September 2022, Unicode contains 149,186 characters, including characters, symbols, and even emojis from various languages. In the vast Unicode character set, commonly used characters occupy 2 bytes, while some rare characters may occupy 3 or even 4 bytes.</p> <p>Unicode is a universal character set that assigns a number (called a \"code point\") to each character, but it does not specify how these character code points should be stored in a computer system. One might ask: How does a system interpret Unicode code points of varying lengths within a text? For example, given a 2-byte code, how does the system determine if it represents a single 2-byte character or two 1-byte characters?</p> <p>A straightforward solution to this problem is to store all characters as equal-length encodings. As shown in the Figure 3-7 , each character in \"Hello\" occupies 1 byte, while each character in \"\u7b97\u6cd5\" (algorithm) occupies 2 bytes. We could encode all characters in \"Hello \u7b97\u6cd5\" as 2 bytes by padding the higher bits with zeros. This method would enable the system to interpret a character every 2 bytes, recovering the content of the phrase.</p> <p></p> <p> Figure 3-7 \u00a0 Unicode Encoding Example </p> <p>However, as ASCII has shown us, encoding English only requires 1 byte. Using the above approach would double the space occupied by English text compared to ASCII encoding, which is a waste of memory space. Therefore, a more efficient Unicode encoding method is needed.</p>"},{"location":"chapter_data_structure/character_encoding/#344-utf-8-encoding","title":"3.4.4 \u00a0 UTF-8 Encoding","text":"<p>Currently, UTF-8 has become the most widely used Unicode encoding method internationally. It is a variable-length encoding, using 1 to 4 bytes to represent a character, depending on the complexity of the character. ASCII characters need only 1 byte, Latin and Greek letters require 2 bytes, commonly used Chinese characters need 3 bytes, and some other rare characters need 4 bytes.</p> <p>The encoding rules for UTF-8 are not complex and can be divided into two cases:</p> <ul> <li>For 1-byte characters, set the highest bit to \\(0\\), and the remaining 7 bits to the Unicode code point. Notably, ASCII characters occupy the first 128 code points in the Unicode set. This means that UTF-8 encoding is backward compatible with ASCII. This implies that UTF-8 can be used to parse ancient ASCII text.</li> <li>For characters of length \\(n\\) bytes (where \\(n &gt; 1\\)), set the highest \\(n\\) bits of the first byte to \\(1\\), and the \\((n + 1)^{\\text{th}}\\) bit to \\(0\\); starting from the second byte, set the highest 2 bits of each byte to \\(10\\); the rest of the bits are used to fill the Unicode code point.</li> </ul> <p>The Figure 3-8  shows the UTF-8 encoding for \"Hello\u7b97\u6cd5\". It can be observed that since the highest \\(n\\) bits are set to \\(1\\), the system can determine the length of the character as \\(n\\) by counting the number of highest bits set to \\(1\\).</p> <p>But why set the highest 2 bits of the remaining bytes to \\(10\\)? Actually, this \\(10\\) serves as a kind of checksum. If the system starts parsing text from an incorrect byte, the \\(10\\) at the beginning of the byte can help the system quickly detect anomalies.</p> <p>The reason for using \\(10\\) as a checksum is that, under UTF-8 encoding rules, it's impossible for the highest two bits of a character to be \\(10\\). This can be proven by contradiction: If the highest two bits of a character are \\(10\\), it indicates that the character's length is \\(1\\), corresponding to ASCII. However, the highest bit of an ASCII character should be \\(0\\), which contradicts the assumption.</p> <p></p> <p> Figure 3-8 \u00a0 UTF-8 Encoding Example </p> <p>Apart from UTF-8, other common encoding methods include:</p> <ul> <li>UTF-16 Encoding: Uses 2 or 4 bytes to represent a character. All ASCII characters and commonly used non-English characters are represented with 2 bytes; a few characters require 4 bytes. For 2-byte characters, the UTF-16 encoding equals the Unicode code point.</li> <li>UTF-32 Encoding: Every character uses 4 bytes. This means UTF-32 occupies more space than UTF-8 and UTF-16, especially for texts with a high proportion of ASCII characters.</li> </ul> <p>From the perspective of storage space, using UTF-8 to represent English characters is very efficient because it only requires 1 byte; using UTF-16 to encode some non-English characters (such as Chinese) can be more efficient because it only requires 2 bytes, while UTF-8 might need 3 bytes.</p> <p>From a compatibility perspective, UTF-8 is the most versatile, with many tools and libraries supporting UTF-8 as a priority.</p>"},{"location":"chapter_data_structure/character_encoding/#345-character-encoding-in-programming-languages","title":"3.4.5 \u00a0 Character Encoding in Programming Languages","text":"<p>Historically, many programming languages utilized fixed-length encodings such as UTF-16 or UTF-32 for processing strings during program execution. This allows strings to be handled as arrays, offering several advantages:</p> <ul> <li>Random Access: Strings encoded in UTF-16 can be accessed randomly with ease. For UTF-8, which is a variable-length encoding, locating the \\(i^{th}\\) character requires traversing the string from the start to the \\(i^{th}\\) position, taking \\(O(n)\\) time.</li> <li>Character Counting: Similar to random access, counting the number of characters in a UTF-16 encoded string is an \\(O(1)\\) operation. However, counting characters in a UTF-8 encoded string requires traversing the entire string.</li> <li>String Operations: Many string operations like splitting, concatenating, inserting, and deleting are easier on UTF-16 encoded strings. These operations generally require additional computation on UTF-8 encoded strings to ensure the validity of the UTF-8 encoding.</li> </ul> <p>The design of character encoding schemes in programming languages is an interesting topic involving various factors:</p> <ul> <li>Java\u2019s <code>String</code> type uses UTF-16 encoding, with each character occupying 2 bytes. This was based on the initial belief that 16 bits were sufficient to represent all possible characters and proven incorrect later. As the Unicode standard expanded beyond 16 bits, characters in Java may now be represented by a pair of 16-bit values, known as \u201csurrogate pairs.\u201d</li> <li>JavaScript and TypeScript use UTF-16 encoding for similar reasons as Java. When JavaScript was first introduced by Netscape in 1995, Unicode was still in its early stages, and 16-bit encoding was sufficient to represent all Unicode characters.</li> <li>C# uses UTF-16 encoding, largely because the .NET platform, designed by Microsoft, and many Microsoft technologies, including the Windows operating system, extensively use UTF-16 encoding.</li> </ul> <p>Due to the underestimation of character counts, these languages had to use \"surrogate pairs\" to represent Unicode characters exceeding 16 bits. This approach has its drawbacks: strings containing surrogate pairs may have characters occupying 2 or 4 bytes, losing the advantage of fixed-length encoding. Additionally, handling surrogate pairs adds complexity and debugging difficulty to programming.</p> <p>Addressing these challenges, some languages have adopted alternative encoding strategies:</p> <ul> <li>Python\u2019s <code>str</code> type uses Unicode encoding with a flexible representation where the storage length of characters depends on the largest Unicode code point in the string. If all characters are ASCII, each character occupies 1 byte, 2 bytes for characters within the Basic Multilingual Plane (BMP), and 4 bytes for characters beyond the BMP.</li> <li>Go\u2019s <code>string</code> type internally uses UTF-8 encoding. Go also provides the <code>rune</code> type for representing individual Unicode code points.</li> <li>Rust\u2019s <code>str</code> and <code>String</code> types use UTF-8 encoding internally. Rust also offers the <code>char</code> type for individual Unicode code points.</li> </ul> <p>It\u2019s important to note that the above discussion pertains to how strings are stored in programming languages, which is different from how strings are stored in files or transmitted over networks. For file storage or network transmission, strings are usually encoded in UTF-8 format for optimal compatibility and space efficiency.</p>"},{"location":"chapter_data_structure/classification_of_data_structure/","title":"3.1 \u00a0 Classification of Data Structures","text":"<p>Common data structures include arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graphs. They can be classified into \"logical structure\" and \"physical structure\".</p>"},{"location":"chapter_data_structure/classification_of_data_structure/#311-logical-structure-linear-and-non-linear","title":"3.1.1 \u00a0 Logical Structure: Linear and Non-Linear","text":"<p>The logical structures reveal the logical relationships between data elements. In arrays and linked lists, data are arranged in a specific sequence, demonstrating the linear relationship between data; while in trees, data are arranged hierarchically from the top down, showing the derived relationship between \"ancestors\" and \"descendants\"; and graphs are composed of nodes and edges, reflecting the intricate network relationship.</p> <p>As shown in the Figure 3-1 , logical structures can be divided into two major categories: \"linear\" and \"non-linear\". Linear structures are more intuitive, indicating data is arranged linearly in logical relationships; non-linear structures, conversely, are arranged non-linearly.</p> <ul> <li>Linear Data Structures: Arrays, Linked Lists, Stacks, Queues, Hash Tables.</li> <li>Non-Linear Data Structures: Trees, Heaps, Graphs, Hash Tables.</li> </ul> <p></p> <p> Figure 3-1 \u00a0 Linear and Non-Linear Data Structures </p> <p>Non-linear data structures can be further divided into tree structures and network structures.</p> <ul> <li>Linear Structures: Arrays, linked lists, queues, stacks, and hash tables, where elements have a one-to-one sequential relationship.</li> <li>Tree Structures: Trees, Heaps, Hash Tables, where elements have a one-to-many relationship.</li> <li>Network Structures: Graphs, where elements have a many-to-many relationships.</li> </ul>"},{"location":"chapter_data_structure/classification_of_data_structure/#312-physical-structure-contiguous-and-dispersed","title":"3.1.2 \u00a0 Physical Structure: Contiguous and Dispersed","text":"<p>During the execution of an algorithm, the data being processed is stored in memory. The Figure 3-2  shows a computer memory stick where each black square is a physical memory space. We can think of memory as a vast Excel spreadsheet, with each cell capable of storing a certain amount of data.</p> <p>The system accesses the data at the target location by means of a memory address. As shown in the Figure 3-2 , the computer assigns a unique identifier to each cell in the table according to specific rules, ensuring that each memory space has a unique memory address. With these addresses, the program can access the data stored in memory.</p> <p></p> <p> Figure 3-2 \u00a0 Memory Stick, Memory Spaces, Memory Addresses </p> <p>Tip</p> <p>It's worth noting that comparing memory to an Excel spreadsheet is a simplified analogy. The actual working mechanism of memory is more complex, involving concepts like address space, memory management, cache mechanisms, virtual memory, and physical memory.</p> <p>Memory is a shared resource for all programs. When a block of memory is occupied by one program, it cannot be simultaneously used by other programs. Therefore, considering memory resources is crucial in designing data structures and algorithms. For instance, the algorithm's peak memory usage should not exceed the remaining free memory of the system; if there is a lack of contiguous memory blocks, then the data structure chosen must be able to be stored in non-contiguous memory blocks.</p> <p>As illustrated in the Figure 3-3 , the physical structure reflects the way data is stored in computer memory and it can be divided into contiguous space storage (arrays) and non-contiguous space storage (linked lists). The two types of physical structures exhibit complementary characteristics in terms of time efficiency and space efficiency.</p> <p></p> <p> Figure 3-3 \u00a0 Contiguous Space Storage and Dispersed Space Storage </p> <p>It is worth noting that all data structures are implemented based on arrays, linked lists, or a combination of both. For example, stacks and queues can be implemented using either arrays or linked lists; while implementations of hash tables may involve both arrays and linked lists. - Array-based implementations: Stacks, Queues, Hash Tables, Trees, Heaps, Graphs, Matrices, Tensors (arrays with dimensions \\(\\geq 3\\)). - Linked-list-based implementations: Stacks, Queues, Hash Tables, Trees, Heaps, Graphs, etc.</p> <p>Data structures implemented based on arrays are also called \u201cStatic Data Structures,\u201d meaning their length cannot be changed after initialization. Conversely, those based on linked lists are called \u201cDynamic Data Structures,\u201d which can still adjust their size during program execution.</p> <p>Tip</p> <p>If you find it challenging to comprehend the physical structure, it is recommended that you read the next chapter, \"Arrays and Linked Lists,\" and revisit this section later.</p>"},{"location":"chapter_data_structure/number_encoding/","title":"3.3 \u00a0 Number Encoding *","text":"<p>Note</p> <p>In this book, chapters marked with an asterisk '*' are optional readings. If you are short on time or find them challenging, you may skip these initially and return to them after completing the essential chapters.</p>"},{"location":"chapter_data_structure/number_encoding/#331-integer-encoding","title":"3.3.1 \u00a0 Integer Encoding","text":"<p>In the table from the previous section, we observed that all integer types can represent one more negative number than positive numbers, such as the <code>byte</code> range of \\([-128, 127]\\). This phenomenon seems counterintuitive, and its underlying reason involves knowledge of sign-magnitude, one's complement, and two's complement encoding.</p> <p>Firstly, it's important to note that numbers are stored in computers using the two's complement form. Before analyzing why this is the case, let's define these three encoding methods:</p> <ul> <li>Sign-magnitude: The highest bit of a binary representation of a number is considered the sign bit, where \\(0\\) represents a positive number and \\(1\\) represents a negative number. The remaining bits represent the value of the number.</li> <li>One's complement: The one's complement of a positive number is the same as its sign-magnitude. For negative numbers, it's obtained by inverting all bits except the sign bit.</li> <li>Two's complement: The two's complement of a positive number is the same as its sign-magnitude. For negative numbers, it's obtained by adding \\(1\\) to their one's complement.</li> </ul> <p>The following diagram illustrates the conversions among sign-magnitude, one's complement, and two's complement:</p> <p></p> <p> Figure 3-4 \u00a0 Conversions between Sign-Magnitude, One's Complement, and Two's Complement </p> <p>Although sign-magnitude is the most intuitive, it has limitations. For one, negative numbers in sign-magnitude cannot be directly used in calculations. For example, in sign-magnitude, calculating \\(1 + (-2)\\) results in \\(-3\\), which is incorrect.</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 + 1000 \\; 0010 \\newline &amp; = 1000 \\; 0011 \\newline &amp; \\rightarrow -3 \\end{aligned} \\] <p>To address this, computers introduced the one's complement. If we convert to one's complement and calculate \\(1 + (-2)\\), then convert the result back to sign-magnitude, we get the correct result of \\(-1\\).</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 \\; \\text{(Sign-magnitude)} + 1000 \\; 0010 \\; \\text{(Sign-magnitude)} \\newline &amp; = 0000 \\; 0001 \\; \\text{(One's complement)} + 1111 \\; 1101 \\; \\text{(One's complement)} \\newline &amp; = 1111 \\; 1110 \\; \\text{(One's complement)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(Sign-magnitude)} \\newline &amp; \\rightarrow -1 \\end{aligned} \\] <p>Additionally, there are two representations of zero in sign-magnitude: \\(+0\\) and \\(-0\\). This means two different binary encodings for zero, which could lead to ambiguity. For example, in conditional checks, not differentiating between positive and negative zero might result in incorrect outcomes. Addressing this ambiguity would require additional checks, potentially reducing computational efficiency.</p> \\[ \\begin{aligned} +0 &amp; \\rightarrow 0000 \\; 0000 \\newline -0 &amp; \\rightarrow 1000 \\; 0000 \\end{aligned} \\] <p>Like sign-magnitude, one's complement also suffers from the positive and negative zero ambiguity. Therefore, computers further introduced the two's complement. Let's observe the conversion process for negative zero in sign-magnitude, one's complement, and two's complement:</p> \\[ \\begin{aligned} -0 \\rightarrow \\; &amp; 1000 \\; 0000 \\; \\text{(Sign-magnitude)} \\newline = \\; &amp; 1111 \\; 1111 \\; \\text{(One's complement)} \\newline = 1 \\; &amp; 0000 \\; 0000 \\; \\text{(Two's complement)} \\newline \\end{aligned} \\] <p>Adding \\(1\\) to the one's complement of negative zero produces a carry, but with <code>byte</code> length being only 8 bits, the carried-over \\(1\\) to the 9<sup>th</sup> bit is discarded. Therefore, the two's complement of negative zero is \\(0000 \\; 0000\\), the same as positive zero, thus resolving the ambiguity.</p> <p>One last puzzle is the \\([-128, 127]\\) range for <code>byte</code>, with an additional negative number, \\(-128\\). We observe that for the interval \\([-127, +127]\\), all integers have corresponding sign-magnitude, one's complement, and two's complement, allowing for mutual conversion between them.</p> <p>However, the two's complement \\(1000 \\; 0000\\) is an exception without a corresponding sign-magnitude. According to the conversion method, its sign-magnitude would be \\(0000 \\; 0000\\), indicating zero. This presents a contradiction because its two's complement should represent itself. Computers designate this special two's complement \\(1000 \\; 0000\\) as representing \\(-128\\). In fact, the calculation of \\((-1) + (-127)\\) in two's complement results in \\(-128\\).</p> \\[ \\begin{aligned} &amp; (-127) + (-1) \\newline &amp; \\rightarrow 1111 \\; 1111 \\; \\text{(Sign-magnitude)} + 1000 \\; 0001 \\; \\text{(Sign-magnitude)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(One's complement)} + 1111 \\; 1110 \\; \\text{(One's complement)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(Two's complement)} + 1111 \\; 1111 \\; \\text{(Two's complement)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(Two's complement)} \\newline &amp; \\rightarrow -128 \\end{aligned} \\] <p>As you might have noticed, all these calculations are additions, hinting at an important fact: computers' internal hardware circuits are primarily designed around addition operations. This is because addition is simpler to implement in hardware compared to other operations like multiplication, division, and subtraction, allowing for easier parallelization and faster computation.</p> <p>It's important to note that this doesn't mean computers can only perform addition. By combining addition with basic logical operations, computers can execute a variety of other mathematical operations. For example, the subtraction \\(a - b\\) can be translated into \\(a + (-b)\\); multiplication and division can be translated into multiple additions or subtractions.</p> <p>We can now summarize the reason for using two's complement in computers: with two's complement representation, computers can use the same circuits and operations to handle both positive and negative number addition, eliminating the need for special hardware circuits for subtraction and avoiding the ambiguity of positive and negative zero. This greatly simplifies hardware design and enhances computational efficiency.</p> <p>The design of two's complement is quite ingenious, and due to space constraints, we'll stop here. Interested readers are encouraged to explore further.</p>"},{"location":"chapter_data_structure/number_encoding/#332-floating-point-number-encoding","title":"3.3.2 \u00a0 Floating-Point Number Encoding","text":"<p>You might have noticed something intriguing: despite having the same length of 4 bytes, why does a <code>float</code> have a much larger range of values compared to an <code>int</code>? This seems counterintuitive, as one would expect the range to shrink for <code>float</code> since it needs to represent fractions.</p> <p>In fact, this is due to the different representation method used by floating-point numbers (<code>float</code>). Let's consider a 32-bit binary number as:</p> \\[ b_{31} b_{30} b_{29} \\ldots b_2 b_1 b_0 \\] <p>According to the IEEE 754 standard, a 32-bit <code>float</code> consists of the following three parts:</p> <ul> <li>Sign bit \\(\\mathrm{S}\\): Occupies 1 bit, corresponding to \\(b_{31}\\).</li> <li>Exponent bit \\(\\mathrm{E}\\): Occupies 8 bits, corresponding to \\(b_{30} b_{29} \\ldots b_{23}\\).</li> <li>Fraction bit \\(\\mathrm{N}\\): Occupies 23 bits, corresponding to \\(b_{22} b_{21} \\ldots b_0\\).</li> </ul> <p>The value of a binary <code>float</code> number is calculated as:</p> \\[ \\text{val} = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2 - 127} \\times \\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\] <p>Converted to a decimal formula, this becomes:</p> \\[ \\text{val} = (-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} - 127} \\times (1 + \\mathrm{N}) \\] <p>The range of each component is:</p> \\[ \\begin{aligned} \\mathrm{S} \\in &amp; \\{ 0, 1\\}, \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = &amp; (1 + \\sum_{i=1}^{23} b_{23-i} \\times 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\] <p></p> <p> Figure 3-5 \u00a0 Example Calculation of a float in IEEE 754 Standard </p> <p>Observing the diagram, given an example data \\(\\mathrm{S} = 0\\), \\(\\mathrm{E} = 124\\), \\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\), we have:</p> \\[ \\text{val} = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\] <p>Now we can answer the initial question: The representation of <code>float</code> includes an exponent bit, leading to a much larger range than <code>int</code>. Based on the above calculation, the maximum positive number representable by <code>float</code> is approximately \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\), and the minimum negative number is obtained by switching the sign bit.</p> <p>However, the trade-off for <code>float</code>'s expanded range is a sacrifice in precision. The integer type <code>int</code> uses all 32 bits to represent the number, with values evenly distributed; but due to the exponent bit, the larger the value of a <code>float</code>, the greater the difference between adjacent numbers.</p> <p>As shown in the Table 3-2 , exponent bits \\(E = 0\\) and \\(E = 255\\) have special meanings, used to represent zero, infinity, \\(\\mathrm{NaN}\\), etc.</p> <p> Table 3-2 \u00a0 Meaning of Exponent Bits </p> Exponent Bit E Fraction Bit \\(\\mathrm{N} = 0\\) Fraction Bit \\(\\mathrm{N} \\ne 0\\) Calculation Formula \\(0\\) \\(\\pm 0\\) Subnormal Numbers \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) Normal Numbers Normal Numbers \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\) <p>It's worth noting that subnormal numbers significantly improve the precision of floating-point numbers. The smallest positive normal number is \\(2^{-126}\\), and the smallest positive subnormal number is \\(2^{-126} \\times 2^{-23}\\).</p> <p>Double-precision <code>double</code> also uses a similar representation method to <code>float</code>, which is not elaborated here for brevity.</p>"},{"location":"chapter_data_structure/summary/","title":"3.5 \u00a0 Summary","text":""},{"location":"chapter_data_structure/summary/#1-key-review","title":"1. \u00a0 Key Review","text":"<ul> <li>Data structures can be categorized from two perspectives: logical structure and physical structure. Logical structure describes the logical relationships between data elements, while physical structure describes how data is stored in computer memory.</li> <li>Common logical structures include linear, tree-like, and network structures. We generally classify data structures into linear (arrays, linked lists, stacks, queues) and non-linear (trees, graphs, heaps) based on their logical structure. The implementation of hash tables may involve both linear and non-linear data structures.</li> <li>When a program runs, data is stored in computer memory. Each memory space has a corresponding memory address, and the program accesses data through these addresses.</li> <li>Physical structures are primarily divided into contiguous space storage (arrays) and dispersed space storage (linked lists). All data structures are implemented using arrays, linked lists, or a combination of both.</li> <li>Basic data types in computers include integers (<code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>), floating-point numbers (<code>float</code>, <code>double</code>), characters (<code>char</code>), and booleans (<code>boolean</code>). Their range depends on the size of the space occupied and the representation method.</li> <li>Original code, complement code, and two's complement code are three methods of encoding numbers in computers, and they can be converted into each other. The highest bit of the original code of an integer is the sign bit, and the remaining bits represent the value of the number.</li> <li>Integers are stored in computers in the form of two's complement. In this representation, the computer can treat the addition of positive and negative numbers uniformly, without the need for special hardware circuits for subtraction, and there is no ambiguity of positive and negative zero.</li> <li>The encoding of floating-point numbers consists of 1 sign bit, 8 exponent bits, and 23 fraction bits. Due to the presence of the exponent bit, the range of floating-point numbers is much greater than that of integers, but at the cost of sacrificing precision.</li> <li>ASCII is the earliest English character set, 1 byte in length, and includes 127 characters. The GBK character set is a commonly used Chinese character set, including more than 20,000 Chinese characters. Unicode strives to provide a complete character set standard, including characters from various languages worldwide, thus solving the problem of garbled characters caused by inconsistent character encoding methods.</li> <li>UTF-8 is the most popular Unicode encoding method, with excellent universality. It is a variable-length encoding method with good scalability and effectively improves the efficiency of space usage. UTF-16 and UTF-32 are fixed-length encoding methods. When encoding Chinese characters, UTF-16 occupies less space than UTF-8. Programming languages like Java and C# use UTF-16 encoding by default.</li> </ul>"},{"location":"chapter_data_structure/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: Why does a hash table contain both linear and non-linear data structures?</p> <p>The underlying structure of a hash table is an array. To resolve hash collisions, we may use \"chaining\": each bucket in the array points to a linked list, which, when exceeding a certain threshold, might be transformed into a tree (usually a red-black tree). From a storage perspective, the foundation of a hash table is an array, where each bucket slot might contain a value, a linked list, or a tree. Therefore, hash tables may contain both linear data structures (arrays, linked lists) and non-linear data structures (trees).</p> <p>Q: Is the length of the <code>char</code> type 1 byte?</p> <p>The length of the <code>char</code> type is determined by the encoding method used by the programming language. For example, Java, JavaScript, TypeScript, and C# all use UTF-16 encoding (to save Unicode code points), so the length of the char type is 2 bytes.</p> <p>Q: Is there ambiguity in calling data structures based on arrays \"static data structures\"? Because operations like push and pop on stacks are \"dynamic\".</p> <p>While stacks indeed allow for dynamic data operations, the data structure itself remains \"static\" (with unchangeable length). Even though data structures based on arrays can dynamically add or remove elements, their capacity is fixed. If the data volume exceeds the pre-allocated size, a new, larger array needs to be created, and the contents of the old array copied into it.</p> <p>Q: When building stacks (queues) without specifying their size, why are they considered \"static data structures\"?</p> <p>In high-level programming languages, we don't need to manually specify the initial capacity of stacks (queues); this task is automatically handled internally by the class. For example, the initial capacity of Java's ArrayList is usually 10. Furthermore, the expansion operation is also implemented automatically. See the subsequent \"List\" chapter for details.</p>"},{"location":"chapter_graph/","title":"Chapter 9. \u00a0 Graph","text":"<p>Abstract</p> <p>In the journey of life, we are like individual nodes, connected by countless invisible edges.</p> <p>Every encountering and parting leaves a unique mark on this vast network graph.</p>"},{"location":"chapter_graph/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>9.1 \u00a0 Graph</li> <li>9.2 \u00a0 Basic Graph Operations</li> <li>9.3 \u00a0 Graph Traversal</li> <li>9.4 \u00a0 Summary</li> </ul>"},{"location":"chapter_graph/graph/","title":"9.1 \u00a0 Graph","text":"<p>A \"graph\" is a type of nonlinear data structure, consisting of \"vertices\" and \"edges\". A graph \\(G\\) can be abstractly represented as a collection of a set of vertices \\(V\\) and a set of edges \\(E\\). The following example shows a graph containing 5 vertices and 7 edges.</p> \\[ \\begin{aligned} V &amp; = \\{ 1, 2, 3, 4, 5 \\} \\newline E &amp; = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G &amp; = \\{ V, E \\} \\newline \\end{aligned} \\] <p>If vertices are viewed as nodes and edges as references (pointers) connecting the nodes, graphs can be seen as a data structure that extends from linked lists. As shown below, compared to linear relationships (linked lists) and divide-and-conquer relationships (trees), network relationships (graphs) are more complex due to their higher degree of freedom.</p> <p></p> <p> Figure 9-1 \u00a0 Relationship between linked lists, trees, and graphs </p>"},{"location":"chapter_graph/graph/#911-common-types-of-graphs","title":"9.1.1 \u00a0 Common types of graphs","text":"<p>Based on whether edges have direction, graphs can be divided into \"undirected graphs\" and \"directed graphs\", as shown below.</p> <ul> <li>In undirected graphs, edges represent a \"bidirectional\" connection between two vertices, for example, the \"friendship\" in WeChat or QQ.</li> <li>In directed graphs, edges have directionality, that is, the edges \\(A \\rightarrow B\\) and \\(A \\leftarrow B\\) are independent of each other, for example, the \"follow\" and \"be followed\" relationship on Weibo or TikTok.</li> </ul> <p></p> <p> Figure 9-2 \u00a0 Directed and undirected graphs </p> <p>Based on whether all vertices are connected, graphs can be divided into \"connected graphs\" and \"disconnected graphs\", as shown below.</p> <ul> <li>For connected graphs, it is possible to reach any other vertex starting from a certain vertex.</li> <li>For disconnected graphs, there is at least one vertex that cannot be reached from a certain starting vertex.</li> </ul> <p></p> <p> Figure 9-3 \u00a0 Connected and disconnected graphs </p> <p>We can also add a \"weight\" variable to edges, resulting in \"weighted graphs\" as shown below. For example, in mobile games like \"Honor of Kings\", the system calculates the \"closeness\" between players based on shared gaming time, and this closeness network can be represented with a weighted graph.</p> <p></p> <p> Figure 9-4 \u00a0 Weighted and unweighted graphs </p> <p>Graph data structures include the following commonly used terms.</p> <ul> <li>\"Adjacency\": When there is an edge connecting two vertices, these two vertices are said to be \"adjacent\". In the above figure, the adjacent vertices of vertex 1 are vertices 2, 3, and 5.</li> <li>\"Path\": The sequence of edges passed from vertex A to vertex B is called a \"path\" from A to B. In the above figure, the edge sequence 1-5-2-4 is a path from vertex 1 to vertex 4.</li> <li>\"Degree\": The number of edges a vertex has. For directed graphs, \"in-degree\" refers to how many edges point to the vertex, and \"out-degree\" refers to how many edges point out from the vertex.</li> </ul>"},{"location":"chapter_graph/graph/#912-representation-of-graphs","title":"9.1.2 \u00a0 Representation of graphs","text":"<p>Common representations of graphs include \"adjacency matrices\" and \"adjacency lists\". The following examples use undirected graphs.</p>"},{"location":"chapter_graph/graph/#1-adjacency-matrix","title":"1. \u00a0 Adjacency matrix","text":"<p>Let the number of vertices in the graph be \\(n\\), the \"adjacency matrix\" uses an \\(n \\times n\\) matrix to represent the graph, where each row (column) represents a vertex, and the matrix elements represent edges, with \\(1\\) or \\(0\\) indicating whether there is an edge between two vertices.</p> <p>As shown below, let the adjacency matrix be \\(M\\), and the list of vertices be \\(V\\), then the matrix element \\(M[i, j] = 1\\) indicates there is an edge between vertex \\(V[i]\\) and vertex \\(V[j]\\), conversely \\(M[i, j] = 0\\) indicates there is no edge between the two vertices.</p> <p></p> <p> Figure 9-5 \u00a0 Representation of a graph with an adjacency matrix </p> <p>Adjacency matrices have the following characteristics.</p> <ul> <li>A vertex cannot be connected to itself, so the elements on the main diagonal of the adjacency matrix are meaningless.</li> <li>For undirected graphs, edges in both directions are equivalent, thus the adjacency matrix is symmetric about the main diagonal.</li> <li>By replacing the elements of the adjacency matrix from \\(1\\) and \\(0\\) to weights, it can represent weighted graphs.</li> </ul> <p>When representing graphs with adjacency matrices, it is possible to directly access matrix elements to obtain edges, thus operations of addition, deletion, lookup, and modification are very efficient, all with a time complexity of \\(O(1)\\). However, the space complexity of the matrix is \\(O(n^2)\\), which consumes more memory.</p>"},{"location":"chapter_graph/graph/#2-adjacency-list","title":"2. \u00a0 Adjacency list","text":"<p>The \"adjacency list\" uses \\(n\\) linked lists to represent the graph, with each linked list node representing a vertex. The \\(i\\)-th linked list corresponds to vertex \\(i\\) and contains all adjacent vertices (vertices connected to that vertex). The Figure 9-6  shows an example of a graph stored using an adjacency list.</p> <p></p> <p> Figure 9-6 \u00a0 Representation of a graph with an adjacency list </p> <p>The adjacency list only stores actual edges, and the total number of edges is often much less than \\(n^2\\), making it more space-efficient. However, finding edges in the adjacency list requires traversing the linked list, so its time efficiency is not as good as that of the adjacency matrix.</p> <p>Observing the above figure, the structure of the adjacency list is very similar to the \"chaining\" in hash tables, hence we can use similar methods to optimize efficiency. For example, when the linked list is long, it can be transformed into an AVL tree or red-black tree, thus optimizing the time efficiency from \\(O(n)\\) to \\(O(\\log n)\\); the linked list can also be transformed into a hash table, thus reducing the time complexity to \\(O(1)\\).</p>"},{"location":"chapter_graph/graph/#913-common-applications-of-graphs","title":"9.1.3 \u00a0 Common applications of graphs","text":"<p>As shown in the Table 9-1 , many real-world systems can be modeled with graphs, and corresponding problems can be reduced to graph computing problems.</p> <p> Table 9-1 \u00a0 Common graphs in real life </p> Vertices Edges Graph Computing Problem Social Networks Users Friendships Potential Friend Recommendations Subway Lines Stations Connectivity Between Stations Shortest Route Recommendations Solar System Celestial Bodies Gravitational Forces Between Celestial Bodies Planetary Orbit Calculations"},{"location":"chapter_graph/graph_operations/","title":"9.2 \u00a0 Basic operations on graphs","text":"<p>The basic operations on graphs can be divided into operations on \"edges\" and operations on \"vertices\". Under the two representation methods of \"adjacency matrix\" and \"adjacency list\", the implementation methods are different.</p>"},{"location":"chapter_graph/graph_operations/#921-implementation-based-on-adjacency-matrix","title":"9.2.1 \u00a0 Implementation based on adjacency matrix","text":"<p>Given an undirected graph with \\(n\\) vertices, the various operations are implemented as shown in the Figure 9-7 .</p> <ul> <li>Adding or removing an edge: Directly modify the specified edge in the adjacency matrix, using \\(O(1)\\) time. Since it is an undirected graph, it is necessary to update the edges in both directions simultaneously.</li> <li>Adding a vertex: Add a row and a column at the end of the adjacency matrix and fill them all with \\(0\\)s, using \\(O(n)\\) time.</li> <li>Removing a vertex: Delete a row and a column in the adjacency matrix. The worst case is when the first row and column are removed, requiring \\((n-1)^2\\) elements to be \"moved up and to the left\", thus using \\(O(n^2)\\) time.</li> <li>Initialization: Pass in \\(n\\) vertices, initialize a vertex list <code>vertices</code> of length \\(n\\), using \\(O(n)\\) time; initialize an \\(n \\times n\\) size adjacency matrix <code>adjMat</code>, using \\(O(n^2)\\) time.</li> </ul> Initialize adjacency matrixAdd an edgeRemove an edgeAdd a vertexRemove a vertex <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-7 \u00a0 Initialization, adding and removing edges, adding and removing vertices in adjacency matrix </p> <p>Below is the implementation code for graphs represented using an adjacency matrix:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_adjacency_matrix.py<pre><code>class GraphAdjMat:\n    \"\"\"\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n\n    def __init__(self, vertices: list[int], edges: list[list[int]]):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        # \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n        self.vertices: list[int] = []\n        # \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n        self.adj_mat: list[list[int]] = []\n        # \u6dfb\u52a0\u9876\u70b9\n        for val in vertices:\n            self.add_vertex(val)\n        # \u6dfb\u52a0\u8fb9\n        # \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for e in edges:\n            self.add_edge(e[0], e[1])\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\n        return len(self.vertices)\n\n    def add_vertex(self, val: int):\n        \"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\n        n = self.size()\n        # \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        self.vertices.append(val)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        new_row = [0] * n\n        self.adj_mat.append(new_row)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for row in self.adj_mat:\n            row.append(0)\n\n    def remove_vertex(self, index: int):\n        \"\"\"\u5220\u9664\u9876\u70b9\"\"\"\n        if index &gt;= self.size():\n            raise IndexError()\n        # \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        self.vertices.pop(index)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        self.adj_mat.pop(index)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for row in self.adj_mat:\n            row.pop(index)\n\n    def add_edge(self, i: int, j: int):\n        \"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n        # \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        # \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        # \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        self.adj_mat[i][j] = 1\n        self.adj_mat[j][i] = 1\n\n    def remove_edge(self, i: int, j: int):\n        \"\"\"\u5220\u9664\u8fb9\"\"\"\n        # \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        # \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        self.adj_mat[i][j] = 0\n        self.adj_mat[j][i] = 0\n\n    def print(self):\n        \"\"\"\u6253\u5370\u90bb\u63a5\u77e9\u9635\"\"\"\n        print(\"\u9876\u70b9\u5217\u8868 =\", self.vertices)\n        print(\"\u90bb\u63a5\u77e9\u9635 =\")\n        print_matrix(self.adj_mat)\n</code></pre> graph_adjacency_matrix.cpp<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    vector&lt;int&gt; vertices;       // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    vector&lt;vector&lt;int&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    GraphAdjMat(const vector&lt;int&gt; &amp;vertices, const vector&lt;vector&lt;int&gt;&gt; &amp;edges) {\n        // \u6dfb\u52a0\u9876\u70b9\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (const vector&lt;int&gt; &amp;edge : edges) {\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int size() const {\n        return vertices.size();\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    void addVertex(int val) {\n        int n = size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.push_back(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        adjMat.emplace_back(vector&lt;int&gt;(n, 0));\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.push_back(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    void removeVertex(int index) {\n        if (index &gt;= size()) {\n            throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.erase(vertices.begin() + index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.erase(adjMat.begin() + index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.erase(row.begin() + index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    void addEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    void removeEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n        }\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    void print() {\n        cout &lt;&lt; \"\u9876\u70b9\u5217\u8868 = \";\n        printVector(vertices);\n        cout &lt;&lt; \"\u90bb\u63a5\u77e9\u9635 =\" &lt;&lt; endl;\n        printVectorMatrix(adjMat);\n    }\n};\n</code></pre> graph_adjacency_matrix.java<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    List&lt;Integer&gt; vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    List&lt;List&lt;Integer&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = new ArrayList&lt;&gt;();\n        this.adjMat = new ArrayList&lt;&gt;();\n        // \u6dfb\u52a0\u9876\u70b9\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (int[] e : edges) {\n            addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    public int size() {\n        return vertices.size();\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void addVertex(int val) {\n        int n = size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.add(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        List&lt;Integer&gt; newRow = new ArrayList&lt;&gt;(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.add(0);\n        }\n        adjMat.add(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.add(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void removeVertex(int index) {\n        if (index &gt;= size())\n            throw new IndexOutOfBoundsException();\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.remove(index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void addEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat.get(i).set(j, 1);\n        adjMat.get(j).set(i, 1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void removeEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        adjMat.get(i).set(j, 0);\n        adjMat.get(j).set(i, 0);\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    public void print() {\n        System.out.print(\"\u9876\u70b9\u5217\u8868 = \");\n        System.out.println(vertices);\n        System.out.println(\"\u90bb\u63a5\u77e9\u9635 =\");\n        PrintUtil.printMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.cs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    List&lt;int&gt; vertices;     // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    List&lt;List&lt;int&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u6dfb\u52a0\u9876\u70b9\n        foreach (int val in vertices) {\n            AddVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        foreach (int[] e in edges) {\n            AddEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int Size() {\n        return vertices.Count;\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void AddVertex(int val) {\n        int n = Size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.Add(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        List&lt;int&gt; newRow = new(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.Add(0);\n        }\n        adjMat.Add(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.Add(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void RemoveVertex(int index) {\n        if (index &gt;= Size())\n            throw new IndexOutOfRangeException();\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.RemoveAt(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.RemoveAt(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.RemoveAt(index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void AddEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void RemoveEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    public void Print() {\n        Console.Write(\"\u9876\u70b9\u5217\u8868 = \");\n        PrintUtil.PrintList(vertices);\n        Console.WriteLine(\"\u90bb\u63a5\u77e9\u9635 =\");\n        PrintUtil.PrintMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.go<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjMat struct {\n    // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    vertices []int\n    // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    adjMat [][]int\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n    // \u6dfb\u52a0\u9876\u70b9\n    n := len(vertices)\n    adjMat := make([][]int, n)\n    for i := range adjMat {\n        adjMat[i] = make([]int, n)\n    }\n    // \u521d\u59cb\u5316\u56fe\n    g := &amp;graphAdjMat{\n        vertices: vertices,\n        adjMat:   adjMat,\n    }\n    // \u6dfb\u52a0\u8fb9\n    // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    for i := range edges {\n        g.addEdge(edges[i][0], edges[i][1])\n    }\n    return g\n}\n\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjMat) size() int {\n    return len(g.vertices)\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjMat) addVertex(val int) {\n    n := g.size()\n    // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n    g.vertices = append(g.vertices, val)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n    newRow := make([]int, n)\n    g.adjMat = append(g.adjMat, newRow)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i], 0)\n    }\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjMat) removeVertex(index int) {\n    if index &gt;= g.size() {\n        return\n    }\n    // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n    g.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n    g.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n    }\n}\n\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) addEdge(i, j int) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n    g.adjMat[i][j] = 1\n    g.adjMat[j][i] = 1\n}\n\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    g.adjMat[i][j] = 0\n    g.adjMat[j][i] = 0\n}\n\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc (g *graphAdjMat) print() {\n    fmt.Printf(\"\\t\u9876\u70b9\u5217\u8868 = %v\\n\", g.vertices)\n    fmt.Printf(\"\\t\u90bb\u63a5\u77e9\u9635 = \\n\")\n    for i := range g.adjMat {\n        fmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n    }\n}\n</code></pre> graph_adjacency_matrix.swift<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    private var vertices: [Int] // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    private var adjMat: [[Int]] // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init(vertices: [Int], edges: [[Int]]) {\n        self.vertices = []\n        adjMat = []\n        // \u6dfb\u52a0\u9876\u70b9\n        for val in vertices {\n            addVertex(val: val)\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for e in edges {\n            addEdge(i: e[0], j: e[1])\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    func size() -&gt; Int {\n        vertices.count\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    func addVertex(val: Int) {\n        let n = size()\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.append(val)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        let newRow = Array(repeating: 0, count: n)\n        adjMat.append(newRow)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for i in adjMat.indices {\n            adjMat[i].append(0)\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    func removeVertex(index: Int) {\n        if index &gt;= size() {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.remove(at: index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.remove(at: index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for i in adjMat.indices {\n            adjMat[i].remove(at: index)\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    func addEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    func removeEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    func print() {\n        Swift.print(\"\u9876\u70b9\u5217\u8868 = \", terminator: \"\")\n        Swift.print(vertices)\n        Swift.print(\"\u90bb\u63a5\u77e9\u9635 =\")\n        PrintUtil.printMatrix(matrix: adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.js<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    constructor(vertices, edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u6dfb\u52a0\u9876\u70b9\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size() {\n        return this.vertices.length;\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(val) {\n        const n = this.size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        this.vertices.push(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        const newRow = [];\n        for (let j = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(index) {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        this.vertices.splice(index, 1);\n\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        this.adjMat.splice(index, 1);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    addEdge(i, j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    removeEdge(i, j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    print() {\n        console.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\n        console.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.ts<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    vertices: number[]; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    adjMat: number[][]; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    constructor(vertices: number[], edges: number[][]) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u6dfb\u52a0\u9876\u70b9\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size(): number {\n        return this.vertices.length;\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(val: number): void {\n        const n: number = this.size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        this.vertices.push(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        const newRow: number[] = [];\n        for (let j: number = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(index: number): void {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        this.vertices.splice(index, 1);\n\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        this.adjMat.splice(index, 1);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    addEdge(i: number, j: number): void {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    removeEdge(i: number, j: number): void {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    print(): void {\n        console.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\n        console.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.dart<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n  List&lt;int&gt; vertices = []; // \u9876\u70b9\u5143\u7d20\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n  List&lt;List&lt;int&gt;&gt; adjMat = []; //\u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  GraphAdjMat(List&lt;int&gt; vertices, List&lt;List&lt;int&gt;&gt; edges) {\n    this.vertices = [];\n    this.adjMat = [];\n    // \u6dfb\u52a0\u9876\u70b9\n    for (int val in vertices) {\n      addVertex(val);\n    }\n    // \u6dfb\u52a0\u8fb9\n    // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    for (List&lt;int&gt; e in edges) {\n      addEdge(e[0], e[1]);\n    }\n  }\n\n  /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n  int size() {\n    return vertices.length;\n  }\n\n  /* \u6dfb\u52a0\u9876\u70b9 */\n  void addVertex(int val) {\n    int n = size();\n    // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n    vertices.add(val);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n    List&lt;int&gt; newRow = List.filled(n, 0, growable: true);\n    adjMat.add(newRow);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n    for (List&lt;int&gt; row in adjMat) {\n      row.add(0);\n    }\n  }\n\n  /* \u5220\u9664\u9876\u70b9 */\n  void removeVertex(int index) {\n    if (index &gt;= size()) {\n      throw IndexError;\n    }\n    // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n    vertices.removeAt(index);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n    adjMat.removeAt(index);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n    for (List&lt;int&gt; row in adjMat) {\n      row.removeAt(index);\n    }\n  }\n\n  /* \u6dfb\u52a0\u8fb9 */\n  // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n  void addEdge(int i, int j) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n    adjMat[i][j] = 1;\n    adjMat[j][i] = 1;\n  }\n\n  /* \u5220\u9664\u8fb9 */\n  // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n  void removeEdge(int i, int j) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    adjMat[i][j] = 0;\n    adjMat[j][i] = 0;\n  }\n\n  /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n  void printAdjMat() {\n    print(\"\u9876\u70b9\u5217\u8868 = $vertices\");\n    print(\"\u90bb\u63a5\u77e9\u9635 = \");\n    printMatrix(adjMat);\n  }\n}\n</code></pre> graph_adjacency_matrix.rs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjMat {\n    // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    pub vertices: Vec&lt;i32&gt;,\n    // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    pub adj_mat: Vec&lt;Vec&lt;i32&gt;&gt;,\n}\n\nimpl GraphAdjMat {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(vertices: Vec&lt;i32&gt;, edges: Vec&lt;[usize; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjMat {\n            vertices: vec![],\n            adj_mat: vec![],\n        };\n        // \u6dfb\u52a0\u9876\u70b9\n        for val in vertices {\n            graph.add_vertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for edge in edges {\n            graph.add_edge(edge[0], edge[1])\n        }\n\n        graph\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    pub fn size(&amp;self) -&gt; usize {\n        self.vertices.len()\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    pub fn add_vertex(&amp;mut self, val: i32) {\n        let n = self.size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        self.vertices.push(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        self.adj_mat.push(vec![0; n]);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for row in &amp;mut self.adj_mat {\n            row.push(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    pub fn remove_vertex(&amp;mut self, index: usize) {\n        if index &gt;= self.size() {\n            panic!(\"index error\")\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        self.vertices.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        self.adj_mat.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for row in &amp;mut self.adj_mat {\n            row.remove(index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    pub fn add_edge(&amp;mut self, i: usize, j: usize) {\n        // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        self.adj_mat[i][j] = 1;\n        self.adj_mat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    pub fn remove_edge(&amp;mut self, i: usize, j: usize) {\n        // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        self.adj_mat[i][j] = 0;\n        self.adj_mat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    pub fn print(&amp;self) {\n        println!(\"\u9876\u70b9\u5217\u8868 = {:?}\", self.vertices);\n        println!(\"\u90bb\u63a5\u77e9\u9635 =\");\n        println!(\"[\");\n        for row in &amp;self.adj_mat {\n            println!(\"  {:?},\", row);\n        }\n        println!(\"]\")\n    }\n}\n</code></pre> graph_adjacency_matrix.c<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7ed3\u6784\u4f53 */\ntypedef struct {\n    int vertices[MAX_SIZE];\n    int adjMat[MAX_SIZE][MAX_SIZE];\n    int size;\n} GraphAdjMat;\n\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjMat *newGraphAdjMat() {\n    GraphAdjMat *graph = (GraphAdjMat *)malloc(sizeof(GraphAdjMat));\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        for (int j = 0; j &lt; MAX_SIZE; j++) {\n            graph-&gt;adjMat[i][j] = 0;\n        }\n    }\n    return graph;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjMat(GraphAdjMat *graph) {\n    free(graph);\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjMat *graph, int val) {\n    if (graph-&gt;size == MAX_SIZE) {\n        fprintf(stderr, \"\u56fe\u7684\u9876\u70b9\u6570\u91cf\u5df2\u8fbe\u6700\u5927\u503c\\n\");\n        return;\n    }\n    // \u6dfb\u52a0\u7b2c n \u4e2a\u9876\u70b9\uff0c\u5e76\u5c06\u7b2c n \u884c\u548c\u5217\u7f6e\u96f6\n    int n = graph-&gt;size;\n    graph-&gt;vertices[n] = val;\n    for (int i = 0; i &lt;= n; i++) {\n        graph-&gt;adjMat[n][i] = graph-&gt;adjMat[i][n] = 0;\n    }\n    graph-&gt;size++;\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjMat *graph, int index) {\n    if (index &lt; 0 || index &gt;= graph-&gt;size) {\n        fprintf(stderr, \"\u9876\u70b9\u7d22\u5f15\u8d8a\u754c\\n\");\n        return;\n    }\n    // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        graph-&gt;vertices[i] = graph-&gt;vertices[i + 1];\n    }\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        for (int j = 0; j &lt; graph-&gt;size; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i + 1][j];\n        }\n    }\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        for (int j = index; j &lt; graph-&gt;size - 1; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i][j + 1];\n        }\n    }\n    graph-&gt;size--;\n}\n\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 1;\n    graph-&gt;adjMat[j][i] = 1;\n}\n\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 0;\n    graph-&gt;adjMat[j][i] = 0;\n}\n\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid printGraphAdjMat(GraphAdjMat *graph) {\n    printf(\"\u9876\u70b9\u5217\u8868 = \");\n    printArray(graph-&gt;vertices, graph-&gt;size);\n    printf(\"\u90bb\u63a5\u77e9\u9635 =\\n\");\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        printArray(graph-&gt;adjMat[i], graph-&gt;size);\n    }\n}\n</code></pre> graph_adjacency_matrix.kt<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat(vertices: IntArray, edges: Array&lt;IntArray&gt;) {\n    val vertices: MutableList&lt;Int&gt; = ArrayList() // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    val adjMat: MutableList&lt;MutableList&lt;Int&gt;&gt; = ArrayList() // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    init {\n        // \u6dfb\u52a0\u9876\u70b9\n        for (vertex in vertices) {\n            addVertex(vertex)\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (edge in edges) {\n            addEdge(edge[0], edge[1])\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    fun size(): Int {\n        return vertices.size\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    fun addVertex(value: Int) {\n        val n = size()\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.add(value)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        val newRow: MutableList&lt;Int&gt; = mutableListOf()\n        for (j in 0..&lt;n) {\n            newRow.add(0)\n        }\n        adjMat.add(newRow)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (row in adjMat) {\n            row.add(0)\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    fun removeVertex(index: Int) {\n        if (index &gt;= size()) throw IndexOutOfBoundsException()\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.removeAt(index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.removeAt(index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (row in adjMat) {\n            row.removeAt(index)\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    fun addEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) throw java.lang.IndexOutOfBoundsException()\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    fun removeEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) throw java.lang.IndexOutOfBoundsException()\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    fun print() {\n        print(\"\u9876\u70b9\u5217\u8868 = \")\n        println(vertices);\n        println(\"\u90bb\u63a5\u77e9\u9635 =\");\n        printMatrix(adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.rb<pre><code>[class]{GraphAdjMat}-[func]{}\n</code></pre> graph_adjacency_matrix.zig<pre><code>[class]{GraphAdjMat}-[func]{}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_graph/graph_operations/#922-implementation-based-on-adjacency-list","title":"9.2.2 \u00a0 Implementation based on adjacency list","text":"<p>Given an undirected graph with a total of \\(n\\) vertices and \\(m\\) edges, the various operations can be implemented as shown in the Figure 9-8 .</p> <ul> <li>Adding an edge: Simply add the edge at the end of the corresponding vertex's linked list, using \\(O(1)\\) time. Because it is an undirected graph, it is necessary to add edges in both directions simultaneously.</li> <li>Removing an edge: Find and remove the specified edge in the corresponding vertex's linked list, using \\(O(m)\\) time. In an undirected graph, it is necessary to remove edges in both directions simultaneously.</li> <li>Adding a vertex: Add a linked list in the adjacency list and make the new vertex the head node of the list, using \\(O(1)\\) time.</li> <li>Removing a vertex: It is necessary to traverse the entire adjacency list, removing all edges that include the specified vertex, using \\(O(n + m)\\) time.</li> <li>Initialization: Create \\(n\\) vertices and \\(2m\\) edges in the adjacency list, using \\(O(n + m)\\) time.</li> </ul> Initialize adjacency listAdd an edgeRemove an edgeAdd a vertexRemove a vertex <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-8 \u00a0 Initialization, adding and removing edges, adding and removing vertices in adjacency list </p> <p>Below is the adjacency list code implementation. Compared to the above diagram, the actual code has the following differences.</p> <ul> <li>For convenience in adding and removing vertices, and to simplify the code, we use lists (dynamic arrays) instead of linked lists.</li> <li>Use a hash table to store the adjacency list, <code>key</code> being the vertex instance, <code>value</code> being the list (linked list) of adjacent vertices of that vertex.</li> </ul> <p>Additionally, we use the <code>Vertex</code> class to represent vertices in the adjacency list. The reason for this is: if, like with the adjacency matrix, list indexes were used to distinguish different vertices, then suppose you want to delete the vertex at index \\(i\\), you would need to traverse the entire adjacency list and decrement all indexes greater than \\(i\\) by \\(1\\), which is very inefficient. However, if each vertex is a unique <code>Vertex</code> instance, then deleting a vertex does not require any changes to other vertices.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_adjacency_list.py<pre><code>class GraphAdjList:\n    \"\"\"\u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n\n    def __init__(self, edges: list[list[Vertex]]):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        # \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        self.adj_list = dict[Vertex, list[Vertex]]()\n        # \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for edge in edges:\n            self.add_vertex(edge[0])\n            self.add_vertex(edge[1])\n            self.add_edge(edge[0], edge[1])\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\n        return len(self.adj_list)\n\n    def add_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # \u6dfb\u52a0\u8fb9 vet1 - vet2\n        self.adj_list[vet1].append(vet2)\n        self.adj_list[vet2].append(vet1)\n\n    def remove_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"\u5220\u9664\u8fb9\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # \u5220\u9664\u8fb9 vet1 - vet2\n        self.adj_list[vet1].remove(vet2)\n        self.adj_list[vet2].remove(vet1)\n\n    def add_vertex(self, vet: Vertex):\n        \"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\n        if vet in self.adj_list:\n            return\n        # \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        self.adj_list[vet] = []\n\n    def remove_vertex(self, vet: Vertex):\n        \"\"\"\u5220\u9664\u9876\u70b9\"\"\"\n        if vet not in self.adj_list:\n            raise ValueError()\n        # \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        self.adj_list.pop(vet)\n        # \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for vertex in self.adj_list:\n            if vet in self.adj_list[vertex]:\n                self.adj_list[vertex].remove(vet)\n\n    def print(self):\n        \"\"\"\u6253\u5370\u90bb\u63a5\u8868\"\"\"\n        print(\"\u90bb\u63a5\u8868 =\")\n        for vertex in self.adj_list:\n            tmp = [v.val for v in self.adj_list[vertex]]\n            print(f\"{vertex.val}: {tmp},\")\n</code></pre> graph_adjacency_list.cpp<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n  public:\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    unordered_map&lt;Vertex *, vector&lt;Vertex *&gt;&gt; adjList;\n\n    /* \u5728 vector \u4e2d\u5220\u9664\u6307\u5b9a\u8282\u70b9 */\n    void remove(vector&lt;Vertex *&gt; &amp;vec, Vertex *vet) {\n        for (int i = 0; i &lt; vec.size(); i++) {\n            if (vec[i] == vet) {\n                vec.erase(vec.begin() + i);\n                break;\n            }\n        }\n    }\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    GraphAdjList(const vector&lt;vector&lt;Vertex *&gt;&gt; &amp;edges) {\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (const vector&lt;Vertex *&gt; &amp;edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int size() {\n        return adjList.size();\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    void addEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1].push_back(vet2);\n        adjList[vet2].push_back(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    void removeEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        remove(adjList[vet1], vet2);\n        remove(adjList[vet2], vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    void addVertex(Vertex *vet) {\n        if (adjList.count(vet))\n            return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList[vet] = vector&lt;Vertex *&gt;();\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    void removeVertex(Vertex *vet) {\n        if (!adjList.count(vet))\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.erase(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (auto &amp;adj : adjList) {\n            remove(adj.second, vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    void print() {\n        cout &lt;&lt; \"\u90bb\u63a5\u8868 =\" &lt;&lt; endl;\n        for (auto &amp;adj : adjList) {\n            const auto &amp;key = adj.first;\n            const auto &amp;vec = adj.second;\n            cout &lt;&lt; key-&gt;val &lt;&lt; \": \";\n            printVector(vetsToVals(vec));\n        }\n    }\n};\n</code></pre> graph_adjacency_list.java<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public GraphAdjList(Vertex[][] edges) {\n        this.adjList = new HashMap&lt;&gt;();\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (Vertex[] edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    public int size() {\n        return adjList.size();\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    public void addEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList.get(vet1).add(vet2);\n        adjList.get(vet2).add(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    public void removeEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList.get(vet1).remove(vet2);\n        adjList.get(vet2).remove(vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void addVertex(Vertex vet) {\n        if (adjList.containsKey(vet))\n            return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList.put(vet, new ArrayList&lt;&gt;());\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void removeVertex(Vertex vet) {\n        if (!adjList.containsKey(vet))\n            throw new IllegalArgumentException();\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.remove(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (List&lt;Vertex&gt; list : adjList.values()) {\n            list.remove(vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    public void print() {\n        System.out.println(\"\u90bb\u63a5\u8868 =\");\n        for (Map.Entry&lt;Vertex, List&lt;Vertex&gt;&gt; pair : adjList.entrySet()) {\n            List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n            for (Vertex vertex : pair.getValue())\n                tmp.add(vertex.val);\n            System.out.println(pair.getKey().val + \": \" + tmp + \",\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.cs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    public Dictionary&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* \u6784\u9020\u51fd\u6570 */\n    public GraphAdjList(Vertex[][] edges) {\n        adjList = [];\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        foreach (Vertex[] edge in edges) {\n            AddVertex(edge[0]);\n            AddVertex(edge[1]);\n            AddEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int Size() {\n        return adjList.Count;\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    public void AddEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1].Add(vet2);\n        adjList[vet2].Add(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    public void RemoveEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList[vet1].Remove(vet2);\n        adjList[vet2].Remove(vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void AddVertex(Vertex vet) {\n        if (adjList.ContainsKey(vet))\n            return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList.Add(vet, []);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void RemoveVertex(Vertex vet) {\n        if (!adjList.ContainsKey(vet))\n            throw new InvalidOperationException();\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.Remove(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        foreach (List&lt;Vertex&gt; list in adjList.Values) {\n            list.Remove(vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    public void Print() {\n        Console.WriteLine(\"\u90bb\u63a5\u8868 =\");\n        foreach (KeyValuePair&lt;Vertex, List&lt;Vertex&gt;&gt; pair in adjList) {\n            List&lt;int&gt; tmp = [];\n            foreach (Vertex vertex in pair.Value)\n                tmp.Add(vertex.val);\n            Console.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.go<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjList struct {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    adjList map[Vertex][]Vertex\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\n    g := &amp;graphAdjList{\n        adjList: make(map[Vertex][]Vertex),\n    }\n    // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n    for _, edge := range edges {\n        g.addVertex(edge[0])\n        g.addVertex(edge[1])\n        g.addEdge(edge[0], edge[1])\n    }\n    return g\n}\n\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjList) size() int {\n    return len(g.adjList)\n}\n\n/* \u6dfb\u52a0\u8fb9 */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // \u6dfb\u52a0\u8fb9 vet1 - vet2, \u6dfb\u52a0\u533f\u540d struct{},\n    g.adjList[vet1] = append(g.adjList[vet1], vet2)\n    g.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n\n/* \u5220\u9664\u8fb9 */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // \u5220\u9664\u8fb9 vet1 - vet2\n    g.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2)\n    g.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1)\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if ok {\n        return\n    }\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n    g.adjList[vet] = make([]Vertex, 0)\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if !ok {\n        panic(\"error\")\n    }\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n    delete(g.adjList, vet)\n    // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n    for v, list := range g.adjList {\n        g.adjList[v] = DeleteSliceElms(list, vet)\n    }\n}\n\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nfunc (g *graphAdjList) print() {\n    var builder strings.Builder\n    fmt.Printf(\"\u90bb\u63a5\u8868 = \\n\")\n    for k, v := range g.adjList {\n        builder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\n        for _, vet := range v {\n            builder.WriteString(strconv.Itoa(vet.Val) + \" \")\n        }\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> graph_adjacency_list.swift<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    public private(set) var adjList: [Vertex: [Vertex]]\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public init(edges: [[Vertex]]) {\n        adjList = [:]\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for edge in edges {\n            addVertex(vet: edge[0])\n            addVertex(vet: edge[1])\n            addEdge(vet1: edge[0], vet2: edge[1])\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    public func size() -&gt; Int {\n        adjList.count\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    public func addEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"\u53c2\u6570\u9519\u8bef\")\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1]?.append(vet2)\n        adjList[vet2]?.append(vet1)\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    public func removeEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"\u53c2\u6570\u9519\u8bef\")\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList[vet1]?.removeAll { $0 == vet2 }\n        adjList[vet2]?.removeAll { $0 == vet1 }\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public func addVertex(vet: Vertex) {\n        if adjList[vet] != nil {\n            return\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList[vet] = []\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public func removeVertex(vet: Vertex) {\n        if adjList[vet] == nil {\n            fatalError(\"\u53c2\u6570\u9519\u8bef\")\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.removeValue(forKey: vet)\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for key in adjList.keys {\n            adjList[key]?.removeAll { $0 == vet }\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    public func print() {\n        Swift.print(\"\u90bb\u63a5\u8868 =\")\n        for (vertex, list) in adjList {\n            let list = list.map { $0.val }\n            Swift.print(\"\\(vertex.val): \\(list),\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.js<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    adjList;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(edges) {\n        this.adjList = new Map();\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size() {\n        return this.adjList.size;\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    addEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    removeEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(vet) {\n        if (this.adjList.has(vet)) return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        this.adjList.set(vet, []);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(vet) {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        this.adjList.delete(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (const set of this.adjList.values()) {\n            const index = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    print() {\n        console.log('\u90bb\u63a5\u8868 =');\n        for (const [key, value] of this.adjList) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.ts<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    adjList: Map&lt;Vertex, Vertex[]&gt;;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(edges: Vertex[][]) {\n        this.adjList = new Map();\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size(): number {\n        return this.adjList.size;\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    addEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    removeEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(vet: Vertex): void {\n        if (this.adjList.has(vet)) return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        this.adjList.set(vet, []);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(vet: Vertex): void {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        this.adjList.delete(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (const set of this.adjList.values()) {\n            const index: number = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    print(): void {\n        console.log('\u90bb\u63a5\u8868 =');\n        for (const [key, value] of this.adjList.entries()) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.dart<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n  // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n  Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList = {};\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  GraphAdjList(List&lt;List&lt;Vertex&gt;&gt; edges) {\n    for (List&lt;Vertex&gt; edge in edges) {\n      addVertex(edge[0]);\n      addVertex(edge[1]);\n      addEdge(edge[0], edge[1]);\n    }\n  }\n\n  /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n  int size() {\n    return adjList.length;\n  }\n\n  /* \u6dfb\u52a0\u8fb9 */\n  void addEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // \u6dfb\u52a0\u8fb9 vet1 - vet2\n    adjList[vet1]!.add(vet2);\n    adjList[vet2]!.add(vet1);\n  }\n\n  /* \u5220\u9664\u8fb9 */\n  void removeEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // \u5220\u9664\u8fb9 vet1 - vet2\n    adjList[vet1]!.remove(vet2);\n    adjList[vet2]!.remove(vet1);\n  }\n\n  /* \u6dfb\u52a0\u9876\u70b9 */\n  void addVertex(Vertex vet) {\n    if (adjList.containsKey(vet)) return;\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n    adjList[vet] = [];\n  }\n\n  /* \u5220\u9664\u9876\u70b9 */\n  void removeVertex(Vertex vet) {\n    if (!adjList.containsKey(vet)) {\n      throw ArgumentError;\n    }\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n    adjList.remove(vet);\n    // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n    adjList.forEach((key, value) {\n      value.remove(vet);\n    });\n  }\n\n  /* \u6253\u5370\u90bb\u63a5\u8868 */\n  void printAdjList() {\n    print(\"\u90bb\u63a5\u8868 =\");\n    adjList.forEach((key, value) {\n      List&lt;int&gt; tmp = [];\n      for (Vertex vertex in value) {\n        tmp.add(vertex.val);\n      }\n      print(\"${key.val}: $tmp,\");\n    });\n  }\n}\n</code></pre> graph_adjacency_list.rs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    pub adj_list: HashMap&lt;Vertex, Vec&lt;Vertex&gt;&gt;,\n}\n\nimpl GraphAdjList {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(edges: Vec&lt;[Vertex; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjList {\n            adj_list: HashMap::new(),\n        };\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for edge in edges {\n            graph.add_vertex(edge[0]);\n            graph.add_vertex(edge[1]);\n            graph.add_edge(edge[0], edge[1]);\n        }\n\n        graph\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    #[allow(unused)]\n    pub fn size(&amp;self) -&gt; usize {\n        self.adj_list.len()\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    pub fn add_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n        {\n            panic!(\"value error\");\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        self.adj_list.get_mut(&amp;vet1).unwrap().push(vet2);\n        self.adj_list.get_mut(&amp;vet2).unwrap().push(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    #[allow(unused)]\n    pub fn remove_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n        {\n            panic!(\"value error\");\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        self.adj_list\n            .get_mut(&amp;vet1)\n            .unwrap()\n            .retain(|&amp;vet| vet != vet2);\n        self.adj_list\n            .get_mut(&amp;vet2)\n            .unwrap()\n            .retain(|&amp;vet| vet != vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    pub fn add_vertex(&amp;mut self, vet: Vertex) {\n        if self.adj_list.contains_key(&amp;vet) {\n            return;\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        self.adj_list.insert(vet, vec![]);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    #[allow(unused)]\n    pub fn remove_vertex(&amp;mut self, vet: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet) {\n            panic!(\"value error\");\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        self.adj_list.remove(&amp;vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for list in self.adj_list.values_mut() {\n            list.retain(|&amp;v| v != vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    pub fn print(&amp;self) {\n        println!(\"\u90bb\u63a5\u8868 =\");\n        for (vertex, list) in &amp;self.adj_list {\n            let list = list.iter().map(|vertex| vertex.val).collect::&lt;Vec&lt;i32&gt;&gt;();\n            println!(\"{}: {:?},\", vertex.val, list);\n        }\n    }\n}\n</code></pre> graph_adjacency_list.c<pre><code>/* \u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct AdjListNode {\n    Vertex *vertex;           // \u9876\u70b9\n    struct AdjListNode *next; // \u540e\u7ee7\u8282\u70b9\n} AdjListNode;\n\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* \u6dfb\u52a0\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid addEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *node = (AdjListNode *)malloc(sizeof(AdjListNode));\n    node-&gt;vertex = vet;\n    // \u5934\u63d2\u6cd5\n    node-&gt;next = head-&gt;next;\n    head-&gt;next = node;\n}\n\n/* \u5220\u9664\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid removeEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *pre = head;\n    AdjListNode *cur = head-&gt;next;\n    // \u5728\u94fe\u8868\u4e2d\u641c\u7d22 vet \u5bf9\u5e94\u8282\u70b9\n    while (cur != NULL &amp;&amp; cur-&gt;vertex != vet) {\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n    if (cur == NULL)\n        return;\n    // \u5c06 vet \u5bf9\u5e94\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\n    pre-&gt;next = cur-&gt;next;\n    // \u91ca\u653e\u5185\u5b58\n    free(cur);\n}\n\n/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntypedef struct {\n    AdjListNode *heads[MAX_SIZE]; // \u8282\u70b9\u6570\u7ec4\n    int size;                     // \u8282\u70b9\u6570\u91cf\n} GraphAdjList;\n\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjList *newGraphAdjList() {\n    GraphAdjList *graph = (GraphAdjList *)malloc(sizeof(GraphAdjList));\n    if (!graph) {\n        return NULL;\n    }\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        graph-&gt;heads[i] = NULL;\n    }\n    return graph;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjList(GraphAdjList *graph) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        AdjListNode *cur = graph-&gt;heads[i];\n        while (cur != NULL) {\n            AdjListNode *next = cur-&gt;next;\n            if (cur != graph-&gt;heads[i]) {\n                free(cur);\n            }\n            cur = next;\n        }\n        free(graph-&gt;heads[i]-&gt;vertex);\n        free(graph-&gt;heads[i]);\n    }\n    free(graph);\n}\n\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL &amp;&amp; head1 != head2);\n    // \u6dfb\u52a0\u8fb9 vet1 - vet2\n    addEdgeHelper(head1, vet2);\n    addEdgeHelper(head2, vet1);\n}\n\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL);\n    // \u5220\u9664\u8fb9 vet1 - vet2\n    removeEdgeHelper(head1, head2-&gt;vertex);\n    removeEdgeHelper(head2, head1-&gt;vertex);\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjList *graph, Vertex *vet) {\n    assert(graph != NULL &amp;&amp; graph-&gt;size &lt; MAX_SIZE);\n    AdjListNode *head = (AdjListNode *)malloc(sizeof(AdjListNode));\n    head-&gt;vertex = vet;\n    head-&gt;next = NULL;\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n    graph-&gt;heads[graph-&gt;size++] = head;\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjList *graph, Vertex *vet) {\n    AdjListNode *node = findNode(graph, vet);\n    assert(node != NULL);\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n    AdjListNode *cur = node, *pre = NULL;\n    while (cur) {\n        pre = cur;\n        cur = cur-&gt;next;\n        free(pre);\n    }\n    // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        cur = graph-&gt;heads[i];\n        pre = NULL;\n        while (cur) {\n            pre = cur;\n            cur = cur-&gt;next;\n            if (cur &amp;&amp; cur-&gt;vertex == vet) {\n                pre-&gt;next = cur-&gt;next;\n                free(cur);\n                break;\n            }\n        }\n    }\n    // \u5c06\u8be5\u9876\u70b9\u4e4b\u540e\u7684\u9876\u70b9\u5411\u524d\u79fb\u52a8\uff0c\u4ee5\u586b\u8865\u7a7a\u7f3a\n    int i;\n    for (i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i] == node)\n            break;\n    }\n    for (int j = i; j &lt; graph-&gt;size - 1; j++) {\n        graph-&gt;heads[j] = graph-&gt;heads[j + 1];\n    }\n    graph-&gt;size--;\n    free(vet);\n}\n</code></pre> graph_adjacency_list.kt<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList(edges: Array&lt;Array&lt;Vertex?&gt;&gt;) {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    val adjList: MutableMap&lt;Vertex, MutableList&lt;Vertex&gt;&gt; = HashMap()\n\n    /* \u6784\u9020\u51fd\u6570 */\n    init {\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (edge in edges) {\n            addVertex(edge[0]!!);\n            addVertex(edge[1]!!);\n            addEdge(edge[0]!!, edge[1]!!);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    fun size(): Int {\n        return adjList.size\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    fun addEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1]?.add(vet2)\n        adjList[vet2]?.add(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    fun removeEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList[vet1]?.remove(vet2);\n        adjList[vet2]?.remove(vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    fun addVertex(vet: Vertex) {\n        if (adjList.containsKey(vet))\n            return\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList[vet] = mutableListOf()\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    fun removeVertex(vet: Vertex) {\n        if (!adjList.containsKey(vet))\n            throw IllegalArgumentException()\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.remove(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (list in adjList.values) {\n            list.remove(vet)\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    fun print() {\n        println(\"\u90bb\u63a5\u8868 =\")\n        for (pair in adjList.entries) {\n            val tmp = ArrayList&lt;Int&gt;()\n            for (vertex in pair.value) {\n                tmp.add(vertex.value)\n            }\n            println(\"${pair.key.value}: $tmp,\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.rb<pre><code>[class]{GraphAdjList}-[func]{}\n</code></pre> graph_adjacency_list.zig<pre><code>[class]{GraphAdjList}-[func]{}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_graph/graph_operations/#923-efficiency-comparison","title":"9.2.3 \u00a0 Efficiency comparison","text":"<p>Assuming there are \\(n\\) vertices and \\(m\\) edges in the graph, the Table 9-2  compares the time efficiency and space efficiency of the adjacency matrix and adjacency list.</p> <p> Table 9-2 \u00a0 Comparison of adjacency matrix and adjacency list </p> Adjacency matrix Adjacency list (Linked list) Adjacency list (Hash table) Determine adjacency \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) Add an edge \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) Remove an edge \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) Add a vertex \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) Remove a vertex \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) Memory space usage \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\) <p>Observing the Table 9-2 , it seems that the adjacency list (hash table) has the best time efficiency and space efficiency. However, in practice, operating on edges in the adjacency matrix is more efficient, requiring only a single array access or assignment operation. Overall, the adjacency matrix exemplifies the principle of \"space for time\", while the adjacency list exemplifies \"time for space\".</p>"},{"location":"chapter_graph/graph_traversal/","title":"9.3 \u00a0 Graph traversal","text":"<p>Trees represent a \"one-to-many\" relationship, while graphs have a higher degree of freedom and can represent any \"many-to-many\" relationship. Therefore, we can consider trees as a special case of graphs. Clearly, tree traversal operations are also a special case of graph traversal operations.</p> <p>Both graphs and trees require the application of search algorithms to implement traversal operations. Graph traversal can be divided into two types: \"Breadth-First Search (BFS)\" and \"Depth-First Search (DFS)\".</p>"},{"location":"chapter_graph/graph_traversal/#931-breadth-first-search","title":"9.3.1 \u00a0 Breadth-first search","text":"<p>Breadth-first search is a near-to-far traversal method, starting from a certain node, always prioritizing the visit to the nearest vertices and expanding outwards layer by layer. As shown in the Figure 9-9 , starting from the top left vertex, first traverse all adjacent vertices of that vertex, then traverse all adjacent vertices of the next vertex, and so on, until all vertices have been visited.</p> <p></p> <p> Figure 9-9 \u00a0 Breadth-first traversal of a graph </p>"},{"location":"chapter_graph/graph_traversal/#1-algorithm-implementation","title":"1. \u00a0 Algorithm implementation","text":"<p>BFS is usually implemented with the help of a queue, as shown in the code below. The queue has a \"first in, first out\" property, which aligns with the BFS idea of traversing \"from near to far\".</p> <ol> <li>Add the starting vertex <code>startVet</code> to the queue and start the loop.</li> <li>In each iteration of the loop, pop the vertex at the front of the queue and record it as visited, then add all adjacent vertices of that vertex to the back of the queue.</li> <li>Repeat step <code>2.</code> until all vertices have been visited.</li> </ol> <p>To prevent revisiting vertices, we use a hash table <code>visited</code> to record which nodes have been visited.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_bfs.py<pre><code>def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\n    # \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    # \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res = []\n    # \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited = set[Vertex]([start_vet])\n    # \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    que = deque[Vertex]([start_vet])\n    # \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while len(que) &gt; 0:\n        vet = que.popleft()  # \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        # \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for adj_vet in graph.adj_list[vet]:\n            if adj_vet in visited:\n                continue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            que.append(adj_vet)  # \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adj_vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    # \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n</code></pre> graph_bfs.cpp<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector&lt;Vertex *&gt; graphBFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    vector&lt;Vertex *&gt; res;\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    unordered_set&lt;Vertex *&gt; visited = {startVet};\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    queue&lt;Vertex *&gt; que;\n    que.push(startVet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!que.empty()) {\n        Vertex *vet = que.front();\n        que.pop();          // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push_back(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (auto adjVet : graph.adjList[vet]) {\n            if (visited.count(adjVet))\n                continue;            // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            que.push(adjVet);        // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.emplace(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.java<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    visited.add(startVet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    Queue&lt;Vertex&gt; que = new LinkedList&lt;&gt;();\n    que.offer(startVet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!que.isEmpty()) {\n        Vertex vet = que.poll(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.add(vet);            // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (Vertex adjVet : graph.adjList.get(vet)) {\n            if (visited.contains(adjVet))\n                continue;        // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            que.offer(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.cs<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; GraphBFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    HashSet&lt;Vertex&gt; visited = [startVet];\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    Queue&lt;Vertex&gt; que = new();\n    que.Enqueue(startVet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (que.Count &gt; 0) {\n        Vertex vet = que.Dequeue(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.Add(vet);               // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        foreach (Vertex adjVet in graph.adjList[vet]) {\n            if (visited.Contains(adjVet)) {\n                continue;          // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.Enqueue(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.Add(adjVet);   // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.go<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res := make([]Vertex, 0)\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited := make(map[Vertex]struct{})\n    visited[startVet] = struct{}{}\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS, \u4f7f\u7528\u5207\u7247\u6a21\u62df\u961f\u5217\n    queue := make([]Vertex, 0)\n    queue = append(queue, startVet)\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    for len(queue) &gt; 0 {\n        // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        vet := queue[0]\n        queue = queue[1:]\n        // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        res = append(res, vet)\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for _, adjVet := range g.adjList[vet] {\n            _, isExist := visited[adjVet]\n            // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            if !isExist {\n                queue = append(queue, adjVet)\n                visited[adjVet] = struct{}{}\n            }\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.swift<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    var res: [Vertex] = []\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    var visited: Set&lt;Vertex&gt; = [startVet]\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    var que: [Vertex] = [startVet]\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while !que.isEmpty {\n        let vet = que.removeFirst() // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for adjVet in graph.adjList[vet] ?? [] {\n            if visited.contains(adjVet) {\n                continue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.append(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.insert(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.js<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph, startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited = new Set();\n    visited.add(startVet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    const que = [startVet];\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (que.length) {\n        const vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.ts<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res: Vertex[] = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited: Set&lt;Vertex&gt; = new Set();\n    visited.add(startVet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    const que = [startVet];\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (que.length) {\n        const vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\n            visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.dart<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n  // \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n  // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n  List&lt;Vertex&gt; res = [];\n  // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n  Set&lt;Vertex&gt; visited = {};\n  visited.add(startVet);\n  // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n  Queue&lt;Vertex&gt; que = Queue();\n  que.add(startVet);\n  // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n  while (que.isNotEmpty) {\n    Vertex vet = que.removeFirst(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n    res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (Vertex adjVet in graph.adjList[vet]!) {\n      if (visited.contains(adjVet)) {\n        continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n      }\n      que.add(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n      visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    }\n  }\n  // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n  return res;\n}\n</code></pre> graph_bfs.rs<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    let mut res = vec![];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    let mut visited = HashSet::new();\n    visited.insert(start_vet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    let mut que = VecDeque::new();\n    que.push_back(start_vet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while !que.is_empty() {\n        let vet = que.pop_front().unwrap(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n            for &amp;adj_vet in adj_vets {\n                if visited.contains(&amp;adj_vet) {\n                    continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n                }\n                que.push_back(adj_vet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n                visited.insert(adj_vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n            }\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res\n}\n</code></pre> graph_bfs.c<pre><code>/* \u8282\u70b9\u961f\u5217\u7ed3\u6784\u4f53 */\ntypedef struct {\n    Vertex *vertices[MAX_SIZE];\n    int front, rear, size;\n} Queue;\n\n/* \u6784\u9020\u51fd\u6570 */\nQueue *newQueue() {\n    Queue *q = (Queue *)malloc(sizeof(Queue));\n    q-&gt;front = q-&gt;rear = q-&gt;size = 0;\n    return q;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nint isEmpty(Queue *q) {\n    return q-&gt;size == 0;\n}\n\n/* \u5165\u961f\u64cd\u4f5c */\nvoid enqueue(Queue *q, Vertex *vet) {\n    q-&gt;vertices[q-&gt;rear] = vet;\n    q-&gt;rear = (q-&gt;rear + 1) % MAX_SIZE;\n    q-&gt;size++;\n}\n\n/* \u51fa\u961f\u64cd\u4f5c */\nVertex *dequeue(Queue *q) {\n    Vertex *vet = q-&gt;vertices[q-&gt;front];\n    q-&gt;front = (q-&gt;front + 1) % MAX_SIZE;\n    q-&gt;size--;\n    return vet;\n}\n\n/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **visited, int size, Vertex *vet) {\n    // \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\n    for (int i = 0; i &lt; size; i++) {\n        if (visited[i] == vet)\n            return 1;\n    }\n    return 0;\n}\n\n/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphBFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize, Vertex **visited, int *visitedSize) {\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    Queue *queue = newQueue();\n    enqueue(queue, startVet);\n    visited[(*visitedSize)++] = startVet;\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!isEmpty(queue)) {\n        Vertex *vet = dequeue(queue); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res[(*resSize)++] = vet;      // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        AdjListNode *node = findNode(graph, vet);\n        while (node != NULL) {\n            // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            if (!isVisited(visited, *visitedSize, node-&gt;vertex)) {\n                enqueue(queue, node-&gt;vertex);             // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n                visited[(*visitedSize)++] = node-&gt;vertex; // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n            }\n            node = node-&gt;next;\n        }\n    }\n    // \u91ca\u653e\u5185\u5b58\n    free(queue);\n}\n</code></pre> graph_bfs.kt<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfun graphBFS(graph: GraphAdjList, startVet: Vertex): List&lt;Vertex&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    val res: MutableList&lt;Vertex&gt; = ArrayList()\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    val visited: MutableSet&lt;Vertex&gt; = HashSet()\n    visited.add(startVet)\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    val que: Queue&lt;Vertex&gt; = LinkedList()\n    que.offer(startVet)\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!que.isEmpty()) {\n        val vet = que.poll() // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.add(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (adjVet in graph.adjList[vet]!!) {\n            if (visited.contains(adjVet)) continue  // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n\n            que.offer(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.rb<pre><code>[class]{}-[func]{graph_bfs}\n</code></pre> graph_bfs.zig<pre><code>[class]{}-[func]{graphBFS}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The code is relatively abstract, it is suggested to compare with the following figure to deepen the understanding.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-10 \u00a0 Steps of breadth-first search of a graph </p> <p>Is the sequence of breadth-first traversal unique?</p> <p>Not unique. Breadth-first traversal only requires traversing in a \"from near to far\" order, and the traversal order of multiple vertices at the same distance can be arbitrarily shuffled. For example, in the above figure, the visitation order of vertices \\(1\\) and \\(3\\) can be switched, as can the order of vertices \\(2\\), \\(4\\), and \\(6\\).</p>"},{"location":"chapter_graph/graph_traversal/#2-complexity-analysis","title":"2. \u00a0 Complexity analysis","text":"<p>Time complexity: All vertices will be enqueued and dequeued once, using \\(O(|V|)\\) time; in the process of traversing adjacent vertices, since it is an undirected graph, all edges will be visited \\(2\\) times, using \\(O(2|E|)\\) time; overall using \\(O(|V| + |E|)\\) time.</p> <p>Space complexity: The maximum number of vertices in list <code>res</code>, hash table <code>visited</code>, and queue <code>que</code> is \\(|V|\\), using \\(O(|V|)\\) space.</p>"},{"location":"chapter_graph/graph_traversal/#932-depth-first-search","title":"9.3.2 \u00a0 Depth-first search","text":"<p>Depth-first search is a traversal method that prioritizes going as far as possible and then backtracks when no further paths are available. As shown in the Figure 9-11 , starting from the top left vertex, visit some adjacent vertex of the current vertex until no further path is available, then return and continue until all vertices are traversed.</p> <p></p> <p> Figure 9-11 \u00a0 Depth-first traversal of a graph </p>"},{"location":"chapter_graph/graph_traversal/#1-algorithm-implementation_1","title":"1. \u00a0 Algorithm implementation","text":"<p>This \"go as far as possible and then return\" algorithm paradigm is usually implemented based on recursion. Similar to breadth-first search, in depth-first search, we also need the help of a hash table <code>visited</code> to record the visited vertices to avoid revisiting.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_dfs.py<pre><code>def dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n    \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570\"\"\"\n    res.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    # \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for adjVet in graph.adj_list[vet]:\n        if adjVet in visited:\n            continue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        # \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet)\n\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\n    # \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    # \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res = []\n    # \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited = set[Vertex]()\n    dfs(graph, visited, res, start_vet)\n    return res\n</code></pre> graph_dfs.cpp<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList &amp;graph, unordered_set&lt;Vertex *&gt; &amp;visited, vector&lt;Vertex *&gt; &amp;res, Vertex *vet) {\n    res.push_back(vet);   // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.emplace(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (Vertex *adjVet : graph.adjList[vet]) {\n        if (visited.count(adjVet))\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector&lt;Vertex *&gt; graphDFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    vector&lt;Vertex *&gt; res;\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    unordered_set&lt;Vertex *&gt; visited;\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.java<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList graph, Set&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (Vertex adjVet : graph.adjList.get(vet)) {\n        if (visited.contains(adjVet))\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.cs<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid DFS(GraphAdjList graph, HashSet&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.Add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.Add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    foreach (Vertex adjVet in graph.adjList[vet]) {\n        if (visited.Contains(adjVet)) {\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9                             \n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        DFS(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; GraphDFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    HashSet&lt;Vertex&gt; visited = [];\n    DFS(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.go<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n    // append \u64cd\u4f5c\u4f1a\u8fd4\u56de\u65b0\u7684\u7684\u5f15\u7528\uff0c\u5fc5\u987b\u8ba9\u539f\u5f15\u7528\u91cd\u65b0\u8d4b\u503c\u4e3a\u65b0slice\u7684\u5f15\u7528\n    *res = append(*res, vet)\n    visited[vet] = struct{}{}\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for _, adjVet := range g.adjList[vet] {\n        _, isExist := visited[adjVet]\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        if !isExist {\n            dfs(g, visited, res, adjVet)\n        }\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res := make([]Vertex, 0)\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited := make(map[Vertex]struct{})\n    dfs(g, visited, &amp;res, startVet)\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_dfs.swift<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfunc dfs(graph: GraphAdjList, visited: inout Set&lt;Vertex&gt;, res: inout [Vertex], vet: Vertex) {\n    res.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.insert(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for adjVet in graph.adjList[vet] ?? [] {\n        if visited.contains(adjVet) {\n            continue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: adjVet)\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    var res: [Vertex] = []\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    var visited: Set&lt;Vertex&gt; = []\n    dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: startVet)\n    return res\n}\n</code></pre> graph_dfs.js<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction dfs(graph, visited, res, vet) {\n    res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph, startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.ts<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfunction dfs(\n    graph: GraphAdjList,\n    visited: Set&lt;Vertex&gt;,\n    res: Vertex[],\n    vet: Vertex\n): void {\n    res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res: Vertex[] = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited: Set&lt;Vertex&gt; = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.dart<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(\n  GraphAdjList graph,\n  Set&lt;Vertex&gt; visited,\n  List&lt;Vertex&gt; res,\n  Vertex vet,\n) {\n  res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n  visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n  // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n  for (Vertex adjVet in graph.adjList[vet]!) {\n    if (visited.contains(adjVet)) {\n      continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n    }\n    // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n    dfs(graph, visited, res, adjVet);\n  }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n  // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n  List&lt;Vertex&gt; res = [];\n  // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n  Set&lt;Vertex&gt; visited = {};\n  dfs(graph, visited, res, startVet);\n  return res;\n}\n</code></pre> graph_dfs.rs<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfn dfs(graph: &amp;GraphAdjList, visited: &amp;mut HashSet&lt;Vertex&gt;, res: &amp;mut Vec&lt;Vertex&gt;, vet: Vertex) {\n    res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.insert(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n                         // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n        for &amp;adj_vet in adj_vets {\n            if visited.contains(&amp;adj_vet) {\n                continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n            dfs(graph, visited, res, adj_vet);\n        }\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    let mut res = vec![];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    let mut visited = HashSet::new();\n    dfs(&amp;graph, &amp;mut visited, &amp;mut res, start_vet);\n\n    res\n}\n</code></pre> graph_dfs.c<pre><code>/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **res, int size, Vertex *vet) {\n    // \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\n    for (int i = 0; i &lt; size; i++) {\n        if (res[i] == vet) {\n            return 1;\n        }\n    }\n    return 0;\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList *graph, Vertex **res, int *resSize, Vertex *vet) {\n    // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    res[(*resSize)++] = vet;\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    AdjListNode *node = findNode(graph, vet);\n    while (node != NULL) {\n        // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        if (!isVisited(res, *resSize, node-&gt;vertex)) {\n            // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n            dfs(graph, res, resSize, node-&gt;vertex);\n        }\n        node = node-&gt;next;\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphDFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize) {\n    dfs(graph, res, resSize, startVet);\n}\n</code></pre> graph_dfs.kt<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfun dfs(\n    graph: GraphAdjList,\n    visited: MutableSet&lt;Vertex?&gt;,\n    res: MutableList&lt;Vertex?&gt;,\n    vet: Vertex?\n) {\n    res.add(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (adjVet in graph.adjList[vet]!!) {\n        if (visited.contains(adjVet)) continue  // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet)\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfun graphDFS(\n    graph: GraphAdjList,\n    startVet: Vertex?\n): List&lt;Vertex?&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    val res: MutableList&lt;Vertex?&gt; = ArrayList()\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    val visited: MutableSet&lt;Vertex?&gt; = HashSet()\n    dfs(graph, visited, res, startVet)\n    return res\n}\n</code></pre> graph_dfs.rb<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{graph_dfs}\n</code></pre> graph_dfs.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{graphDFS}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The algorithm process of depth-first search is shown in the following figure.</p> <ul> <li>Dashed lines represent downward recursion, indicating that a new recursive method has been initiated to visit a new vertex.</li> <li>Curved dashed lines represent upward backtracking, indicating that this recursive method has returned to the position where this method was initiated.</li> </ul> <p>To deepen the understanding, it is suggested to combine the following figure with the code to simulate (or draw) the entire DFS process in your mind, including when each recursive method is initiated and when it returns.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-12 \u00a0 Steps of depth-first search of a graph </p> <p>Is the sequence of depth-first traversal unique?</p> <p>Similar to breadth-first traversal, the order of the depth-first traversal sequence is also not unique. Given a certain vertex, exploring in any direction first is possible, that is, the order of adjacent vertices can be arbitrarily shuffled, all being part of depth-first traversal.</p> <p>Taking tree traversal as an example, \"root \\(\\rightarrow\\) left \\(\\rightarrow\\) right\", \"left \\(\\rightarrow\\) root \\(\\rightarrow\\) right\", \"left \\(\\rightarrow\\) right \\(\\rightarrow\\) root\" correspond to preorder, inorder, and postorder traversals, respectively. They showcase three types of traversal priorities, yet all three are considered depth-first traversal.</p>"},{"location":"chapter_graph/graph_traversal/#2-complexity-analysis_1","title":"2. \u00a0 Complexity analysis","text":"<p>Time complexity: All vertices will be visited once, using \\(O(|V|)\\) time; all edges will be visited twice, using \\(O(2|E|)\\) time; overall using \\(O(|V| + |E|)\\) time.</p> <p>Space complexity: The maximum number of vertices in list <code>res</code>, hash table <code>visited</code> is \\(|V|\\), and the maximum recursion depth is \\(|V|\\), therefore using \\(O(|V|)\\) space.</p>"},{"location":"chapter_graph/summary/","title":"9.4 \u00a0 Summary","text":""},{"location":"chapter_graph/summary/#1-key-review","title":"1. \u00a0 Key review","text":"<ul> <li>A graph consists of vertices and edges and can be represented as a set comprising a group of vertices and a group of edges.</li> <li>Compared to linear relationships (linked lists) and divide-and-conquer relationships (trees), network relationships (graphs) have a higher degree of freedom and are therefore more complex.</li> <li>The edges of a directed graph have directionality, any vertex in a connected graph is reachable, and each edge in a weighted graph contains a weight variable.</li> <li>Adjacency matrices use matrices to represent graphs, with each row (column) representing a vertex and matrix elements representing edges, using \\(1\\) or \\(0\\) to indicate the presence or absence of an edge between two vertices. Adjacency matrices are highly efficient for add, delete, find, and modify operations, but they consume more space.</li> <li>Adjacency lists use multiple linked lists to represent graphs, with the \\(i^{th}\\) list corresponding to vertex \\(i\\), containing all its adjacent vertices. Adjacency lists save more space compared to adjacency matrices, but since it is necessary to traverse the list to find edges, their time efficiency is lower.</li> <li>When the linked lists in the adjacency list are too long, they can be converted into red-black trees or hash tables to improve query efficiency.</li> <li>From the perspective of algorithmic thinking, adjacency matrices embody the principle of \"space for time,\" while adjacency lists embody \"time for space.\"</li> <li>Graphs can be used to model various real systems, such as social networks, subway routes, etc.</li> <li>A tree is a special case of a graph, and tree traversal is also a special case of graph traversal.</li> <li>Breadth-first traversal of a graph is a search method that expands layer by layer from near to far, usually implemented with a queue.</li> <li>Depth-first traversal of a graph is a search method that prefers to go as deep as possible and backtracks when no further paths are available, often based on recursion.</li> </ul>"},{"location":"chapter_graph/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: Is a path defined as a sequence of vertices or a sequence of edges?</p> <p>Definitions vary between different language versions on Wikipedia: the English version defines a path as \"a sequence of edges,\" while the Chinese version defines it as \"a sequence of vertices.\" Here is the original text from the English version: In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.</p> <p>In this document, a path is considered a sequence of edges, rather than a sequence of vertices. This is because there might be multiple edges connecting two vertices, in which case each edge corresponds to a path.</p> <p>Q: In a disconnected graph, are there points that cannot be traversed to?</p> <p>In a disconnected graph, starting from a certain vertex, there is at least one vertex that cannot be reached. Traversing a disconnected graph requires setting multiple starting points to traverse all connected components of the graph.</p> <p>Q: In an adjacency list, does the order of \"all vertices connected to that vertex\" matter?</p> <p>It can be in any order. However, in practical applications, it might be necessary to sort according to certain rules, such as the order in which vertices are added, or the order of vertex values, etc., to facilitate the quick search for vertices with certain extremal values.</p>"},{"location":"chapter_hashing/","title":"Chapter 6. \u00a0 Hash Table","text":"<p>Abstract</p> <p>In the world of computing, a hash table is akin to an intelligent librarian.</p> <p>It understands how to compute index numbers, enabling swift retrieval of the desired book.</p>"},{"location":"chapter_hashing/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>6.1 \u00a0 Hash Table</li> <li>6.2 \u00a0 Hash Collision</li> <li>6.3 \u00a0 Hash Algorithm</li> <li>6.4 \u00a0 Summary</li> </ul>"},{"location":"chapter_hashing/hash_algorithm/","title":"6.3 \u00a0 Hash Algorithms","text":"<p>The previous two sections introduced the working principle of hash tables and the methods to handle hash collisions. However, both open addressing and chaining can only ensure that the hash table functions normally when collisions occur, but cannot reduce the frequency of hash collisions.</p> <p>If hash collisions occur too frequently, the performance of the hash table will deteriorate drastically. As shown in the Figure 6-8 , for a chaining hash table, in the ideal case, the key-value pairs are evenly distributed across the buckets, achieving optimal query efficiency; in the worst case, all key-value pairs are stored in the same bucket, degrading the time complexity to \\(O(n)\\).</p> <p></p> <p> Figure 6-8 \u00a0 Ideal and Worst Cases of Hash Collisions </p> <p>The distribution of key-value pairs is determined by the hash function. Recalling the steps of calculating a hash function, first compute the hash value, then modulo it by the array length:</p> <pre><code>index = hash(key) % capacity\n</code></pre> <p>Observing the above formula, when the hash table capacity <code>capacity</code> is fixed, the hash algorithm <code>hash()</code> determines the output value, thereby determining the distribution of key-value pairs in the hash table.</p> <p>This means that, to reduce the probability of hash collisions, we should focus on the design of the hash algorithm <code>hash()</code>.</p>"},{"location":"chapter_hashing/hash_algorithm/#631-goals-of-hash-algorithms","title":"6.3.1 \u00a0 Goals of Hash Algorithms","text":"<p>To achieve a \"fast and stable\" hash table data structure, hash algorithms should have the following characteristics:</p> <ul> <li>Determinism: For the same input, the hash algorithm should always produce the same output. Only then can the hash table be reliable.</li> <li>High Efficiency: The process of computing the hash value should be fast enough. The smaller the computational overhead, the more practical the hash table.</li> <li>Uniform Distribution: The hash algorithm should ensure that key-value pairs are evenly distributed in the hash table. The more uniform the distribution, the lower the probability of hash collisions.</li> </ul> <p>In fact, hash algorithms are not only used to implement hash tables but are also widely applied in other fields.</p> <ul> <li>Password Storage: To protect the security of user passwords, systems usually do not store the plaintext passwords but rather the hash values of the passwords. When a user enters a password, the system calculates the hash value of the input and compares it with the stored hash value. If they match, the password is considered correct.</li> <li>Data Integrity Check: The data sender can calculate the hash value of the data and send it along; the receiver can recalculate the hash value of the received data and compare it with the received hash value. If they match, the data is considered intact.</li> </ul> <p>For cryptographic applications, to prevent reverse engineering such as deducing the original password from the hash value, hash algorithms need higher-level security features.</p> <ul> <li>Unidirectionality: It should be impossible to deduce any information about the input data from the hash value.</li> <li>Collision Resistance: It should be extremely difficult to find two different inputs that produce the same hash value.</li> <li>Avalanche Effect: Minor changes in the input should lead to significant and unpredictable changes in the output.</li> </ul> <p>Note that \"Uniform Distribution\" and \"Collision Resistance\" are two separate concepts. Satisfying uniform distribution does not necessarily mean collision resistance. For example, under random input <code>key</code>, the hash function <code>key % 100</code> can produce a uniformly distributed output. However, this hash algorithm is too simple, and all <code>key</code> with the same last two digits will have the same output, making it easy to deduce a usable <code>key</code> from the hash value, thereby cracking the password.</p>"},{"location":"chapter_hashing/hash_algorithm/#632-design-of-hash-algorithms","title":"6.3.2 \u00a0 Design of Hash Algorithms","text":"<p>The design of hash algorithms is a complex issue that requires consideration of many factors. However, for some less demanding scenarios, we can also design some simple hash algorithms.</p> <ul> <li>Additive Hash: Add up the ASCII codes of each character in the input and use the total sum as the hash value.</li> <li>Multiplicative Hash: Utilize the non-correlation of multiplication, multiplying each round by a constant, accumulating the ASCII codes of each character into the hash value.</li> <li>XOR Hash: Accumulate the hash value by XORing each element of the input data.</li> <li>Rotating Hash: Accumulate the ASCII code of each character into a hash value, performing a rotation operation on the hash value before each accumulation.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig simple_hash.py<pre><code>def add_hash(key: str) -&gt; int:\n    \"\"\"\u52a0\u6cd5\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash += ord(c)\n    return hash % modulus\n\ndef mul_hash(key: str) -&gt; int:\n    \"\"\"\u4e58\u6cd5\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = 31 * hash + ord(c)\n    return hash % modulus\n\ndef xor_hash(key: str) -&gt; int:\n    \"\"\"\u5f02\u6216\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash ^= ord(c)\n    return hash % modulus\n\ndef rot_hash(key: str) -&gt; int:\n    \"\"\"\u65cb\u8f6c\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ ord(c)\n    return hash % modulus\n</code></pre> simple_hash.cpp<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (31 * hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash ^= (int)c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.java<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (31 * hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\n    int hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash ^= (int) c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n</code></pre> simple_hash.cs<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint AddHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint MulHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (31 * hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint XorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash ^= c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint RotHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.go<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (31*hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key string) int {\n    hash := 0\n    modulus := 1000000007\n    for _, b := range []byte(key) {\n        fmt.Println(int(b))\n        hash ^= int(b)\n        hash = (31*hash + int(b)) % modulus\n    }\n    return hash &amp; modulus\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ int64(b)) % modulus\n    }\n    return int(hash)\n}\n</code></pre> simple_hash.swift<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (31 * hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash ^= Int(scalar.value)\n        }\n    }\n    return hash &amp; MODULUS\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n</code></pre> simple_hash.js<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.ts<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.dart<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (31 * hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash ^= key.codeUnitAt(i);\n  }\n  return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n</code></pre> simple_hash.rs<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfn add_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfn mul_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (31 * hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfn xor_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash ^= c as i64;\n    }\n\n    (hash &amp; MODULUS) as i32\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfn rot_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n</code></pre> simple_hash.c<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (31 * hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(char *key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash ^= (unsigned char)key[i];\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (unsigned char)key[i]) % MODULUS;\n    }\n\n    return (int)hash;\n}\n</code></pre> simple_hash.kt<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfun addHash(key: String): Int {\n    var hash = 0L\n    for (c in key.toCharArray()) {\n        hash = (hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfun mulHash(key: String): Int {\n    var hash = 0L\n    for (c in key.toCharArray()) {\n        hash = (31 * hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfun xorHash(key: String): Int {\n    var hash = 0\n    for (c in key.toCharArray()) {\n        hash = hash xor c.code\n    }\n    return hash and MODULUS\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfun rotHash(key: String): Int {\n    var hash = 0L\n    for (c in key.toCharArray()) {\n        hash = ((hash shl 4) xor (hash shr 28) xor c.code.toLong()) % MODULUS\n    }\n    return hash.toInt()\n}\n</code></pre> simple_hash.rb<pre><code>[class]{}-[func]{add_hash}\n\n[class]{}-[func]{mul_hash}\n\n[class]{}-[func]{xor_hash}\n\n[class]{}-[func]{rot_hash}\n</code></pre> simple_hash.zig<pre><code>[class]{}-[func]{addHash}\n\n[class]{}-[func]{mulHash}\n\n[class]{}-[func]{xorHash}\n\n[class]{}-[func]{rotHash}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>It is observed that the last step of each hash algorithm is to take the modulus of the large prime number \\(1000000007\\) to ensure that the hash value is within an appropriate range. It is worth pondering why emphasis is placed on modulo a prime number, or what are the disadvantages of modulo a composite number? This is an interesting question.</p> <p>To conclude: Using a large prime number as the modulus can maximize the uniform distribution of hash values. Since a prime number does not share common factors with other numbers, it can reduce the periodic patterns caused by the modulo operation, thus avoiding hash collisions.</p> <p>For example, suppose we choose the composite number \\(9\\) as the modulus, which can be divided by \\(3\\), then all <code>key</code> divisible by \\(3\\) will be mapped to hash values \\(0\\), \\(3\\), \\(6\\).</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 9 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\dots \\} \\end{aligned} \\] <p>If the input <code>key</code> happens to have this kind of arithmetic sequence distribution, then the hash values will cluster, thereby exacerbating hash collisions. Now, suppose we replace <code>modulus</code> with the prime number \\(13\\), since there are no common factors between <code>key</code> and <code>modulus</code>, the uniformity of the output hash values will be significantly improved.</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 13 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\dots \\} \\end{aligned} \\] <p>It is worth noting that if the <code>key</code> is guaranteed to be randomly and uniformly distributed, then choosing a prime number or a composite number as the modulus can both produce uniformly distributed hash values. However, when the distribution of <code>key</code> has some periodicity, modulo a composite number is more likely to result in clustering.</p> <p>In summary, we usually choose a prime number as the modulus, and this prime number should be large enough to eliminate periodic patterns as much as possible, enhancing the robustness of the hash algorithm.</p>"},{"location":"chapter_hashing/hash_algorithm/#633-common-hash-algorithms","title":"6.3.3 \u00a0 Common Hash Algorithms","text":"<p>It is not hard to see that the simple hash algorithms mentioned above are quite \"fragile\" and far from reaching the design goals of hash algorithms. For example, since addition and XOR obey the commutative law, additive hash and XOR hash cannot distinguish strings with the same content but in different order, which may exacerbate hash collisions and cause security issues.</p> <p>In practice, we usually use some standard hash algorithms, such as MD5, SHA-1, SHA-2, and SHA-3. They can map input data of any length to a fixed-length hash value.</p> <p>Over the past century, hash algorithms have been in a continuous process of upgrading and optimization. Some researchers strive to improve the performance of hash algorithms, while others, including hackers, are dedicated to finding security issues in hash algorithms. The Table 6-2  shows hash algorithms commonly used in practical applications.</p> <ul> <li>MD5 and SHA-1 have been successfully attacked multiple times and are thus abandoned in various security applications.</li> <li>SHA-2 series, especially SHA-256, is one of the most secure hash algorithms to date, with no successful attacks reported, hence commonly used in various security applications and protocols.</li> <li>SHA-3 has lower implementation costs and higher computational efficiency compared to SHA-2, but its current usage coverage is not as extensive as the SHA-2 series.</li> </ul> <p> Table 6-2 \u00a0 Common Hash Algorithms </p> MD5 SHA-1 SHA-2 SHA-3 Release Year 1992 1995 2002 2008 Output Length 128 bit 160 bit 256/512 bit 224/256/384/512 bit Hash Collisions Frequent Frequent Rare Rare Security Level Low, has been successfully attacked Low, has been successfully attacked High High Applications Abandoned, still used for data integrity checks Abandoned Cryptocurrency transaction verification, digital signatures, etc. Can be used to replace SHA-2"},{"location":"chapter_hashing/hash_algorithm/#hash-values-in-data-structures","title":"Hash Values in Data Structures","text":"<p>We know that the keys in a hash table can be of various data types such as integers, decimals, or strings. Programming languages usually provide built-in hash algorithms for these data types to calculate the bucket indices in the hash table. Taking Python as an example, we can use the <code>hash()</code> function to compute the hash values for various data types.</p> <ul> <li>The hash values of integers and booleans are their own values.</li> <li>The calculation of hash values for floating-point numbers and strings is more complex, and interested readers are encouraged to study this on their own.</li> <li>The hash value of a tuple is a combination of the hash values of each of its elements, resulting in a single hash value.</li> <li>The hash value of an object is generated based on its memory address. By overriding the hash method of an object, hash values can be generated based on content.</li> </ul> <p>Tip</p> <p>Be aware that the definition and methods of the built-in hash value calculation functions in different programming languages vary.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig built_in_hash.py<pre><code>num = 3\nhash_num = hash(num)\n# Hash value of integer 3 is 3\n\nbol = True\nhash_bol = hash(bol)\n# Hash value of boolean True is 1\n\ndec = 3.14159\nhash_dec = hash(dec)\n# Hash value of decimal 3.14159 is 326484311674566659\n\nstr = \"Hello \u7b97\u6cd5\"\nhash_str = hash(str)\n# Hash value of string \"Hello \u7b97\u6cd5\" is 4617003410720528961\n\ntup = (12836, \"\u5c0f\u54c8\")\nhash_tup = hash(tup)\n# Hash value of tuple (12836, '\u5c0f\u54c8') is 1029005403108185979\n\nobj = ListNode(0)\nhash_obj = hash(obj)\n# Hash value of ListNode object at 0x1058fd810 is 274267521\n</code></pre> built_in_hash.cpp<pre><code>int num = 3;\nsize_t hashNum = hash&lt;int&gt;()(num);\n// Hash value of integer 3 is 3\n\nbool bol = true;\nsize_t hashBol = hash&lt;bool&gt;()(bol);\n// Hash value of boolean 1 is 1\n\ndouble dec = 3.14159;\nsize_t hashDec = hash&lt;double&gt;()(dec);\n// Hash value of decimal 3.14159 is 4614256650576692846\n\nstring str = \"Hello \u7b97\u6cd5\";\nsize_t hashStr = hash&lt;string&gt;()(str);\n// Hash value of string \"Hello \u7b97\u6cd5\" is 15466937326284535026\n\n// In C++, built-in std::hash() only provides hash values for basic data types\n// Hash values for arrays and objects need to be implemented separately\n</code></pre> built_in_hash.java<pre><code>int num = 3;\nint hashNum = Integer.hashCode(num);\n// Hash value of integer 3 is 3\n\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// Hash value of boolean true is 1231\n\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// Hash value of decimal 3.14159 is -1340954729\n\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode();\n// Hash value of string \"Hello \u7b97\u6cd5\" is -727081396\n\nObject[] arr = { 12836, \"\u5c0f\u54c8\" };\nint hashTup = Arrays.hashCode(arr);\n// Hash value of array [12836, \u5c0f\u54c8] is 1151158\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// Hash value of ListNode object utils.ListNode@7dc5e7b4 is 2110121908\n</code></pre> built_in_hash.cs<pre><code>int num = 3;\nint hashNum = num.GetHashCode();\n// Hash value of integer 3 is 3;\n\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// Hash value of boolean true is 1;\n\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// Hash value of decimal 3.14159 is -1340954729;\n\nstring str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.GetHashCode();\n// Hash value of string \"Hello \u7b97\u6cd5\" is -586107568;\n\nobject[] arr = [12836, \"\u5c0f\u54c8\"];\nint hashTup = arr.GetHashCode();\n// Hash value of array [12836, \u5c0f\u54c8] is 42931033;\n\nListNode obj = new(0);\nint hashObj = obj.GetHashCode();\n// Hash value of ListNode object 0 is 39053774;\n</code></pre> built_in_hash.go<pre><code>// Go does not provide built-in hash code functions\n</code></pre> built_in_hash.swift<pre><code>let num = 3\nlet hashNum = num.hashValue\n// Hash value of integer 3 is 9047044699613009734\n\nlet bol = true\nlet hashBol = bol.hashValue\n// Hash value of boolean true is -4431640247352757451\n\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// Hash value of decimal 3.14159 is -2465384235396674631\n\nlet str = \"Hello \u7b97\u6cd5\"\nlet hashStr = str.hashValue\n// Hash value of string \"Hello \u7b97\u6cd5\" is -7850626797806988787\n\nlet arr = [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")]\nlet hashTup = arr.hashValue\n// Hash value of array [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")] is -2308633508154532996\n\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// Hash value of ListNode object utils.ListNode is -2434780518035996159\n</code></pre> built_in_hash.js<pre><code>// JavaScript does not provide built-in hash code functions\n</code></pre> built_in_hash.ts<pre><code>// TypeScript does not provide built-in hash code functions\n</code></pre> built_in_hash.dart<pre><code>int num = 3;\nint hashNum = num.hashCode;\n// Hash value of integer 3 is 34803\n\nbool bol = true;\nint hashBol = bol.hashCode;\n// Hash value of boolean true is 1231\n\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// Hash value of decimal 3.14159 is 2570631074981783\n\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode;\n// Hash value of string \"Hello \u7b97\u6cd5\" is 468167534\n\nList arr = [12836, \"\u5c0f\u54c8\"];\nint hashArr = arr.hashCode;\n// Hash value of array [12836, \u5c0f\u54c8] is 976512528\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// Hash value of ListNode object Instance of 'ListNode' is 1033450432\n</code></pre> built_in_hash.rs<pre><code>use std::collections::hash_map::DefaultHasher;\nuse std::hash::{Hash, Hasher};\n\nlet num = 3;\nlet mut num_hasher = DefaultHasher::new();\nnum.hash(&amp;mut num_hasher);\nlet hash_num = num_hasher.finish();\n// Hash value of integer 3 is 568126464209439262\n\nlet bol = true;\nlet mut bol_hasher = DefaultHasher::new();\nbol.hash(&amp;mut bol_hasher);\nlet hash_bol = bol_hasher.finish();\n// Hash value of boolean true is 4952851536318644461\n\nlet dec: f32 = 3.14159;\nlet mut dec_hasher = DefaultHasher::new();\ndec.to_bits().hash(&amp;mut dec_hasher);\nlet hash_dec = dec_hasher.finish();\n// Hash value of decimal 3.14159 is 2566941990314602357\n\nlet str = \"Hello \u7b97\u6cd5\";\nlet mut str_hasher = DefaultHasher::new();\nstr.hash(&amp;mut str_hasher);\nlet hash_str = str_hasher.finish();\n// Hash value of string \"Hello \u7b97\u6cd5\" is 16092673739211250988\n\nlet arr = (&amp;12836, &amp;\"\u5c0f\u54c8\");\nlet mut tup_hasher = DefaultHasher::new();\narr.hash(&amp;mut tup_hasher);\nlet hash_tup = tup_hasher.finish();\n// Hash value of tuple (12836, \"\u5c0f\u54c8\") is 1885128010422702749\n\nlet node = ListNode::new(42);\nlet mut hasher = DefaultHasher::new();\nnode.borrow().val.hash(&amp;mut hasher);\nlet hash = hasher.finish();\n// Hash value of ListNode object RefCell { value: ListNode { val: 42, next: None } } is 15387811073369036852\n</code></pre> built_in_hash.c<pre><code>// C does not provide built-in hash code functions\n</code></pre> built_in_hash.kt<pre><code>\n</code></pre> built_in_hash.zig<pre><code>\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>In many programming languages, only immutable objects can serve as the <code>key</code> in a hash table. If we use a list (dynamic array) as a <code>key</code>, when the contents of the list change, its hash value also changes, and we would no longer be able to find the original <code>value</code> in the hash table.</p> <p>Although the member variables of a custom object (such as a linked list node) are mutable, it is hashable. This is because the hash value of an object is usually generated based on its memory address, and even if the contents of the object change, the memory address remains the same, so the hash value remains unchanged.</p> <p>You might have noticed that the hash values output in different consoles are different. This is because the Python interpreter adds a random salt to the string hash function each time it starts up. This approach effectively prevents HashDoS attacks and enhances the security of the hash algorithm.</p>"},{"location":"chapter_hashing/hash_collision/","title":"6.2 \u00a0 Hash Collision","text":"<p>As mentioned in the previous section, usually the input space of a hash function is much larger than its output space, making hash collisions theoretically inevitable. For example, if the input space consists of all integers and the output space is the size of the array capacity, multiple integers will inevitably map to the same bucket index.</p> <p>Hash collisions can lead to incorrect query results, severely affecting the usability of hash tables. To solve this problem, we expand the hash table whenever a hash collision occurs, until the collision is resolved. This method is simple and effective but inefficient due to the extensive data transfer and hash value computation involved in resizing the hash table. To improve efficiency, we can adopt the following strategies:</p> <ol> <li>Improve the data structure of the hash table, allowing it to function normally in the event of a hash collision.</li> <li>Only perform resizing when necessary, i.e., when hash collisions are severe.</li> </ol> <p>There are mainly two methods for improving the structure of hash tables: \"Separate Chaining\" and \"Open Addressing\".</p>"},{"location":"chapter_hashing/hash_collision/#621-separate-chaining","title":"6.2.1 \u00a0 Separate Chaining","text":"<p>In the original hash table, each bucket can store only one key-value pair. \"Separate chaining\" transforms individual elements into a linked list, with key-value pairs as list nodes, storing all colliding key-value pairs in the same list. The Figure 6-5  shows an example of a hash table with separate chaining.</p> <p></p> <p> Figure 6-5 \u00a0 Separate Chaining Hash Table </p> <p>The operations of a hash table implemented with separate chaining have changed as follows:</p> <ul> <li>Querying Elements: Input <code>key</code>, pass through the hash function to obtain the bucket index, access the head node of the list, then traverse the list and compare <code>key</code> to find the target key-value pair.</li> <li>Adding Elements: First access the list head node via the hash function, then add the node (key-value pair) to the list.</li> <li>Deleting Elements: Access the list head based on the hash function's result, then traverse the list to find and remove the target node.</li> </ul> <p>Separate chaining has the following limitations:</p> <ul> <li>Increased Space Usage: The linked list contains node pointers, which consume more memory space than arrays.</li> <li>Reduced Query Efficiency: Due to the need for linear traversal of the list to find the corresponding element.</li> </ul> <p>The code below provides a simple implementation of a separate chaining hash table, with two things to note:</p> <ul> <li>Lists (dynamic arrays) are used instead of linked lists for simplicity. In this setup, the hash table (array) contains multiple buckets, each of which is a list.</li> <li>This implementation includes a method for resizing the hash table. When the load factor exceeds \\(\\frac{2}{3}\\), we resize the hash table to twice its original size.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map_chaining.py<pre><code>class HashMapChaining:\n    \"\"\"\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\n        self.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\n        self.load_thres = 2.0 / 3.0  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        self.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\n        self.buckets = [[] for _ in range(self.capacity)]  # \u6876\u6570\u7ec4\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\n        return self.size / self.capacity\n\n    def get(self, key: int) -&gt; str | None:\n        \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for pair in bucket:\n            if pair.key == key:\n                return pair.val\n        # \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n        # \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for pair in bucket:\n            if pair.key == key:\n                pair.val = val\n                return\n        # \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        pair = Pair(key, val)\n        bucket.append(pair)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for pair in bucket:\n            if pair.key == key:\n                bucket.remove(pair)\n                self.size -= 1\n                break\n\n    def extend(self):\n        \"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n        # \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        buckets = self.buckets\n        # \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio\n        self.buckets = [[] for _ in range(self.capacity)]\n        self.size = 0\n        # \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for bucket in buckets:\n            for pair in bucket:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n        for bucket in self.buckets:\n            res = []\n            for pair in bucket:\n                res.append(str(pair.key) + \" -&gt; \" + pair.val)\n            print(res)\n</code></pre> hash_map_chaining.cpp<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n  private:\n    int size;                       // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity;                   // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres;               // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio;                // \u6269\u5bb9\u500d\u6570\n    vector&lt;vector&lt;Pair *&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    HashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3.0), extendRatio(2) {\n        buckets.resize(capacity);\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~HashMapChaining() {\n        for (auto &amp;bucket : buckets) {\n            for (Pair *pair : bucket) {\n                // \u91ca\u653e\u5185\u5b58\n                delete pair;\n            }\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double loadFactor() {\n        return (double)size / (double)capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    string get(int key) {\n        int index = hashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                return pair-&gt;val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n        return \"\";\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                pair-&gt;val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        buckets[index].push_back(new Pair(key, val));\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        auto &amp;bucket = buckets[index];\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (int i = 0; i &lt; bucket.size(); i++) {\n            if (bucket[i]-&gt;key == key) {\n                Pair *tmp = bucket[i];\n                bucket.erase(bucket.begin() + i); // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n                delete tmp;                       // \u91ca\u653e\u5185\u5b58\n                size--;\n                return;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        vector&lt;vector&lt;Pair *&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets.clear();\n        buckets.resize(capacity);\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (auto &amp;bucket : bucketsTmp) {\n            for (Pair *pair : bucket) {\n                put(pair-&gt;key, pair-&gt;val);\n                // \u91ca\u653e\u5185\u5b58\n                delete pair;\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (auto &amp;bucket : buckets) {\n            cout &lt;&lt; \"[\";\n            for (Pair *pair : bucket) {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; \", \";\n            }\n            cout &lt;&lt; \"]\\n\";\n        }\n    }\n};\n</code></pre> hash_map_chaining.java<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio; // \u6269\u5bb9\u500d\u6570\n    List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    String get(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, String val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        Pair pair = new Pair(key, val);\n        bucket.add(pair);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (List&lt;Pair&gt; bucket : bucketsTmp) {\n            for (Pair pair : bucket) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (List&lt;Pair&gt; bucket : buckets) {\n            List&lt;String&gt; res = new ArrayList&lt;&gt;();\n            for (Pair pair : bucket) {\n                res.add(pair.key + \" -&gt; \" + pair.val);\n            }\n            System.out.println(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.cs<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio; // \u6269\u5bb9\u500d\u6570\n    List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        int index = HashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        buckets[index].Add(new Pair(key, val));\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        foreach (Pair pair in buckets[index].ToList()) {\n            if (pair.key == key) {\n                buckets[index].Remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void Extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        foreach (List&lt;Pair&gt; bucket in bucketsTmp) {\n            foreach (Pair pair in bucket) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void Print() {\n        foreach (List&lt;Pair&gt; bucket in buckets) {\n            List&lt;string&gt; res = [];\n            foreach (Pair pair in bucket) {\n                res.Add(pair.key + \" -&gt; \" + pair.val);\n            }\n            foreach (string kv in res) {\n                Console.WriteLine(kv);\n            }\n        }\n    }\n}\n</code></pre> hash_map_chaining.go<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntype hashMapChaining struct {\n    size        int      // \u952e\u503c\u5bf9\u6570\u91cf\n    capacity    int      // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    loadThres   float64  // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    extendRatio int      // \u6269\u5bb9\u500d\u6570\n    buckets     [][]pair // \u6876\u6570\u7ec4\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapChaining() *hashMapChaining {\n    buckets := make([][]pair, 4)\n    for i := 0; i &lt; 4; i++ {\n        buckets[i] = make([]pair, 0)\n    }\n    return &amp;hashMapChaining{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     buckets,\n    }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapChaining) hashFunc(key int) int {\n    return key % m.capacity\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapChaining) loadFactor() float64 {\n    return float64(m.size) / float64(m.capacity)\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapChaining) get(key int) string {\n    idx := m.hashFunc(key)\n    bucket := m.buckets[idx]\n    // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    for _, p := range bucket {\n        if p.key == key {\n            return p.val\n        }\n    }\n    // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n    return \"\"\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapChaining) put(key int, val string) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if m.loadFactor() &gt; m.loadThres {\n        m.extend()\n    }\n    idx := m.hashFunc(key)\n    // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n    for i := range m.buckets[idx] {\n        if m.buckets[idx][i].key == key {\n            m.buckets[idx][i].val = val\n            return\n        }\n    }\n    // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n    p := pair{\n        key: key,\n        val: val,\n    }\n    m.buckets[idx] = append(m.buckets[idx], p)\n    m.size += 1\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapChaining) remove(key int) {\n    idx := m.hashFunc(key)\n    // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n    for i, p := range m.buckets[idx] {\n        if p.key == key {\n            // \u5207\u7247\u5220\u9664\n            m.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\n            m.size -= 1\n            break\n        }\n    }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    tmpBuckets := make([][]pair, len(m.buckets))\n    for i := 0; i &lt; len(m.buckets); i++ {\n        tmpBuckets[i] = make([]pair, len(m.buckets[i]))\n        copy(tmpBuckets[i], m.buckets[i])\n    }\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    m.capacity *= m.extendRatio\n    m.buckets = make([][]pair, m.capacity)\n    for i := 0; i &lt; m.capacity; i++ {\n        m.buckets[i] = make([]pair, 0)\n    }\n    m.size = 0\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for _, bucket := range tmpBuckets {\n        for _, p := range bucket {\n            m.put(p.key, p.val)\n        }\n    }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) print() {\n    var builder strings.Builder\n\n    for _, bucket := range m.buckets {\n        builder.WriteString(\"[\")\n        for _, p := range bucket {\n            builder.WriteString(strconv.Itoa(p.key) + \" -&gt; \" + p.val + \" \")\n        }\n        builder.WriteString(\"]\")\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> hash_map_chaining.swift<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n    var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    var loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    var extendRatio: Int // \u6269\u5bb9\u500d\u6570\n    var buckets: [[Pair]] // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: [], count: capacity)\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for pair in bucket {\n            if pair.key == key {\n                return pair.val\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de nil\n        return nil\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val\n                return\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        let pair = Pair(key: key, val: val)\n        buckets[index].append(pair)\n        size += 1\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (pairIndex, pair) in bucket.enumerated() {\n            if pair.key == key {\n                buckets[index].remove(at: pairIndex)\n                size -= 1\n                break\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    func extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        buckets = Array(repeating: [], count: capacity)\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for bucket in bucketsTmp {\n            for pair in bucket {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    func print() {\n        for bucket in buckets {\n            let res = bucket.map { \"\\($0.key) -&gt; \\($0.val)\" }\n            Swift.print(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.js<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    #size; // \u952e\u503c\u5bf9\u6570\u91cf\n    #capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    #loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    #extendRatio; // \u6269\u5bb9\u500d\u6570\n    #buckets; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key) {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key, val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key) {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    #extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print() {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.ts<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    #size: number; // \u952e\u503c\u5bf9\u6570\u91cf\n    #capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    #loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    #extendRatio: number; // \u6269\u5bb9\u500d\u6570\n    #buckets: Pair[][]; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key: number): number {\n        return key % this.#capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    #loadFactor(): number {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key: number): string | null {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key: number, val: string): void {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key: number): void {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    #extend(): void {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print(): void {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.dart<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n  late int size; // \u952e\u503c\u5bf9\u6570\u91cf\n  late int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n  late double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n  late int extendRatio; // \u6269\u5bb9\u500d\u6570\n  late List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  HashMapChaining() {\n    size = 0;\n    capacity = 4;\n    loadThres = 2.0 / 3.0;\n    extendRatio = 2;\n    buckets = List.generate(capacity, (_) =&gt; []);\n  }\n\n  /* \u54c8\u5e0c\u51fd\u6570 */\n  int hashFunc(int key) {\n    return key % capacity;\n  }\n\n  /* \u8d1f\u8f7d\u56e0\u5b50 */\n  double loadFactor() {\n    return size / capacity;\n  }\n\n  /* \u67e5\u8be2\u64cd\u4f5c */\n  String? get(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        return pair.val;\n      }\n    }\n    // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n    return null;\n  }\n\n  /* \u6dfb\u52a0\u64cd\u4f5c */\n  void put(int key, String val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor() &gt; loadThres) {\n      extend();\n    }\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        pair.val = val;\n        return;\n      }\n    }\n    // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n    Pair pair = Pair(key, val);\n    bucket.add(pair);\n    size++;\n  }\n\n  /* \u5220\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        bucket.remove(pair);\n        size--;\n        break;\n      }\n    }\n  }\n\n  /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n  void extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    capacity *= extendRatio;\n    buckets = List.generate(capacity, (_) =&gt; []);\n    size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (List&lt;Pair&gt; bucket in bucketsTmp) {\n      for (Pair pair in bucket) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* \u6253\u5370\u54c8\u5e0c\u8868 */\n  void printHashMap() {\n    for (List&lt;Pair&gt; bucket in buckets) {\n      List&lt;String&gt; res = [];\n      for (Pair pair in bucket) {\n        res.add(\"${pair.key} -&gt; ${pair.val}\");\n      }\n      print(res);\n    }\n  }\n}\n</code></pre> hash_map_chaining.rs<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapChaining {\n    size: i32,\n    capacity: i32,\n    load_thres: f32,\n    extend_ratio: i32,\n    buckets: Vec&lt;Vec&lt;Pair&gt;&gt;,\n}\n\nimpl HashMapChaining {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![vec![]; 4],\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % self.capacity as usize\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fn load_factor(&amp;self) -&gt; f32 {\n        self.size as f32 / self.capacity as f32\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fn remove(&amp;mut self, key: i32) -&gt; Option&lt;String&gt; {\n        let index = self.hash_func(key);\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for i in 0..bucket.len() {\n            if bucket[i].key == key {\n                let pair = bucket.remove(i);\n                self.size -= 1;\n                return Some(pair.val);\n            }\n        }\n\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de None\n        None\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fn extend(&amp;mut self) {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let buckets_tmp = std::mem::replace(&amp;mut self.buckets, vec![]);\n\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![Vec::new(); self.capacity as usize];\n        self.size = 0;\n\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for bucket in buckets_tmp {\n            for pair in bucket {\n                self.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fn print(&amp;self) {\n        for bucket in &amp;self.buckets {\n            let mut res = Vec::new();\n            for pair in bucket {\n                res.push(format!(\"{} -&gt; {}\", pair.key, pair.val));\n            }\n            println!(\"{:?}\", res);\n        }\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n\n        let index = self.hash_func(key);\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val.clone();\n                return;\n            }\n        }\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        let pair = Pair {\n            key,\n            val: val.clone(),\n        };\n        bucket.push(pair);\n        self.size += 1;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        let index = self.hash_func(key);\n        let bucket = &amp;self.buckets[index];\n\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for pair in bucket {\n            if pair.key == key {\n                return Some(&amp;pair.val);\n            }\n        }\n\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de None\n        None\n    }\n}\n</code></pre> hash_map_chaining.c<pre><code>/* \u94fe\u8868\u8282\u70b9 */\ntypedef struct Node {\n    Pair *pair;\n    struct Node *next;\n} Node;\n\n/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\n    int size;         // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity;     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio;  // \u6269\u5bb9\u500d\u6570\n    Node **buckets;   // \u6876\u6570\u7ec4\n} HashMapChaining;\n\n/* \u6784\u9020\u51fd\u6570 */\nHashMapChaining *newHashMapChaining() {\n    HashMapChaining *hashMap = (HashMapChaining *)malloc(sizeof(HashMapChaining));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    return hashMap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapChaining(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        while (cur) {\n            Node *tmp = cur;\n            cur = cur-&gt;next;\n            free(tmp-&gt;pair);\n            free(tmp);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap);\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapChaining *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapChaining *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            return cur-&gt;pair-&gt;val;\n        }\n        cur = cur-&gt;next;\n    }\n    return \"\"; // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapChaining *hashMap, int key, const char *val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    int index = hashFunc(hashMap, key);\n    // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            strcpy(cur-&gt;pair-&gt;val, val); // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n            return;\n        }\n        cur = cur-&gt;next;\n    }\n    // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n    Pair *newPair = (Pair *)malloc(sizeof(Pair));\n    newPair-&gt;key = key;\n    strcpy(newPair-&gt;val, val);\n    Node *newNode = (Node *)malloc(sizeof(Node));\n    newNode-&gt;pair = newPair;\n    newNode-&gt;next = hashMap-&gt;buckets[index];\n    hashMap-&gt;buckets[index] = newNode;\n    hashMap-&gt;size++;\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapChaining *hashMap) {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    int oldCapacity = hashMap-&gt;capacity;\n    Node **oldBuckets = hashMap-&gt;buckets;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    hashMap-&gt;size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Node *cur = oldBuckets[i];\n        while (cur) {\n            put(hashMap, cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            Node *temp = cur;\n            cur = cur-&gt;next;\n            // \u91ca\u653e\u5185\u5b58\n            free(temp-&gt;pair);\n            free(temp);\n        }\n    }\n\n    free(oldBuckets);\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    Node *cur = hashMap-&gt;buckets[index];\n    Node *pre = NULL;\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n            if (pre) {\n                pre-&gt;next = cur-&gt;next;\n            } else {\n                hashMap-&gt;buckets[index] = cur-&gt;next;\n            }\n            // \u91ca\u653e\u5185\u5b58\n            free(cur-&gt;pair);\n            free(cur);\n            hashMap-&gt;size--;\n            return;\n        }\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        printf(\"[\");\n        while (cur) {\n            printf(\"%d -&gt; %s, \", cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            cur = cur-&gt;next;\n        }\n        printf(\"]\\n\");\n    }\n}\n</code></pre> hash_map_chaining.kt<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining() {\n    var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n    var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    val loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    val extendRatio: Int // \u6269\u5bb9\u500d\u6570\n    var buckets: MutableList&lt;MutableList&lt;Pair&gt;&gt; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = ArrayList(capacity)\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (pair in bucket) {\n            if (pair.key == key) return pair.value\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (pair in bucket) {\n            if (pair.key == key) {\n                pair.value = value\n                return\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        val pair = Pair(key, value)\n        bucket.add(pair)\n        size++\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (pair in bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair)\n                size--\n                break\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fun extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        val bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        // mutablelist \u65e0\u56fa\u5b9a\u5927\u5c0f\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (bucket in bucketsTmp) {\n            for (pair in bucket) {\n                put(pair.key, pair.value)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (bucket in buckets) {\n            val res = mutableListOf&lt;String&gt;()\n            for (pair in bucket) {\n                val k = pair.key\n                val v = pair.value\n                res.add(\"$k -&gt; $v\")\n            }\n            println(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.rb<pre><code>[class]{HashMapChaining}-[func]{}\n</code></pre> hash_map_chaining.zig<pre><code>[class]{HashMapChaining}-[func]{}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>It's worth noting that when the list is very long, the query efficiency \\(O(n)\\) is poor. At this point, the list can be converted to an \"AVL tree\" or \"Red-Black tree\" to optimize the time complexity of the query operation to \\(O(\\log n)\\).</p>"},{"location":"chapter_hashing/hash_collision/#622-open-addressing","title":"6.2.2 \u00a0 Open Addressing","text":"<p>\"Open addressing\" does not introduce additional data structures but uses \"multiple probes\" to handle hash collisions. The probing methods mainly include linear probing, quadratic probing, and double hashing.</p> <p>Let's use linear probing as an example to introduce the mechanism of open addressing hash tables.</p>"},{"location":"chapter_hashing/hash_collision/#1-linear-probing","title":"1. \u00a0 Linear Probing","text":"<p>Linear probing uses a fixed-step linear search for probing, differing from ordinary hash tables.</p> <ul> <li>Inserting Elements: Calculate the bucket index using the hash function. If the bucket already contains an element, linearly traverse forward from the conflict position (usually with a step size of \\(1\\)) until an empty bucket is found, then insert the element.</li> <li>Searching for Elements: If a hash collision is found, use the same step size to linearly traverse forward until the corresponding element is found and return <code>value</code>; if an empty bucket is encountered, it means the target element is not in the hash table, so return <code>None</code>.</li> </ul> <p>The Figure 6-6  shows the distribution of key-value pairs in an open addressing (linear probing) hash table. According to this hash function, keys with the same last two digits will be mapped to the same bucket. Through linear probing, they are stored consecutively in that bucket and the buckets below it.</p> <p></p> <p> Figure 6-6 \u00a0 Distribution of Key-Value Pairs in Open Addressing (Linear Probing) Hash Table </p> <p>However, linear probing tends to create \"clustering\". Specifically, the longer a continuous position in the array is occupied, the more likely these positions are to encounter hash collisions, further promoting the growth of these clusters and eventually leading to deterioration in the efficiency of operations.</p> <p>It's important to note that we cannot directly delete elements in an open addressing hash table. Deleting an element creates an empty bucket <code>None</code> in the array. When searching for elements, if linear probing encounters this empty bucket, it will return, making the elements below this bucket inaccessible. The program may incorrectly assume these elements do not exist, as shown in the Figure 6-7 .</p> <p></p> <p> Figure 6-7 \u00a0 Query Issues Caused by Deletion in Open Addressing </p> <p>To solve this problem, we can use a \"lazy deletion\" mechanism: instead of directly removing elements from the hash table, use a constant <code>TOMBSTONE</code> to mark the bucket. In this mechanism, both <code>None</code> and <code>TOMBSTONE</code> represent empty buckets and can hold key-value pairs. However, when linear probing encounters <code>TOMBSTONE</code>, it should continue traversing since there may still be key-value pairs below it.</p> <p>However, lazy deletion may accelerate the degradation of hash table performance. Every deletion operation produces a delete mark, and as <code>TOMBSTONE</code> increases, so does the search time, as linear probing may have to skip multiple <code>TOMBSTONE</code> to find the target element.</p> <p>Therefore, consider recording the index of the first <code>TOMBSTONE</code> encountered during linear probing and swapping the target element found with this <code>TOMBSTONE</code>. The advantage of this is that each time a query or addition is performed, the element is moved to a bucket closer to the ideal position (starting point of probing), thereby optimizing the query efficiency.</p> <p>The code below implements an open addressing (linear probing) hash table with lazy deletion. To make fuller use of the hash table space, we treat the hash table as a \"circular array,\" continuing to traverse from the beginning when the end of the array is passed.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map_open_addressing.py<pre><code>class HashMapOpenAddressing:\n    \"\"\"\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\n        self.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\n        self.load_thres = 2.0 / 3.0  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        self.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\n        self.buckets: list[Pair | None] = [None] * self.capacity  # \u6876\u6570\u7ec4\n        self.TOMBSTONE = Pair(-1, \"-1\")  # \u5220\u9664\u6807\u8bb0\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\n        return self.size / self.capacity\n\n    def find_bucket(self, key: int) -&gt; int:\n        \"\"\"\u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\"\"\"\n        index = self.hash_func(key)\n        first_tombstone = -1\n        # \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while self.buckets[index] is not None:\n            # \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if self.buckets[index].key == key:\n                # \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if first_tombstone != -1:\n                    self.buckets[first_tombstone] = self.buckets[index]\n                    self.buckets[index] = self.TOMBSTONE\n                    return first_tombstone  # \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                return index  # \u8fd4\u56de\u6876\u7d22\u5f15\n            # \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if first_tombstone == -1 and self.buckets[index] is self.TOMBSTONE:\n                first_tombstone = index\n            # \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % self.capacity\n        # \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return index if first_tombstone == -1 else first_tombstone\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n        # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            return self.buckets[index].val\n        # \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n        # \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index].val = val\n            return\n        # \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        self.buckets[index] = Pair(key, val)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n        # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index] = self.TOMBSTONE\n            self.size -= 1\n\n    def extend(self):\n        \"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n        # \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        buckets_tmp = self.buckets\n        # \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio\n        self.buckets = [None] * self.capacity\n        self.size = 0\n        # \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for pair in buckets_tmp:\n            if pair not in [None, self.TOMBSTONE]:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n        for pair in self.buckets:\n            if pair is None:\n                print(\"None\")\n            elif pair is self.TOMBSTONE:\n                print(\"TOMBSTONE\")\n            else:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> hash_map_open_addressing.cpp<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n  private:\n    int size;                             // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity = 4;                     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    const double loadThres = 2.0 / 3.0;     // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    const int extendRatio = 2;            // \u6269\u5bb9\u500d\u6570\n    vector&lt;Pair *&gt; buckets;               // \u6876\u6570\u7ec4\n    Pair *TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    HashMapOpenAddressing() : size(0), buckets(capacity, nullptr) {\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~HashMapOpenAddressing() {\n        for (Pair *pair : buckets) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                delete pair;\n            }\n        }\n        delete TOMBSTONE;\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double loadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != nullptr) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index]-&gt;key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    string get(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]-&gt;val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n        return \"\";\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]-&gt;val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            delete buckets[index];\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        vector&lt;Pair *&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = vector&lt;Pair *&gt;(capacity, nullptr);\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (Pair *pair : bucketsTmp) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                put(pair-&gt;key, pair-&gt;val);\n                delete pair;\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (Pair *pair : buckets) {\n            if (pair == nullptr) {\n                cout &lt;&lt; \"nullptr\" &lt;&lt; endl;\n            } else if (pair == TOMBSTONE) {\n                cout &lt;&lt; \"TOMBSTONE\" &lt;&lt; endl;\n            } else {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; endl;\n            }\n        }\n    }\n};\n</code></pre> hash_map_open_addressing.java<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    private int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    private int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    private final double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    private final int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n    private Pair[] buckets; // \u6876\u6570\u7ec4\n    private final Pair TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    private double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    private int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index].key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public String get(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void put(int key, String val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void remove(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    private void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        Pair[] bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (Pair pair : bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void print() {\n        for (Pair pair : buckets) {\n            if (pair == null) {\n                System.out.println(\"null\");\n            } else if (pair == TOMBSTONE) {\n                System.out.println(\"TOMBSTONE\");\n            } else {\n                System.out.println(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.cs<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n    Pair[] buckets; // \u6876\u6570\u7ec4\n    Pair TOMBSTONE = new(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    int FindBucket(int key) {\n        int index = HashFunc(key);\n        int firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index].key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public string? Get(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void Extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        Pair[] bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        foreach (Pair pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void Print() {\n        foreach (Pair pair in buckets) {\n            if (pair == null) {\n                Console.WriteLine(\"null\");\n            } else if (pair == TOMBSTONE) {\n                Console.WriteLine(\"TOMBSTONE\");\n            } else {\n                Console.WriteLine(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.go<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntype hashMapOpenAddressing struct {\n    size        int     // \u952e\u503c\u5bf9\u6570\u91cf\n    capacity    int     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    loadThres   float64 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    extendRatio int     // \u6269\u5bb9\u500d\u6570\n    buckets     []*pair // \u6876\u6570\u7ec4\n    TOMBSTONE   *pair   // \u5220\u9664\u6807\u8bb0\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\n    return &amp;hashMapOpenAddressing{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     make([]*pair, 4),\n        TOMBSTONE:   &amp;pair{-1, \"-1\"},\n    }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (h *hashMapOpenAddressing) hashFunc(key int) int {\n    return key % h.capacity // \u6839\u636e\u952e\u8ba1\u7b97\u54c8\u5e0c\u503c\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (h *hashMapOpenAddressing) loadFactor() float64 {\n    return float64(h.size) / float64(h.capacity) // \u8ba1\u7b97\u5f53\u524d\u8d1f\u8f7d\u56e0\u5b50\n}\n\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nfunc (h *hashMapOpenAddressing) findBucket(key int) int {\n    index := h.hashFunc(key) // \u83b7\u53d6\u521d\u59cb\u7d22\u5f15\n    firstTombstone := -1     // \u8bb0\u5f55\u9047\u5230\u7684\u7b2c\u4e00\u4e2aTOMBSTONE\u7684\u4f4d\u7f6e\n    for h.buckets[index] != nil {\n        if h.buckets[index].key == key {\n            if firstTombstone != -1 {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                h.buckets[firstTombstone] = h.buckets[index]\n                h.buckets[index] = h.TOMBSTONE\n                return firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n            }\n            return index // \u8fd4\u56de\u627e\u5230\u7684\u7d22\u5f15\n        }\n        if firstTombstone == -1 &amp;&amp; h.buckets[index] == h.TOMBSTONE {\n            firstTombstone = index // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\u7684\u4f4d\u7f6e\n        }\n        index = (index + 1) % h.capacity // \u7ebf\u6027\u63a2\u6d4b\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n    if firstTombstone != -1 {\n        return firstTombstone\n    }\n    return index\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) get(key int) string {\n    index := h.findBucket(key) // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        return h.buckets[index].val // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    }\n    return \"\" // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de \"\"\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) put(key int, val string) {\n    if h.loadFactor() &gt; h.loadThres {\n        h.extend() // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    }\n    index := h.findBucket(key) // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] == nil || h.buckets[index] == h.TOMBSTONE {\n        h.buckets[index] = &amp;pair{key, val} // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        h.size++\n    } else {\n        h.buckets[index].val = val // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val\n    }\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) remove(key int) {\n    index := h.findBucket(key) // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        h.buckets[index] = h.TOMBSTONE // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        h.size--\n    }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (h *hashMapOpenAddressing) extend() {\n    oldBuckets := h.buckets               // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    h.capacity *= h.extendRatio           // \u66f4\u65b0\u5bb9\u91cf\n    h.buckets = make([]*pair, h.capacity) // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    h.size = 0                            // \u91cd\u7f6e\u5927\u5c0f\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for _, pair := range oldBuckets {\n        if pair != nil &amp;&amp; pair != h.TOMBSTONE {\n            h.put(pair.key, pair.val)\n        }\n    }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (h *hashMapOpenAddressing) print() {\n    for _, pair := range h.buckets {\n        if pair == nil {\n            fmt.Println(\"nil\")\n        } else if pair == h.TOMBSTONE {\n            fmt.Println(\"TOMBSTONE\")\n        } else {\n            fmt.Printf(\"%d -&gt; %s\\n\", pair.key, pair.val)\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.swift<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n    var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    var loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    var extendRatio: Int // \u6269\u5bb9\u500d\u6570\n    var buckets: [Pair?] // \u6876\u6570\u7ec4\n    var TOMBSTONE: Pair // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: nil, count: capacity)\n        TOMBSTONE = Pair(key: -1, val: \"-1\")\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    func findBucket(key: Int) -&gt; Int {\n        var index = hashFunc(key: key)\n        var firstTombstone = -1\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while buckets[index] != nil {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if buckets[index]!.key == key {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if firstTombstone != -1 {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE {\n                firstTombstone = index\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            return buckets[index]!.val\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return nil\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index]!.val = val\n            return\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = Pair(key: key, val: val)\n        size += 1\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index] = TOMBSTONE\n            size -= 1\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    func extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        buckets = Array(repeating: nil, count: capacity)\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for pair in bucketsTmp {\n            if let pair, pair != TOMBSTONE {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    func print() {\n        for pair in buckets {\n            if pair == nil {\n                Swift.print(\"null\")\n            } else if pair == TOMBSTONE {\n                Swift.print(\"TOMBSTONE\")\n            } else {\n                Swift.print(\"\\(pair!.key) -&gt; \\(pair!.val)\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.js<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    #size; // \u952e\u503c\u5bf9\u6570\u91cf\n    #capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    #loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    #extendRatio; // \u6269\u5bb9\u500d\u6570\n    #buckets; // \u6876\u6570\u7ec4\n    #TOMBSTONE; // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\n        this.#capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n        this.#loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        this.#extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n        this.#buckets = Array(this.#capacity).fill(null); // \u6876\u6570\u7ec4\n        this.#TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    #findBucket(key) {\n        let index = this.#hashFunc(key);\n        let firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (this.#buckets[index] !== null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (this.#buckets[index].key === key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone !== -1) {\n                    this.#buckets[firstTombstone] = this.#buckets[index];\n                    this.#buckets[index] = this.#TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.#buckets[index] === this.#TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % this.#capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            return this.#buckets[index].val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key, val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        this.#buckets[index] = new Pair(key, val);\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index] = this.#TOMBSTONE;\n            this.#size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    #extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = Array(this.#capacity).fill(null);\n        this.#size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.#TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print() {\n        for (const pair of this.#buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.#TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.ts<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    private size: number; // \u952e\u503c\u5bf9\u6570\u91cf\n    private capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    private loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    private extendRatio: number; // \u6269\u5bb9\u500d\u6570\n    private buckets: Array&lt;Pair | null&gt;; // \u6876\u6570\u7ec4\n    private TOMBSTONE: Pair; // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\n        this.capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n        this.loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        this.extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n        this.buckets = Array(this.capacity).fill(null); // \u6876\u6570\u7ec4\n        this.TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private hashFunc(key: number): number {\n        return key % this.capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    private loadFactor(): number {\n        return this.size / this.capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    private findBucket(key: number): number {\n        let index = this.hashFunc(key);\n        let firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (this.buckets[index] !== null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (this.buckets[index]!.key === key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone !== -1) {\n                    this.buckets[firstTombstone] = this.buckets[index];\n                    this.buckets[index] = this.TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.buckets[index] === this.TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % this.capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key: number): string | null {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            return this.buckets[index]!.val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key: number, val: string): void {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.loadFactor() &gt; this.loadThres) {\n            this.extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index]!.val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        this.buckets[index] = new Pair(key, val);\n        this.size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key: number): void {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index] = this.TOMBSTONE;\n            this.size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    private extend(): void {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.capacity *= this.extendRatio;\n        this.buckets = Array(this.capacity).fill(null);\n        this.size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print(): void {\n        for (const pair of this.buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.dart<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n  late int _size; // \u952e\u503c\u5bf9\u6570\u91cf\n  int _capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n  double _loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n  int _extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n  late List&lt;Pair?&gt; _buckets; // \u6876\u6570\u7ec4\n  Pair _TOMBSTONE = Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  HashMapOpenAddressing() {\n    _size = 0;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n  }\n\n  /* \u54c8\u5e0c\u51fd\u6570 */\n  int hashFunc(int key) {\n    return key % _capacity;\n  }\n\n  /* \u8d1f\u8f7d\u56e0\u5b50 */\n  double loadFactor() {\n    return _size / _capacity;\n  }\n\n  /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n  int findBucket(int key) {\n    int index = hashFunc(key);\n    int firstTombstone = -1;\n    // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n    while (_buckets[index] != null) {\n      // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n      if (_buckets[index]!.key == key) {\n        // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n        if (firstTombstone != -1) {\n          _buckets[firstTombstone] = _buckets[index];\n          _buckets[index] = _TOMBSTONE;\n          return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n        }\n        return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n      }\n      // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n      if (firstTombstone == -1 &amp;&amp; _buckets[index] == _TOMBSTONE) {\n        firstTombstone = index;\n      }\n      // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n      index = (index + 1) % _capacity;\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n    return firstTombstone == -1 ? index : firstTombstone;\n  }\n\n  /* \u67e5\u8be2\u64cd\u4f5c */\n  String? get(int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      return _buckets[index]!.val;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n    return null;\n  }\n\n  /* \u6dfb\u52a0\u64cd\u4f5c */\n  void put(int key, String val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor() &gt; _loadThres) {\n      extend();\n    }\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index]!.val = val;\n      return;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n    _buckets[index] = new Pair(key, val);\n    _size++;\n  }\n\n  /* \u5220\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index] = _TOMBSTONE;\n      _size--;\n    }\n  }\n\n  /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n  void extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    List&lt;Pair?&gt; bucketsTmp = _buckets;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    _capacity *= _extendRatio;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n    _size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (Pair? pair in bucketsTmp) {\n      if (pair != null &amp;&amp; pair != _TOMBSTONE) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* \u6253\u5370\u54c8\u5e0c\u8868 */\n  void printHashMap() {\n    for (Pair? pair in _buckets) {\n      if (pair == null) {\n        print(\"null\");\n      } else if (pair == _TOMBSTONE) {\n        print(\"TOMBSTONE\");\n      } else {\n        print(\"${pair.key} -&gt; ${pair.val}\");\n      }\n    }\n  }\n}\n</code></pre> hash_map_open_addressing.rs<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapOpenAddressing {\n    size: usize,                // \u952e\u503c\u5bf9\u6570\u91cf\n    capacity: usize,            // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    load_thres: f64,            // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    extend_ratio: usize,        // \u6269\u5bb9\u500d\u6570\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;, // \u6876\u6570\u7ec4\n    TOMBSTONE: Option&lt;Pair&gt;,    // \u5220\u9664\u6807\u8bb0\n}\n\nimpl HashMapOpenAddressing {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![None; 4],\n            TOMBSTONE: Some(Pair {\n                key: -1,\n                val: \"-1\".to_string(),\n            }),\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        (key % self.capacity as i32) as usize\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fn load_factor(&amp;self) -&gt; f64 {\n        self.size as f64 / self.capacity as f64\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    fn find_bucket(&amp;mut self, key: i32) -&gt; usize {\n        let mut index = self.hash_func(key);\n        let mut first_tombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while self.buckets[index].is_some() {\n            // \u82e5\u9047\u5230 key\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if self.buckets[index].as_ref().unwrap().key == key {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u5efa\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n                if first_tombstone != -1 {\n                    self.buckets[first_tombstone as usize] = self.buckets[index].take();\n                    self.buckets[index] = self.TOMBSTONE.clone();\n                    return first_tombstone as usize; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if first_tombstone == -1 &amp;&amp; self.buckets[index] == self.TOMBSTONE {\n                first_tombstone = index as i32;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % self.capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        if first_tombstone == -1 {\n            index\n        } else {\n            first_tombstone as usize\n        }\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fn get(&amp;mut self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            return self.buckets[index].as_ref().map(|pair| &amp;pair.val as &amp;str);\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        None\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index].as_mut().unwrap().val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        self.buckets[index] = Some(Pair { key, val });\n        self.size += 1;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fn remove(&amp;mut self, key: i32) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index] = self.TOMBSTONE.clone();\n            self.size -= 1;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fn extend(&amp;mut self) {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let buckets_tmp = self.buckets.clone();\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![None; self.capacity];\n        self.size = 0;\n\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for pair in buckets_tmp {\n            if pair.is_none() || pair == self.TOMBSTONE {\n                continue;\n            }\n            let pair = pair.unwrap();\n\n            self.put(pair.key, pair.val);\n        }\n    }\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fn print(&amp;self) {\n        for pair in &amp;self.buckets {\n            if pair.is_none() {\n                println!(\"null\");\n            } else if pair == &amp;self.TOMBSTONE {\n                println!(\"TOMBSTONE\");\n            } else {\n                let pair = pair.as_ref().unwrap();\n                println!(\"{} -&gt; {}\", pair.key, pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.c<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\n    int size;         // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity;     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio;  // \u6269\u5bb9\u500d\u6570\n    Pair **buckets;   // \u6876\u6570\u7ec4\n    Pair *TOMBSTONE;  // \u5220\u9664\u6807\u8bb0\n} HashMapOpenAddressing;\n\n/* \u6784\u9020\u51fd\u6570 */\nHashMapOpenAddressing *newHashMapOpenAddressing() {\n    HashMapOpenAddressing *hashMap = (HashMapOpenAddressing *)malloc(sizeof(HashMapOpenAddressing));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Pair **)malloc(sizeof(Pair *) * hashMap-&gt;capacity);\n    hashMap-&gt;TOMBSTONE = (Pair *)malloc(sizeof(Pair));\n    hashMap-&gt;TOMBSTONE-&gt;key = -1;\n    hashMap-&gt;TOMBSTONE-&gt;val = \"-1\";\n\n    return hashMap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapOpenAddressing(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapOpenAddressing *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapOpenAddressing *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nint findBucket(HashMapOpenAddressing *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    int firstTombstone = -1;\n    // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n    while (hashMap-&gt;buckets[index] != NULL) {\n        // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        if (hashMap-&gt;buckets[index]-&gt;key == key) {\n            // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n            if (firstTombstone != -1) {\n                hashMap-&gt;buckets[firstTombstone] = hashMap-&gt;buckets[index];\n                hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n                return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n            }\n            return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n        }\n        // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n        if (firstTombstone == -1 &amp;&amp; hashMap-&gt;buckets[index] == hashMap-&gt;TOMBSTONE) {\n            firstTombstone = index;\n        }\n        // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n        index = (index + 1) % hashMap-&gt;capacity;\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n    return firstTombstone == -1 ? index : firstTombstone;\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapOpenAddressing *hashMap, int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        return hashMap-&gt;buckets[index]-&gt;val;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n    return \"\";\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapOpenAddressing *hashMap, int key, char *val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        free(hashMap-&gt;buckets[index]-&gt;val);\n        hashMap-&gt;buckets[index]-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n        strcpy(hashMap-&gt;buckets[index]-&gt;val, val);\n        hashMap-&gt;buckets[index]-&gt;val[strlen(val)] = '\\0';\n        return;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n    Pair *pair = (Pair *)malloc(sizeof(Pair));\n    pair-&gt;key = key;\n    pair-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n    strcpy(pair-&gt;val, val);\n    pair-&gt;val[strlen(val)] = '\\0';\n\n    hashMap-&gt;buckets[index] = pair;\n    hashMap-&gt;size++;\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapOpenAddressing *hashMap, int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        Pair *pair = hashMap-&gt;buckets[index];\n        free(pair-&gt;val);\n        free(pair);\n        hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n        hashMap-&gt;size--;\n    }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapOpenAddressing *hashMap) {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    Pair **bucketsTmp = hashMap-&gt;buckets;\n    int oldCapacity = hashMap-&gt;capacity;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Pair **)malloc(sizeof(Pair *) * hashMap-&gt;capacity);\n    hashMap-&gt;size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Pair *pair = bucketsTmp[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            put(hashMap, pair-&gt;key, pair-&gt;val);\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(bucketsTmp);\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair == NULL) {\n            printf(\"NULL\\n\");\n        } else if (pair == hashMap-&gt;TOMBSTONE) {\n            printf(\"TOMBSTONE\\n\");\n        } else {\n            printf(\"%d -&gt; %s\\n\", pair-&gt;key, pair-&gt;val);\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.kt<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    private var size: Int = 0 // \u952e\u503c\u5bf9\u6570\u91cf\n    private var capacity = 4 // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    private val loadThres: Double = 2.0 / 3.0 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    private val extendRatio = 2 // \u6269\u5bb9\u500d\u6570\n    private var buckets: Array&lt;Pair?&gt; // \u6876\u6570\u7ec4\n    private val TOMBSTONE = Pair(-1, \"-1\") // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init {\n        buckets = arrayOfNulls(capacity)\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    fun findBucket(key: Int): Int {\n        var index = hashFunc(key)\n        var firstTombstone = -1\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index]?.key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return if (firstTombstone == -1) index else firstTombstone\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]?.value\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]!!.value = value\n            return\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = Pair(key, value)\n        size++\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE\n            size--\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fun extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        val bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        buckets = arrayOfNulls(capacity)\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair.value)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (pair in buckets) {\n            if (pair == null) {\n                println(\"null\")\n            } else if (pair == TOMBSTONE) {\n                println(\"TOMESTOME\")\n            } else {\n                println(\"${pair.key} -&gt; ${pair.value}\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.rb<pre><code>[class]{HashMapOpenAddressing}-[func]{}\n</code></pre> hash_map_open_addressing.zig<pre><code>[class]{HashMapOpenAddressing}-[func]{}\n</code></pre>"},{"location":"chapter_hashing/hash_collision/#2-quadratic-probing","title":"2. \u00a0 Quadratic Probing","text":"<p>Quadratic probing is similar to linear probing and is one of the common strategies of open addressing. When a collision occurs, quadratic probing does not simply skip a fixed number of steps but skips \"the square of the number of probes,\" i.e., \\(1, 4, 9, \\dots\\) steps.</p> <p>Quadratic probing has the following advantages:</p> <ul> <li>Quadratic probing attempts to alleviate the clustering effect of linear probing by skipping the distance of the square of the number of probes.</li> <li>Quadratic probing skips larger distances to find empty positions, helping to distribute data more evenly.</li> </ul> <p>However, quadratic probing is not perfect:</p> <ul> <li>Clustering still exists, i.e., some positions are more likely to be occupied than others.</li> <li>Due to the growth of squares, quadratic probing may not probe the entire hash table, meaning it might not access empty buckets even if they exist in the hash table.</li> </ul>"},{"location":"chapter_hashing/hash_collision/#3-double-hashing","title":"3. \u00a0 Double Hashing","text":"<p>As the name suggests, the double hashing method uses multiple hash functions \\(f_1(x)\\), \\(f_2(x)\\), \\(f_3(x)\\), \\(\\dots\\) for probing.</p> <ul> <li>Inserting Elements: If hash function \\(f_1(x)\\) encounters a conflict, try \\(f_2(x)\\), and so on, until an empty position is found and the element is inserted.</li> <li>Searching for Elements: Search in the same order of hash functions until the target element is found and returned; if an empty position is encountered or all hash functions have been tried, it indicates the element is not in the hash table, then return <code>None</code>.</li> </ul> <p>Compared to linear probing, double hashing is less prone to clustering but involves additional computation for multiple hash functions.</p> <p>Tip</p> <p>Please note that open addressing (linear probing, quadratic probing, and double hashing) hash tables all have the issue of \"not being able to directly delete elements.\"</p>"},{"location":"chapter_hashing/hash_collision/#623-choice-of-programming-languages","title":"6.2.3 \u00a0 Choice of Programming Languages","text":"<p>Various programming languages have adopted different hash table implementation strategies, here are a few examples:</p> <ul> <li>Python uses open addressing. The <code>dict</code> dictionary uses pseudo-random numbers for probing.</li> <li>Java uses separate chaining. Since JDK 1.8, when the array length in <code>HashMap</code> reaches 64 and the length of a linked list reaches 8, the linked list is converted to a red-black tree to improve search performance.</li> <li>Go uses separate chaining. Go stipulates that each bucket can store up to 8 key-value pairs, and if the capacity is exceeded, an overflow bucket is connected; when there are too many overflow buckets, a special equal-size expansion operation is performed to ensure performance.</li> </ul>"},{"location":"chapter_hashing/hash_map/","title":"6.1 \u00a0 Hash Table","text":"<p>A \"hash table\", also known as a \"hash map\", achieves efficient element querying by establishing a mapping between keys and values. Specifically, when we input a <code>key</code> into the hash table, we can retrieve the corresponding <code>value</code> in \\(O(1)\\) time.</p> <p>As shown in the Figure 6-1 , given \\(n\\) students, each with two pieces of data: \"name\" and \"student number\". If we want to implement a query feature that returns the corresponding name when given a student number, we can use the hash table shown in the Figure 6-1 .</p> <p></p> <p> Figure 6-1 \u00a0 Abstract representation of a hash table </p> <p>Apart from hash tables, arrays and linked lists can also be used to implement querying functions. Their efficiency is compared in the Table 6-1 .</p> <ul> <li>Adding Elements: Simply add the element to the end of the array (or linked list), using \\(O(1)\\) time.</li> <li>Querying Elements: Since the array (or linked list) is unordered, it requires traversing all the elements, using \\(O(n)\\) time.</li> <li>Deleting Elements: First, locate the element, then delete it from the array (or linked list), using \\(O(n)\\) time.</li> </ul> <p> Table 6-1 \u00a0 Comparison of Element Query Efficiency </p> Array Linked List Hash Table Find Element \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) Add Element \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) Delete Element \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) <p>Observations reveal that the time complexity for adding, deleting, and querying in a hash table is \\(O(1)\\), which is highly efficient.</p>"},{"location":"chapter_hashing/hash_map/#611-common-operations-of-hash-table","title":"6.1.1 \u00a0 Common Operations of Hash Table","text":"<p>Common operations of a hash table include initialization, querying, adding key-value pairs, and deleting key-value pairs, etc. Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig hash_map.py<pre><code># Initialize hash table\nhmap: dict = {}\n\n# Add operation\n# Add key-value pair (key, value) to the hash table\nhmap[12836] = \"Xiao Ha\"\nhmap[15937] = \"Xiao Luo\"\nhmap[16750] = \"Xiao Suan\"\nhmap[13276] = \"Xiao Fa\"\nhmap[10583] = \"Xiao Ya\"\n\n# Query operation\n# Input key into hash table, get value\nname: str = hmap[15937]\n\n# Delete operation\n# Delete key-value pair (key, value) from hash table\nhmap.pop(10583)\n</code></pre> hash_map.cpp<pre><code>/* Initialize hash table */\nunordered_map&lt;int, string&gt; map;\n\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nmap[12836] = \"Xiao Ha\";\nmap[15937] = \"Xiao Luo\";\nmap[16750] = \"Xiao Suan\";\nmap[13276] = \"Xiao Fa\";\nmap[10583] = \"Xiao Ya\";\n\n/* Query operation */\n// Input key into hash table, get value\nstring name = map[15937];\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.erase(10583);\n</code></pre> hash_map.java<pre><code>/* Initialize hash table */\nMap&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nmap.put(12836, \"Xiao Ha\");   \nmap.put(15937, \"Xiao Luo\");   \nmap.put(16750, \"Xiao Suan\");   \nmap.put(13276, \"Xiao Fa\");\nmap.put(10583, \"Xiao Ya\");\n\n/* Query operation */\n// Input key into hash table, get value\nString name = map.get(15937);\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.remove(10583);\n</code></pre> hash_map.cs<pre><code>/* Initialize hash table */\nDictionary&lt;int, string&gt; map = new() {\n    /* Add operation */\n    // Add key-value pair (key, value) to the hash table\n    { 12836, \"Xiao Ha\" },\n    { 15937, \"Xiao Luo\" },\n    { 16750, \"Xiao Suan\" },\n    { 13276, \"Xiao Fa\" },\n    { 10583, \"Xiao Ya\" }\n};\n\n/* Query operation */\n// Input key into hash table, get value\nstring name = map[15937];\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.Remove(10583);\n</code></pre> hash_map_test.go<pre><code>/* Initialize hash table */\nhmap := make(map[int]string)\n\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nhmap[12836] = \"Xiao Ha\"\nhmap[15937] = \"Xiao Luo\"\nhmap[16750] = \"Xiao Suan\"\nhmap[13276] = \"Xiao Fa\"\nhmap[10583] = \"Xiao Ya\"\n\n/* Query operation */\n// Input key into hash table, get value\nname := hmap[15937]\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\ndelete(hmap, 10583)\n</code></pre> hash_map.swift<pre><code>/* Initialize hash table */\nvar map: [Int: String] = [:]\n\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nmap[12836] = \"Xiao Ha\"\nmap[15937] = \"Xiao Luo\"\nmap[16750] = \"Xiao Suan\"\nmap[13276] = \"Xiao Fa\"\nmap[10583] = \"Xiao Ya\"\n\n/* Query operation */\n// Input key into hash table, get value\nlet name = map[15937]!\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.removeValue(forKey: 10583)\n</code></pre> hash_map.js<pre><code>/* Initialize hash table */\nconst map = new Map();\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nmap.set(12836, 'Xiao Ha');\nmap.set(15937, 'Xiao Luo');\nmap.set(16750, 'Xiao Suan');\nmap.set(13276, 'Xiao Fa');\nmap.set(10583, 'Xiao Ya');\n\n/* Query operation */\n// Input key into hash table, get value\nlet name = map.get(15937);\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.delete(10583);\n</code></pre> hash_map.ts<pre><code>/* Initialize hash table */\nconst map = new Map&lt;number, string&gt;();\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nmap.set(12836, 'Xiao Ha');\nmap.set(15937, 'Xiao Luo');\nmap.set(16750, 'Xiao Suan');\nmap.set(13276, 'Xiao Fa');\nmap.set(10583, 'Xiao Ya');\nconsole.info('\\nAfter adding, the hash table is\\nKey -&gt; Value');\nconsole.info(map);\n\n/* Query operation */\n// Input key into hash table, get value\nlet name = map.get(15937);\nconsole.info('\\nInput student number 15937, query name ' + name);\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.delete(10583);\nconsole.info('\\nAfter deleting 10583, the hash table is\\nKey -&gt; Value');\nconsole.info(map);\n</code></pre> hash_map.dart<pre><code>/* Initialize hash table */\nMap&lt;int, String&gt; map = {};\n\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nmap[12836] = \"Xiao Ha\";\nmap[15937] = \"Xiao Luo\";\nmap[16750] = \"Xiao Suan\";\nmap[13276] = \"Xiao Fa\";\nmap[10583] = \"Xiao Ya\";\n\n/* Query operation */\n// Input key into hash table, get value\nString name = map[15937];\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.remove(10583);\n</code></pre> hash_map.rs<pre><code>use std::collections::HashMap;\n\n/* Initialize hash table */\nlet mut map: HashMap&lt;i32, String&gt; = HashMap::new();\n\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nmap.insert(12836, \"Xiao Ha\".to_string());\nmap.insert(15937, \"Xiao Luo\".to_string());\nmap.insert(16750, \"Xiao Suan\".to_string());\nmap.insert(13279, \"Xiao Fa\".to_string());\nmap.insert(10583, \"Xiao Ya\".to_string());\n\n/* Query operation */\n// Input key into hash table, get value\nlet _name: Option&lt;&amp;String&gt; = map.get(&amp;15937);\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nlet _removed_value: Option&lt;String&gt; = map.remove(&amp;10583);\n</code></pre> hash_map.c<pre><code>// C does not provide a built-in hash table\n</code></pre> hash_map.kt<pre><code>\n</code></pre> hash_map.zig<pre><code>\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>There are three common ways to traverse a hash table: traversing key-value pairs, keys, and values. Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig hash_map.py<pre><code># Traverse hash table\n# Traverse key-value pairs key-&gt;value\nfor key, value in hmap.items():\n    print(key, \"-&gt;\", value)\n# Traverse keys only\nfor key in hmap.keys():\n    print(key)\n# Traverse values only\nfor value in hmap.values():\n    print(value)\n</code></pre> hash_map.cpp<pre><code>/* Traverse hash table */\n// Traverse key-value pairs key-&gt;value\nfor (auto kv: map) {\n    cout &lt;&lt; kv.first &lt;&lt; \" -&gt; \" &lt;&lt; kv.second &lt;&lt; endl;\n}\n// Traverse using iterator key-&gt;value\nfor (auto iter = map.begin(); iter != map.end(); iter++) {\n    cout &lt;&lt; iter-&gt;first &lt;&lt; \"-&gt;\" &lt;&lt; iter-&gt;second &lt;&lt; endl;\n}\n</code></pre> hash_map.java<pre><code>/* Traverse hash table */\n// Traverse key-value pairs key-&gt;value\nfor (Map.Entry&lt;Integer, String&gt; kv: map.entrySet()) {\n    System.out.println(kv.getKey() + \" -&gt; \" + kv.getValue());\n}\n// Traverse keys only\nfor (int key: map.keySet()) {\n    System.out.println(key);\n}\n// Traverse values only\nfor (String val: map.values()) {\n    System.out.println(val);\n}\n</code></pre> hash_map.cs<pre><code>/* Traverse hash table */\n// Traverse key-value pairs Key-&gt;Value\nforeach (var kv in map) {\n    Console.WriteLine(kv.Key + \" -&gt; \" + kv.Value);\n}\n// Traverse keys only\nforeach (int key in map.Keys) {\n    Console.WriteLine(key);\n}\n// Traverse values only\nforeach (string val in map.Values) {\n    Console.WriteLine(val);\n}\n</code></pre> hash_map_test.go<pre><code>/* Traverse hash table */\n// Traverse key-value pairs key-&gt;value\nfor key, value := range hmap {\n    fmt.Println(key, \"-&gt;\", value)\n}\n// Traverse keys only\nfor key := range hmap {\n    fmt.Println(key)\n}\n// Traverse values only\nfor _, value := range hmap {\n    fmt.Println(value)\n}\n</code></pre> hash_map.swift<pre><code>/* Traverse hash table */\n// Traverse key-value pairs Key-&gt;Value\nfor (key, value) in map {\n    print(\"\\(key) -&gt; \\(value)\")\n}\n// Traverse keys only\nfor key in map.keys {\n    print(key)\n}\n// Traverse values only\nfor value in map.values {\n    print(value)\n}\n</code></pre> hash_map.js<pre><code>/* Traverse hash table */\nconsole.info('\\nTraverse key-value pairs Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\nTraverse keys only Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\nTraverse values only Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.ts<pre><code>/* Traverse hash table */\nconsole.info('\\nTraverse key-value pairs Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\nTraverse keys only Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\nTraverse values only Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.dart<pre><code>/* Traverse hash table */\n// Traverse key-value pairs Key-&gt;Value\nmap.forEach((key, value) {\nprint('$key -&gt; $value');\n});\n\n// Traverse keys only Key\nmap.keys.forEach((key) {\nprint(key);\n});\n\n// Traverse values only Value\nmap.values.forEach((value) {\nprint(value);\n});\n</code></pre> hash_map.rs<pre><code>/* Traverse hash table */\n// Traverse key-value pairs Key-&gt;Value\nfor (key, value) in &amp;map {\n    println!(\"{key} -&gt; {value}\");\n}\n\n// Traverse keys only Key\nfor key in map.keys() {\n    println!(\"{key}\"); \n}\n\n// Traverse values only Value\nfor value in map.values() {\n    println!(\"{value}\");\n}\n</code></pre> hash_map.c<pre><code>// C does not provide a built-in hash table\n</code></pre> hash_map.kt<pre><code>\n</code></pre> hash_map.zig<pre><code>// Zig example is not provided\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_hashing/hash_map/#612-simple-implementation-of-hash-table","title":"6.1.2 \u00a0 Simple Implementation of Hash Table","text":"<p>First, let's consider the simplest case: implementing a hash table using just an array. In the hash table, each empty slot in the array is called a \"bucket\", and each bucket can store one key-value pair. Therefore, the query operation involves finding the bucket corresponding to the <code>key</code> and retrieving the <code>value</code> from it.</p> <p>So, how do we locate the appropriate bucket based on the <code>key</code>? This is achieved through a \"hash function\". The role of the hash function is to map a larger input space to a smaller output space. In a hash table, the input space is all possible keys, and the output space is all buckets (array indices). In other words, input a <code>key</code>, and we can use the hash function to determine the storage location of the corresponding key-value pair in the array.</p> <p>The calculation process of the hash function for a given <code>key</code> is divided into the following two steps:</p> <ol> <li>Calculate the hash value using a certain hash algorithm <code>hash()</code>.</li> <li>Take the modulus of the hash value with the number of buckets (array length) <code>capacity</code> to obtain the array index <code>index</code>.</li> </ol> <pre><code>index = hash(key) % capacity\n</code></pre> <p>Afterward, we can use <code>index</code> to access the corresponding bucket in the hash table and thereby retrieve the <code>value</code>.</p> <p>Assuming array length <code>capacity = 100</code> and hash algorithm <code>hash(key) = key</code>, the hash function is <code>key % 100</code>. The Figure 6-2  uses <code>key</code> as the student number and <code>value</code> as the name to demonstrate the working principle of the hash function.</p> <p></p> <p> Figure 6-2 \u00a0 Working principle of hash function </p> <p>The following code implements a simple hash table. Here, we encapsulate <code>key</code> and <code>value</code> into a class <code>Pair</code> to represent the key-value pair.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_hash_map.py<pre><code>class Pair:\n    \"\"\"\u952e\u503c\u5bf9\"\"\"\n\n    def __init__(self, key: int, val: str):\n        self.key = key\n        self.val = val\n\nclass ArrayHashMap:\n    \"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        # \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        self.buckets: list[Pair | None] = [None] * 100\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n        index = key % 100\n        return index\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n        index: int = self.hash_func(key)\n        pair: Pair = self.buckets[index]\n        if pair is None:\n            return None\n        return pair.val\n\n    def put(self, key: int, val: str):\n        \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n        pair = Pair(key, val)\n        index: int = self.hash_func(key)\n        self.buckets[index] = pair\n\n    def remove(self, key: int):\n        \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n        index: int = self.hash_func(key)\n        # \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\n        self.buckets[index] = None\n\n    def entry_set(self) -&gt; list[Pair]:\n        \"\"\"\u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\"\"\"\n        result: list[Pair] = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair)\n        return result\n\n    def key_set(self) -&gt; list[int]:\n        \"\"\"\u83b7\u53d6\u6240\u6709\u952e\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.key)\n        return result\n\n    def value_set(self) -&gt; list[str]:\n        \"\"\"\u83b7\u53d6\u6240\u6709\u503c\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.val)\n        return result\n\n    def print(self):\n        \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n        for pair in self.buckets:\n            if pair is not None:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> array_hash_map.cpp<pre><code>/* \u952e\u503c\u5bf9 */\nstruct Pair {\n  public:\n    int key;\n    string val;\n    Pair(int key, string val) {\n        this-&gt;key = key;\n        this-&gt;val = val;\n    }\n};\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n  private:\n    vector&lt;Pair *&gt; buckets;\n\n  public:\n    ArrayHashMap() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = vector&lt;Pair *&gt;(100);\n    }\n\n    ~ArrayHashMap() {\n        // \u91ca\u653e\u5185\u5b58\n        for (const auto &amp;bucket : buckets) {\n            delete bucket;\n        }\n        buckets.clear();\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    string get(int key) {\n        int index = hashFunc(key);\n        Pair *pair = buckets[index];\n        if (pair == nullptr)\n            return \"\";\n        return pair-&gt;val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, string val) {\n        Pair *pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        // \u91ca\u653e\u5185\u5b58\u5e76\u7f6e\u4e3a nullptr\n        delete buckets[index];\n        buckets[index] = nullptr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    vector&lt;Pair *&gt; pairSet() {\n        vector&lt;Pair *&gt; pairSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                pairSet.push_back(pair);\n            }\n        }\n        return pairSet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    vector&lt;int&gt; keySet() {\n        vector&lt;int&gt; keySet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                keySet.push_back(pair-&gt;key);\n            }\n        }\n        return keySet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    vector&lt;string&gt; valueSet() {\n        vector&lt;string&gt; valueSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                valueSet.push_back(pair-&gt;val);\n            }\n        }\n        return valueSet;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (Pair *kv : pairSet()) {\n            cout &lt;&lt; kv-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; kv-&gt;val &lt;&lt; endl;\n        }\n    }\n};\n</code></pre> array_hash_map.java<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair {\n    public int key;\n    public String val;\n\n    public Pair(int key, String val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private List&lt;Pair&gt; buckets;\n\n    public ArrayHashMap() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = new ArrayList&lt;&gt;();\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.add(null);\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public String get(int key) {\n        int index = hashFunc(key);\n        Pair pair = buckets.get(index);\n        if (pair == null)\n            return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void put(int key, String val) {\n        Pair pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets.set(index, pair);\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void remove(int key) {\n        int index = hashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets.set(index, null);\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    public List&lt;Pair&gt; pairSet() {\n        List&lt;Pair&gt; pairSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                pairSet.add(pair);\n        }\n        return pairSet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    public List&lt;Integer&gt; keySet() {\n        List&lt;Integer&gt; keySet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                keySet.add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    public List&lt;String&gt; valueSet() {\n        List&lt;String&gt; valueSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                valueSet.add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void print() {\n        for (Pair kv : pairSet()) {\n            System.out.println(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.cs<pre><code>/* \u952e\u503c\u5bf9 int-&gt;string */\nclass Pair(int key, string val) {\n    public int key = key;\n    public string val = val;\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    List&lt;Pair?&gt; buckets;\n    public ArrayHashMap() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = [];\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.Add(null);\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int HashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        Pair? pair = buckets[index];\n        if (pair == null) return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        Pair pair = new(key, val);\n        int index = HashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = null;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    public List&lt;Pair&gt; PairSet() {\n        List&lt;Pair&gt; pairSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                pairSet.Add(pair);\n        }\n        return pairSet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    public List&lt;int&gt; KeySet() {\n        List&lt;int&gt; keySet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                keySet.Add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    public List&lt;string&gt; ValueSet() {\n        List&lt;string&gt; valueSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                valueSet.Add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void Print() {\n        foreach (Pair kv in PairSet()) {\n            Console.WriteLine(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.go<pre><code>/* \u952e\u503c\u5bf9 */\ntype pair struct {\n    key int\n    val string\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntype arrayHashMap struct {\n    buckets []*pair\n}\n\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nfunc newArrayHashMap() *arrayHashMap {\n    // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n    buckets := make([]*pair, 100)\n    return &amp;arrayHashMap{buckets: buckets}\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (a *arrayHashMap) hashFunc(key int) int {\n    index := key % 100\n    return index\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (a *arrayHashMap) get(key int) string {\n    index := a.hashFunc(key)\n    pair := a.buckets[index]\n    if pair == nil {\n        return \"Not Found\"\n    }\n    return pair.val\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (a *arrayHashMap) put(key int, val string) {\n    pair := &amp;pair{key: key, val: val}\n    index := a.hashFunc(key)\n    a.buckets[index] = pair\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (a *arrayHashMap) remove(key int) {\n    index := a.hashFunc(key)\n    // \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\n    a.buckets[index] = nil\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e\u5bf9 */\nfunc (a *arrayHashMap) pairSet() []*pair {\n    var pairs []*pair\n    for _, pair := range a.buckets {\n        if pair != nil {\n            pairs = append(pairs, pair)\n        }\n    }\n    return pairs\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc (a *arrayHashMap) keySet() []int {\n    var keys []int\n    for _, pair := range a.buckets {\n        if pair != nil {\n            keys = append(keys, pair.key)\n        }\n    }\n    return keys\n}\n\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc (a *arrayHashMap) valueSet() []string {\n    var values []string\n    for _, pair := range a.buckets {\n        if pair != nil {\n            values = append(values, pair.val)\n        }\n    }\n    return values\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (a *arrayHashMap) print() {\n    for _, pair := range a.buckets {\n        if pair != nil {\n            fmt.Println(pair.key, \"-&gt;\", pair.val)\n        }\n    }\n}\n</code></pre> array_hash_map.swift<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair: Equatable {\n    public var key: Int\n    public var val: String\n\n    public init(key: Int, val: String) {\n        self.key = key\n        self.val = val\n    }\n\n    public static func == (lhs: Pair, rhs: Pair) -&gt; Bool {\n        lhs.key == rhs.key &amp;&amp; lhs.val == rhs.val\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private var buckets: [Pair?]\n\n    init() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = Array(repeating: nil, count: 100)\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private func hashFunc(key: Int) -&gt; Int {\n        let index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let pair = buckets[index]\n        return pair?.val\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        let pair = Pair(key: key, val: val)\n        let index = hashFunc(key: key)\n        buckets[index] = pair\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        // \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = nil\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    func pairSet() -&gt; [Pair] {\n        buckets.compactMap { $0 }\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    func keySet() -&gt; [Int] {\n        buckets.compactMap { $0?.key }\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    func valueSet() -&gt; [String] {\n        buckets.compactMap { $0?.val }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    func print() {\n        for pair in pairSet() {\n            Swift.print(\"\\(pair.key) -&gt; \\(pair.val)\")\n        }\n    }\n}\n</code></pre> array_hash_map.js<pre><code>/* \u952e\u503c\u5bf9 Number -&gt; String */\nclass Pair {\n    constructor(key, val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    #buckets;\n    constructor() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        this.#buckets = new Array(100).fill(null);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key) {\n        return key % 100;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key) {\n        let index = this.#hashFunc(key);\n        let pair = this.#buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    set(key, val) {\n        let index = this.#hashFunc(key);\n        this.#buckets[index] = new Pair(key, val);\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    delete(key) {\n        let index = this.#hashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        this.#buckets[index] = null;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    entries() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    keys() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    values() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.ts<pre><code>/* \u952e\u503c\u5bf9 Number -&gt; String */\nclass Pair {\n    public key: number;\n    public val: string;\n\n    constructor(key: number, val: string) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private readonly buckets: (Pair | null)[];\n\n    constructor() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        this.buckets = new Array(100).fill(null);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private hashFunc(key: number): number {\n        return key % 100;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public get(key: number): string | null {\n        let index = this.hashFunc(key);\n        let pair = this.buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public set(key: number, val: string) {\n        let index = this.hashFunc(key);\n        this.buckets[index] = new Pair(key, val);\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public delete(key: number) {\n        let index = this.hashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        this.buckets[index] = null;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    public entries(): (Pair | null)[] {\n        let arr: (Pair | null)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    public keys(): (number | undefined)[] {\n        let arr: (number | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    public values(): (string | undefined)[] {\n        let arr: (string | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.dart<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair {\n  int key;\n  String val;\n  Pair(this.key, this.val);\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n  late List&lt;Pair?&gt; _buckets;\n\n  ArrayHashMap() {\n    // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n    _buckets = List.filled(100, null);\n  }\n\n  /* \u54c8\u5e0c\u51fd\u6570 */\n  int _hashFunc(int key) {\n    final int index = key % 100;\n    return index;\n  }\n\n  /* \u67e5\u8be2\u64cd\u4f5c */\n  String? get(int key) {\n    final int index = _hashFunc(key);\n    final Pair? pair = _buckets[index];\n    if (pair == null) {\n      return null;\n    }\n    return pair.val;\n  }\n\n  /* \u6dfb\u52a0\u64cd\u4f5c */\n  void put(int key, String val) {\n    final Pair pair = Pair(key, val);\n    final int index = _hashFunc(key);\n    _buckets[index] = pair;\n  }\n\n  /* \u5220\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    final int index = _hashFunc(key);\n    _buckets[index] = null;\n  }\n\n  /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n  List&lt;Pair&gt; pairSet() {\n    List&lt;Pair&gt; pairSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        pairSet.add(pair);\n      }\n    }\n    return pairSet;\n  }\n\n  /* \u83b7\u53d6\u6240\u6709\u952e */\n  List&lt;int&gt; keySet() {\n    List&lt;int&gt; keySet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        keySet.add(pair.key);\n      }\n    }\n    return keySet;\n  }\n\n  /* \u83b7\u53d6\u6240\u6709\u503c */\n  List&lt;String&gt; values() {\n    List&lt;String&gt; valueSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        valueSet.add(pair.val);\n      }\n    }\n    return valueSet;\n  }\n\n  /* \u6253\u5370\u54c8\u5e0c\u8868 */\n  void printHashMap() {\n    for (final Pair kv in pairSet()) {\n      print(\"${kv.key} -&gt; ${kv.val}\");\n    }\n  }\n}\n</code></pre> array_hash_map.rs<pre><code>/* \u952e\u503c\u5bf9 */\n#[derive(Debug, Clone, PartialEq)]\npub struct Pair {\n    pub key: i32,\n    pub val: String,\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\npub struct ArrayHashMap {\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;,\n}\n\nimpl ArrayHashMap {\n    pub fn new() -&gt; ArrayHashMap {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        Self {\n            buckets: vec![None; 100],\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % 100\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    pub fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;String&gt; {\n        let index = self.hash_func(key);\n        self.buckets[index].as_ref().map(|pair| &amp;pair.val)\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    pub fn put(&amp;mut self, key: i32, val: &amp;str) {\n        let index = self.hash_func(key);\n        self.buckets[index] = Some(Pair {\n            key,\n            val: val.to_string(),\n        });\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    pub fn remove(&amp;mut self, key: i32) {\n        let index = self.hash_func(key);\n        // \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\n        self.buckets[index] = None;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    pub fn entry_set(&amp;self) -&gt; Vec&lt;&amp;Pair&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref())\n            .collect()\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    pub fn key_set(&amp;self) -&gt; Vec&lt;&amp;i32&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.key))\n            .collect()\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    pub fn value_set(&amp;self) -&gt; Vec&lt;&amp;String&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.val))\n            .collect()\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    pub fn print(&amp;self) {\n        for pair in self.entry_set() {\n            println!(\"{} -&gt; {}\", pair.key, pair.val);\n        }\n    }\n}\n</code></pre> array_hash_map.c<pre><code>/* \u952e\u503c\u5bf9 int-&gt;string */\ntypedef struct {\n    int key;\n    char *val;\n} Pair;\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntypedef struct {\n    Pair *buckets[HASHTABLE_CAPACITY];\n} ArrayHashMap;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayHashMap *newArrayHashMap() {\n    ArrayHashMap *hmap = malloc(sizeof(ArrayHashMap));\n    return hmap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayHashMap(ArrayHashMap *hmap) {\n    for (int i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            free(hmap-&gt;buckets[i]-&gt;val);\n            free(hmap-&gt;buckets[i]);\n        }\n    }\n    free(hmap);\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(ArrayHashMap *hmap, const int key, const char *val) {\n    Pair *Pair = malloc(sizeof(Pair));\n    Pair-&gt;key = key;\n    Pair-&gt;val = malloc(strlen(val) + 1);\n    strcpy(Pair-&gt;val, val);\n\n    int index = hashFunc(key);\n    hmap-&gt;buckets[index] = Pair;\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(ArrayHashMap *hmap, const int key) {\n    int index = hashFunc(key);\n    free(hmap-&gt;buckets[index]-&gt;val);\n    free(hmap-&gt;buckets[index]);\n    hmap-&gt;buckets[index] = NULL;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nvoid pairSet(ArrayHashMap *hmap, MapSet *set) {\n    Pair *entries;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    entries = malloc(sizeof(Pair) * total);\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            entries[index].key = hmap-&gt;buckets[i]-&gt;key;\n            entries[index].val = malloc(strlen(hmap-&gt;buckets[i]-&gt;val) + 1);\n            strcpy(entries[index].val, hmap-&gt;buckets[i]-&gt;val);\n            index++;\n        }\n    }\n    set-&gt;set = entries;\n    set-&gt;len = total;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e */\nvoid keySet(ArrayHashMap *hmap, MapSet *set) {\n    int *keys;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    keys = malloc(total * sizeof(int));\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            keys[index] = hmap-&gt;buckets[i]-&gt;key;\n            index++;\n        }\n    }\n    set-&gt;set = keys;\n    set-&gt;len = total;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvoid valueSet(ArrayHashMap *hmap, MapSet *set) {\n    char **vals;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    vals = malloc(total * sizeof(char *));\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            vals[index] = hmap-&gt;buckets[i]-&gt;val;\n            index++;\n        }\n    }\n    set-&gt;set = vals;\n    set-&gt;len = total;\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(ArrayHashMap *hmap) {\n    int i;\n    MapSet set;\n    pairSet(hmap, &amp;set);\n    Pair *entries = (Pair *)set.set;\n    for (i = 0; i &lt; set.len; i++) {\n        printf(\"%d -&gt; %s\\n\", entries[i].key, entries[i].val);\n    }\n    free(set.set);\n}\n</code></pre> array_hash_map.kt<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair(\n    var key: Int,\n    var value: String\n)\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private val buckets = arrayOfNulls&lt;Pair&gt;(100)\n\n    init {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        for (i in 0..&lt;100) {\n            buckets[i] = null\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        val index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val pair = buckets[index] ?: return null\n        return pair.value\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        val pair = Pair(key, value)\n        val index = hashFunc(key)\n        buckets[index] = pair\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = null\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    fun pairSet(): MutableList&lt;Pair&gt; {\n        val pairSet = ArrayList&lt;Pair&gt;()\n        for (pair in buckets) {\n            if (pair != null) pairSet.add(pair)\n        }\n        return pairSet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    fun keySet(): MutableList&lt;Int&gt; {\n        val keySet = ArrayList&lt;Int&gt;()\n        for (pair in buckets) {\n            if (pair != null) keySet.add(pair.key)\n        }\n        return keySet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    fun valueSet(): MutableList&lt;String&gt; {\n        val valueSet = ArrayList&lt;String&gt;()\n        for (pair in buckets) {\n            pair?.let { valueSet.add(it.value) }\n        }\n        return valueSet\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (kv in pairSet()) {\n            val key = kv.key\n            val value = kv.value\n            println(\"${key}-&gt;${value}\")\n        }\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private val buckets = arrayOfNulls&lt;Pair&gt;(100)\n\n    init {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        for (i in 0..&lt;100) {\n            buckets[i] = null\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        val index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val pair = buckets[index] ?: return null\n        return pair.value\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        val pair = Pair(key, value)\n        val index = hashFunc(key)\n        buckets[index] = pair\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = null\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    fun pairSet(): MutableList&lt;Pair&gt; {\n        val pairSet = ArrayList&lt;Pair&gt;()\n        for (pair in buckets) {\n            if (pair != null) pairSet.add(pair)\n        }\n        return pairSet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    fun keySet(): MutableList&lt;Int&gt; {\n        val keySet = ArrayList&lt;Int&gt;()\n        for (pair in buckets) {\n            if (pair != null) keySet.add(pair.key)\n        }\n        return keySet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    fun valueSet(): MutableList&lt;String&gt; {\n        val valueSet = ArrayList&lt;String&gt;()\n        for (pair in buckets) {\n            pair?.let { valueSet.add(it.value) }\n        }\n        return valueSet\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (kv in pairSet()) {\n            val key = kv.key\n            val value = kv.value\n            println(\"${key}-&gt;${value}\")\n        }\n    }\n}\n</code></pre> array_hash_map.rb<pre><code>[class]{Pair}-[func]{}\n\n[class]{ArrayHashMap}-[func]{}\n</code></pre> array_hash_map.zig<pre><code>// \u952e\u503c\u5bf9\nconst Pair = struct {\n    key: usize = undefined,\n    val: []const u8 = undefined,\n\n   pub fn init(key: usize, val: []const u8) Pair {\n        return Pair {\n            .key = key,\n            .val = val,\n        };\n    }\n};\n\n// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\nfn ArrayHashMap(comptime T: type) type {\n    return struct {\n        bucket: ?std.ArrayList(?T) = null,\n        mem_allocator: std.mem.Allocator = undefined,\n\n        const Self = @This();\n\n        // \u6784\u9020\u51fd\u6570\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            self.mem_allocator = allocator;\n            // \u521d\u59cb\u5316\u4e00\u4e2a\u957f\u5ea6\u4e3a 100 \u7684\u6876\uff08\u6570\u7ec4\uff09\n            self.bucket = std.ArrayList(?T).init(self.mem_allocator);\n            var i: i32 = 0;\n            while (i &lt; 100) : (i += 1) {\n                try self.bucket.?.append(null);\n            }\n        }\n\n        // \u6790\u6784\u51fd\u6570\n        pub fn deinit(self: *Self) void {\n            if (self.bucket != null) self.bucket.?.deinit();\n        }\n\n        // \u54c8\u5e0c\u51fd\u6570\n        fn hashFunc(key: usize) usize {\n            var index = key % 100;\n            return index;\n        }\n\n        // \u67e5\u8be2\u64cd\u4f5c\n        pub fn get(self: *Self, key: usize) []const u8 {\n            var index = hashFunc(key);\n            var pair = self.bucket.?.items[index];\n            return pair.?.val;\n        }\n\n        // \u6dfb\u52a0\u64cd\u4f5c\n        pub fn put(self: *Self, key: usize, val: []const u8) !void {\n            var pair = Pair.init(key, val);\n            var index = hashFunc(key);\n            self.bucket.?.items[index] = pair;\n        }\n\n        // \u5220\u9664\u64cd\u4f5c\n        pub fn remove(self: *Self, key: usize) !void {\n            var index = hashFunc(key);\n            // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n            self.bucket.?.items[index] = null;\n        }       \n\n        // \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\n        pub fn pairSet(self: *Self) !std.ArrayList(T) {\n            var entry_set = std.ArrayList(T).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try entry_set.append(item.?);\n            }\n            return entry_set;\n        }  \n\n        // \u83b7\u53d6\u6240\u6709\u952e\n        pub fn keySet(self: *Self) !std.ArrayList(usize) {\n            var key_set = std.ArrayList(usize).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try key_set.append(item.?.key);\n            }\n            return key_set;\n        }  \n\n        // \u83b7\u53d6\u6240\u6709\u503c\n        pub fn valueSet(self: *Self) !std.ArrayList([]const u8) {\n            var value_set = std.ArrayList([]const u8).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try value_set.append(item.?.val);\n            }\n            return value_set;\n        }\n\n        // \u6253\u5370\u54c8\u5e0c\u8868\n        pub fn print(self: *Self) !void {\n            var entry_set = try self.pairSet();\n            defer entry_set.deinit();\n            for (entry_set.items) |item| {\n                std.debug.print(\"{} -&gt; {s}\\n\", .{item.key, item.val});\n            }\n        }\n    };\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_hashing/hash_map/#613-hash-collision-and-resizing","title":"6.1.3 \u00a0 Hash Collision and Resizing","text":"<p>Fundamentally, the role of the hash function is to map the entire input space of all keys to the output space of all array indices. However, the input space is often much larger than the output space. Therefore, theoretically, there must be situations where \"multiple inputs correspond to the same output\".</p> <p>For the hash function in the above example, if the last two digits of the input <code>key</code> are the same, the output of the hash function will also be the same. For example, when querying for students with student numbers 12836 and 20336, we find:</p> <pre><code>12836 % 100 = 36\n20336 % 100 = 36\n</code></pre> <p>As shown in the Figure 6-3 , both student numbers point to the same name, which is obviously incorrect. This situation where multiple inputs correspond to the same output is known as \"hash collision\".</p> <p></p> <p> Figure 6-3 \u00a0 Example of hash collision </p> <p>It is easy to understand that the larger the capacity \\(n\\) of the hash table, the lower the probability of multiple keys being allocated to the same bucket, and the fewer the collisions. Therefore, expanding the capacity of the hash table can reduce hash collisions.</p> <p>As shown in the Figure 6-4 , before expansion, key-value pairs <code>(136, A)</code> and <code>(236, D)</code> collided; after expansion, the collision is resolved.</p> <p></p> <p> Figure 6-4 \u00a0 Hash table expansion </p> <p>Similar to array expansion, resizing a hash table requires migrating all key-value pairs from the original hash table to the new one, which is time-consuming. Furthermore, since the capacity <code>capacity</code> of the hash table changes, we need to recalculate the storage positions of all key-value pairs using the hash function, which adds to the computational overhead of the resizing process. Therefore, programming languages often reserve a sufficiently large capacity for the hash table to prevent frequent resizing.</p> <p>The \"load factor\" is an important concept for hash tables. It is defined as the ratio of the number of elements in the hash table to the number of buckets. It is used to measure the severity of hash collisions and is often used as a trigger for resizing the hash table. For example, in Java, when the load factor exceeds \\(0.75\\), the system will resize the hash table to twice its original size.</p>"},{"location":"chapter_hashing/summary/","title":"6.4 \u00a0 Summary","text":""},{"location":"chapter_hashing/summary/#1-key-review","title":"1. \u00a0 Key Review","text":"<ul> <li>Given an input <code>key</code>, a hash table can retrieve the corresponding <code>value</code> in \\(O(1)\\) time, which is highly efficient.</li> <li>Common hash table operations include querying, adding key-value pairs, deleting key-value pairs, and traversing the hash table.</li> <li>The hash function maps a <code>key</code> to an array index, allowing access to the corresponding bucket to retrieve the <code>value</code>.</li> <li>Two different keys may end up with the same array index after hashing, leading to erroneous query results. This phenomenon is known as hash collision.</li> <li>The larger the capacity of the hash table, the lower the probability of hash collisions. Therefore, hash table resizing can mitigate hash collisions. Similar to array resizing, hash table resizing is costly.</li> <li>Load factor, defined as the ratio of the number of elements to the number of buckets in the hash table, reflects the severity of hash collisions and is often used as a trigger for resizing the hash table.</li> <li>Chaining addresses hash collisions by converting each element into a linked list, storing all colliding elements in the same list. However, excessively long lists can reduce query efficiency, which can be improved by converting the lists into red-black trees.</li> <li>Open addressing handles hash collisions through multiple probes. Linear probing uses a fixed step size but cannot delete elements and is prone to clustering. Multiple hashing uses several hash functions for probing, making it less susceptible to clustering but increasing computational load.</li> <li>Different programming languages adopt various hash table implementations. For example, Java's <code>HashMap</code> uses chaining, while Python's <code>dict</code> employs open addressing.</li> <li>In hash tables, we desire hash algorithms with determinism, high efficiency, and uniform distribution. In cryptography, hash algorithms should also possess collision resistance and the avalanche effect.</li> <li>Hash algorithms typically use large prime numbers as moduli to ensure uniform distribution of hash values and reduce hash collisions.</li> <li>Common hash algorithms include MD5, SHA-1, SHA-2, and SHA-3. MD5 is often used for file integrity checks, while SHA-2 is commonly used in secure applications and protocols.</li> <li>Programming languages usually provide built-in hash algorithms for data types to calculate bucket indices in hash tables. Generally, only immutable objects are hashable.</li> </ul>"},{"location":"chapter_hashing/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: When does the time complexity of a hash table degrade to \\(O(n)\\)?</p> <p>The time complexity of a hash table can degrade to \\(O(n)\\) when hash collisions are severe. When the hash function is well-designed, the capacity is set appropriately, and collisions are evenly distributed, the time complexity is \\(O(1)\\). We usually consider the time complexity to be \\(O(1)\\) when using built-in hash tables in programming languages.</p> <p>Q: Why not use the hash function \\(f(x) = x\\)? This would eliminate collisions.</p> <p>Under the hash function \\(f(x) = x\\), each element corresponds to a unique bucket index, which is equivalent to an array. However, the input space is usually much larger than the output space (array length), so the last step of a hash function is often to take the modulo of the array length. In other words, the goal of a hash table is to map a larger state space to a smaller one while providing \\(O(1)\\) query efficiency.</p> <p>Q: Why can hash tables be more efficient than arrays, linked lists, or binary trees, even though they are implemented using these structures?</p> <p>Firstly, hash tables have higher time efficiency but lower space efficiency. A significant portion of memory in hash tables remains unused.</p> <p>Secondly, they are only more efficient in specific use cases. If a feature can be implemented with the same time complexity using an array or a linked list, it's usually faster than using a hash table. This is because the computation of the hash function incurs overhead, making the constant factor in the time complexity larger.</p> <p>Lastly, the time complexity of hash tables can degrade. For example, in chaining, we perform search operations in a linked list or red-black tree, which still risks degrading to \\(O(n)\\) time.</p> <p>Q: Does multiple hashing also have the flaw of not being able to delete elements directly? Can space marked as deleted be reused?</p> <p>Multiple hashing is a form of open addressing, and all open addressing methods have the drawback of not being able to delete elements directly; they require marking elements as deleted. Marked spaces can be reused. When inserting new elements into the hash table, and the hash function points to a position marked as deleted, that position can be used by the new element. This maintains the probing sequence of the hash table while ensuring efficient use of space.</p> <p>Q: Why do hash collisions occur during the search process in linear probing?</p> <p>During the search process, the hash function points to the corresponding bucket and key-value pair. If the <code>key</code> doesn't match, it indicates a hash collision. Therefore, linear probing will search downwards at a predetermined step size until the correct key-value pair is found or the search fails.</p> <p>Q: Why can resizing a hash table alleviate hash collisions?</p> <p>The last step of a hash function often involves taking the modulo of the array length \\(n\\), to keep the output within the array index range. When resizing, the array length \\(n\\) changes, and the indices corresponding to the keys may also change. Keys that were previously mapped to the same bucket might be distributed across multiple buckets after resizing, thereby mitigating hash collisions.</p>"},{"location":"chapter_heap/","title":"Chapter 8. \u00a0 Heap","text":"<p>Abstract</p> <p>The heap is like mountain peaks, stacked and undulating, each with its unique shape.</p> <p>Among these peaks, the highest one always catches the eye first.</p>"},{"location":"chapter_heap/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>8.1 \u00a0 Heap</li> <li>8.2 \u00a0 Building a Heap</li> <li>8.3 \u00a0 Top-k Problem</li> <li>8.4 \u00a0 Summary</li> </ul>"},{"location":"chapter_heap/build_heap/","title":"8.2 \u00a0 Heap construction operation","text":"<p>In some cases, we want to build a heap using all elements of a list, and this process is known as \"heap construction operation.\"</p>"},{"location":"chapter_heap/build_heap/#821-implementing-with-heap-insertion-operation","title":"8.2.1 \u00a0 Implementing with heap insertion operation","text":"<p>First, we create an empty heap and then iterate through the list, performing the \"heap insertion operation\" on each element in turn. This means adding the element to the end of the heap and then \"heapifying\" it from bottom to top.</p> <p>Each time an element is added to the heap, the length of the heap increases by one. Since nodes are added to the binary tree from top to bottom, the heap is constructed \"from top to bottom.\"</p> <p>Let the number of elements be \\(n\\), and each element's insertion operation takes \\(O(\\log{n})\\) time, thus the time complexity of this heap construction method is \\(O(n \\log n)\\).</p>"},{"location":"chapter_heap/build_heap/#822-implementing-by-heapifying-through-traversal","title":"8.2.2 \u00a0 Implementing by heapifying through traversal","text":"<p>In fact, we can implement a more efficient method of heap construction in two steps.</p> <ol> <li>Add all elements of the list as they are into the heap, at this point the properties of the heap are not yet satisfied.</li> <li>Traverse the heap in reverse order (reverse of level-order traversal), and perform \"top to bottom heapify\" on each non-leaf node.</li> </ol> <p>After heapifying a node, the subtree with that node as the root becomes a valid sub-heap. Since the traversal is in reverse order, the heap is built \"from bottom to top.\"</p> <p>The reason for choosing reverse traversal is that it ensures the subtree below the current node is already a valid sub-heap, making the heapification of the current node effective.</p> <p>It's worth mentioning that since leaf nodes have no children, they naturally form valid sub-heaps and do not need to be heapified. As shown in the following code, the last non-leaf node is the parent of the last node; we start from it and traverse in reverse order to perform heapification:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def __init__(self, nums: list[int]):\n    \"\"\"\u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\"\"\"\n    # \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    self.max_heap = nums\n    # \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in range(self.parent(self.size() - 1), -1, -1):\n        self.sift_down(i)\n</code></pre> my_heap.cpp<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(vector&lt;int&gt; nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = nums;\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List&lt;Integer&gt; nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = new ArrayList&lt;&gt;(nums);\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(IEnumerable&lt;int&gt; nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = new List&lt;int&gt;(nums);\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    var size = Parent(this.Size() - 1);\n    for (int i = size; i &gt;= 0; i--) {\n        SiftDown(i);\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nfunc newMaxHeap(nums []any) *maxHeap {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    h := &amp;maxHeap{data: nums}\n    for i := h.parent(len(h.data) - 1); i &gt;= 0; i-- {\n        // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n        h.siftDown(i)\n    }\n    return h\n}\n</code></pre> my_heap.swift<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\ninit(nums: [Int]) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = nums\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in (0 ... parent(i: size() - 1)).reversed() {\n        siftDown(i: i)\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    this.#maxHeap = nums === undefined ? [] : [...nums];\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = this.#parent(this.size() - 1); i &gt;= 0; i--) {\n        this.#siftDown(i);\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums?: number[]) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    this.maxHeap = nums === undefined ? [] : [...nums];\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = this.parent(this.size() - 1); i &gt;= 0; i--) {\n        this.siftDown(i);\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List&lt;int&gt; nums) {\n  // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n  _maxHeap = nums;\n  // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n  for (int i = _parent(size() - 1); i &gt;= 0; i--) {\n    siftDown(i);\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nfn new(nums: Vec&lt;i32&gt;) -&gt; Self {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    let mut heap = MaxHeap { max_heap: nums };\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in (0..=Self::parent(heap.size() - 1)).rev() {\n        heap.sift_down(i);\n    }\n    heap\n}\n</code></pre> my_heap.c<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nMaxHeap *newMaxHeap(int nums[], int size) {\n    // \u6240\u6709\u5143\u7d20\u5165\u5806\n    MaxHeap *maxHeap = (MaxHeap *)malloc(sizeof(MaxHeap));\n    maxHeap-&gt;size = size;\n    memcpy(maxHeap-&gt;data, nums, size * sizeof(int));\n    for (int i = parent(maxHeap, size - 1); i &gt;= 0; i--) {\n        // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n        siftDown(maxHeap, i);\n    }\n    return maxHeap;\n}\n</code></pre> my_heap.kt<pre><code>/* \u5927\u9876\u5806 */\nclass MaxHeap(nums: List&lt;Int&gt;?) {\n    // \u4f7f\u7528\u5217\u8868\u800c\u975e\u6570\u7ec4\uff0c\u8fd9\u6837\u65e0\u987b\u8003\u8651\u6269\u5bb9\u95ee\u9898\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    private val maxHeap = ArrayList(nums!!)\n\n    /* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\n    init {\n        // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n        for (i in parent(size() - 1) downTo 0) {\n            siftDown(i)\n        }\n    }\n\n    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    private fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    private fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    private fun parent(i: Int): Int {\n        return (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u4ea4\u6362\u5143\u7d20 */\n    private fun swap(i: Int, j: Int) {\n        maxHeap[i] = maxHeap[j].also { maxHeap[j] = maxHeap[i] }\n    }\n\n    /* \u83b7\u53d6\u5806\u5927\u5c0f */\n    fun size(): Int {\n        return maxHeap.size\n    }\n\n    /* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        /* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\n        return size() == 0\n    }\n\n    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\n    fun peek(): Int {\n        return maxHeap[0]\n    }\n\n    /* \u5143\u7d20\u5165\u5806 */\n    fun push(value: Int) {\n        // \u6dfb\u52a0\u8282\u70b9\n        maxHeap.add(value)\n        // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n        siftUp(size() - 1)\n    }\n\n    /* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n    private fun siftUp(it: Int) {\n        // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n        var i = it\n        while (true) {\n            // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n            val p = parent(i)\n            // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n            if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n            // \u4ea4\u6362\u4e24\u8282\u70b9\n            swap(i, p)\n            // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n            i = p\n        }\n    }\n\n    /* \u5143\u7d20\u51fa\u5806 */\n    fun pop(): Int {\n        // \u5224\u7a7a\u5904\u7406\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        swap(0, size() - 1)\n        // \u5220\u9664\u8282\u70b9\n        val value = maxHeap.removeAt(size() - 1)\n        // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n        siftDown(0)\n        // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n        return value\n    }\n\n    /* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n    private fun siftDown(it: Int) {\n        // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n        var i = it\n        while (true) {\n            // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n            val l = left(i)\n            val r = right(i)\n            var ma = i\n            if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n            if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n            // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n            if (ma == i) break\n            // \u4ea4\u6362\u4e24\u8282\u70b9\n            swap(i, ma)\n            // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n            i = ma\n        }\n    }\n\n    /* \u6253\u5370\u5806\uff08\u4e8c\u53c9\u6811\uff09 */\n    fun print() {\n        val queue = PriorityQueue { a: Int, b: Int -&gt; b - a }\n        queue.addAll(maxHeap)\n        printHeap(queue)\n    }\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{__init__}\n</code></pre> my_heap.zig<pre><code>// \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\nfn init(self: *Self, allocator: std.mem.Allocator, nums: []const T) !void {\n    if (self.max_heap != null) return;\n    self.max_heap = std.ArrayList(T).init(allocator);\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    try self.max_heap.?.appendSlice(nums);\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    var i: usize = parent(self.size() - 1) + 1;\n    while (i &gt; 0) : (i -= 1) {\n        try self.siftDown(i - 1);\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_heap/build_heap/#823-complexity-analysis","title":"8.2.3 \u00a0 Complexity analysis","text":"<p>Next, let's attempt to calculate the time complexity of this second method of heap construction.</p> <ul> <li>Assuming the number of nodes in the complete binary tree is \\(n\\), then the number of leaf nodes is \\((n + 1) / 2\\), where \\(/\\) is integer division. Therefore, the number of nodes that need to be heapified is \\((n - 1) / 2\\).</li> <li>In the process of \"top to bottom heapification,\" each node is heapified to the leaf nodes at most, so the maximum number of iterations is the height of the binary tree \\(\\log n\\).</li> </ul> <p>Multiplying the two, we get the time complexity of the heap construction process as \\(O(n \\log n)\\). But this estimate is not accurate, because it does not take into account the nature of the binary tree having far more nodes at the lower levels than at the top.</p> <p>Let's perform a more accurate calculation. To simplify the calculation, assume a \"perfect binary tree\" with \\(n\\) nodes and height \\(h\\); this assumption does not affect the correctness of the result.</p> <p></p> <p> Figure 8-5 \u00a0 Node counts at each level of a perfect binary tree </p> <p>As shown in the Figure 8-5 , the maximum number of iterations for a node \"to be heapified from top to bottom\" is equal to the distance from that node to the leaf nodes, which is precisely \"node height.\" Therefore, we can sum the \"number of nodes \\(\\times\\) node height\" at each level, to get the total number of heapification iterations for all nodes.</p> \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{(h-1)}\\times1 \\] <p>To simplify the above equation, we need to use knowledge of sequences from high school, first multiply \\(T(h)\\) by \\(2\\), to get:</p> \\[ \\begin{aligned} T(h) &amp; = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{h-1}\\times1 \\newline 2T(h) &amp; = 2^1h + 2^2(h-1) + 2^3(h-2) + \\dots + 2^h\\times1 \\newline \\end{aligned} \\] <p>By subtracting \\(T(h)\\) from \\(2T(h)\\) using the method of displacement, we get:</p> \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\dots + 2^{h-1} + 2^h \\] <p>Observing the equation, \\(T(h)\\) is an geometric series, which can be directly calculated using the sum formula, resulting in a time complexity of:</p> \\[ \\begin{aligned} T(h) &amp; = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline &amp; = 2^{h+1} - h - 2 \\newline &amp; = O(2^h) \\end{aligned} \\] <p>Further, a perfect binary tree with height \\(h\\) has \\(n = 2^{h+1} - 1\\) nodes, thus the complexity is \\(O(2^h) = O(n)\\). This calculation shows that the time complexity of inputting a list and constructing a heap is \\(O(n)\\), which is very efficient.</p>"},{"location":"chapter_heap/heap/","title":"8.1 \u00a0 Heap","text":"<p>A \"heap\" is a complete binary tree that satisfies specific conditions and can be mainly divided into two types, as shown in the Figure 8-1 .</p> <ul> <li>\"Min heap\": The value of any node \\(\\leq\\) the values of its child nodes.</li> <li>\"Max heap\": The value of any node \\(\\geq\\) the values of its child nodes.</li> </ul> <p></p> <p> Figure 8-1 \u00a0 Min heap and max heap </p> <p>As a special case of a complete binary tree, heaps have the following characteristics:</p> <ul> <li>The bottom layer nodes are filled from left to right, and nodes in other layers are fully filled.</li> <li>The root node of the binary tree is called the \"heap top,\" and the bottom-rightmost node is called the \"heap bottom.\"</li> <li>For max heaps (min heaps), the value of the heap top element (root node) is the largest (smallest).</li> </ul>"},{"location":"chapter_heap/heap/#811-common-operations-on-heaps","title":"8.1.1 \u00a0 Common operations on heaps","text":"<p>It should be noted that many programming languages provide a \"priority queue,\" which is an abstract data structure defined as a queue with priority sorting.</p> <p>In fact, heaps are often used to implement priority queues, with max heaps equivalent to priority queues where elements are dequeued in descending order. From a usage perspective, we can consider \"priority queue\" and \"heap\" as equivalent data structures. Therefore, this book does not make a special distinction between the two, uniformly referring to them as \"heap.\"</p> <p>Common operations on heaps are shown in the Table 8-1 , and the method names depend on the programming language.</p> <p> Table 8-1 \u00a0 Efficiency of Heap Operations </p> Method name Description Time complexity <code>push()</code> Add an element to the heap \\(O(\\log n)\\) <code>pop()</code> Remove the top element from the heap \\(O(\\log n)\\) <code>peek()</code> Access the top element (for max/min heap, the max/min value) \\(O(1)\\) <code>size()</code> Get the number of elements in the heap \\(O(1)\\) <code>isEmpty()</code> Check if the heap is empty \\(O(1)\\) <p>In practice, we can directly use the heap class (or priority queue class) provided by programming languages.</p> <p>Similar to sorting algorithms where we have \"ascending order\" and \"descending order,\" we can switch between \"min heap\" and \"max heap\" by setting a <code>flag</code> or modifying the <code>Comparator</code>. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig heap.py<pre><code># \u521d\u59cb\u5316\u5c0f\u9876\u5806\nmin_heap, flag = [], 1\n# \u521d\u59cb\u5316\u5927\u9876\u5806\nmax_heap, flag = [], -1\n\n# Python \u7684 heapq \u6a21\u5757\u9ed8\u8ba4\u5b9e\u73b0\u5c0f\u9876\u5806\n# \u8003\u8651\u5c06\u201c\u5143\u7d20\u53d6\u8d1f\u201d\u540e\u518d\u5165\u5806\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5c06\u5927\u5c0f\u5173\u7cfb\u98a0\u5012\uff0c\u4ece\u800c\u5b9e\u73b0\u5927\u9876\u5806\n# \u5728\u672c\u793a\u4f8b\u4e2d\uff0cflag = 1 \u65f6\u5bf9\u5e94\u5c0f\u9876\u5806\uff0cflag = -1 \u65f6\u5bf9\u5e94\u5927\u9876\u5806\n\n# \u5143\u7d20\u5165\u5806\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n\n# \u83b7\u53d6\u5806\u9876\u5143\u7d20\npeek: int = flag * max_heap[0] # 5\n\n# \u5806\u9876\u5143\u7d20\u51fa\u5806\n# \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n\n# \u83b7\u53d6\u5806\u5927\u5c0f\nsize: int = len(max_heap)\n\n# \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = not max_heap\n\n# \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\nmin_heap: list[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n</code></pre> heap.cpp<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap;\n// \u521d\u59cb\u5316\u5927\u9876\u5806\npriority_queue&lt;int, vector&lt;int&gt;, less&lt;int&gt;&gt; maxHeap;\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.top(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.empty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nvector&lt;int&gt; input{1, 3, 2, 5, 4};\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap(input.begin(), input.end());\n</code></pre> heap.java<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nQueue&lt;Integer&gt; minHeap = new PriorityQueue&lt;&gt;();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nQueue&lt;Integer&gt; maxHeap = new PriorityQueue&lt;&gt;((a, b) -&gt; b - a);\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.peek(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll(); // 5\npeek = maxHeap.poll(); // 4\npeek = maxHeap.poll(); // 3\npeek = maxHeap.poll(); // 2\npeek = maxHeap.poll(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = maxHeap.isEmpty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue&lt;&gt;(Arrays.asList(1, 3, 2, 5, 4));\n</code></pre> heap.cs<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nPriorityQueue&lt;int, int&gt; minHeap = new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nPriorityQueue&lt;int, int&gt; maxHeap = new(Comparer&lt;int&gt;.Create((x, y) =&gt; y - x));\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.Peek();//5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.Count;\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.Count == 0;\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue&lt;int, int&gt;([(1, 1), (3, 3), (2, 2), (5, 5), (4, 4)]);\n</code></pre> heap.go<pre><code>// Go \u8bed\u8a00\u4e2d\u53ef\u4ee5\u901a\u8fc7\u5b9e\u73b0 heap.Interface \u6765\u6784\u5efa\u6574\u6570\u5927\u9876\u5806\n// \u5b9e\u73b0 heap.Interface \u9700\u8981\u540c\u65f6\u5b9e\u73b0 sort.Interface\ntype intHeap []any\n\n// Push heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u63a8\u5165\u5143\u7d20\u5230\u5806\nfunc (h *intHeap) Push(x any) {\n    // Push \u548c Pop \u4f7f\u7528 pointer receiver \u4f5c\u4e3a\u53c2\u6570\n    // \u56e0\u4e3a\u5b83\u4eec\u4e0d\u4ec5\u4f1a\u5bf9\u5207\u7247\u7684\u5185\u5bb9\u8fdb\u884c\u8c03\u6574\uff0c\u8fd8\u4f1a\u4fee\u6539\u5207\u7247\u7684\u957f\u5ea6\u3002\n    *h = append(*h, x.(int))\n}\n\n// Pop heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u5f39\u51fa\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Pop() any {\n    // \u5f85\u51fa\u5806\u5143\u7d20\u5b58\u653e\u5728\u6700\u540e\n    last := (*h)[len(*h)-1]\n    *h = (*h)[:len(*h)-1]\n    return last\n}\n\n// Len sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Len() int {\n    return len(*h)\n}\n\n// Less sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Less(i, j int) bool {\n    // \u5982\u679c\u5b9e\u73b0\u5c0f\u9876\u5806\uff0c\u5219\u9700\u8981\u8c03\u6574\u4e3a\u5c0f\u4e8e\u53f7\n    return (*h)[i].(int) &gt; (*h)[j].(int)\n}\n\n// Swap sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Swap(i, j int) {\n    (*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n\n// Top \u83b7\u53d6\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Top() any {\n    return (*h)[0]\n}\n\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n    /* \u521d\u59cb\u5316\u5806 */\n    // \u521d\u59cb\u5316\u5927\u9876\u5806\n    maxHeap := &amp;intHeap{}\n    heap.Init(maxHeap)\n    /* \u5143\u7d20\u5165\u5806 */\n    // \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u6dfb\u52a0\u5143\u7d20\n    heap.Push(maxHeap, 1)\n    heap.Push(maxHeap, 3)\n    heap.Push(maxHeap, 2)\n    heap.Push(maxHeap, 4)\n    heap.Push(maxHeap, 5)\n\n    /* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\n    top := maxHeap.Top()\n    fmt.Printf(\"\u5806\u9876\u5143\u7d20\u4e3a %d\\n\", top)\n\n    /* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n    // \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u79fb\u9664\u5143\u7d20\n    heap.Pop(maxHeap) // 5\n    heap.Pop(maxHeap) // 4\n    heap.Pop(maxHeap) // 3\n    heap.Pop(maxHeap) // 2\n    heap.Pop(maxHeap) // 1\n\n    /* \u83b7\u53d6\u5806\u5927\u5c0f */\n    size := len(*maxHeap)\n    fmt.Printf(\"\u5806\u5143\u7d20\u6570\u91cf\u4e3a %d\\n\", size)\n\n    /* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\n    isEmpty := len(*maxHeap) == 0\n    fmt.Printf(\"\u5806\u662f\u5426\u4e3a\u7a7a %t\\n\", isEmpty)\n}\n</code></pre> heap.swift<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// Swift \u7684 Heap \u7c7b\u578b\u540c\u65f6\u652f\u6301\u6700\u5927\u5806\u548c\u6700\u5c0f\u5806\uff0c\u4e14\u9700\u8981\u5f15\u5165 swift-collections\nvar heap = Heap&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u5806 */\nheap.insert(1)\nheap.insert(3)\nheap.insert(2)\nheap.insert(5)\nheap.insert(4)\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nvar peek = heap.max()!\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\npeek = heap.removeMax() // 5\npeek = heap.removeMax() // 4\npeek = heap.removeMax() // 3\npeek = heap.removeMax() // 2\npeek = heap.removeMax() // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nlet size = heap.count\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = heap.isEmpty\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nlet heap2 = Heap([1, 3, 2, 5, 4])\n</code></pre> heap.js<pre><code>// JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.ts<pre><code>// TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.dart<pre><code>// Dart \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.rs<pre><code>use std::collections::BinaryHeap;\nuse std::cmp::Reverse;\n\n/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nlet mut min_heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\nlet mut max_heap = BinaryHeap::new();\n\n/* \u5143\u7d20\u5165\u5806 */\nmax_heap.push(1);\nmax_heap.push(3);\nmax_heap.push(2);\nmax_heap.push(5);\nmax_heap.push(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nlet peek = max_heap.peek().unwrap();  // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nlet peek = max_heap.pop().unwrap();   // 5\nlet peek = max_heap.pop().unwrap();   // 4\nlet peek = max_heap.pop().unwrap();   // 3\nlet peek = max_heap.pop().unwrap();   // 2\nlet peek = max_heap.pop().unwrap();   // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nlet size = max_heap.len();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = max_heap.is_empty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nlet min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]);\n</code></pre> heap.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.kt<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nvar minHeap = PriorityQueue&lt;Int&gt;()\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nval maxHeap = PriorityQueue { a: Int, b: Int -&gt; b - a }\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1)\nmaxHeap.offer(3)\nmaxHeap.offer(2)\nmaxHeap.offer(5)\nmaxHeap.offer(4)\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nvar peek = maxHeap.peek() // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll() // 5\npeek = maxHeap.poll() // 4\npeek = maxHeap.poll() // 3\npeek = maxHeap.poll() // 2\npeek = maxHeap.poll() // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nval size = maxHeap.size\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nval isEmpty = maxHeap.isEmpty()\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = PriorityQueue(mutableListOf(1, 3, 2, 5, 4))\n</code></pre> heap.rb<pre><code>\n</code></pre> heap.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p>https://pythontutor.com/render.html#code=import%20heapq%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%B0%8F%E9%A1%B6%E5%A0%86%0A%20%20%20%20min_heap,%20flag%20%3D%20%5B%5D,%201%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%A4%A7%E9%A1%B6%E5%A0%86%0A%20%20%20%20max_heap,%20flag%20%3D%20%5B%5D,%20-1%0A%20%20%20%20%0A%20%20%20%20%23%20Python%20%E7%9A%84%20heapq%20%E6%A8%A1%E5%9D%97%E9%BB%98%E8%AE%A4%E5%AE%9E%E7%8E%B0%E5%B0%8F%E9%A1%B6%E5%A0%86%0A%20%20%20%20%23%20%E8%80%83%E8%99%91%E5%B0%86%E2%80%9C%E5%85%83%E7%B4%A0%E5%8F%96%E8%B4%9F%E2%80%9D%E5%90%8E%E5%86%8D%E5%85%A5%E5%A0%86%EF%BC%8C%E8%BF%99%E6%A0%B7%E5%B0%B1%E5%8F%AF%E4%BB%A5%E5%B0%86%E5%A4%A7%E5%B0%8F%E5%85%B3%E7%B3%BB%E9%A2%A0%E5%80%92%EF%BC%8C%E4%BB%8E%E8%80%8C%E5%AE%9E%E7%8E%B0%E5%A4%A7%E9%A1%B6%E5%A0%86%0A%20%20%20%20%23%20%E5%9C%A8%E6%9C%AC%E7%A4%BA%E4%BE%8B%E4%B8%AD%EF%BC%8Cflag%20%3D%201%20%E6%97%B6%E5%AF%B9%E5%BA%94%E5%B0%8F%E9%A1%B6%E5%A0%86%EF%BC%8Cflag%20%3D%20-1%20%E6%97%B6%E5%AF%B9%E5%BA%94%E5%A4%A7%E9%A1%B6%E5%A0%86%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%85%A5%E5%A0%86%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%201%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%203%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%202%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%205%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%204%29%0A%20%20%20%20%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%A0%86%E9%A1%B6%E5%85%83%E7%B4%A0%0A%20%20%20%20peek%20%3D%20flag%20*%20max_heap%5B0%5D%20%23%205%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%A0%86%E9%A1%B6%E5%85%83%E7%B4%A0%E5%87%BA%E5%A0%86%0A%20%20%20%20%23%20%E5%87%BA%E5%A0%86%E5%85%83%E7%B4%A0%E4%BC%9A%E5%BD%A2%E6%88%90%E4%B8%80%E4%B8%AA%E4%BB%8E%E5%A4%A7%E5%88%B0%E5%B0%8F%E7%9A%84%E5%BA%8F%E5%88%97%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%205%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%204%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%203%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%202%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%201%0A%20%20%20%20%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%A0%86%E5%A4%A7%E5%B0%8F%0A%20%20%20%20size%20%3D%20len%28max_heap%29%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%88%A4%E6%96%AD%E5%A0%86%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%0A%20%20%20%20is_empty%20%3D%20not%20max_heap%0A%20%20%20%20%0A%20%20%20%20%23%20%E8%BE%93%E5%85%A5%E5%88%97%E8%A1%A8%E5%B9%B6%E5%BB%BA%E5%A0%86%0A%20%20%20%20min_heap%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20heapq.heapify%28min_heap%29&amp;cumulative=false&amp;curInstr=3&amp;heapPrimitives=nevernest&amp;mode=display&amp;origin=opt-frontend.js&amp;py=311&amp;rawInputLstJSON=%5B%5D&amp;textReferences=false</p>"},{"location":"chapter_heap/heap/#812-implementation-of-heaps","title":"8.1.2 \u00a0 Implementation of heaps","text":"<p>The following implementation is of a max heap. To convert it into a min heap, simply invert all size logic comparisons (for example, replace \\(\\geq\\) with \\(\\leq\\)). Interested readers are encouraged to implement it on their own.</p>"},{"location":"chapter_heap/heap/#1-storage-and-representation-of-heaps","title":"1. \u00a0 Storage and representation of heaps","text":"<p>As mentioned in the \"Binary Trees\" section, complete binary trees are well-suited for array representation. Since heaps are a type of complete binary tree, we will use arrays to store heaps.</p> <p>When using an array to represent a binary tree, elements represent node values, and indexes represent node positions in the binary tree. Node pointers are implemented through an index mapping formula.</p> <p>As shown in the Figure 8-2 , given an index \\(i\\), the index of its left child is \\(2i + 1\\), the index of its right child is \\(2i + 2\\), and the index of its parent is \\((i - 1) / 2\\) (floor division). When the index is out of bounds, it signifies a null node or the node does not exist.</p> <p></p> <p> Figure 8-2 \u00a0 Representation and storage of heaps </p> <p>We can encapsulate the index mapping formula into functions for convenient later use:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def left(self, i: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n    return 2 * i + 1\n\ndef right(self, i: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n    return 2 * i + 2\n\ndef parent(self, i: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n    return (i - 1) // 2  # \u5411\u4e0b\u6574\u9664\n</code></pre> my_heap.cpp<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.java<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.cs<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint Left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint Right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint Parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.go<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) left(i int) int {\n    return 2*i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) right(i int) int {\n    return 2*i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) parent(i int) int {\n    // \u5411\u4e0b\u6574\u9664\n    return (i - 1) / 2\n}\n</code></pre> my_heap.swift<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc left(i: Int) -&gt; Int {\n    2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc right(i: Int) -&gt; Int {\n    2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc parent(i: Int) -&gt; Int {\n    (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.js<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n#left(i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n#right(i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n#parent(i) {\n    return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.ts<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nleft(i: number): number {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nright(i: number): number {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nparent(i: number): number {\n    return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.dart<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint _left(int i) {\n  return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint _right(int i) {\n  return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint _parent(int i) {\n  return (i - 1) ~/ 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.rs<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfn left(i: usize) -&gt; usize {\n    2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfn right(i: usize) -&gt; usize {\n    2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfn parent(i: usize) -&gt; usize {\n    (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.c<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint left(MaxHeap *maxHeap, int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint right(MaxHeap *maxHeap, int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint parent(MaxHeap *maxHeap, int i) {\n    return (i - 1) / 2;\n}\n</code></pre> my_heap.kt<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfun left(i: Int): Int {\n    return 2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfun right(i: Int): Int {\n    return 2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfun parent(i: Int): Int {\n    return (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{left}\n\n[class]{MaxHeap}-[func]{right}\n\n[class]{MaxHeap}-[func]{parent}\n</code></pre> my_heap.zig<pre><code>// \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\nfn left(i: usize) usize {\n    return 2 * i + 1;\n}\n\n// \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\nfn right(i: usize) usize {\n    return 2 * i + 2;\n}\n\n// \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15\nfn parent(i: usize) usize {\n    // return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n    return @divFloor(i - 1, 2);\n}\n</code></pre>"},{"location":"chapter_heap/heap/#2-accessing-the-top-element-of-the-heap","title":"2. \u00a0 Accessing the top element of the heap","text":"<p>The top element of the heap is the root node of the binary tree, which is also the first element of the list:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def peek(self) -&gt; int:\n    \"\"\"\u8bbf\u95ee\u5806\u9876\u5143\u7d20\"\"\"\n    return self.max_heap[0]\n</code></pre> my_heap.cpp<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.java<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n    return maxHeap.get(0);\n}\n</code></pre> my_heap.cs<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint Peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.go<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc (h *maxHeap) peek() any {\n    return h.data[0]\n}\n</code></pre> my_heap.swift<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc peek() -&gt; Int {\n    maxHeap[0]\n}\n</code></pre> my_heap.js<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek() {\n    return this.#maxHeap[0];\n}\n</code></pre> my_heap.ts<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek(): number {\n    return this.maxHeap[0];\n}\n</code></pre> my_heap.dart<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n  return _maxHeap[0];\n}\n</code></pre> my_heap.rs<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfn peek(&amp;self) -&gt; Option&lt;i32&gt; {\n    self.max_heap.first().copied()\n}\n</code></pre> my_heap.c<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek(MaxHeap *maxHeap) {\n    return maxHeap-&gt;data[0];\n}\n</code></pre> my_heap.kt<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfun peek(): Int {\n    return maxHeap[0]\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{peek}\n</code></pre> my_heap.zig<pre><code>// \u8bbf\u95ee\u5806\u9876\u5143\u7d20\nfn peek(self: *Self) T {\n    return self.max_heap.?.items[0];\n}  \n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_heap/heap/#3-inserting-an-element-into-the-heap","title":"3. \u00a0 Inserting an element into the heap","text":"<p>Given an element <code>val</code>, we first add it to the bottom of the heap. After addition, since <code>val</code> may be larger than other elements in the heap, the heap's integrity might be compromised, thus it's necessary to repair the path from the inserted node to the root node. This operation is called \"heapifying\".</p> <p>Considering starting from the node inserted, perform heapify from bottom to top. As shown in the Figure 8-3 , we compare the value of the inserted node with its parent node, and if the inserted node is larger, we swap them. Then continue this operation, repairing each node in the heap from bottom to top until passing the root node or encountering a node that does not need to be swapped.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 8-3 \u00a0 Steps of element insertion into the heap </p> <p>Given a total of \\(n\\) nodes, the height of the tree is \\(O(\\log n)\\). Hence, the loop iterations for the heapify operation are at most \\(O(\\log n)\\), making the time complexity of the element insertion operation \\(O(\\log n)\\). The code is as shown:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def push(self, val: int):\n    \"\"\"\u5143\u7d20\u5165\u5806\"\"\"\n    # \u6dfb\u52a0\u8282\u70b9\n    self.max_heap.append(val)\n    # \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    self.sift_up(self.size() - 1)\n\ndef sift_up(self, i: int):\n    \"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\"\"\"\n    while True:\n        # \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        p = self.parent(i)\n        # \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if p &lt; 0 or self.max_heap[i] &lt;= self.max_heap[p]:\n            break\n        # \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, p)\n        # \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.push_back(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(maxHeap[i], maxHeap[p]);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.add(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap.get(i) &lt;= maxHeap.get(p))\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid Push(int val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.Add(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    SiftUp(Size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid SiftUp(int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = Parent(i);\n        // \u82e5\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        Swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunc (h *maxHeap) push(val any) {\n    // \u6dfb\u52a0\u8282\u70b9\n    h.data = append(h.data, val)\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    h.siftUp(len(h.data) - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc (h *maxHeap) siftUp(i int) {\n    for true {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        p := h.parent(i)\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if p &lt; 0 || h.data[i].(int) &lt;= h.data[p].(int) {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        h.swap(i, p)\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunc push(val: Int) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.append(val)\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(i: size() - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc siftUp(i: Int) {\n    var i = i\n    while true {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        let p = parent(i: i)\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if p &lt; 0 || maxHeap[i] &lt;= maxHeap[p] {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i: i, j: p)\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u5165\u5806 */\npush(val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    this.#maxHeap.push(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    this.#siftUp(this.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n#siftUp(i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        const p = this.#parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || this.#maxHeap[i] &lt;= this.#maxHeap[p]) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.#swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u5165\u5806 */\npush(val: number): void {\n    // \u6dfb\u52a0\u8282\u70b9\n    this.maxHeap.push(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    this.siftUp(this.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nsiftUp(i: number): void {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        const p = this.parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || this.maxHeap[i] &lt;= this.maxHeap[p]) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n  // \u6dfb\u52a0\u8282\u70b9\n  _maxHeap.add(val);\n  // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n  siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n  while (true) {\n    // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n    int p = _parent(i);\n    // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n    if (p &lt; 0 || _maxHeap[i] &lt;= _maxHeap[p]) {\n      break;\n    }\n    // \u4ea4\u6362\u4e24\u8282\u70b9\n    _swap(i, p);\n    // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n    i = p;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfn push(&amp;mut self, val: i32) {\n    // \u6dfb\u52a0\u8282\u70b9\n    self.max_heap.push(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    self.sift_up(self.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfn sift_up(&amp;mut self, mut i: usize) {\n    loop {\n        // \u8282\u70b9 i \u5df2\u7ecf\u662f\u5806\u9876\u8282\u70b9\u4e86\uff0c\u7ed3\u675f\u5806\u5316\n        if i == 0 {\n            break;\n        }\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        let p = Self::parent(i);\n        // \u5f53\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if self.max_heap[i] &lt;= self.max_heap[p] {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(MaxHeap *maxHeap, int val) {\n    // \u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c\u4e0d\u5e94\u8be5\u6dfb\u52a0\u8fd9\u4e48\u591a\u8282\u70b9\n    if (maxHeap-&gt;size == MAX_SIZE) {\n        printf(\"heap is full!\");\n        return;\n    }\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap-&gt;data[maxHeap-&gt;size] = val;\n    maxHeap-&gt;size++;\n\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(maxHeap, maxHeap-&gt;size - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = parent(maxHeap, i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap-&gt;data[i] &lt;= maxHeap-&gt;data[p]) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(maxHeap, i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfun push(value: Int) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.add(value)\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(size() - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfun siftUp(it: Int) {\n    // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n    var i = it\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        val p = parent(i)\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, p)\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{push}\n\n[class]{MaxHeap}-[func]{sift_up}\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u5165\u5806\nfn push(self: *Self, val: T) !void {\n    // \u6dfb\u52a0\u8282\u70b9\n    try self.max_heap.?.append(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    try self.siftUp(self.size() - 1);\n}  \n\n// \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\nfn siftUp(self: *Self, i_: usize) !void {\n    var i = i_;\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        var p = parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 or self.max_heap.?.items[i] &lt;= self.max_heap.?.items[p]) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        try self.swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_heap/heap/#4-removing-the-top-element-from-the-heap","title":"4. \u00a0 Removing the top element from the heap","text":"<p>The top element of the heap is the root node of the binary tree, that is, the first element of the list. If we directly remove the first element from the list, all node indexes in the binary tree would change, making it difficult to use heapify for repairs subsequently. To minimize changes in element indexes, we use the following steps.</p> <ol> <li>Swap the top element with the bottom element of the heap (swap the root node with the rightmost leaf node).</li> <li>After swapping, remove the bottom of the heap from the list (note, since it has been swapped, what is actually being removed is the original top element).</li> <li>Starting from the root node, perform heapify from top to bottom.</li> </ol> <p>As shown in the Figure 8-4 , the direction of \"heapify from top to bottom\" is opposite to \"heapify from bottom to top\". We compare the value of the root node with its two children and swap it with the largest child. Then repeat this operation until passing the leaf node or encountering a node that does not need to be swapped.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 8-4 \u00a0 Steps of removing the top element from the heap </p> <p>Similar to the element insertion operation, the time complexity of the top element removal operation is also \\(O(\\log n)\\). The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def pop(self) -&gt; int:\n    \"\"\"\u5143\u7d20\u51fa\u5806\"\"\"\n    # \u5224\u7a7a\u5904\u7406\n    if self.is_empty():\n        raise IndexError(\"\u5806\u4e3a\u7a7a\")\n    # \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    self.swap(0, self.size() - 1)\n    # \u5220\u9664\u8282\u70b9\n    val = self.max_heap.pop()\n    # \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    self.sift_down(0)\n    # \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val\n\ndef sift_down(self, i: int):\n    \"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\n    while True:\n        # \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        l, r, ma = self.left(i), self.right(i), i\n        if l &lt; self.size() and self.max_heap[l] &gt; self.max_heap[ma]:\n            ma = l\n        if r &lt; self.size() and self.max_heap[r] &gt; self.max_heap[ma]:\n            ma = r\n        # \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i:\n            break\n        # \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, ma)\n        # \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nvoid pop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty()) {\n        throw out_of_range(\"\u5806\u4e3a\u7a7a\");\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(maxHeap[0], maxHeap[size() - 1]);\n    // \u5220\u9664\u8282\u70b9\n    maxHeap.pop_back();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(0);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        swap(maxHeap[i], maxHeap[ma]);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty())\n        throw new IndexOutOfBoundsException();\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(0, size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    int val = maxHeap.remove(size() - 1);\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap.get(l) &gt; maxHeap.get(ma))\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap.get(r) &gt; maxHeap.get(ma))\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint Pop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (IsEmpty())\n        throw new IndexOutOfRangeException();\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    Swap(0, Size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    int val = maxHeap.Last();\n    maxHeap.RemoveAt(Size() - 1);\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    SiftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid SiftDown(int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = Left(i), r = Right(i), ma = i;\n        if (l &lt; Size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; Size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // \u82e5\u201c\u8282\u70b9 i \u6700\u5927\u201d\u6216\u201c\u8d8a\u8fc7\u53f6\u8282\u70b9\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\n        if (ma == i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        Swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfunc (h *maxHeap) pop() any {\n    // \u5224\u7a7a\u5904\u7406\n    if h.isEmpty() {\n        fmt.Println(\"error\")\n        return nil\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    h.swap(0, h.size()-1)\n    // \u5220\u9664\u8282\u70b9\n    val := h.data[len(h.data)-1]\n    h.data = h.data[:len(h.data)-1]\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    h.siftDown(0)\n\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc (h *maxHeap) siftDown(i int) {\n    for true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\n        l, r, max := h.left(i), h.right(i), i\n        if l &lt; h.size() &amp;&amp; h.data[l].(int) &gt; h.data[max].(int) {\n            max = l\n        }\n        if r &lt; h.size() &amp;&amp; h.data[r].(int) &gt; h.data[max].(int) {\n            max = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if max == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        h.swap(i, max)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = max\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfunc pop() -&gt; Int {\n    // \u5224\u7a7a\u5904\u7406\n    if isEmpty() {\n        fatalError(\"\u5806\u4e3a\u7a7a\")\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(i: 0, j: size() - 1)\n    // \u5220\u9664\u8282\u70b9\n    let val = maxHeap.remove(at: size() - 1)\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(i: 0)\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(i: Int) {\n    var i = i\n    while true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = left(i: i)\n        let r = right(i: i)\n        var ma = i\n        if l &lt; size(), maxHeap[l] &gt; maxHeap[ma] {\n            ma = l\n        }\n        if r &lt; size(), maxHeap[r] &gt; maxHeap[ma] {\n            ma = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i: i, j: ma)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u51fa\u5806 */\npop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (this.isEmpty()) throw new Error('\u5806\u4e3a\u7a7a');\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    this.#swap(0, this.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    const val = this.#maxHeap.pop();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    this.#siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n#siftDown(i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        const l = this.#left(i),\n            r = this.#right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.#maxHeap[l] &gt; this.#maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.#maxHeap[r] &gt; this.#maxHeap[ma]) ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.#swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u51fa\u5806 */\npop(): number {\n    // \u5224\u7a7a\u5904\u7406\n    if (this.isEmpty()) throw new RangeError('Heap is empty.');\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    this.swap(0, this.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    const val = this.maxHeap.pop();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    this.siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nsiftDown(i: number): void {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        const l = this.left(i),\n            r = this.right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.maxHeap[l] &gt; this.maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.maxHeap[r] &gt; this.maxHeap[ma]) ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n  // \u5224\u7a7a\u5904\u7406\n  if (isEmpty()) throw Exception('\u5806\u4e3a\u7a7a');\n  // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n  _swap(0, size() - 1);\n  // \u5220\u9664\u8282\u70b9\n  int val = _maxHeap.removeLast();\n  // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n  siftDown(0);\n  // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n  return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n  while (true) {\n    // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n    int l = _left(i);\n    int r = _right(i);\n    int ma = i;\n    if (l &lt; size() &amp;&amp; _maxHeap[l] &gt; _maxHeap[ma]) ma = l;\n    if (r &lt; size() &amp;&amp; _maxHeap[r] &gt; _maxHeap[ma]) ma = r;\n    // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n    if (ma == i) break;\n    // \u4ea4\u6362\u4e24\u8282\u70b9\n    _swap(i, ma);\n    // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n    i = ma;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfn pop(&amp;mut self) -&gt; i32 {\n    // \u5224\u7a7a\u5904\u7406\n    if self.is_empty() {\n        panic!(\"index out of bounds\");\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    self.swap(0, self.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    let val = self.max_heap.remove(self.size() - 1);\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    self.sift_down(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfn sift_down(&amp;mut self, mut i: usize) {\n    loop {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let (l, r, mut ma) = (Self::left(i), Self::right(i), i);\n        if l &lt; self.size() &amp;&amp; self.max_heap[l] &gt; self.max_heap[ma] {\n            ma = l;\n        }\n        if r &lt; self.size() &amp;&amp; self.max_heap[r] &gt; self.max_heap[ma] {\n            ma = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop(MaxHeap *maxHeap) {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty(maxHeap)) {\n        printf(\"heap is empty!\");\n        return INT_MAX;\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(maxHeap, 0, size(maxHeap) - 1);\n    // \u5220\u9664\u8282\u70b9\n    int val = maxHeap-&gt;data[maxHeap-&gt;size - 1];\n    maxHeap-&gt;size--;\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(maxHeap, 0);\n\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\n        int l = left(maxHeap, i);\n        int r = right(maxHeap, i);\n        int max = i;\n        if (l &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[l] &gt; maxHeap-&gt;data[max]) {\n            max = l;\n        }\n        if (r &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[r] &gt; maxHeap-&gt;data[max]) {\n            max = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (max == i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(maxHeap, i, max);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = max;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfun pop(): Int {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty()) throw IndexOutOfBoundsException()\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(0, size() - 1)\n    // \u5220\u9664\u8282\u70b9\n    val value = maxHeap.removeAt(size() - 1)\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(0)\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return value\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfun siftDown(it: Int) {\n    // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n    var i = it\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        val l = left(i)\n        val r = right(i)\n        var ma = i\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) break\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, ma)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{pop}\n\n[class]{MaxHeap}-[func]{sift_down}\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u51fa\u5806\nfn pop(self: *Self) !T {\n    // \u5224\u65ad\u5904\u7406\n    if (self.isEmpty()) unreachable;\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    try self.swap(0, self.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    var val = self.max_heap.?.pop();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    try self.siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n} \n\n// \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\nfn siftDown(self: *Self, i_: usize) !void {\n    var i = i_;\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        var l = left(i);\n        var r = right(i);\n        var ma = i;\n        if (l &lt; self.size() and self.max_heap.?.items[l] &gt; self.max_heap.?.items[ma]) ma = l;\n        if (r &lt; self.size() and self.max_heap.?.items[r] &gt; self.max_heap.?.items[ma]) ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        try self.swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_heap/heap/#813-common-applications-of-heaps","title":"8.1.3 \u00a0 Common applications of heaps","text":"<ul> <li>Priority Queue: Heaps are often the preferred data structure for implementing priority queues, with both enqueue and dequeue operations having a time complexity of \\(O(\\log n)\\), and building a queue having a time complexity of \\(O(n)\\), all of which are very efficient.</li> <li>Heap Sort: Given a set of data, we can create a heap from them and then continually perform element removal operations to obtain ordered data. However, we usually use a more elegant method to implement heap sort, as detailed in the \"Heap Sort\" section.</li> <li>Finding the Largest \\(k\\) Elements: This is a classic algorithm problem and also a typical application, such as selecting the top 10 hot news for Weibo hot search, picking the top 10 selling products, etc.</li> </ul>"},{"location":"chapter_heap/summary/","title":"8.4 \u00a0 Summary","text":""},{"location":"chapter_heap/summary/#1-key-review","title":"1. \u00a0 Key review","text":"<ul> <li>A heap is a complete binary tree, which can be divided into a max heap and a min heap based on its property. The top element of a max (min) heap is the largest (smallest).</li> <li>A priority queue is defined as a queue with dequeue priority, usually implemented using a heap.</li> <li>Common operations of a heap and their corresponding time complexities include: element insertion into the heap \\(O(\\log n)\\), removing the top element from the heap \\(O(\\log n)\\), and accessing the top element of the heap \\(O(1)\\).</li> <li>A complete binary tree is well-suited to be represented by an array, thus heaps are commonly stored using arrays.</li> <li>Heapify operations are used to maintain the properties of the heap and are used in both heap insertion and removal operations.</li> <li>The time complexity of inserting \\(n\\) elements into a heap and building the heap can be optimized to \\(O(n)\\), which is highly efficient.</li> <li>Top-k is a classic algorithm problem that can be efficiently solved using the heap data structure, with a time complexity of \\(O(n \\log k)\\).</li> </ul>"},{"location":"chapter_heap/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: Is the \"heap\" in data structures the same concept as the \"heap\" in memory management?</p> <p>The two are not the same concept, even though they are both referred to as \"heap\". The heap in computer system memory is part of dynamic memory allocation, where the program can use it to store data during execution. The program can request a certain amount of heap memory to store complex structures like objects and arrays. When these data are no longer needed, the program needs to release this memory to prevent memory leaks. Compared to stack memory, the management and usage of heap memory need to be more cautious, as improper use may lead to memory leaks and dangling pointers.</p>"},{"location":"chapter_heap/top_k/","title":"8.3 \u00a0 Top-k problem","text":"<p>Question</p> <p>Given an unordered array <code>nums</code> of length \\(n\\), return the largest \\(k\\) elements in the array.</p> <p>For this problem, we will first introduce two straightforward solutions, then explain a more efficient heap-based method.</p>"},{"location":"chapter_heap/top_k/#831-method-1-iterative-selection","title":"8.3.1 \u00a0 Method 1: Iterative selection","text":"<p>We can perform \\(k\\) rounds of iterations as shown in the Figure 8-6 , extracting the \\(1^{st}\\), \\(2^{nd}\\), \\(\\dots\\), \\(k^{th}\\) largest elements in each round, with a time complexity of \\(O(nk)\\).</p> <p>This method is only suitable when \\(k \\ll n\\), as the time complexity approaches \\(O(n^2)\\) when \\(k\\) is close to \\(n\\), which is very time-consuming.</p> <p></p> <p> Figure 8-6 \u00a0 Iteratively finding the largest k elements </p> <p>Tip</p> <p>When \\(k = n\\), we can obtain a complete ordered sequence, which is equivalent to the \"selection sort\" algorithm.</p>"},{"location":"chapter_heap/top_k/#832-method-2-sorting","title":"8.3.2 \u00a0 Method 2: Sorting","text":"<p>As shown in the Figure 8-7 , we can first sort the array <code>nums</code> and then return the last \\(k\\) elements, with a time complexity of \\(O(n \\log n)\\).</p> <p>Clearly, this method \"overachieves\" the task, as we only need to find the largest \\(k\\) elements, without the need to sort the other elements.</p> <p></p> <p> Figure 8-7 \u00a0 Sorting to find the largest k elements </p>"},{"location":"chapter_heap/top_k/#833-method-3-heap","title":"8.3.3 \u00a0 Method 3: Heap","text":"<p>We can solve the Top-k problem more efficiently based on heaps, as shown in the following process.</p> <ol> <li>Initialize a min heap, where the top element is the smallest.</li> <li>First, insert the first \\(k\\) elements of the array into the heap.</li> <li>Starting from the \\(k + 1^{th}\\) element, if the current element is greater than the top element of the heap, remove the top element of the heap and insert the current element into the heap.</li> <li>After completing the traversal, the heap contains the largest \\(k\\) elements.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 8-8 \u00a0 Find the largest k elements based on heap </p> <p>Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig top_k.py<pre><code>def top_k_heap(nums: list[int], k: int) -&gt; list[int]:\n    \"\"\"\u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\"\"\"\n    # \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    heap = []\n    # \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for i in range(k):\n        heapq.heappush(heap, nums[i])\n    # \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for i in range(k, len(nums)):\n        # \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if nums[i] &gt; heap[0]:\n            heapq.heappop(heap)\n            heapq.heappush(heap, nums[i])\n    return heap\n</code></pre> top_k.cpp<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; topKHeap(vector&lt;int&gt; &amp;nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    priority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; heap;\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        heap.push(nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; nums.size(); i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.top()) {\n            heap.pop();\n            heap.push(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.java<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nQueue&lt;Integer&gt; topKHeap(int[] nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    Queue&lt;Integer&gt; heap = new PriorityQueue&lt;Integer&gt;();\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        heap.offer(nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; nums.length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll();\n            heap.offer(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.cs<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nPriorityQueue&lt;int, int&gt; TopKHeap(int[] nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    PriorityQueue&lt;int, int&gt; heap = new();\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        heap.Enqueue(nums[i], nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; nums.Length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.Peek()) {\n            heap.Dequeue();\n            heap.Enqueue(nums[i], nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.go<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums []int, k int) *minHeap {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    h := &amp;minHeap{}\n    heap.Init(h)\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for i := 0; i &lt; k; i++ {\n        heap.Push(h, nums[i])\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for i := k; i &lt; len(nums); i++ {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if nums[i] &gt; h.Top().(int) {\n            heap.Pop(h)\n            heap.Push(h, nums[i])\n        }\n    }\n    return h\n}\n</code></pre> top_k.swift<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums: [Int], k: Int) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5e76\u5c06\u524d k \u4e2a\u5143\u7d20\u5efa\u5806\n    var heap = Heap(nums.prefix(k))\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for i in nums.indices.dropFirst(k) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if nums[i] &gt; heap.min()! {\n            _ = heap.removeMin()\n            heap.insert(nums[i])\n        }\n    }\n    return heap.unordered\n}\n</code></pre> top_k.js<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap, val) {\n    // \u5143\u7d20\u53d6\u53cd\n    maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.pop();\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return maxHeap.getMaxHeap().map((num) =&gt; -num);\n}\n\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums, k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n    const maxHeap = new MaxHeap([]);\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (let i = k; i &lt; nums.length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.ts<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap: MaxHeap, val: number): void {\n    // \u5143\u7d20\u53d6\u53cd\n    maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap: MaxHeap): number {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.pop();\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap: MaxHeap): number {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap: MaxHeap): number[] {\n    // \u5143\u7d20\u53d6\u53cd\n    return maxHeap.getMaxHeap().map((num: number) =&gt; -num);\n}\n\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums: number[], k: number): number[] {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n    const maxHeap = new MaxHeap([]);\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (let i = k; i &lt; nums.length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.dart<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nMinHeap topKHeap(List&lt;int&gt; nums, int k) {\n  // \u521d\u59cb\u5316\u5c0f\u9876\u5806\uff0c\u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n  MinHeap heap = MinHeap(nums.sublist(0, k));\n  // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n  for (int i = k; i &lt; nums.length; i++) {\n    // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n    if (nums[i] &gt; heap.peek()) {\n      heap.pop();\n      heap.push(nums[i]);\n    }\n  }\n  return heap;\n}\n</code></pre> top_k.rs<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfn top_k_heap(nums: Vec&lt;i32&gt;, k: usize) -&gt; BinaryHeap&lt;Reverse&lt;i32&gt;&gt; {\n    // BinaryHeap \u662f\u5927\u9876\u5806\uff0c\u4f7f\u7528 Reverse \u5c06\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u5b9e\u73b0\u5c0f\u9876\u5806\n    let mut heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for &amp;num in nums.iter().take(k) {\n        heap.push(Reverse(num));\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for &amp;num in nums.iter().skip(k) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if num &gt; heap.peek().unwrap().0 {\n            heap.pop();\n            heap.push(Reverse(num));\n        }\n    }\n    heap\n}\n</code></pre> top_k.c<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid pushMinHeap(MaxHeap *maxHeap, int val) {\n    // \u5143\u7d20\u53d6\u53cd\n    push(maxHeap, -val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nint popMinHeap(MaxHeap *maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -pop(maxHeap);\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peekMinHeap(MaxHeap *maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -peek(maxHeap);\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n// \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\u7684\u51fd\u6570\nint *topKHeap(int *nums, int sizeNums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n    int *empty = (int *)malloc(0);\n    MaxHeap *maxHeap = newMaxHeap(empty, 0);\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; sizeNums; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    int *res = getMinHeap(maxHeap);\n    // \u91ca\u653e\u5185\u5b58\n    delMaxHeap(maxHeap);\n    return res;\n}\n</code></pre> top_k.kt<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfun topKHeap(nums: IntArray, k: Int): Queue&lt;Int&gt; {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    val heap = PriorityQueue&lt;Int&gt;()\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (i in 0..&lt;k) {\n        heap.offer(nums[i])\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (i in k..&lt;nums.size) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll()\n            heap.offer(nums[i])\n        }\n    }\n    return heap\n}\n</code></pre> top_k.rb<pre><code>[class]{}-[func]{top_k_heap}\n</code></pre> top_k.zig<pre><code>[class]{}-[func]{topKHeap}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>A total of \\(n\\) rounds of heap insertions and deletions are performed, with the maximum heap size being \\(k\\), hence the time complexity is \\(O(n \\log k)\\). This method is very efficient; when \\(k\\) is small, the time complexity tends towards \\(O(n)\\); when \\(k\\) is large, the time complexity will not exceed \\(O(n \\log n)\\).</p> <p>Additionally, this method is suitable for scenarios with dynamic data streams. By continuously adding data, we can maintain the elements within the heap, thereby achieving dynamic updates of the largest \\(k\\) elements.</p>"},{"location":"chapter_introduction/","title":"Chapter 1. \u00a0 Introduction to Algorithms","text":"<p>Abstract</p> <p>A graceful maiden dances, intertwined with the data, her skirt swaying to the melody of algorithms.</p> <p>She invites you to a dance, follow her steps, and enter the world of algorithms full of logic and beauty.</p>"},{"location":"chapter_introduction/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>1.1 \u00a0 Algorithms are Everywhere</li> <li>1.2 \u00a0 What is an Algorithm</li> <li>1.3 \u00a0 Summary</li> </ul>"},{"location":"chapter_introduction/algorithms_are_everywhere/","title":"1.1 \u00a0 Algorithms are Everywhere","text":"<p>When we hear the word \"algorithm,\" we naturally think of mathematics. However, many algorithms do not involve complex mathematics but rely more on basic logic, which can be seen everywhere in our daily lives.</p> <p>Before formally discussing algorithms, there's an interesting fact worth sharing: you have already unconsciously learned many algorithms and have become accustomed to applying them in your daily life. Here, I will give a few specific examples to prove this point.</p> <p>Example 1: Looking Up a Dictionary. In an English dictionary, words are listed alphabetically. Suppose we're searching for a word that starts with the letter \\(r\\). This is typically done in the following way:</p> <ol> <li>Open the dictionary to about halfway and check the first letter on the page, let's say the letter is \\(m\\).</li> <li>Since \\(r\\) comes after \\(m\\) in the alphabet, we can ignore the first half of the dictionary and focus on the latter half.</li> <li>Repeat steps <code>1.</code> and <code>2.</code> until you find the page where the word starts with \\(r\\).</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 1-1 \u00a0 Process of Looking Up a Dictionary </p> <p>This essential skill for elementary students, looking up a dictionary, is actually the famous \"Binary Search\" algorithm. From a data structure perspective, we can consider the dictionary as a sorted \"array\"; from an algorithmic perspective, the series of actions taken to look up a word in the dictionary can be viewed as \"Binary Search.\"</p> <p>Example 2: Organizing Playing Cards. When playing cards, we need to arrange the cards in our hand in ascending order, as shown in the following process.</p> <ol> <li>Divide the playing cards into \"ordered\" and \"unordered\" sections, assuming initially the leftmost card is already in order.</li> <li>Take out a card from the unordered section and insert it into the correct position in the ordered section; after this, the leftmost two cards are in order.</li> <li>Continue to repeat step <code>2.</code> until all cards are in order.</li> </ol> <p></p> <p> Figure 1-2 \u00a0 Playing Cards Sorting Process </p> <p>The above method of organizing playing cards is essentially the \"Insertion Sort\" algorithm, which is very efficient for small datasets. Many programming languages' sorting functions include the insertion sort.</p> <p>Example 3: Making Change. Suppose we buy goods worth \\(69\\) yuan at a supermarket and give the cashier \\(100\\) yuan, then the cashier needs to give us \\(31\\) yuan in change. They would naturally complete the thought process as shown below.</p> <ol> <li>The options are currencies smaller than \\(31\\), including \\(1\\), \\(5\\), \\(10\\), and \\(20\\).</li> <li>Take out the largest \\(20\\) from the options, leaving \\(31 - 20 = 11\\).</li> <li>Take out the largest \\(10\\) from the remaining options, leaving \\(11 - 10 = 1\\).</li> <li>Take out the largest \\(1\\) from the remaining options, leaving \\(1 - 1 = 0\\).</li> <li>Complete the change-making, with the solution being \\(20 + 10 + 1 = 31\\).</li> </ol> <p></p> <p> Figure 1-3 \u00a0 Change making process </p> <p>In the above steps, we make the best choice at each step (using the largest denomination possible), ultimately resulting in a feasible change-making plan. From the perspective of data structures and algorithms, this method is essentially a \"Greedy\" algorithm.</p> <p>From cooking a meal to interstellar travel, almost all problem-solving involves algorithms. The advent of computers allows us to store data structures in memory and write code to call the CPU and GPU to execute algorithms. In this way, we can transfer real-life problems to computers, solving various complex issues more efficiently.</p> <p>Tip</p> <p>If concepts such as data structures, algorithms, arrays, and binary search still seem somewhat obsecure, I encourage you to continue reading. This book will gently guide you into the realm of understanding data structures and algorithms.</p>"},{"location":"chapter_introduction/summary/","title":"1.3 \u00a0 Summary","text":"<ul> <li>Algorithms are ubiquitous in daily life and are not as inaccessible and complex as they might seem. In fact, we have already unconsciously learned many algorithms to solve various problems in life.</li> <li>The principle of looking up a word in a dictionary is consistent with the binary search algorithm. The binary search algorithm embodies the important algorithmic concept of divide and conquer.</li> <li>The process of organizing playing cards is very similar to the insertion sort algorithm. The insertion sort algorithm is suitable for sorting small datasets.</li> <li>The steps of making change in currency essentially follow the greedy algorithm, where each step involves making the best possible choice at the moment.</li> <li>An algorithm is a set of instructions or steps used to solve a specific problem within a finite amount of time, while a data structure is the way data is organized and stored in a computer.</li> <li>Data structures and algorithms are closely linked. Data structures are the foundation of algorithms, and algorithms are the stage to utilize the functions of data structures.</li> <li>We can liken data structures and algorithms to building blocks. The blocks represent data, the shape and connection method of the blocks represent data structures, and the steps of assembling the blocks correspond to algorithms.</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/","title":"1.2 \u00a0 What is an Algorithm","text":""},{"location":"chapter_introduction/what_is_dsa/#121-definition-of-an-algorithm","title":"1.2.1 \u00a0 Definition of an Algorithm","text":"<p>An \"algorithm\" is a set of instructions or steps to solve a specific problem within a finite amount of time. It has the following characteristics:</p> <ul> <li>The problem is clearly defined, including unambiguous definitions of input and output.</li> <li>The algorithm is feasible, meaning it can be completed within a finite number of steps, time, and memory space.</li> <li>Each step has a definitive meaning. The output is consistently the same under the same inputs and conditions.</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#122-definition-of-a-data-structure","title":"1.2.2 \u00a0 Definition of a Data Structure","text":"<p>A \"data structure\" is a way of organizing and storing data in a computer, with the following design goals:</p> <ul> <li>Minimize space occupancy to save computer memory.</li> <li>Make data operations as fast as possible, covering data access, addition, deletion, updating, etc.</li> <li>Provide concise data representation and logical information to enable efficient algorithm execution.</li> </ul> <p>Designing data structures is a balancing act, often requiring trade-offs. If you want to improve in one aspect, you often need to compromise in another. Here are two examples:</p> <ul> <li>Compared to arrays, linked lists offer more convenience in data addition and deletion but sacrifice data access speed.</li> <li>Graphs, compared to linked lists, provide richer logical information but require more memory space.</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#123-relationship-between-data-structures-and-algorithms","title":"1.2.3 \u00a0 Relationship Between Data Structures and Algorithms","text":"<p>As shown in the Figure 1-4 , data structures and algorithms are highly related and closely integrated, specifically in the following three aspects:</p> <ul> <li>Data structures are the foundation of algorithms. They provide structured data storage and methods for manipulating data for algorithms.</li> <li>Algorithms are the stage where data structures come into play. The data structure alone only stores data information; it is through the application of algorithms that specific problems can be solved.</li> <li>Algorithms can often be implemented based on different data structures, but their execution efficiency can vary greatly. Choosing the right data structure is key.</li> </ul> <p></p> <p> Figure 1-4 \u00a0 Relationship between data structures and algorithms </p> <p>Data structures and algorithms can be likened to a set of building blocks, as illustrated in the Figure 1-5 . A building block set includes numerous pieces, accompanied by detailed assembly instructions. Following these instructions step by step allows us to construct an intricate block model.</p> <p></p> <p> Figure 1-5 \u00a0 Assembling blocks </p> <p>The detailed correspondence between the two is shown in the Table 1-1 .</p> <p> Table 1-1 \u00a0 Comparing Data Structures and Algorithms to Building Blocks </p> Data Structures and Algorithms Building Blocks Input data Unassembled blocks Data structure Organization of blocks, including shape, size, connections, etc Algorithm A series of steps to assemble the blocks into the desired shape Output data Completed Block model <p>It's worth noting that data structures and algorithms are independent of programming languages. For this reason, this book is able to provide implementations in multiple programming languages.</p> <p>Conventional Abbreviation</p> <p>In real-life discussions, we often refer to \"Data Structures and Algorithms\" simply as \"Algorithms\". For example, the well-known LeetCode algorithm problems actually test both data structure and algorithm knowledge.</p>"},{"location":"chapter_preface/","title":"Chapter 0. \u00a0 Preface","text":"<p>Abstract</p> <p>Algorithms are like a beautiful symphony, with each line of code flowing like a rhythm.</p> <p>May this book ring softly in your mind, leaving a unique and profound melody.</p>"},{"location":"chapter_preface/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>0.1 \u00a0 About This Book</li> <li>0.2 \u00a0 How to Read</li> <li>0.3 \u00a0 Summary</li> </ul>"},{"location":"chapter_preface/about_the_book/","title":"0.1 \u00a0 About This Book","text":"<p>This open-source project aims to create a free, and beginner-friendly crash course on data structures and algorithms.</p> <ul> <li>Using animated illustrations, it delivers structured insights into data structures and algorithmic concepts, ensuring comprehensibility and a smooth learning curve.</li> <li>Run code with just one click, supporting Java, C++, Python, Go, JS, TS, C#, Swift, Rust, Dart, Zig and other languages.</li> <li>Readers are encouraged to engage with each other in the discussion area for each section, questions and comments are usually answered within two days.</li> </ul>"},{"location":"chapter_preface/about_the_book/#011-target-audience","title":"0.1.1 \u00a0 Target Audience","text":"<p>If you are new to algorithms with limited exposure, or you have accumulated some experience in algorithms, but you only have a vague understanding of data structures and algorithms, and you are constantly jumping between \"yep\" and \"hmm\", then this book is for you!</p> <p>If you have already accumulated a certain amount of problem-solving experience, and are familiar with most types of problems, then this book can help you review and organize your algorithm knowledge system. The repository's source code can be used as a \"problem-solving toolkit\" or an \"algorithm cheat sheet\".</p> <p>If you are an algorithm expert, we look forward to receiving your valuable suggestions, or join us and collaborate.</p> <p>Prerequisites</p> <p>You should know how to write and read simple code in at least one programming language.</p>"},{"location":"chapter_preface/about_the_book/#012-content-structure","title":"0.1.2 \u00a0 Content Structure","text":"<p>The main content of the book is shown in the following figure.</p> <ul> <li>Complexity Analysis: explores aspects and methods for evaluating data structures and algorithms. Covers methods of deriving time complexity and space complexity, along with common types and examples.</li> <li>Data Structures: focuses on fundamental data types, classification methods, definitions, pros and cons, common operations, types, applications, and implementation methods of data structures such as array, linked list, stack, queue, hash table, tree, heap, graph, etc.</li> <li>Algorithms: defines algorithms, discusses their pros and cons, efficiency, application scenarios, problem-solving steps, and includes sample questions for various algorithms such as search, sorting, divide and conquer, backtracking, dynamic programming, greedy algorithms, and more.</li> </ul> <p></p> <p> Figure 0-1 \u00a0 Main Content of the Book </p>"},{"location":"chapter_preface/about_the_book/#013-acknowledgements","title":"0.1.3 \u00a0 Acknowledgements","text":"<p>This book is continuously improved with the joint efforts of many contributors from the open-source community. Thanks to each writer who invested their time and energy, listed in the order generated by GitHub: krahets, codingonion, nuomi1, Gonglja, Reanon, justin-tse, danielsss, hpstory, S-N-O-R-L-A-X, night-cruise, msk397, gvenusleo, RiverTwilight, gyt95, zhuoqinyue, Zuoxun, Xia-Sang, mingXta, FangYuan33, GN-Yu, IsChristina, xBLACKICEx, guowei-gong, Cathay-Chen, mgisr, JoseHung, qualifier1024, pengchzn, Guanngxu, longsizhuo, L-Super, what-is-me, yuan0221, lhxsm, Slone123c, WSL0809, longranger2, theNefelibatas, xiongsp, JeffersonHuang, hongyun-robot, K3v123, yuelinxin, a16su, gaofer, malone6, Wonderdch, xjr7670, DullSword, Horbin-Magician, NI-SW, reeswell, XC-Zero, XiaChuerwu, yd-j, iron-irax, huawuque404, MolDuM, Nigh, KorsChen, foursevenlove, 52coder, bubble9um, youshaoXG, curly210102, gltianwen, fanchenggang, Transmigration-zhou, FloranceYeh, FreddieLi, ShiMaRing, lipusheng, Javesun99, JackYang-hellobobo, shanghai-Jerry, 0130w, Keynman, psychelzh, logan-qiu, ZnYang2018, MwumLi, 1ch0, Phoenix0415, qingpeng9802, Richard-Zhang1019, QiLOL, Suremotoo, Turing-1024-Lee, Evilrabbit520, GaochaoZhu, ZJKung, linzeyan, hezhizhen, ZongYangL, beintentional, czruby, coderlef, dshlstarr, szu17dmy, fbigm, gledfish, hts0000, boloboloda, iStig, jiaxianhua, wenjianmin, keshida, kilikilikid, lclc6, lwbaptx, liuxjerry, lucaswangdev, lyl625760, chadyi, noobcodemaker, selear, siqyka, syd168, 4yDX3906, tao363, wangwang105, weibk, yabo083, yi427, yishangzhang, zhouLion, baagod, ElaBosak233, xb534, luluxia, yanedie, thomasq0, YangXuanyi and th1nk3r-ing.</p> <p>The code review work for this book was completed by codingonion, Gonglja, gvenusleo, hpstory, justin\u2010tse, krahets, night-cruise, nuomi1, and Reanon (listed in alphabetical order). Thanks to them for their time and effort, ensuring the standardization and uniformity of the code in various languages.</p> <p>Throughout the creation of this book, numerous individuals provided invaluable assistance, including but not limited to:</p> <ul> <li>Thanks to my mentor at the company, Dr. Xi Li, who encouraged me in a conversation to \"get moving fast,\" which solidified my determination to write this book;</li> <li>Thanks to my girlfriend Bubble, as the first reader of this book, for offering many valuable suggestions from the perspective of a beginner in algorithms, making this book more suitable for newbies;</li> <li>Thanks to Tengbao, Qibao, and Feibao for coming up with a creative name for this book, evoking everyone's fond memories of writing their first line of code \"Hello World!\";</li> <li>Thanks to Xiaoquan for providing professional help in intellectual property, which has played a significant role in the development of this open-source book;</li> <li>Thanks to Sutong for designing a beautiful cover and logo for this book, and for patiently making multiple revisions under my insistence;</li> <li>Thanks to @squidfunk for providing writing and typesetting suggestions, as well as his developed open-source documentation theme Material-for-MkDocs.</li> </ul> <p>Throughout the writing journey, I delved into numerous textbooks and articles on data structures and algorithms. These works served as exemplary models, ensuring the accuracy and quality of this book's content. I extend my gratitude to all who preceded me for their invaluable contributions!</p> <p>This book advocates a combination of hands-on and minds-on learning, inspired in this regard by \"Dive into Deep Learning\". I highly recommend this excellent book to all readers.</p> <p>Heartfelt thanks to my parents, whose ongoing support and encouragement have allowed me to do this interesting work.</p>"},{"location":"chapter_preface/suggestions/","title":"0.2 \u00a0 How to Read","text":"<p>Tip</p> <p>For the best reading experience, it is recommended that you read through this section.</p>"},{"location":"chapter_preface/suggestions/#021-writing-conventions","title":"0.2.1 \u00a0 Writing Conventions","text":"<ul> <li>Chapters marked with '*' after the title are optional and contain relatively challenging content. If you are short on time, it is advisable to skip them.</li> <li>Key technical terms and their English equivalents are enclosed in Bold + italics brackets, for example, array. It's advisable to familiarize yourself with these for better comprehension of technical texts.</li> <li>Proprietary terms and words with specific meanings are indicated with \u201cquotation marks\u201d to avoid ambiguity.</li> <li>Bolded text indicates key content or summary statements, which deserve special attention.</li> <li>When it comes to terms that are inconsistent between programming languages, this book follows Python, for example using \\(\\text{None}\\) to mean \"null\".</li> <li>This book partially ignores the comment conventions for programming languages in exchange for a more compact layout of the content. The comments primarily consist of three types: title comments, content comments, and multi-line comments.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>\"\"\"Header comments for labeling functions, classes, test samples, etc\"\"\"\"\n\n# Comments for explaining details\n\n\"\"\"\nMultiline\ncomments\n\"\"\"\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>// Header comments for labeling functions, classes, test samples, etc\n\n// Comments for explaining details.\n\n// Multiline\n// comments\n</code></pre>"},{"location":"chapter_preface/suggestions/#022-efficient-learning-via-animated-illustrations","title":"0.2.2 \u00a0 Efficient Learning via Animated Illustrations","text":"<p>Compared with text, videos and pictures have a higher density of information and are more structured, making them easier to understand. In this book, key and difficult concepts are mainly presented through animations and illustrations, with text serving as explanations and supplements.</p> <p>When encountering content with animations or illustrations as shown in the Figure 0-2 , prioritize understanding the figure, with text as supplementary, integrating both for a comprehensive understanding.</p> <p></p> <p> Figure 0-2 \u00a0 Animated Illustration Example </p>"},{"location":"chapter_preface/suggestions/#023-deepen-understanding-through-coding-practice","title":"0.2.3 \u00a0 Deepen Understanding through Coding Practice","text":"<p>The source code of this book is hosted on the GitHub Repository. As shown in the Figure 0-3 , the source code comes with test examples and can be executed with just a single click.</p> <p>If time permits, it's recommended to type out the code yourself. If pressed for time, at least read and run all the codes.</p> <p>Compared to just reading code, writing code often yields more learning. Learning by doing is the real way to learn.</p> <p></p> <p> Figure 0-3 \u00a0 Running Code Example </p> <p>Setting up to run the code involves three main steps.</p> <p>Step 1: Install a local programming environment. Follow the tutorial in the appendix for installation, or skip this step if already installed.</p> <p>Step 2: Clone or download the code repository. Visit the GitHub Repository.</p> <p>If Git is installed, use the following command to clone the repository:</p> <pre><code>git clone https://github.com/krahets/hello-algo.git\n</code></pre> <p>Alternatively, you can also click the \"Download ZIP\" button at the location shown in the Figure 0-4  to directly download the code as a compressed ZIP file. Then, you can simply extract it locally.</p> <p></p> <p> Figure 0-4 \u00a0 Cloning Repository and Downloading Code </p> <p>Step 3: Run the source code. As shown in the Figure 0-5 , for the code block labeled with the file name at the top, we can find the corresponding source code file in the <code>codes</code> folder of the repository. These files can be executed with a single click, which will help you save unnecessary debugging time and allow you to focus on learning.</p> <p></p> <p> Figure 0-5 \u00a0 Code Block and Corresponding Source Code File </p>"},{"location":"chapter_preface/suggestions/#024-learning-together-in-discussion","title":"0.2.4 \u00a0 Learning Together in Discussion","text":"<p>While reading this book, please don't skip over the points that you didn't learn. Feel free to post your questions in the comment section. We will be happy to answer them and can usually respond within two days.</p> <p>As illustrated in the Figure 0-6 , each chapter features a comment section at the bottom. I encourage you to pay attention to these comments. They not only expose you to others' encountered problems, aiding in identifying knowledge gaps and sparking deeper contemplation, but also invite you to generously contribute by answering fellow readers' inquiries, sharing insights, and fostering mutual improvement.</p> <p></p> <p> Figure 0-6 \u00a0 Comment Section Example </p>"},{"location":"chapter_preface/suggestions/#025-algorithm-learning-path","title":"0.2.5 \u00a0 Algorithm Learning Path","text":"<p>Overall, the journey of mastering data structures and algorithms can be divided into three stages:</p> <ol> <li>Stage 1: Introduction to algorithms. We need to familiarize ourselves with the characteristics and usage of various data structures and learn about the principles, processes, uses, and efficiency of different algorithms.</li> <li>Stage 2: Practicing algorithm problems. It is recommended to start from popular problems, such as Sword for Offer and LeetCode Hot 100, and accumulate at least 100 questions to familiarize yourself with mainstream algorithmic problems. Forgetfulness can be a challenge when you start practicing, but rest assured that this is normal. We can follow the \"Ebbinghaus Forgetting Curve\" to review the questions, and usually after 3~5 rounds of repetitions, we will be able to memorize them.</li> <li>Stage 3: Building the knowledge system. In terms of learning, we can read algorithm column articles, solution frameworks, and algorithm textbooks to continuously enrich the knowledge system. In terms of practicing, we can try advanced strategies, such as categorizing by topic, multiple solutions for a single problem,  and one solution for multiple problems, etc. Insights on these strategies can be found in various communities.</li> </ol> <p>As shown in the Figure 0-7 , this book mainly covers \u201cStage 1,\u201d aiming to help you more efficiently embark on Stages 2 and 3.</p> <p></p> <p> Figure 0-7 \u00a0 Algorithm Learning Path </p>"},{"location":"chapter_preface/summary/","title":"0.3 \u00a0 Summary","text":"<ul> <li>The main audience of this book is beginners in algorithm. If you already have some basic knowledge, this book can help you systematically review your algorithm knowledge, and the source code in this book can also be used as a \"Coding Toolkit\".</li> <li>The book consists of three main sections, Complexity Analysis, Data Structures, and Algorithms, covering most of the topics in the field.</li> <li>For newcomers to algorithms, it is crucial to read an introductory book in the beginning stages to avoid many detours or common pitfalls.</li> <li>Animations and figures within the book are usually used to introduce key points and difficult knowledge. These should be given more attention when reading the book.</li> <li>Practice is the best way to learn programming. It is highly recommended that you run the source code and type in the code yourself.</li> <li>Each chapter in the web version of this book features a discussion section, and you are welcome to share your questions and insights at any time.</li> </ul>"},{"location":"chapter_stack_and_queue/","title":"Chapter 5. \u00a0 Stack and Queue","text":"<p>Abstract</p> <p>A stack is like cats placed on top of each other, while a queue is like cats lined up one by one.</p> <p>They represent the logical relationships of Last-In-First-Out (LIFO) and First-In-First-Out (FIFO), respectively.</p>"},{"location":"chapter_stack_and_queue/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>5.1 \u00a0 Stack</li> <li>5.2 \u00a0 Queue</li> <li>5.3 \u00a0 Double-ended Queue</li> <li>5.4 \u00a0 Summary</li> </ul>"},{"location":"chapter_stack_and_queue/deque/","title":"5.3 \u00a0 Double-Ended Queue","text":"<p>In a queue, we can only delete elements from the head or add elements to the tail. As shown in the following diagram, a \"double-ended queue (deque)\" offers more flexibility, allowing the addition or removal of elements at both the head and the tail.</p> <p></p> <p> Figure 5-7 \u00a0 Operations in Double-Ended Queue </p>"},{"location":"chapter_stack_and_queue/deque/#531-common-operations-in-double-ended-queue","title":"5.3.1 \u00a0 Common Operations in Double-Ended Queue","text":"<p>The common operations in a double-ended queue are listed below, and the names of specific methods depend on the programming language used.</p> <p> Table 5-3 \u00a0 Efficiency of Double-Ended Queue Operations </p> Method Name Description Time Complexity <code>pushFirst()</code> Add an element to the head \\(O(1)\\) <code>pushLast()</code> Add an element to the tail \\(O(1)\\) <code>popFirst()</code> Remove the first element \\(O(1)\\) <code>popLast()</code> Remove the last element \\(O(1)\\) <code>peekFirst()</code> Access the first element \\(O(1)\\) <code>peekLast()</code> Access the last element \\(O(1)\\) <p>Similarly, we can directly use the double-ended queue classes implemented in programming languages:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig deque.py<pre><code>from collections import deque\n\n# Initialize the deque\ndeque: deque[int] = deque()\n\n# Enqueue elements\ndeque.append(2)      # Add to the tail\ndeque.append(5)\ndeque.append(4)\ndeque.appendleft(3)  # Add to the head\ndeque.appendleft(1)\n\n# Access elements\nfront: int = deque[0]  # The first element\nrear: int = deque[-1]  # The last element\n\n# Dequeue elements\npop_front: int = deque.popleft()  # The first element dequeued\npop_rear: int = deque.pop()       # The last element dequeued\n\n# Get the length of the deque\nsize: int = len(deque)\n\n# Check if the deque is empty\nis_empty: bool = len(deque) == 0\n</code></pre> deque.cpp<pre><code>/* Initialize the deque */\ndeque&lt;int&gt; deque;\n\n/* Enqueue elements */\ndeque.push_back(2);   // Add to the tail\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // Add to the head\ndeque.push_front(1);\n\n/* Access elements */\nint front = deque.front(); // The first element\nint back = deque.back();   // The last element\n\n/* Dequeue elements */\ndeque.pop_front();  // The first element dequeued\ndeque.pop_back();   // The last element dequeued\n\n/* Get the length of the deque */\nint size = deque.size();\n\n/* Check if the deque is empty */\nbool empty = deque.empty();\n</code></pre> deque.java<pre><code>/* Initialize the deque */\nDeque&lt;Integer&gt; deque = new LinkedList&lt;&gt;();\n\n/* Enqueue elements */\ndeque.offerLast(2);   // Add to the tail\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // Add to the head\ndeque.offerFirst(1);\n\n/* Access elements */\nint peekFirst = deque.peekFirst();  // The first element\nint peekLast = deque.peekLast();    // The last element\n\n/* Dequeue elements */\nint popFirst = deque.pollFirst();  // The first element dequeued\nint popLast = deque.pollLast();    // The last element dequeued\n\n/* Get the length of the deque */\nint size = deque.size();\n\n/* Check if the deque is empty */\nboolean isEmpty = deque.isEmpty();\n</code></pre> deque.cs<pre><code>/* Initialize the deque */\n// In C#, LinkedList is used as a deque\nLinkedList&lt;int&gt; deque = new();\n\n/* Enqueue elements */\ndeque.AddLast(2);   // Add to the tail\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // Add to the head\ndeque.AddFirst(1);\n\n/* Access elements */\nint peekFirst = deque.First.Value;  // The first element\nint peekLast = deque.Last.Value;    // The last element\n\n/* Dequeue elements */\ndeque.RemoveFirst();  // The first element dequeued\ndeque.RemoveLast();   // The last element dequeued\n\n/* Get the length of the deque */\nint size = deque.Count;\n\n/* Check if the deque is empty */\nbool isEmpty = deque.Count == 0;\n</code></pre> deque_test.go<pre><code>/* Initialize the deque */\n// In Go, use list as a deque\ndeque := list.New()\n\n/* Enqueue elements */\ndeque.PushBack(2)      // Add to the tail\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // Add to the head\ndeque.PushFront(1)\n\n/* Access elements */\nfront := deque.Front() // The first element\nrear := deque.Back()   // The last element\n\n/* Dequeue elements */\ndeque.Remove(front)    // The first element dequeued\ndeque.Remove(rear)     // The last element dequeued\n\n/* Get the length of the deque */\nsize := deque.Len()\n\n/* Check if the deque is empty */\nisEmpty := deque.Len() == 0\n</code></pre> deque.swift<pre><code>/* Initialize the deque */\n// Swift does not have a built-in deque class, so Array can be used as a deque\nvar deque: [Int] = []\n\n/* Enqueue elements */\ndeque.append(2) // Add to the tail\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // Add to the head\ndeque.insert(1, at: 0)\n\n/* Access elements */\nlet peekFirst = deque.first! // The first element\nlet peekLast = deque.last!   // The last element\n\n/* Dequeue elements */\n// Using Array, popFirst has a complexity of O(n)\nlet popFirst = deque.removeFirst() // The first element dequeued\nlet popLast = deque.removeLast()   // The last element dequeued\n\n/* Get the length of the deque */\nlet size = deque.count\n\n/* Check if the deque is empty */\nlet isEmpty = deque.isEmpty\n</code></pre> deque.js<pre><code>/* Initialize the deque */\n// JavaScript does not have a built-in deque, so Array is used as a deque\nconst deque = [];\n\n/* Enqueue elements */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// Note that unshift() has a time complexity of O(n) as it's an array\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* Access elements */\nconst peekFirst = deque[0]; // The first element\nconst peekLast = deque[deque.length - 1]; // The last element\n\n/* Dequeue elements */\n// Note that shift() has a time complexity of O(n) as it's an array\nconst popFront = deque.shift(); // The first element dequeued\nconst popBack = deque.pop();    // The last element dequeued\n\n/* Get the length of the deque */\nconst size = deque.length;\n\n/* Check if the deque is empty */\nconst isEmpty = size === 0;\n</code></pre> deque.ts<pre><code>/* Initialize the deque */\n// TypeScript does not have a built-in deque, so Array is used as a deque\nconst deque: number[] = [];\n\n/* Enqueue elements */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// Note that unshift() has a time complexity of O(n) as it's an array\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* Access elements */\nconst peekFirst: number = deque[0]; // The first element\nconst peekLast: number = deque[deque.length - 1]; // The last element\n\n/* Dequeue elements */\n// Note that shift() has a time complexity of O(n) as it's an array\nconst popFront: number = deque.shift() as number; // The first element dequeued\nconst popBack: number = deque.pop() as number;    // The last element dequeued\n\n/* Get the length of the deque */\nconst size: number = deque.length;\n\n/* Check if the deque is empty */\nconst isEmpty: boolean = size === 0;\n</code></pre> deque.dart<pre><code>/* Initialize the deque */\n// In Dart, Queue is defined as a deque\nQueue&lt;int&gt; deque = Queue&lt;int&gt;();\n\n/* Enqueue elements */\ndeque.addLast(2);  // Add to the tail\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // Add to the head\ndeque.addFirst(1);\n\n/* Access elements */\nint peekFirst = deque.first; // The first element\nint peekLast = deque.last;   // The last element\n\n/* Dequeue elements */\nint popFirst = deque.removeFirst(); // The first element dequeued\nint popLast = deque.removeLast();   // The last element dequeued\n\n/* Get the length of the deque */\nint size = deque.length;\n\n/* Check if the deque is empty */\nbool isEmpty = deque.isEmpty;\n</code></pre> deque.rs<pre><code>/* Initialize the deque */\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* Enqueue elements */\ndeque.push_back(2);  // Add to the tail\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // Add to the head\ndeque.push_front(1);\n\n/* Access elements */\nif let Some(front) = deque.front() { // The first element\n}\nif let Some(rear) = deque.back() {   // The last element\n}\n\n/* Dequeue elements */\nif let Some(pop_front) = deque.pop_front() { // The first element dequeued\n}\nif let Some(pop_rear) = deque.pop_back() {   // The last element dequeued\n}\n\n/* Get the length of the deque */\nlet size = deque.len();\n\n/* Check if the deque is empty */\nlet is_empty = deque.is_empty();\n</code></pre> deque.c<pre><code>// C does not provide a built-in deque\n</code></pre> deque.kt<pre><code>\n</code></pre> deque.zig<pre><code>\n</code></pre> Visualizing Code <p>https://pythontutor.com/render.html#code=from%20collections%20import%20deque%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%0A%20%20%20%20deq%20%3D%20deque%28%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%85%A5%E9%98%9F%0A%20%20%20%20deq.append%282%29%20%20%23%20%E6%B7%BB%E5%8A%A0%E8%87%B3%E9%98%9F%E5%B0%BE%0A%20%20%20%20deq.append%285%29%0A%20%20%20%20deq.append%284%29%0A%20%20%20%20deq.appendleft%283%29%20%20%23%20%E6%B7%BB%E5%8A%A0%E8%87%B3%E9%98%9F%E9%A6%96%0A%20%20%20%20deq.appendleft%281%29%0A%20%20%20%20print%28%22%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%20deque%20%3D%22,%20deq%29%0A%0A%20%20%20%20%23%20%E8%AE%BF%E9%97%AE%E5%85%83%E7%B4%A0%0A%20%20%20%20front%20%3D%20deq%5B0%5D%20%20%23%20%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%20front%20%3D%22,%20front%29%0A%20%20%20%20rear%20%3D%20deq%5B-1%5D%20%20%23%20%E9%98%9F%E5%B0%BE%E5%85%83%E7%B4%A0%0A%20%20%20%20print%28%22%E9%98%9F%E5%B0%BE%E5%85%83%E7%B4%A0%20rear%20%3D%22,%20rear%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20pop_front%20%3D%20deq.popleft%28%29%20%20%23%20%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%87%BA%E9%98%9F%E5%85%83%E7%B4%A0%20%20pop_front%20%3D%22,%20pop_front%29%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%87%BA%E9%98%9F%E5%90%8E%20deque%20%3D%22,%20deq%29%0A%20%20%20%20pop_rear%20%3D%20deq.pop%28%29%20%20%23%20%E9%98%9F%E5%B0%BE%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20print%28%22%E9%98%9F%E5%B0%BE%E5%87%BA%E9%98%9F%E5%85%83%E7%B4%A0%20%20pop_rear%20%3D%22,%20pop_rear%29%0A%20%20%20%20print%28%22%E9%98%9F%E5%B0%BE%E5%87%BA%E9%98%9F%E5%90%8E%20deque%20%3D%22,%20deq%29%0A%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E7%9A%84%E9%95%BF%E5%BA%A6%0A%20%20%20%20size%20%3D%20len%28deq%29%0A%20%20%20%20print%28%22%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E9%95%BF%E5%BA%A6%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20%E5%88%A4%E6%96%AD%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%0A%20%20%20%20is_empty%20%3D%20len%28deq%29%20%3D%3D%200%0A%20%20%20%20print%28%22%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%20%3D%22,%20is_empty%29&amp;cumulative=false&amp;curInstr=3&amp;heapPrimitives=nevernest&amp;mode=display&amp;origin=opt-frontend.js&amp;py=311&amp;rawInputLstJSON=%5B%5D&amp;textReferences=false</p>"},{"location":"chapter_stack_and_queue/deque/#532-implementing-a-double-ended-queue","title":"5.3.2 \u00a0 Implementing a Double-Ended Queue *","text":"<p>The implementation of a double-ended queue is similar to that of a regular queue, it can be based on either a linked list or an array as the underlying data structure.</p>"},{"location":"chapter_stack_and_queue/deque/#1-implementation-based-on-doubly-linked-list","title":"1. \u00a0 Implementation Based on Doubly Linked List","text":"<p>Recall from the previous section that we used a regular singly linked list to implement a queue, as it conveniently allows for deleting from the head (corresponding to the dequeue operation) and adding new elements after the tail (corresponding to the enqueue operation).</p> <p>For a double-ended queue, both the head and the tail can perform enqueue and dequeue operations. In other words, a double-ended queue needs to implement operations in the opposite direction as well. For this, we use a \"doubly linked list\" as the underlying data structure of the double-ended queue.</p> <p>As shown in the Figure 5-8 , we treat the head and tail nodes of the doubly linked list as the front and rear of the double-ended queue, respectively, and implement the functionality to add and remove nodes at both ends.</p> LinkedListDequepushLast()pushFirst()popLast()popFirst() <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 5-8 \u00a0 Implementing Double-Ended Queue with Doubly Linked List for Enqueue and Dequeue Operations </p> <p>The implementation code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_deque.py<pre><code>class ListNode:\n    \"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\"\"\"\n\n    def __init__(self, val: int):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self.val: int = val\n        self.next: ListNode | None = None  # \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n        self.prev: ListNode | None = None  # \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\nclass LinkedListDeque:\n    \"\"\"\u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._front: ListNode | None = None  # \u5934\u8282\u70b9 front\n        self._rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\n        self._size: int = 0  # \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self.size() == 0\n\n    def push(self, num: int, is_front: bool):\n        \"\"\"\u5165\u961f\u64cd\u4f5c\"\"\"\n        node = ListNode(num)\n        # \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if self.is_empty():\n            self._front = self._rear = node\n        # \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        elif is_front:\n            # \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            self._front.prev = node\n            node.next = self._front\n            self._front = node  # \u66f4\u65b0\u5934\u8282\u70b9\n        # \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else:\n            # \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            self._rear.next = node\n            node.prev = self._rear\n            self._rear = node  # \u66f4\u65b0\u5c3e\u8282\u70b9\n        self._size += 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n\n    def push_first(self, num: int):\n        \"\"\"\u961f\u9996\u5165\u961f\"\"\"\n        self.push(num, True)\n\n    def push_last(self, num: int):\n        \"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\n        self.push(num, False)\n\n    def pop(self, is_front: bool) -&gt; int:\n        \"\"\"\u51fa\u961f\u64cd\u4f5c\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        # \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if is_front:\n            val: int = self._front.val  # \u6682\u5b58\u5934\u8282\u70b9\u503c\n            # \u5220\u9664\u5934\u8282\u70b9\n            fnext: ListNode | None = self._front.next\n            if fnext != None:\n                fnext.prev = None\n                self._front.next = None\n            self._front = fnext  # \u66f4\u65b0\u5934\u8282\u70b9\n        # \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else:\n            val: int = self._rear.val  # \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            # \u5220\u9664\u5c3e\u8282\u70b9\n            rprev: ListNode | None = self._rear.prev\n            if rprev != None:\n                rprev.next = None\n                self._rear.prev = None\n            self._rear = rprev  # \u66f4\u65b0\u5c3e\u8282\u70b9\n        self._size -= 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val\n\n    def pop_first(self) -&gt; int:\n        \"\"\"\u961f\u9996\u51fa\u961f\"\"\"\n        return self.pop(True)\n\n    def pop_last(self) -&gt; int:\n        \"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\n        return self.pop(False)\n\n    def peek_first(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        return self._front.val\n\n    def peek_last(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        return self._rear.val\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n        node = self._front\n        res = [0] * self.size()\n        for i in range(self.size()):\n            res[i] = node.val\n            node = node.next\n        return res\n</code></pre> linkedlist_deque.cpp<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nstruct DoublyListNode {\n    int val;              // \u8282\u70b9\u503c\n    DoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n    DoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\u6307\u9488\n    DoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n    }\n};\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n  private:\n    DoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    int queSize = 0;              // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    LinkedListDeque() : front(nullptr), rear(nullptr) {\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~LinkedListDeque() {\n        // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n        DoublyListNode *pre, *cur = front;\n        while (cur != nullptr) {\n            pre = cur;\n            cur = cur-&gt;next;\n            delete pre;\n        }\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    void push(int num, bool isFront) {\n        DoublyListNode *node = new DoublyListNode(num);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty())\n            front = rear = node;\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front-&gt;prev = node;\n            node-&gt;next = front;\n            front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear-&gt;next = node;\n            node-&gt;prev = rear;\n            rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    int pop(bool isFront) {\n        if (isEmpty())\n            throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n        int val;\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            val = front-&gt;val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            DoublyListNode *fNext = front-&gt;next;\n            if (fNext != nullptr) {\n                fNext-&gt;prev = nullptr;\n                front-&gt;next = nullptr;\n            }\n            delete front;\n            front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        } else {\n            val = rear-&gt;val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            DoublyListNode *rPrev = rear-&gt;prev;\n            if (rPrev != nullptr) {\n                rPrev-&gt;next = nullptr;\n                rear-&gt;prev = nullptr;\n            }\n            delete rear;\n            rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    int popFirst() {\n        return pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    int popLast() {\n        return pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        return front-&gt;val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        return rear-&gt;val;\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    vector&lt;int&gt; toVector() {\n        DoublyListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_deque.java<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    int val; // \u8282\u70b9\u503c\n    ListNode next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    ListNode prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n    ListNode(int val) {\n        this.val = val;\n        prev = next = null;\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private int queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    public LinkedListDeque() {\n        front = rear = null;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    private void push(int num, boolean isFront) {\n        ListNode node = new ListNode(num);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty())\n            front = rear = node;\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front.prev = node;\n            node.next = front;\n            front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear.next = node;\n            node.prev = rear;\n            rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    private int pop(boolean isFront) {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        int val;\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            val = front.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            ListNode fNext = front.next;\n            if (fNext != null) {\n                fNext.prev = null;\n                front.next = null;\n            }\n            front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        } else {\n            val = rear.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            ListNode rPrev = rear.prev;\n            if (rPrev != null) {\n                rPrev.next = null;\n                rear.prev = null;\n            }\n            rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int popFirst() {\n        return pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int popLast() {\n        return pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return rear.val;\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.cs<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode(int val) {\n    public int val = val;       // \u8282\u70b9\u503c\n    public ListNode? next = null; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    public ListNode? prev = null; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    ListNode? front, rear; // \u5934\u8282\u70b9 front, \u5c3e\u8282\u70b9 rear\n    int queSize = 0;      // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    public LinkedListDeque() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    void Push(int num, bool isFront) {\n        ListNode node = new(num);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (IsEmpty()) {\n            front = node;\n            rear = node;\n        }\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front!.prev = node;\n            node.next = front;\n            front = node; // \u66f4\u65b0\u5934\u8282\u70b9                           \n        }\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear!.next = node;\n            node.prev = rear;\n            rear = node;  // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n\n        queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void PushFirst(int num) {\n        Push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void PushLast(int num) {\n        Push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    int? Pop(bool isFront) {\n        if (IsEmpty())\n            throw new Exception();\n        int? val;\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            val = front?.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            ListNode? fNext = front?.next;\n            if (fNext != null) {\n                fNext.prev = null;\n                front!.next = null;\n            }\n            front = fNext;   // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else {\n            val = rear?.val;  // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            ListNode? rPrev = rear?.prev;\n            if (rPrev != null) {\n                rPrev.next = null;\n                rear!.prev = null;\n            }\n            rear = rPrev;    // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n\n        queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int? PopFirst() {\n        return Pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int? PopLast() {\n        return Pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int? PeekFirst() {\n        if (IsEmpty())\n            throw new Exception();\n        return front?.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int? PeekLast() {\n        if (IsEmpty())\n            throw new Exception();\n        return rear?.val;\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int?[] ToArray() {\n        ListNode? node = front;\n        int?[] res = new int?[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node?.val;\n            node = node?.next;\n        }\n\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.go<pre><code>/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype linkedListDeque struct {\n    // \u4f7f\u7528\u5185\u7f6e\u5305 list\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u53cc\u7aef\u961f\u5217 */\nfunc newLinkedListDeque() *linkedListDeque {\n    return &amp;linkedListDeque{\n        data: list.New(),\n    }\n}\n\n/* \u961f\u9996\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushFirst(value any) {\n    s.data.PushFront(value)\n}\n\n/* \u961f\u5c3e\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushLast(value any) {\n    s.data.PushBack(value)\n}\n\n/* \u961f\u9996\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u961f\u5c3e\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListDeque) peekFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (s *linkedListDeque) peekLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListDeque) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListDeque) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListDeque) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_deque.swift<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    var val: Int // \u8282\u70b9\u503c\n    var next: ListNode? // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    weak var prev: ListNode? // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n    init(val: Int) {\n        self.val = val\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private var front: ListNode? // \u5934\u8282\u70b9 front\n    private var rear: ListNode? // \u5c3e\u8282\u70b9 rear\n    private var _size: Int // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    init() {\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    private func push(num: Int, isFront: Bool) {\n        let node = ListNode(val: num)\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if isEmpty() {\n            front = node\n            rear = node\n        }\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if isFront {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front?.prev = node\n            node.next = front\n            front = node // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear?.next = node\n            node.prev = rear\n            rear = node // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        _size += 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    func pushFirst(num: Int) {\n        push(num: num, isFront: true)\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    func pushLast(num: Int) {\n        push(num: num, isFront: false)\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    private func pop(isFront: Bool) -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        let val: Int\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if isFront {\n            val = front!.val // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            let fNext = front?.next\n            if fNext != nil {\n                fNext?.prev = nil\n                front?.next = nil\n            }\n            front = fNext // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else {\n            val = rear!.val // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            let rPrev = rear?.prev\n            if rPrev != nil {\n                rPrev?.next = nil\n                rear?.prev = nil\n            }\n            rear = rPrev // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        _size -= 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    func popFirst() -&gt; Int {\n        pop(isFront: true)\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    func popLast() -&gt; Int {\n        pop(isFront: false)\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return front!.val\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return rear!.val\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.js<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    val; // \u8282\u70b9\u503c\n\n    constructor(val) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    #front; // \u5934\u8282\u70b9 front\n    #rear; // \u5c3e\u8282\u70b9 rear\n    #queSize; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n        this.#queSize = 0;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\n    pushLast(val) {\n        const node = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            this.#rear.next = node;\n            node.prev = this.#rear;\n            this.#rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        this.#queSize++;\n    }\n\n    /* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\n    pushFirst(val) {\n        const node = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            this.#front.prev = node;\n            node.next = this.#front;\n            this.#front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        this.#queSize++;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\n    popLast() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5c3e\u8282\u70b9\n        let temp = this.#rear.prev;\n        if (temp !== null) {\n            temp.next = null;\n            this.#rear.prev = null;\n        }\n        this.#rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        this.#queSize--;\n        return value;\n    }\n\n    /* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\n    popFirst() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5934\u8282\u70b9\n        let temp = this.#front.next;\n        if (temp !== null) {\n            temp.prev = null;\n            this.#front.next = null;\n        }\n        this.#front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\n        this.#queSize--;\n        return value;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast() {\n        return this.#queSize === 0 ? null : this.#rear.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst() {\n        return this.#queSize === 0 ? null : this.#front.val;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u6253\u5370\u53cc\u5411\u961f\u5217 */\n    print() {\n        const arr = [];\n        let temp = this.#front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.ts<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    prev: ListNode; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    next: ListNode; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    val: number; // \u8282\u70b9\u503c\n\n    constructor(val: number) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private front: ListNode; // \u5934\u8282\u70b9 front\n    private rear: ListNode; // \u5c3e\u8282\u70b9 rear\n    private queSize: number; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n        this.queSize = 0;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\n    pushLast(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            this.rear.next = node;\n            node.prev = this.rear;\n            this.rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        this.queSize++;\n    }\n\n    /* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\n    pushFirst(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            this.front.prev = node;\n            node.next = this.front;\n            this.front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        this.queSize++;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\n    popLast(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5c3e\u8282\u70b9\n        let temp: ListNode = this.rear.prev;\n        if (temp !== null) {\n            temp.next = null;\n            this.rear.prev = null;\n        }\n        this.rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        this.queSize--;\n        return value;\n    }\n\n    /* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\n    popFirst(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5934\u8282\u70b9\n        let temp: ListNode = this.front.next;\n        if (temp !== null) {\n            temp.prev = null;\n            this.front.next = null;\n        }\n        this.front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\n        this.queSize--;\n        return value;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast(): number {\n        return this.queSize === 0 ? null : this.rear.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst(): number {\n        return this.queSize === 0 ? null : this.front.val;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u6253\u5370\u53cc\u5411\u961f\u5217 */\n    print(): void {\n        const arr: number[] = [];\n        let temp: ListNode = this.front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.dart<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n  int val; // \u8282\u70b9\u503c\n  ListNode? next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n  ListNode? prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n  ListNode(this.val, {this.next, this.prev});\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u5bf9\u5217 */\nclass LinkedListDeque {\n  late ListNode? _front; // \u5934\u8282\u70b9 _front\n  late ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\n  int _queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n  LinkedListDeque() {\n    this._front = null;\n    this._rear = null;\n  }\n\n  /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u957f\u5ea6 */\n  int size() {\n    return this._queSize;\n  }\n\n  /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return size() == 0;\n  }\n\n  /* \u5165\u961f\u64cd\u4f5c */\n  void push(int _num, bool isFront) {\n    final ListNode node = ListNode(_num);\n    if (isEmpty()) {\n      // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 _front \u548c _rear \u90fd\u6307\u5411 node\n      _front = _rear = node;\n    } else if (isFront) {\n      // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n      // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n      _front!.prev = node;\n      node.next = _front;\n      _front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n    } else {\n      // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n      // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n      _rear!.next = node;\n      node.prev = _rear;\n      _rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n    }\n    _queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n  }\n\n  /* \u961f\u9996\u5165\u961f */\n  void pushFirst(int _num) {\n    push(_num, true);\n  }\n\n  /* \u961f\u5c3e\u5165\u961f */\n  void pushLast(int _num) {\n    push(_num, false);\n  }\n\n  /* \u51fa\u961f\u64cd\u4f5c */\n  int? pop(bool isFront) {\n    // \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\n    if (isEmpty()) {\n      return null;\n    }\n    final int val;\n    if (isFront) {\n      // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n      val = _front!.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n      // \u5220\u9664\u5934\u8282\u70b9\n      ListNode? fNext = _front!.next;\n      if (fNext != null) {\n        fNext.prev = null;\n        _front!.next = null;\n      }\n      _front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n    } else {\n      // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n      val = _rear!.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n      // \u5220\u9664\u5c3e\u8282\u70b9\n      ListNode? rPrev = _rear!.prev;\n      if (rPrev != null) {\n        rPrev.next = null;\n        _rear!.prev = null;\n      }\n      _rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n    }\n    _queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    return val;\n  }\n\n  /* \u961f\u9996\u51fa\u961f */\n  int? popFirst() {\n    return pop(true);\n  }\n\n  /* \u961f\u5c3e\u51fa\u961f */\n  int? popLast() {\n    return pop(false);\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int? peekFirst() {\n    return _front?.val;\n  }\n\n  /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n  int? peekLast() {\n    return _rear?.val;\n  }\n\n  /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; _queSize; i++) {\n      res.add(node!.val);\n      node = node.next;\n    }\n    return res;\n  }\n}\n</code></pre> linkedlist_deque.rs<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\npub struct ListNode&lt;T&gt; {\n    pub val: T,                                 // \u8282\u70b9\u503c\n    pub next: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n    pub prev: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u524d\u9a71\u8282\u70b9\u6307\u9488\n}\n\nimpl&lt;T&gt; ListNode&lt;T&gt; {\n    pub fn new(val: T) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n        Rc::new(RefCell::new(ListNode {\n            val,\n            next: None,\n            prev: None,\n        }))\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListDeque&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5934\u8282\u70b9 front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // \u5c3e\u8282\u70b9 rear\n    que_size: usize,                         // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListDeque&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    pub fn push(&amp;mut self, num: T, is_front: bool) {\n        let node = ListNode::new(num);\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        if is_front {\n            match self.front.take() {\n                // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n                None =&gt; {\n                    self.rear = Some(node.clone());\n                    self.front = Some(node);\n                }\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n                Some(old_front) =&gt; {\n                    old_front.borrow_mut().prev = Some(node.clone());\n                    node.borrow_mut().next = Some(old_front);\n                    self.front = Some(node); // \u66f4\u65b0\u5934\u8282\u70b9\n                }\n            }\n        }\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else {\n            match self.rear.take() {\n                // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n                None =&gt; {\n                    self.front = Some(node.clone());\n                    self.rear = Some(node);\n                }\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n                Some(old_rear) =&gt; {\n                    old_rear.borrow_mut().next = Some(node.clone());\n                    node.borrow_mut().prev = Some(old_rear);\n                    self.rear = Some(node); // \u66f4\u65b0\u5c3e\u8282\u70b9\n                }\n            }\n        }\n        self.que_size += 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pub fn push_first(&amp;mut self, num: T) {\n        self.push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pub fn push_last(&amp;mut self, num: T) {\n        self.push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    pub fn pop(&amp;mut self, is_front: bool) -&gt; Option&lt;T&gt; {\n        // \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de None\n        if self.is_empty() {\n            return None;\n        };\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if is_front {\n            self.front.take().map(|old_front| {\n                match old_front.borrow_mut().next.take() {\n                    Some(new_front) =&gt; {\n                        new_front.borrow_mut().prev.take();\n                        self.front = Some(new_front); // \u66f4\u65b0\u5934\u8282\u70b9\n                    }\n                    None =&gt; {\n                        self.rear.take();\n                    }\n                }\n                self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n                Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n            })\n        }\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else {\n            self.rear.take().map(|old_rear| {\n                match old_rear.borrow_mut().prev.take() {\n                    Some(new_rear) =&gt; {\n                        new_rear.borrow_mut().next.take();\n                        self.rear = Some(new_rear); // \u66f4\u65b0\u5c3e\u8282\u70b9\n                    }\n                    None =&gt; {\n                        self.front.take();\n                    }\n                }\n                self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n                Rc::try_unwrap(old_rear).ok().unwrap().into_inner().val\n            })\n        }\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    pub fn pop_first(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    pub fn pop_last(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    pub fn peek_first(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    pub fn peek_last(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.rear.as_ref()\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.insert(0, node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_deque.c<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\ntypedef struct DoublyListNode {\n    int val;                     // \u8282\u70b9\u503c\n    struct DoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\n    struct DoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\n} DoublyListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nDoublyListNode *newDoublyListNode(int num) {\n    DoublyListNode *new = (DoublyListNode *)malloc(sizeof(DoublyListNode));\n    new-&gt;val = num;\n    new-&gt;next = NULL;\n    new-&gt;prev = NULL;\n    return new;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delDoublyListNode(DoublyListNode *node) {\n    free(node);\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\n    DoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    int queSize;                  // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n} LinkedListDeque;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListDeque *newLinkedListDeque() {\n    LinkedListDeque *deque = (LinkedListDeque *)malloc(sizeof(LinkedListDeque));\n    deque-&gt;front = NULL;\n    deque-&gt;rear = NULL;\n    deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListdeque(LinkedListDeque *deque) {\n    // \u91ca\u653e\u6240\u6709\u8282\u70b9\n    for (int i = 0; i &lt; deque-&gt;queSize &amp;&amp; deque-&gt;front != NULL; i++) {\n        DoublyListNode *tmp = deque-&gt;front;\n        deque-&gt;front = deque-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // \u91ca\u653e deque \u7ed3\u6784\u4f53\n    free(deque);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListDeque *deque) {\n    return (size(deque) == 0);\n}\n\n/* \u5165\u961f */\nvoid push(LinkedListDeque *deque, int num, bool isFront) {\n    DoublyListNode *node = newDoublyListNode(num);\n    // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411node\n    if (empty(deque)) {\n        deque-&gt;front = deque-&gt;rear = node;\n    }\n    // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n    else if (isFront) {\n        // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n        deque-&gt;front-&gt;prev = node;\n        node-&gt;next = deque-&gt;front;\n        deque-&gt;front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n    }\n    // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n    else {\n        // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n        deque-&gt;rear-&gt;next = node;\n        node-&gt;prev = deque-&gt;rear;\n        deque-&gt;rear = node;\n    }\n    deque-&gt;queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(LinkedListDeque *deque, int num) {\n    push(deque, num, true);\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(LinkedListDeque *deque, int num) {\n    push(deque, num, false);\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;front);\n    return deque-&gt;front-&gt;val;\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;rear);\n    return deque-&gt;rear-&gt;val;\n}\n\n/* \u51fa\u961f */\nint pop(LinkedListDeque *deque, bool isFront) {\n    if (empty(deque))\n        return -1;\n    int val;\n    // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n    if (isFront) {\n        val = peekFirst(deque); // \u6682\u5b58\u5934\u8282\u70b9\u503c\n        DoublyListNode *fNext = deque-&gt;front-&gt;next;\n        if (fNext) {\n            fNext-&gt;prev = NULL;\n            deque-&gt;front-&gt;next = NULL;\n            delDoublyListNode(deque-&gt;front);\n        }\n        deque-&gt;front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n    }\n    // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n    else {\n        val = peekLast(deque); // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n        DoublyListNode *rPrev = deque-&gt;rear-&gt;prev;\n        if (rPrev) {\n            rPrev-&gt;next = NULL;\n            deque-&gt;rear-&gt;prev = NULL;\n            delDoublyListNode(deque-&gt;rear);\n        }\n        deque-&gt;rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n    }\n    deque-&gt;queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    return val;\n}\n\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(LinkedListDeque *deque) {\n    return pop(deque, true);\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(LinkedListDeque *deque) {\n    return pop(deque, false);\n}\n\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListDeque(LinkedListDeque *deque) {\n    int *arr = malloc(sizeof(int) * deque-&gt;queSize);\n    // \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\n    int i;\n    DoublyListNode *node;\n    for (i = 0, node = deque-&gt;front; i &lt; deque-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, deque-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_deque.kt<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode(var value: Int) {\n    // \u8282\u70b9\u503c\n    var next: ListNode? = null // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    var prev: ListNode? = null // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private var front: ListNode? = null // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private var rear: ListNode? = null\n    private var queSize = 0 // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    fun push(num: Int, isFront: Boolean) {\n        val node = ListNode(num)\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty()) {\n            rear = node\n            front = rear\n            // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        } else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front?.prev = node\n            node.next = front\n            front = node // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear?.next = node\n            node.prev = rear\n            rear = node // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize++ // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    fun pushFirst(num: Int) {\n        push(num, true)\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    fun pushLast(num: Int) {\n        push(num, false)\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    fun pop(isFront: Boolean): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n\n        val value: Int\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            value = front!!.value // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            val fNext = front!!.next\n            if (fNext != null) {\n                fNext.prev = null\n                front!!.next = null\n            }\n            front = fNext // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        } else {\n            value = rear!!.value // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            val rPrev = rear!!.prev\n            if (rPrev != null) {\n                rPrev.next = null\n                rear!!.prev = null\n            }\n            rear = rPrev // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize-- // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return value\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    fun popFirst(): Int {\n        return pop(true)\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    fun popLast(): Int {\n        return pop(false)\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peekFirst(): Int {\n        if (isEmpty()) {\n            throw IndexOutOfBoundsException()\n\n        }\n        return front!!.value\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return rear!!.value\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!.value\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.rb<pre><code>[class]{ListNode}-[func]{}\n\n[class]{LinkedListDeque}-[func]{}\n</code></pre> linkedlist_deque.zig<pre><code>// \u53cc\u5411\u94fe\u8868\u8282\u70b9\nfn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = undefined,     // \u8282\u70b9\u503c\n        next: ?*Self = null,    // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n        prev: ?*Self = null,    // \u524d\u9a71\u8282\u70b9\u6307\u9488\n\n        // Initialize a list node with specific value\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n            self.prev = null;\n        }\n    };\n}\n\n// \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\nfn LinkedListDeque(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        front: ?*ListNode(T) = null,                    // \u5934\u8282\u70b9 front\n        rear: ?*ListNode(T) = null,                     // \u5c3e\u8282\u70b9 rear\n        que_size: usize = 0,                             // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.front = null;\n            self.rear = null;\n            self.que_size = 0;\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.que_size;\n        }\n\n        // \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u5165\u961f\u64cd\u4f5c\n        pub fn push(self: *Self, num: T, is_front: bool) !void {\n            var node = try self.mem_allocator.create(ListNode(T));\n            node.init(num);\n            // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n            if (self.isEmpty()) {\n                self.front = node;\n                self.rear = node;\n            // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n            } else if (is_front) {\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n                self.front.?.prev = node;\n                node.next = self.front;\n                self.front = node;  // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n            } else {\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n                self.rear.?.next = node;\n                node.prev = self.rear;\n                self.rear = node;   // \u66f4\u65b0\u5c3e\u8282\u70b9\n            }\n            self.que_size += 1;      // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        } \n\n        // \u961f\u9996\u5165\u961f\n        pub fn pushFirst(self: *Self, num: T) !void {\n            try self.push(num, true);\n        } \n\n        // \u961f\u5c3e\u5165\u961f\n        pub fn pushLast(self: *Self, num: T) !void {\n            try self.push(num, false);\n        } \n\n        // \u51fa\u961f\u64cd\u4f5c\n        pub fn pop(self: *Self, is_front: bool) T {\n            if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n            var val: T = undefined;\n            // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n            if (is_front) {\n                val = self.front.?.val;     // \u6682\u5b58\u5934\u8282\u70b9\u503c\n                // \u5220\u9664\u5934\u8282\u70b9\n                var fNext = self.front.?.next;\n                if (fNext != null) {\n                    fNext.?.prev = null;\n                    self.front.?.next = null;\n                }\n                self.front = fNext;         // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n            } else {\n                val = self.rear.?.val;      // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n                // \u5220\u9664\u5c3e\u8282\u70b9\n                var rPrev = self.rear.?.prev;\n                if (rPrev != null) {\n                    rPrev.?.next = null;\n                    self.rear.?.prev = null;\n                }\n                self.rear = rPrev;          // \u66f4\u65b0\u5c3e\u8282\u70b9\n            }\n            self.que_size -= 1;              // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n            return val;\n        } \n\n        // \u961f\u9996\u51fa\u961f\n        pub fn popFirst(self: *Self) T {\n            return self.pop(true);\n        } \n\n        // \u961f\u5c3e\u51fa\u961f\n        pub fn popLast(self: *Self) T {\n            return self.pop(false);\n        } \n\n        // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n        pub fn peekFirst(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n            return self.front.?.val;\n        }  \n\n        // \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\n        pub fn peekLast(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n            return self.rear.?.val;\n        }\n\n        // \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.front;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[i] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#2-implementation-based-on-array","title":"2. \u00a0 Implementation Based on Array","text":"<p>As shown in the Figure 5-9 , similar to implementing a queue with an array, we can also use a circular array to implement a double-ended queue.</p> ArrayDequepushLast()pushFirst()popLast()popFirst() <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 5-9 \u00a0 Implementing Double-Ended Queue with Array for Enqueue and Dequeue Operations </p> <p>The implementation only needs to add methods for \"front enqueue\" and \"rear dequeue\":</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_deque.py<pre><code>class ArrayDeque:\n    \"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\n\n    def __init__(self, capacity: int):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._nums: list[int] = [0] * capacity\n        self._front: int = 0\n        self._size: int = 0\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self._size == 0\n\n    def index(self, i: int) -&gt; int:\n        \"\"\"\u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15\"\"\"\n        # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        # \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        # \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + self.capacity()) % self.capacity()\n\n    def push_first(self, num: int):\n        \"\"\"\u961f\u9996\u5165\u961f\"\"\"\n        if self._size == self.capacity():\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        # \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        self._front = self.index(self._front - 1)\n        # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        self._nums[self._front] = num\n        self._size += 1\n\n    def push_last(self, num: int):\n        \"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\n        if self._size == self.capacity():\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        # \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        rear = self.index(self._front + self._size)\n        # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop_first(self) -&gt; int:\n        \"\"\"\u961f\u9996\u51fa\u961f\"\"\"\n        num = self.peek_first()\n        # \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        self._front = self.index(self._front + 1)\n        self._size -= 1\n        return num\n\n    def pop_last(self) -&gt; int:\n        \"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\n        num = self.peek_last()\n        self._size -= 1\n        return num\n\n    def peek_first(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        return self._nums[self._front]\n\n    def peek_last(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        # \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        last = self.index(self._front + self._size - 1)\n        return self._nums[last]\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n        # \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        res = []\n        for i in range(self._size):\n            res.append(self._nums[self.index(self._front + i)])\n        return res\n</code></pre> array_deque.cpp<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n  private:\n    vector&lt;int&gt; nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;        // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;      // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    ArrayDeque(int capacity) {\n        nums.resize(capacity);\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    int capacity() {\n        return nums.size();\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    int index(int i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    void pushFirst(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    void pushLast(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        int rear = index(front + queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    int popFirst() {\n        int num = peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        return nums[front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    vector&lt;int&gt; toVector() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        vector&lt;int&gt; res(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n};\n</code></pre> array_deque.java<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    private int[] nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public ArrayDeque(int capacity) {\n        this.nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    private int index(int i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void pushFirst(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void pushLast(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        int rear = index(front + queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int popFirst() {\n        int num = peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int[] toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.cs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public ArrayDeque(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    int Index(int i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + Capacity()) % Capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void PushFirst(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = Index(front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void PushLast(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        int rear = Index(front + queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int PopFirst() {\n        int num = PeekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = Index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int PopLast() {\n        int num = PeekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int PeekFirst() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        return nums[front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int PeekLast() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = Index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int[] ToArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[Index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.go<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype arrayDeque struct {\n    nums        []int // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    queSize     int   // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n    queCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\n    return &amp;arrayDeque{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayDeque) size() int {\n    return q.queSize\n}\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayDeque) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nfunc (q *arrayDeque) index(i int) int {\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n    return (i + q.queCapacity) % q.queCapacity\n}\n\n/* \u961f\u9996\u5165\u961f */\nfunc (q *arrayDeque) pushFirst(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n        return\n    }\n    // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n    q.front = q.index(q.front - 1)\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n    q.nums[q.front] = num\n    q.queSize++\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nfunc (q *arrayDeque) pushLast(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n        return\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    rear := q.index(q.front + q.queSize)\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* \u961f\u9996\u51fa\u961f */\nfunc (q *arrayDeque) popFirst() any {\n    num := q.peekFirst()\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    q.front = q.index(q.front + 1)\n    q.queSize--\n    return num\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nfunc (q *arrayDeque) popLast() any {\n    num := q.peekLast()\n    q.queSize--\n    return num\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayDeque) peekFirst() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (q *arrayDeque) peekLast() any {\n    if q.isEmpty() {\n        return nil\n    }\n    // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    last := q.index(q.front + q.queSize - 1)\n    return q.nums[last]\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayDeque) toSlice() []int {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    res := make([]int, q.queSize)\n    for i, j := 0, q.front; i &lt; q.queSize; i++ {\n        res[i] = q.nums[q.index(j)]\n        j++\n    }\n    return res\n}\n</code></pre> array_deque.swift<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    private var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var _size: Int // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init(capacity: Int) {\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    private func index(i: Int) -&gt; Int {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        (i + capacity()) % capacity()\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    func pushFirst(num: Int) {\n        if size() == capacity() {\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(i: front - 1)\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num\n        _size += 1\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    func pushLast(num: Int) {\n        if size() == capacity() {\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        let rear = index(i: front + size())\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    func popFirst() -&gt; Int {\n        let num = peekFirst()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(i: front + 1)\n        _size -= 1\n        return num\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    func popLast() -&gt; Int {\n        let num = peekLast()\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return nums[front]\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        let last = index(i: front + size() - 1)\n        return nums[last]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    func toArray() -&gt; [Int] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        (front ..&lt; front + size()).map { nums[index(i: $0)] }\n    }\n}\n</code></pre> array_deque.js<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    #nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    #front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    #queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n        this.#front = 0;\n        this.#queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    capacity() {\n        return this.#nums.length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    index(i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pushFirst(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        this.#front = this.index(this.#front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        this.#nums[this.#front] = num;\n        this.#queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pushLast(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        const rear = this.index(this.#front + this.#queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    popFirst() {\n        const num = this.peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        this.#front = this.index(this.#front + 1);\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    popLast() {\n        const num = this.peekLast();\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.#nums[this.#front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        const last = this.index(this.#front + this.#queSize - 1);\n        return this.#nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const res = [];\n        for (let i = 0, j = this.#front; i &lt; this.#queSize; i++, j++) {\n            res[i] = this.#nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.ts<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    private nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private queSize: number; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = 0;\n        this.queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    capacity(): number {\n        return this.nums.length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    index(i: number): number {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pushFirst(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        this.front = this.index(this.front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        this.nums[this.front] = num;\n        this.queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pushLast(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        const rear: number = this.index(this.front + this.queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    popFirst(): number {\n        const num: number = this.peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        this.front = this.index(this.front + 1);\n        this.queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    popLast(): number {\n        const num: number = this.peekLast();\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.nums[this.front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        const last = this.index(this.front + this.queSize - 1);\n        return this.nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    toArray(): number[] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const res: number[] = [];\n        for (let i = 0, j = this.front; i &lt; this.queSize; i++, j++) {\n            res[i] = this.nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.dart<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n  late List&lt;int&gt; _nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n  late int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n  late int _queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  ArrayDeque(int capacity) {\n    this._nums = List.filled(capacity, 0);\n    this._front = this._queSize = 0;\n  }\n\n  /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n  int capacity() {\n    return _nums.length;\n  }\n\n  /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n  int index(int i) {\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n    return (i + capacity()) % capacity();\n  }\n\n  /* \u961f\u9996\u5165\u961f */\n  void pushFirst(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n    }\n    // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 _front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n    _front = index(_front - 1);\n    // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u9996\n    _nums[_front] = _num;\n    _queSize++;\n  }\n\n  /* \u961f\u5c3e\u5165\u961f */\n  void pushLast(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    int rear = index(_front + _queSize);\n    // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* \u961f\u9996\u51fa\u961f */\n  int popFirst() {\n    int _num = peekFirst();\n    // \u961f\u9996\u6307\u9488\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n    _front = index(_front + 1);\n    _queSize--;\n    return _num;\n  }\n\n  /* \u961f\u5c3e\u51fa\u961f */\n  int popLast() {\n    int _num = peekLast();\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int peekFirst() {\n    if (isEmpty()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n    }\n    return _nums[_front];\n  }\n\n  /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n  int peekLast() {\n    if (isEmpty()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n    }\n    // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    int last = index(_front + _queSize - 1);\n    return _nums[last];\n  }\n\n  /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n  List&lt;int&gt; toArray() {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[index(j)];\n    }\n    return res;\n  }\n}\n</code></pre> array_deque.rs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nstruct ArrayDeque {\n    nums: Vec&lt;i32&gt;,  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front: usize,    // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    que_size: usize, // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n}\n\nimpl ArrayDeque {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(capacity: usize) -&gt; Self {\n        Self {\n            nums: vec![0; capacity],\n            front: 0,\n            que_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    pub fn capacity(&amp;self) -&gt; usize {\n        self.nums.len()\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        self.que_size\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    fn index(&amp;self, i: i32) -&gt; usize {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return ((i + self.capacity() as i32) % self.capacity() as i32) as usize;\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pub fn push_first(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        self.front = self.index(self.front as i32 - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        self.nums[self.front] = num;\n        self.que_size += 1;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pub fn push_last(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        let rear = self.index(self.front as i32 + self.que_size as i32);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self.nums[rear] = num;\n        self.que_size += 1;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    fn pop_first(&amp;mut self) -&gt; i32 {\n        let num = self.peek_first();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        self.front = self.index(self.front as i32 + 1);\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    fn pop_last(&amp;mut self) -&gt; i32 {\n        let num = self.peek_last();\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fn peek_first(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        };\n        self.nums[self.front]\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fn peek_last(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        };\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        let last = self.index(self.front as i32 + self.que_size as i32 - 1);\n        self.nums[last]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    fn to_array(&amp;self) -&gt; Vec&lt;i32&gt; {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        let mut res = vec![0; self.que_size];\n        let mut j = self.front;\n        for i in 0..self.que_size {\n            res[i] = self.nums[self.index(j as i32)];\n            j += 1;\n        }\n        res\n    }\n}\n</code></pre> array_deque.c<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\n    int *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n    int queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayDeque;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayDeque *newArrayDeque(int capacity) {\n    ArrayDeque *deque = (ArrayDeque *)malloc(sizeof(ArrayDeque));\n    // \u521d\u59cb\u5316\u6570\u7ec4\n    deque-&gt;queCapacity = capacity;\n    deque-&gt;nums = (int *)malloc(sizeof(int) * deque-&gt;queCapacity);\n    deque-&gt;front = deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayDeque(ArrayDeque *deque) {\n    free(deque-&gt;nums);\n    free(deque);\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayDeque *deque) {\n    return deque-&gt;queCapacity;\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayDeque *deque) {\n    return deque-&gt;queSize == 0;\n}\n\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint dequeIndex(ArrayDeque *deque, int i) {\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n    return ((i + capacity(deque)) % capacity(deque));\n}\n\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\n        return;\n    }\n    // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u56de\u5230\u5c3e\u90e8\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front - 1);\n    // \u5c06 num \u6dfb\u52a0\u5230\u961f\u9996\n    deque-&gt;nums[deque-&gt;front] = num;\n    deque-&gt;queSize++;\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\n        return;\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    int rear = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize);\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    deque-&gt;nums[rear] = num;\n    deque-&gt;queSize++;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(ArrayDeque *deque) {\n    // \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\n    assert(empty(deque) == 0);\n    return deque-&gt;nums[deque-&gt;front];\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(ArrayDeque *deque) {\n    // \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\n    assert(empty(deque) == 0);\n    int last = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize - 1);\n    return deque-&gt;nums[last];\n}\n\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(ArrayDeque *deque) {\n    int num = peekFirst(deque);\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front + 1);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(ArrayDeque *deque) {\n    int num = peekLast(deque);\n    deque-&gt;queSize--;\n    return num;\n}\n</code></pre> array_deque.kt<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque(capacity: Int) {\n    private var nums = IntArray(capacity) // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var queSize = 0 // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    private fun index(i: Int): Int {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity()\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    fun pushFirst(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1)\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num\n        queSize++\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    fun pushLast(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        val rear = index(front + queSize)\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        queSize++\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    fun popFirst(): Int {\n        val num = peekFirst()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(front + 1)\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fun popLast(): Int {\n        val num = peekLast()\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        val last = index(front + queSize - 1)\n        return nums[last]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    fun toArray(): IntArray {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[index(j)]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_deque.rb<pre><code>[class]{ArrayDeque}-[func]{}\n</code></pre> array_deque.zig<pre><code>[class]{ArrayDeque}-[func]{}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#533-applications-of-double-ended-queue","title":"5.3.3 \u00a0 Applications of Double-Ended Queue","text":"<p>The double-ended queue combines the logic of both stacks and queues, thus, it can implement all their respective use cases while offering greater flexibility.</p> <p>We know that software's \"undo\" feature is typically implemented using a stack: the system <code>pushes</code> each change operation onto the stack and then <code>pops</code> to implement undoing. However, considering the limitations of system resources, software often restricts the number of undo steps (for example, only allowing the last 50 steps). When the stack length exceeds 50, the software needs to perform a deletion operation at the bottom of the stack (the front of the queue). But a regular stack cannot perform this function, where a double-ended queue becomes necessary. Note that the core logic of \"undo\" still follows the Last-In-First-Out principle of a stack, but a double-ended queue can more flexibly implement some additional logic.</p>"},{"location":"chapter_stack_and_queue/queue/","title":"5.2 \u00a0 Queue","text":"<p>\"Queue\" is a linear data structure that follows the First-In-First-Out (FIFO) rule. As the name suggests, a queue simulates the phenomenon of lining up, where newcomers join the queue at the rear, and the person at the front leaves the queue first.</p> <p>As shown in the Figure 5-4 , we call the front of the queue the \"head\" and the back the \"tail.\" The operation of adding elements to the rear of the queue is termed \"enqueue,\" and the operation of removing elements from the front is termed \"dequeue.\"</p> <p></p> <p> Figure 5-4 \u00a0 Queue's First-In-First-Out Rule </p>"},{"location":"chapter_stack_and_queue/queue/#521-common-operations-on-queue","title":"5.2.1 \u00a0 Common Operations on Queue","text":"<p>The common operations on a queue are shown in the Table 5-2 . Note that method names may vary across different programming languages. Here, we use the same naming convention as that used for stacks.</p> <p> Table 5-2 \u00a0 Efficiency of Queue Operations </p> Method Name Description Time Complexity <code>push()</code> Enqueue an element, add it to the tail \\(O(1)\\) <code>pop()</code> Dequeue the head element \\(O(1)\\) <code>peek()</code> Access the head element \\(O(1)\\) <p>We can directly use the ready-made queue classes in programming languages:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig queue.py<pre><code>from collections import deque\n\n# Initialize the queue\n# In Python, we generally use the deque class as a queue\n# Although queue.Queue() is a pure queue class, it's not very user-friendly, so it's not recommended\nque: deque[int] = deque()\n\n# Enqueue elements\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n\n# Access the first element\nfront: int = que[0]\n\n# Dequeue an element\npop: int = que.popleft()\n\n# Get the length of the queue\nsize: int = len(que)\n\n# Check if the queue is empty\nis_empty: bool = len(que) == 0\n</code></pre> queue.cpp<pre><code>/* Initialize the queue */\nqueue&lt;int&gt; queue;\n\n/* Enqueue elements */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* Access the first element*/\nint front = queue.front();\n\n/* Dequeue an element */\nqueue.pop();\n\n/* Get the length of the queue */\nint size = queue.size();\n\n/* Check if the queue is empty */\nbool empty = queue.empty();\n</code></pre> queue.java<pre><code>/* Initialize the queue */\nQueue&lt;Integer&gt; queue = new LinkedList&lt;&gt;();\n\n/* Enqueue elements */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n\n/* Access the first element */\nint peek = queue.peek();\n\n/* Dequeue an element */\nint pop = queue.poll();\n\n/* Get the length of the queue */\nint size = queue.size();\n\n/* Check if the queue is empty */\nboolean isEmpty = queue.isEmpty();\n</code></pre> queue.cs<pre><code>/* Initialize the queue */\nQueue&lt;int&gt; queue = new();\n\n/* Enqueue elements */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n\n/* Access the first element */\nint peek = queue.Peek();\n\n/* Dequeue an element */\nint pop = queue.Dequeue();\n\n/* Get the length of the queue */\nint size = queue.Count;\n\n/* Check if the queue is empty */\nbool isEmpty = queue.Count == 0;\n</code></pre> queue_test.go<pre><code>/* Initialize the queue */\n// In Go, use list as a queue\nqueue := list.New()\n\n/* Enqueue elements */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n\n/* Access the first element */\npeek := queue.Front()\n\n/* Dequeue an element */\npop := queue.Front()\nqueue.Remove(pop)\n\n/* Get the length of the queue */\nsize := queue.Len()\n\n/* Check if the queue is empty */\nisEmpty := queue.Len() == 0\n</code></pre> queue.swift<pre><code>/* Initialize the queue */\n// Swift does not have a built-in queue class, so Array can be used as a queue\nvar queue: [Int] = []\n\n/* Enqueue elements */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n\n/* Access the first element */\nlet peek = queue.first!\n\n/* Dequeue an element */\n// Since it's an array, removeFirst has a complexity of O(n)\nlet pool = queue.removeFirst()\n\n/* Get the length of the queue */\nlet size = queue.count\n\n/* Check if the queue is empty */\nlet isEmpty = queue.isEmpty\n</code></pre> queue.js<pre><code>/* Initialize the queue */\n// JavaScript does not have a built-in queue, so Array can be used as a queue\nconst queue = [];\n\n/* Enqueue elements */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* Access the first element */\nconst peek = queue[0];\n\n/* Dequeue an element */\n// Since the underlying structure is an array, shift() method has a time complexity of O(n)\nconst pop = queue.shift();\n\n/* Get the length of the queue */\nconst size = queue.length;\n\n/* Check if the queue is empty */\nconst empty = queue.length === 0;\n</code></pre> queue.ts<pre><code>/* Initialize the queue */\n// TypeScript does not have a built-in queue, so Array can be used as a queue \nconst queue: number[] = [];\n\n/* Enqueue elements */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* Access the first element */\nconst peek = queue[0];\n\n/* Dequeue an element */\n// Since the underlying structure is an array, shift() method has a time complexity of O(n)\nconst pop = queue.shift();\n\n/* Get the length of the queue */\nconst size = queue.length;\n\n/* Check if the queue is empty */\nconst empty = queue.length === 0;\n</code></pre> queue.dart<pre><code>/* Initialize the queue */\n// In Dart, the Queue class is a double-ended queue but can be used as a queue\nQueue&lt;int&gt; queue = Queue();\n\n/* Enqueue elements */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n\n/* Access the first element */\nint peek = queue.first;\n\n/* Dequeue an element */\nint pop = queue.removeFirst();\n\n/* Get the length of the queue */\nint size = queue.length;\n\n/* Check if the queue is empty */\nbool isEmpty = queue.isEmpty;\n</code></pre> queue.rs<pre><code>/* Initialize the double-ended queue */\n// In Rust, use a double-ended queue as a regular queue\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* Enqueue elements */\ndeque.push_back(1);\ndeque.push_back(3);\ndeque.push_back(2);\ndeque.push_back(5);\ndeque.push_back(4);\n\n/* Access the first element */\nif let Some(front) = deque.front() {\n}\n\n/* Dequeue an element */\nif let Some(pop) = deque.pop_front() {\n}\n\n/* Get the length of the queue */\nlet size = deque.len();\n\n/* Check if the queue is empty */\nlet is_empty = deque.is_empty();\n</code></pre> queue.c<pre><code>// C does not provide a built-in queue\n</code></pre> queue.kt<pre><code>\n</code></pre> queue.zig<pre><code>\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_stack_and_queue/queue/#522-implementing-a-queue","title":"5.2.2 \u00a0 Implementing a Queue","text":"<p>To implement a queue, we need a data structure that allows adding elements at one end and removing them at the other. Both linked lists and arrays meet this requirement.</p>"},{"location":"chapter_stack_and_queue/queue/#1-implementation-based-on-a-linked-list","title":"1. \u00a0 Implementation Based on a Linked List","text":"<p>As shown in the Figure 5-5 , we can consider the \"head node\" and \"tail node\" of a linked list as the \"front\" and \"rear\" of the queue, respectively. It is stipulated that nodes can only be added at the rear and removed at the front.</p> LinkedListQueuepush()pop() <p></p> <p></p> <p></p> <p> Figure 5-5 \u00a0 Implementing Queue with Linked List for Enqueue and Dequeue Operations </p> <p>Below is the code for implementing a queue using a linked list:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_queue.py<pre><code>class LinkedListQueue:\n    \"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._front: ListNode | None = None  # \u5934\u8282\u70b9 front\n        self._rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return not self._front\n\n    def push(self, num: int):\n        \"\"\"\u5165\u961f\"\"\"\n        # \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        node = ListNode(num)\n        # \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if self._front is None:\n            self._front = node\n            self._rear = node\n        # \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        else:\n            self._rear.next = node\n            self._rear = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u961f\"\"\"\n        num = self.peek()\n        # \u5220\u9664\u5934\u8282\u70b9\n        self._front = self._front.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\n        return self._front.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        queue = []\n        temp = self._front\n        while temp:\n            queue.append(temp.val)\n            temp = temp.next\n        return queue\n</code></pre> linkedlist_queue.cpp<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n  private:\n    ListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    int queSize;\n\n  public:\n    LinkedListQueue() {\n        front = nullptr;\n        rear = nullptr;\n        queSize = 0;\n    }\n\n    ~LinkedListQueue() {\n        // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n        freeMemoryLinkedList(front);\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u961f */\n    void push(int num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        ListNode *node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == nullptr) {\n            front = node;\n            rear = node;\n        }\n        // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        else {\n            rear-&gt;next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    int pop() {\n        int num = peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        ListNode *tmp = front;\n        front = front-&gt;next;\n        // \u91ca\u653e\u5185\u5b58\n        delete tmp;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peek() {\n        if (size() == 0)\n            throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n        return front-&gt;val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_queue.java<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    private ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f */\n    public void push(int num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        ListNode node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == null) {\n            front = node;\n            rear = node;\n        // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int pop() {\n        int num = peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front.next;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.cs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    ListNode? front, rear;  // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear \n    int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u961f */\n    public void Push(int num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        ListNode node = new(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == null) {\n            front = node;\n            rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else if (rear != null) {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int Pop() {\n        int num = Peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front?.next;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return front!.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] ToArray() {\n        if (front == null)\n            return [];\n\n        ListNode? node = front;\n        int[] res = new int[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.go<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntype linkedListQueue struct {\n    // \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u961f\u5217\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newLinkedListQueue() *linkedListQueue {\n    return &amp;linkedListQueue{\n        data: list.New(),\n    }\n}\n\n/* \u5165\u961f */\nfunc (s *linkedListQueue) push(value any) {\n    s.data.PushBack(value)\n}\n\n/* \u51fa\u961f */\nfunc (s *linkedListQueue) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListQueue) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListQueue) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListQueue) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListQueue) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_queue.swift<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    private var front: ListNode? // \u5934\u8282\u70b9\n    private var rear: ListNode? // \u5c3e\u8282\u70b9\n    private var _size: Int\n\n    init() {\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u961f */\n    func push(num: Int) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        let node = ListNode(x: num)\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if front == nil {\n            front = node\n            rear = node\n        }\n        // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        else {\n            rear?.next = node\n            rear = node\n        }\n        _size += 1\n    }\n\n    /* \u51fa\u961f */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front?.next\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return front!.val\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.js<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    #front; // \u5934\u8282\u70b9 #front\n    #rear; // \u5c3e\u8282\u70b9 #rear\n    #queSize = 0;\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        const node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (!this.#front) {\n            this.#front = node;\n            this.#rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            this.#rear.next = node;\n            this.#rear = node;\n        }\n        this.#queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop() {\n        const num = this.peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        this.#front = this.#front.next;\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek() {\n        if (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.#front.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray() {\n        let node = this.#front;\n        const res = new Array(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.ts<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    private front: ListNode | null; // \u5934\u8282\u70b9 front\n    private rear: ListNode | null; // \u5c3e\u8282\u70b9 rear\n    private queSize: number = 0;\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num: number): void {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        const node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (!this.front) {\n            this.front = node;\n            this.rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            this.rear!.next = node;\n            this.rear = node;\n        }\n        this.queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop(): number {\n        const num = this.peek();\n        if (!this.front) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        // \u5220\u9664\u5934\u8282\u70b9\n        this.front = this.front.next;\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek(): number {\n        if (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.front!.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray(): number[] {\n        let node = this.front;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.dart<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n  ListNode? _front; // \u5934\u8282\u70b9 _front\n  ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\n  int _queSize = 0; // \u961f\u5217\u957f\u5ea6\n\n  LinkedListQueue() {\n    _front = null;\n    _rear = null;\n  }\n\n  /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u5165\u961f */\n  void push(int _num) {\n    // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 _num\n    final node = ListNode(_num);\n    // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n    if (_front == null) {\n      _front = node;\n      _rear = node;\n    } else {\n      // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n      _rear!.next = node;\n      _rear = node;\n    }\n    _queSize++;\n  }\n\n  /* \u51fa\u961f */\n  int pop() {\n    final int _num = peek();\n    // \u5220\u9664\u5934\u8282\u70b9\n    _front = _front!.next;\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int peek() {\n    if (_queSize == 0) {\n      throw Exception('\u961f\u5217\u4e3a\u7a7a');\n    }\n    return _front!.val;\n  }\n\n  /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; queue = [];\n    while (node != null) {\n      queue.add(node.val);\n      node = node.next;\n    }\n    return queue;\n  }\n}\n</code></pre> linkedlist_queue.rs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListQueue&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5934\u8282\u70b9 front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // \u5c3e\u8282\u70b9 rear\n    que_size: usize,                         // \u961f\u5217\u7684\u957f\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListQueue&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u961f */\n    pub fn push(&amp;mut self, num: T) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        let new_rear = ListNode::new(num);\n        match self.rear.take() {\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n            Some(old_rear) =&gt; {\n                old_rear.borrow_mut().next = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n            // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n            None =&gt; {\n                self.front = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n        }\n        self.que_size += 1;\n    }\n\n    /* \u51fa\u961f */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.front.take().map(|old_front| {\n            match old_front.borrow_mut().next.take() {\n                Some(new_front) =&gt; {\n                    self.front = Some(new_front);\n                }\n                None =&gt; {\n                    self.rear.take();\n                }\n            }\n            self.que_size -= 1;\n            Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n        })\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.insert(0, node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_queue.c<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\n    ListNode *front, *rear;\n    int queSize;\n} LinkedListQueue;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListQueue *newLinkedListQueue() {\n    LinkedListQueue *queue = (LinkedListQueue *)malloc(sizeof(LinkedListQueue));\n    queue-&gt;front = NULL;\n    queue-&gt;rear = NULL;\n    queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListQueue(LinkedListQueue *queue) {\n    // \u91ca\u653e\u6240\u6709\u8282\u70b9\n    while (queue-&gt;front != NULL) {\n        ListNode *tmp = queue-&gt;front;\n        queue-&gt;front = queue-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // \u91ca\u653e queue \u7ed3\u6784\u4f53\n    free(queue);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListQueue *queue) {\n    return (size(queue) == 0);\n}\n\n/* \u5165\u961f */\nvoid push(LinkedListQueue *queue, int num) {\n    // \u5c3e\u8282\u70b9\u5904\u6dfb\u52a0 node\n    ListNode *node = newListNode(num);\n    // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n    if (queue-&gt;front == NULL) {\n        queue-&gt;front = node;\n        queue-&gt;rear = node;\n    }\n    // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n    else {\n        queue-&gt;rear-&gt;next = node;\n        queue-&gt;rear = node;\n    }\n    queue-&gt;queSize++;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(LinkedListQueue *queue) {\n    assert(size(queue) &amp;&amp; queue-&gt;front);\n    return queue-&gt;front-&gt;val;\n}\n\n/* \u51fa\u961f */\nint pop(LinkedListQueue *queue) {\n    int num = peek(queue);\n    ListNode *tmp = queue-&gt;front;\n    queue-&gt;front = queue-&gt;front-&gt;next;\n    free(tmp);\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListQueue(LinkedListQueue *queue) {\n    int *arr = malloc(sizeof(int) * queue-&gt;queSize);\n    // \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\n    int i;\n    ListNode *node;\n    for (i = 0, node = queue-&gt;front; i &lt; queue-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, queue-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_queue.kt<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue(\n    // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private var front: ListNode? = null,\n    private var rear: ListNode? = null,\n    private var queSize: Int = 0\n) {\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u961f */\n    fun push(num: Int) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        val node = ListNode(num)\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == null) {\n            front = node\n            rear = node\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            rear?.next = node\n            rear = node\n        }\n        queSize++\n    }\n\n    /* \u51fa\u961f */\n    fun pop(): Int {\n        val num = peek()\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front?.next\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!.value\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!.value\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.rb<pre><code>[class]{LinkedListQueue}-[func]{}\n</code></pre> linkedlist_queue.zig<pre><code>// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\nfn LinkedListQueue(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        front: ?*inc.ListNode(T) = null,                // \u5934\u8282\u70b9 front\n        rear: ?*inc.ListNode(T) = null,                 // \u5c3e\u8282\u70b9 rear\n        que_size: usize = 0,                            // \u961f\u5217\u7684\u957f\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.front = null;\n            self.rear = null;\n            self.que_size = 0;\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.que_size;\n        }\n\n        // \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.size() == 0) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\n            return self.front.?.val;\n        }  \n\n        // \u5165\u961f\n        pub fn push(self: *Self, num: T) !void {\n            // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n            var node = try self.mem_allocator.create(inc.ListNode(T));\n            node.init(num);\n            // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n            if (self.front == null) {\n                self.front = node;\n                self.rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n            } else {\n                self.rear.?.next = node;\n                self.rear = node;\n            }\n            self.que_size += 1;\n        } \n\n        // \u51fa\u961f\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            // \u5220\u9664\u5934\u8282\u70b9\n            self.front = self.front.?.next;\n            self.que_size -= 1;\n            return num;\n        } \n\n        // \u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.front;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[i] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_stack_and_queue/queue/#2-implementation-based-on-an-array","title":"2. \u00a0 Implementation Based on an Array","text":"<p>Deleting the first element in an array has a time complexity of \\(O(n)\\), which would make the dequeue operation inefficient. However, this problem can be cleverly avoided as follows.</p> <p>We use a variable <code>front</code> to indicate the index of the front element and maintain a variable <code>size</code> to record the queue's length. Define <code>rear = front + size</code>, which points to the position immediately following the tail element.</p> <p>With this design, the effective interval of elements in the array is <code>[front, rear - 1]</code>. The implementation methods for various operations are shown in the Figure 5-6 .</p> <ul> <li>Enqueue operation: Assign the input element to the <code>rear</code> index and increase <code>size</code> by 1.</li> <li>Dequeue operation: Simply increase <code>front</code> by 1 and decrease <code>size</code> by 1.</li> </ul> <p>Both enqueue and dequeue operations only require a single operation, each with a time complexity of \\(O(1)\\).</p> ArrayQueuepush()pop() <p></p> <p></p> <p></p> <p> Figure 5-6 \u00a0 Implementing Queue with Array for Enqueue and Dequeue Operations </p> <p>You might notice a problem: as enqueue and dequeue operations are continuously performed, both <code>front</code> and <code>rear</code> move to the right and will eventually reach the end of the array and can't move further. To resolve this, we can treat the array as a \"circular array\" where connecting the end of the array back to its beginning.</p> <p>In a circular array, <code>front</code> or <code>rear</code> needs to loop back to the start of the array upon reaching the end. This cyclical pattern can be achieved with a \"modulo operation\" as shown in the code below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_queue.py<pre><code>class ArrayQueue:\n    \"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\n\n    def __init__(self, size: int):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._nums: list[int] = [0] * size  # \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n        self._front: int = 0  # \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n        self._size: int = 0  # \u961f\u5217\u957f\u5ea6\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"\u5165\u961f\"\"\"\n        if self._size == self.capacity():\n            raise IndexError(\"\u961f\u5217\u5df2\u6ee1\")\n        # \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        rear: int = (self._front + self._size) % self.capacity()\n        # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u961f\"\"\"\n        num: int = self.peek()\n        # \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        self._front = (self._front + 1) % self.capacity()\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\n        return self._nums[self._front]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        res = [0] * self.size()\n        j: int = self._front\n        for i in range(self.size()):\n            res[i] = self._nums[(j % self.capacity())]\n            j += 1\n        return res\n</code></pre> array_queue.cpp<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n  private:\n    int *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;     // \u961f\u5217\u957f\u5ea6\n    int queCapacity; // \u961f\u5217\u5bb9\u91cf\n\n  public:\n    ArrayQueue(int capacity) {\n        // \u521d\u59cb\u5316\u6570\u7ec4\n        nums = new int[capacity];\n        queCapacity = capacity;\n        front = queSize = 0;\n    }\n\n    ~ArrayQueue() {\n        delete[] nums;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    int capacity() {\n        return queCapacity;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f */\n    void push(int num) {\n        if (queSize == queCapacity) {\n            cout &lt;&lt; \"\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        int rear = (front + queSize) % queCapacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    int pop() {\n        int num = peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % queCapacity;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peek() {\n        if (isEmpty())\n            throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n        return nums[front];\n    }\n\n    /* \u5c06\u6570\u7ec4\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        vector&lt;int&gt; arr(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            arr[i] = nums[j % queCapacity];\n        }\n        return arr;\n    }\n};\n</code></pre> array_queue.java<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    private int[] nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private int queSize; // \u961f\u5217\u957f\u5ea6\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u961f */\n    public void push(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        int rear = (front + queSize) % capacity();\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int pop() {\n        int num = peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % capacity();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    public int[] toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.cs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize; // \u961f\u5217\u957f\u5ea6\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u961f */\n    public void Push(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        int rear = (front + queSize) % Capacity();\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int Pop() {\n        int num = Peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % Capacity();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return nums[front];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    public int[] ToArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % this.Capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.go<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntype arrayQueue struct {\n    nums        []int // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    queSize     int   // \u961f\u5217\u957f\u5ea6\n    queCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\n    return &amp;arrayQueue{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayQueue) size() int {\n    return q.queSize\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayQueue) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* \u5165\u961f */\nfunc (q *arrayQueue) push(num int) {\n    // \u5f53 rear == queCapacity \u8868\u793a\u961f\u5217\u5df2\u6ee1\n    if q.queSize == q.queCapacity {\n        return\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n    rear := (q.front + q.queSize) % q.queCapacity\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* \u51fa\u961f */\nfunc (q *arrayQueue) pop() any {\n    num := q.peek()\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n    q.front = (q.front + 1) % q.queCapacity\n    q.queSize--\n    return num\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayQueue) peek() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayQueue) toSlice() []int {\n    rear := (q.front + q.queSize)\n    if rear &gt;= q.queCapacity {\n        rear %= q.queCapacity\n        return append(q.nums[q.front:], q.nums[:rear]...)\n    }\n    return q.nums[q.front:rear]\n}\n</code></pre> array_queue.swift<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    private var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var _size: Int // \u961f\u5217\u957f\u5ea6\n\n    init(capacity: Int) {\n        // \u521d\u59cb\u5316\u6570\u7ec4\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u961f */\n    func push(num: Int) {\n        if size() == capacity() {\n            print(\"\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        let rear = (front + size()) % capacity()\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* \u51fa\u961f */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % capacity()\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return nums[front]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    func toArray() -&gt; [Int] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        (front ..&lt; front + size()).map { nums[$0 % capacity()] }\n    }\n}\n</code></pre> array_queue.js<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    #nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    #front = 0; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    #queSize = 0; // \u961f\u5217\u957f\u5ea6\n\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    get capacity() {\n        return this.#nums.length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num) {\n        if (this.size === this.capacity) {\n            console.log('\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        const rear = (this.#front + this.size) % this.capacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop() {\n        const num = this.peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        this.#front = (this.#front + 1) % this.capacity;\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek() {\n        if (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.#nums[this.#front];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.#front; i &lt; this.size; i++, j++) {\n            arr[i] = this.#nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.ts<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    private nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private queSize: number; // \u961f\u5217\u957f\u5ea6\n\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = this.queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    get capacity(): number {\n        return this.nums.length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num: number): void {\n        if (this.size === this.capacity) {\n            console.log('\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        const rear = (this.front + this.queSize) % this.capacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop(): number {\n        const num = this.peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        this.front = (this.front + 1) % this.capacity;\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek(): number {\n        if (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.nums[this.front];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray(): number[] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.front; i &lt; this.size; i++, j++) {\n            arr[i] = this.nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.dart<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n  late List&lt;int&gt; _nums; // \u7528\u4e8e\u50a8\u5b58\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n  late int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n  late int _queSize; // \u961f\u5217\u957f\u5ea6\n\n  ArrayQueue(int capacity) {\n    _nums = List.filled(capacity, 0);\n    _front = _queSize = 0;\n  }\n\n  /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n  int capaCity() {\n    return _nums.length;\n  }\n\n  /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u5165\u961f */\n  void push(int _num) {\n    if (_queSize == capaCity()) {\n      throw Exception(\"\u961f\u5217\u5df2\u6ee1\");\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n    int rear = (_front + _queSize) % capaCity();\n    // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* \u51fa\u961f */\n  int pop() {\n    int _num = peek();\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n    _front = (_front + 1) % capaCity();\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"\u961f\u5217\u4e3a\u7a7a\");\n    }\n    return _nums[_front];\n  }\n\n  /* \u8fd4\u56de Array */\n  List&lt;int&gt; toArray() {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    final List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[j % capaCity()];\n    }\n    return res;\n  }\n}\n</code></pre> array_queue.rs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nstruct ArrayQueue {\n    nums: Vec&lt;i32&gt;,    // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front: i32,        // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    que_size: i32,     // \u961f\u5217\u957f\u5ea6\n    que_capacity: i32, // \u961f\u5217\u5bb9\u91cf\n}\n\nimpl ArrayQueue {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(capacity: i32) -&gt; ArrayQueue {\n        ArrayQueue {\n            nums: vec![0; capacity as usize],\n            front: 0,\n            que_size: 0,\n            que_capacity: capacity,\n        }\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    fn capacity(&amp;self) -&gt; i32 {\n        self.que_capacity\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    fn size(&amp;self) -&gt; i32 {\n        self.que_size\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* \u5165\u961f */\n    fn push(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        let rear = (self.front + self.que_size) % self.que_capacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self.nums[rear as usize] = num;\n        self.que_size += 1;\n    }\n\n    /* \u51fa\u961f */\n    fn pop(&amp;mut self) -&gt; i32 {\n        let num = self.peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        self.front = (self.front + 1) % self.que_capacity;\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fn peek(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"index out of bounds\");\n        }\n        self.nums[self.front as usize]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    fn to_vector(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let cap = self.que_capacity;\n        let mut j = self.front;\n        let mut arr = vec![0; self.que_size as usize];\n        for i in 0..self.que_size {\n            arr[i as usize] = self.nums[(j % cap) as usize];\n            j += 1;\n        }\n        arr\n    }\n}\n</code></pre> array_queue.c<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\n    int *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n    int queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayQueue;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayQueue *newArrayQueue(int capacity) {\n    ArrayQueue *queue = (ArrayQueue *)malloc(sizeof(ArrayQueue));\n    // \u521d\u59cb\u5316\u6570\u7ec4\n    queue-&gt;queCapacity = capacity;\n    queue-&gt;nums = (int *)malloc(sizeof(int) * queue-&gt;queCapacity);\n    queue-&gt;front = queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayQueue(ArrayQueue *queue) {\n    free(queue-&gt;nums);\n    free(queue);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayQueue *queue) {\n    return queue-&gt;queCapacity;\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayQueue *queue) {\n    return queue-&gt;queSize == 0;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(ArrayQueue *queue) {\n    assert(size(queue) != 0);\n    return queue-&gt;nums[queue-&gt;front];\n}\n\n/* \u5165\u961f */\nvoid push(ArrayQueue *queue, int num) {\n    if (size(queue) == capacity(queue)) {\n        printf(\"\u961f\u5217\u5df2\u6ee1\\r\\n\");\n        return;\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n    int rear = (queue-&gt;front + queue-&gt;queSize) % queue-&gt;queCapacity;\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    queue-&gt;nums[rear] = num;\n    queue-&gt;queSize++;\n}\n\n/* \u51fa\u961f */\nint pop(ArrayQueue *queue) {\n    int num = peek(queue);\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n    queue-&gt;front = (queue-&gt;front + 1) % queue-&gt;queCapacity;\n    queue-&gt;queSize--;\n    return num;\n}\n</code></pre> array_queue.kt<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue(capacity: Int) {\n    private val nums = IntArray(capacity) // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var queSize = 0 // \u961f\u5217\u957f\u5ea6\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* \u5165\u961f */\n    fun push(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        val rear = (front + queSize) % capacity()\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        queSize++\n    }\n\n    /* \u51fa\u961f */\n    fun pop(): Int {\n        val num = peek()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % capacity()\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    fun toArray(): IntArray {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[j % capacity()]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_queue.rb<pre><code>[class]{ArrayQueue}-[func]{}\n</code></pre> array_queue.zig<pre><code>// \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\nfn ArrayQueue(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        nums: []T = undefined,                          // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4     \n        cap: usize = 0,                                 // \u961f\u5217\u5bb9\u91cf\n        front: usize = 0,                               // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n        queSize: usize = 0,                             // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6570\u7ec4\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator, cap: usize) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.cap = cap;\n            self.nums = try self.mem_allocator.alloc(T, self.cap);\n            @memset(self.nums, @as(T, 0));\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\n        pub fn capacity(self: *Self) usize {\n            return self.cap;\n        }\n\n        // \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.queSize;\n        }\n\n        // \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.queSize == 0;\n        }\n\n        // \u5165\u961f\n        pub fn push(self: *Self, num: T) !void {\n            if (self.size() == self.capacity()) {\n                std.debug.print(\"\u961f\u5217\u5df2\u6ee1\\n\", .{});\n                return;\n            }\n            // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n            // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n            var rear = (self.front + self.queSize) % self.capacity();\n            // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n            self.nums[rear] = num;\n            self.queSize += 1;\n        } \n\n        // \u51fa\u961f\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n            self.front = (self.front + 1) % self.capacity();\n            self.queSize -= 1;\n            return num;\n        } \n\n        // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\n            return self.nums[self.front];\n        } \n\n        // \u8fd4\u56de\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            var j: usize = self.front;\n            while (i &lt; self.size()) : ({ i += 1; j += 1; }) {\n                res[i] = self.nums[j % self.capacity()];\n            }\n            return res;\n        }\n    };\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The above implementation of the queue still has its limitations: its length is fixed. However, this issue is not difficult to resolve. We can replace the array with a dynamic array that can expand itself if needed. Interested readers can try to implement this themselves.</p> <p>The comparison of the two implementations is consistent with that of the stack and is not repeated here.</p>"},{"location":"chapter_stack_and_queue/queue/#523-typical-applications-of-queue","title":"5.2.3 \u00a0 Typical Applications of Queue","text":"<ul> <li>Amazon Orders. After shoppers place orders, these orders join a queue, and the system processes them in order. During events like Singles' Day, a massive number of orders are generated in a short time, making high concurrency a key challenge for engineers.</li> <li>Various To-Do Lists. Any scenario requiring a \"first-come, first-served\" functionality, such as a printer's task queue or a restaurant's food delivery queue, can effectively maintain the order of processing with a queue.</li> </ul>"},{"location":"chapter_stack_and_queue/stack/","title":"5.1 \u00a0 Stack","text":"<p>A \"Stack\" is a linear data structure that follows the principle of Last-In-First-Out (LIFO).</p> <p>We can compare a stack to a pile of plates on a table. To access the bottom plate, one must first remove the plates on top. By replacing the plates with various types of elements (such as integers, characters, objects, etc.), we obtain the data structure known as a stack.</p> <p>As shown in the Figure 5-1 , we refer to the top of the pile of elements as the \"top of the stack\" and the bottom as the \"bottom of the stack.\" The operation of adding elements to the top of the stack is called \"push,\" and the operation of removing the top element is called \"pop.\"</p> <p></p> <p> Figure 5-1 \u00a0 Stack's Last-In-First-Out Rule </p>"},{"location":"chapter_stack_and_queue/stack/#511-common-operations-on-stack","title":"5.1.1 \u00a0 Common Operations on Stack","text":"<p>The common operations on a stack are shown in the Table 5-1 . The specific method names depend on the programming language used. Here, we use <code>push()</code>, <code>pop()</code>, and <code>peek()</code> as examples.</p> <p> Table 5-1 \u00a0 Efficiency of Stack Operations </p> Method Description Time Complexity <code>push()</code> Push an element onto the stack (add to the top) \\(O(1)\\) <code>pop()</code> Pop the top element from the stack \\(O(1)\\) <code>peek()</code> Access the top element of the stack \\(O(1)\\) <p>Typically, we can directly use the stack class built into the programming language. However, some languages may not specifically provide a stack class. In these cases, we can use the language's \"array\" or \"linked list\" as a stack and ignore operations that are not related to stack logic in the program.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig stack.py<pre><code># Initialize the stack\n# Python does not have a built-in stack class, so a list can be used as a stack\nstack: list[int] = []\n\n# Push elements onto the stack\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n# Access the top element of the stack\npeek: int = stack[-1]\n\n# Pop an element from the stack\npop: int = stack.pop()\n\n# Get the length of the stack\nsize: int = len(stack)\n\n# Check if the stack is empty\nis_empty: bool = len(stack) == 0\n</code></pre> stack.cpp<pre><code>/* Initialize the stack */\nstack&lt;int&gt; stack;\n\n/* Push elements onto the stack */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access the top element of the stack */\nint top = stack.top();\n\n/* Pop an element from the stack */\nstack.pop(); // No return value\n\n/* Get the length of the stack */\nint size = stack.size();\n\n/* Check if the stack is empty */\nbool empty = stack.empty();\n</code></pre> stack.java<pre><code>/* Initialize the stack */\nStack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n\n/* Push elements onto the stack */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access the top element of the stack */\nint peek = stack.peek();\n\n/* Pop an element from the stack */\nint pop = stack.pop();\n\n/* Get the length of the stack */\nint size = stack.size();\n\n/* Check if the stack is empty */\nboolean isEmpty = stack.isEmpty();\n</code></pre> stack.cs<pre><code>/* Initialize the stack */\nStack&lt;int&gt; stack = new();\n\n/* Push elements onto the stack */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n\n/* Access the top element of the stack */\nint peek = stack.Peek();\n\n/* Pop an element from the stack */\nint pop = stack.Pop();\n\n/* Get the length of the stack */\nint size = stack.Count;\n\n/* Check if the stack is empty */\nbool isEmpty = stack.Count == 0;\n</code></pre> stack_test.go<pre><code>/* Initialize the stack */\n// In Go, it is recommended to use a Slice as a stack\nvar stack []int\n\n/* Push elements onto the stack */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n\n/* Access the top element of the stack */\npeek := stack[len(stack)-1]\n\n/* Pop an element from the stack */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n\n/* Get the length of the stack */\nsize := len(stack)\n\n/* Check if the stack is empty */\nisEmpty := len(stack) == 0\n</code></pre> stack.swift<pre><code>/* Initialize the stack */\n// Swift does not have a built-in stack class, so Array can be used as a stack\nvar stack: [Int] = []\n\n/* Push elements onto the stack */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n/* Access the top element of the stack */\nlet peek = stack.last!\n\n/* Pop an element from the stack */\nlet pop = stack.removeLast()\n\n/* Get the length of the stack */\nlet size = stack.count\n\n/* Check if the stack is empty */\nlet isEmpty = stack.isEmpty\n</code></pre> stack.js<pre><code>/* Initialize the stack */\n// JavaScript does not have a built-in stack class, so Array can be used as a stack\nconst stack = [];\n\n/* Push elements onto the stack */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access the top element of the stack */\nconst peek = stack[stack.length-1];\n\n/* Pop an element from the stack */\nconst pop = stack.pop();\n\n/* Get the length of the stack */\nconst size = stack.length;\n\n/* Check if the stack is empty */\nconst is_empty = stack.length === 0;\n</code></pre> stack.ts<pre><code>/* Initialize the stack */\n// TypeScript does not have a built-in stack class, so Array can be used as a stack\nconst stack: number[] = [];\n\n/* Push elements onto the stack */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access the top element of the stack */\nconst peek = stack[stack.length - 1];\n\n/* Pop an element from the stack */\nconst pop = stack.pop();\n\n/* Get the length of the stack */\nconst size = stack.length;\n\n/* Check if the stack is empty */\nconst is_empty = stack.length === 0;\n</code></pre> stack.dart<pre><code>/* Initialize the stack */\n// Dart does not have a built-in stack class, so List can be used as a stack\nList&lt;int&gt; stack = [];\n\n/* Push elements onto the stack */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n\n/* Access the top element of the stack */\nint peek = stack.last;\n\n/* Pop an element from the stack */\nint pop = stack.removeLast();\n\n/* Get the length of the stack */\nint size = stack.length;\n\n/* Check if the stack is empty */\nbool isEmpty = stack.isEmpty;\n</code></pre> stack.rs<pre><code>/* Initialize the stack */\n// Use Vec as a stack\nlet mut stack: Vec&lt;i32&gt; = Vec::new();\n\n/* Push elements onto the stack */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access the top element of the stack */\nlet top = stack.last().unwrap();\n\n/* Pop an element from the stack */\nlet pop = stack.pop().unwrap();\n\n/* Get the length of the stack */\nlet size = stack.len();\n\n/* Check if the stack is empty */\nlet is_empty = stack.is_empty();\n</code></pre> stack.c<pre><code>// C does not provide a built-in stack\n</code></pre> stack.kt<pre><code>\n</code></pre> stack.zig<pre><code>\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#512-implementing-a-stack","title":"5.1.2 \u00a0 Implementing a Stack","text":"<p>To gain a deeper understanding of how a stack operates, let's try implementing a stack class ourselves.</p> <p>A stack follows the principle of Last-In-First-Out, which means we can only add or remove elements at the top of the stack. However, both arrays and linked lists allow adding and removing elements at any position, therefore a stack can be seen as a restricted array or linked list. In other words, we can \"shield\" certain irrelevant operations of an array or linked list, aligning their external behavior with the characteristics of a stack.</p>"},{"location":"chapter_stack_and_queue/stack/#1-implementation-based-on-linked-list","title":"1. \u00a0 Implementation Based on Linked List","text":"<p>When implementing a stack using a linked list, we can consider the head node of the list as the top of the stack and the tail node as the bottom of the stack.</p> <p>As shown in the Figure 5-2 , for the push operation, we simply insert elements at the head of the linked list. This method of node insertion is known as \"head insertion.\" For the pop operation, we just need to remove the head node from the list.</p> LinkedListStackpush()pop() <p></p> <p></p> <p></p> <p> Figure 5-2 \u00a0 Implementing Stack with Linked List for Push and Pop Operations </p> <p>Below is an example code for implementing a stack based on a linked list:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_stack.py<pre><code>class LinkedListStack:\n    \"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._peek: ListNode | None = None\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return not self._peek\n\n    def push(self, val: int):\n        \"\"\"\u5165\u6808\"\"\"\n        node = ListNode(val)\n        node.next = self._peek\n        self._peek = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u6808\"\"\"\n        num = self.peek()\n        self._peek = self._peek.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u6808\u4e3a\u7a7a\")\n        return self._peek.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        arr = []\n        node = self._peek\n        while node:\n            arr.append(node.val)\n            node = node.next\n        arr.reverse()\n        return arr\n</code></pre> linkedlist_stack.cpp<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n  private:\n    ListNode *stackTop; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    int stkSize;        // \u6808\u7684\u957f\u5ea6\n\n  public:\n    LinkedListStack() {\n        stackTop = nullptr;\n        stkSize = 0;\n    }\n\n    ~LinkedListStack() {\n        // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n        freeMemoryLinkedList(stackTop);\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    int size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    void push(int num) {\n        ListNode *node = new ListNode(num);\n        node-&gt;next = stackTop;\n        stackTop = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    int pop() {\n        int num = top();\n        ListNode *tmp = stackTop;\n        stackTop = stackTop-&gt;next;\n        // \u91ca\u653e\u5185\u5b58\n        delete tmp;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"\u6808\u4e3a\u7a7a\");\n        return stackTop-&gt;val;\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = stackTop;\n        vector&lt;int&gt; res(size());\n        for (int i = res.size() - 1; i &gt;= 0; i--) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_stack.java<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    private ListNode stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private int stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void push(int num) {\n        ListNode node = new ListNode(num);\n        node.next = stackPeek;\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    public int pop() {\n        int num = peek();\n        stackPeek = stackPeek.next;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stackPeek.val;\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] toArray() {\n        ListNode node = stackPeek;\n        int[] res = new int[size()];\n        for (int i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.cs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    ListNode? stackPeek;  // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    int stkSize = 0;   // \u6808\u7684\u957f\u5ea6\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int Size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void Push(int num) {\n        ListNode node = new(num) {\n            next = stackPeek\n        };\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    public int Pop() {\n        int num = Peek();\n        stackPeek = stackPeek!.next;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stackPeek!.val;\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] ToArray() {\n        if (stackPeek == null)\n            return [];\n\n        ListNode? node = stackPeek;\n        int[] res = new int[Size()];\n        for (int i = res.Length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.go<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntype linkedListStack struct {\n    // \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u6808\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u6808 */\nfunc newLinkedListStack() *linkedListStack {\n    return &amp;linkedListStack{\n        data: list.New(),\n    }\n}\n\n/* \u5165\u6808 */\nfunc (s *linkedListStack) push(value int) {\n    s.data.PushBack(value)\n}\n\n/* \u51fa\u6808 */\nfunc (s *linkedListStack) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc (s *linkedListStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc (s *linkedListStack) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListStack) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListStack) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_stack.swift<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    private var _peek: ListNode? // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private var _size: Int // \u6808\u7684\u957f\u5ea6\n\n    init() {\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u6808 */\n    func push(num: Int) {\n        let node = ListNode(x: num)\n        node.next = _peek\n        _peek = node\n        _size += 1\n    }\n\n    /* \u51fa\u6808 */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        _peek = _peek?.next\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u6808\u4e3a\u7a7a\")\n        }\n        return _peek!.val\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        var node = _peek\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices.reversed() {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.js<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    #stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    #stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n    constructor() {\n        this.#stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num) {\n        const node = new ListNode(num);\n        node.next = this.#stackPeek;\n        this.#stackPeek = node;\n        this.#stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    pop() {\n        const num = this.peek();\n        this.#stackPeek = this.#stackPeek.next;\n        this.#stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    peek() {\n        if (!this.#stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.#stackPeek.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray() {\n        let node = this.#stackPeek;\n        const res = new Array(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.ts<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    private stackPeek: ListNode | null; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private stkSize: number = 0; // \u6808\u7684\u957f\u5ea6\n\n    constructor() {\n        this.stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num: number): void {\n        const node = new ListNode(num);\n        node.next = this.stackPeek;\n        this.stackPeek = node;\n        this.stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    pop(): number {\n        const num = this.peek();\n        if (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n        this.stackPeek = this.stackPeek.next;\n        this.stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    peek(): number {\n        if (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.stackPeek.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray(): number[] {\n        let node = this.stackPeek;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.dart<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u7c7b\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n  ListNode? _stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n  int _stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n  LinkedListStack() {\n    _stackPeek = null;\n  }\n\n  /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n  int size() {\n    return _stkSize;\n  }\n\n  /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _stkSize == 0;\n  }\n\n  /* \u5165\u6808 */\n  void push(int _num) {\n    final ListNode node = ListNode(_num);\n    node.next = _stackPeek;\n    _stackPeek = node;\n    _stkSize++;\n  }\n\n  /* \u51fa\u6808 */\n  int pop() {\n    final int _num = peek();\n    _stackPeek = _stackPeek!.next;\n    _stkSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n  int peek() {\n    if (_stackPeek == null) {\n      throw Exception(\"\u6808\u4e3a\u7a7a\");\n    }\n    return _stackPeek!.val;\n  }\n\n  /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a List \u5e76\u8fd4\u56de */\n  List&lt;int&gt; toList() {\n    ListNode? node = _stackPeek;\n    List&lt;int&gt; list = [];\n    while (node != null) {\n      list.add(node.val);\n      node = node.next;\n    }\n    list = list.reversed.toList();\n    return list;\n  }\n}\n</code></pre> linkedlist_stack.rs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\n#[allow(dead_code)]\npub struct LinkedListStack&lt;T&gt; {\n    stack_peek: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    stk_size: usize,                              // \u6808\u7684\u957f\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListStack&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            stack_peek: None,\n            stk_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.stk_size;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    pub fn push(&amp;mut self, num: T) {\n        let node = ListNode::new(num);\n        node.borrow_mut().next = self.stack_peek.take();\n        self.stack_peek = Some(node);\n        self.stk_size += 1;\n    }\n\n    /* \u51fa\u6808 */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack_peek.take().map(|old_head| {\n            match old_head.borrow_mut().next.take() {\n                Some(new_head) =&gt; {\n                    self.stack_peek = Some(new_head);\n                }\n                None =&gt; {\n                    self.stack_peek = None;\n                }\n            }\n            self.stk_size -= 1;\n            Rc::try_unwrap(old_head).ok().unwrap().into_inner().val\n        })\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.stack_peek.as_ref()\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.push(node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_stack.c<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\n    ListNode *top; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    int size;      // \u6808\u7684\u957f\u5ea6\n} LinkedListStack;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListStack *newLinkedListStack() {\n    LinkedListStack *s = malloc(sizeof(LinkedListStack));\n    s-&gt;top = NULL;\n    s-&gt;size = 0;\n    return s;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListStack(LinkedListStack *s) {\n    while (s-&gt;top) {\n        ListNode *n = s-&gt;top-&gt;next;\n        free(s-&gt;top);\n        s-&gt;top = n;\n    }\n    free(s);\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(LinkedListStack *s) {\n    return s-&gt;size;\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(LinkedListStack *s) {\n    return size(s) == 0;\n}\n\n/* \u5165\u6808 */\nvoid push(LinkedListStack *s, int num) {\n    ListNode *node = (ListNode *)malloc(sizeof(ListNode));\n    node-&gt;next = s-&gt;top; // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6307\u9488\u57df\n    node-&gt;val = num;     // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6570\u636e\u57df\n    s-&gt;top = node;       // \u66f4\u65b0\u6808\u9876\n    s-&gt;size++;           // \u66f4\u65b0\u6808\u5927\u5c0f\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(LinkedListStack *s) {\n    if (s-&gt;size == 0) {\n        printf(\"\u6808\u4e3a\u7a7a\\n\");\n        return INT_MAX;\n    }\n    return s-&gt;top-&gt;val;\n}\n\n/* \u51fa\u6808 */\nint pop(LinkedListStack *s) {\n    int val = peek(s);\n    ListNode *tmp = s-&gt;top;\n    s-&gt;top = s-&gt;top-&gt;next;\n    // \u91ca\u653e\u5185\u5b58\n    free(tmp);\n    s-&gt;size--;\n    return val;\n}\n</code></pre> linkedlist_stack.kt<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack(\n    private var stackPeek: ListNode? = null, // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private var stkSize: Int = 0 // \u6808\u7684\u957f\u5ea6\n) {\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return stkSize\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u6808 */\n    fun push(num: Int) {\n        val node = ListNode(num)\n        node.next = stackPeek\n        stackPeek = node\n        stkSize++\n    }\n\n    /* \u51fa\u6808 */\n    fun pop(): Int? {\n        val num = peek()\n        stackPeek = stackPeek?.next\n        stkSize--;\n        return num\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    fun peek(): Int? {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stackPeek?.value\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    fun toArray(): IntArray {\n        var node = stackPeek\n        val res = IntArray(size())\n        for (i in res.size - 1 downTo 0) {\n            res[i] = node?.value!!\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.rb<pre><code>[class]{LinkedListStack}-[func]{}\n</code></pre> linkedlist_stack.zig<pre><code>// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\nfn LinkedListStack(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        stack_top: ?*inc.ListNode(T) = null,             // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n        stk_size: usize = 0,                             // \u6808\u7684\u957f\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,    // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.stack_top = null;\n            self.stk_size = 0;\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u6808\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.stk_size;\n        }\n\n        // \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8bbf\u95ee\u6808\u9876\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.size() == 0) @panic(\"\u6808\u4e3a\u7a7a\");\n            return self.stack_top.?.val;\n        }  \n\n        // \u5165\u6808\n        pub fn push(self: *Self, num: T) !void {\n            var node = try self.mem_allocator.create(inc.ListNode(T));\n            node.init(num);\n            node.next = self.stack_top;\n            self.stack_top = node;\n            self.stk_size += 1;\n        } \n\n        // \u51fa\u6808\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            self.stack_top = self.stack_top.?.next;\n            self.stk_size -= 1;\n            return num;\n        } \n\n        // \u5c06\u6808\u8f6c\u6362\u4e3a\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.stack_top;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[res.len - i - 1] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#2-implementation-based-on-array","title":"2. \u00a0 Implementation Based on Array","text":"<p>When implementing a stack using an array, we can consider the end of the array as the top of the stack. As shown in the Figure 5-3 , push and pop operations correspond to adding and removing elements at the end of the array, respectively, both with a time complexity of \\(O(1)\\).</p> ArrayStackpush()pop() <p></p> <p></p> <p></p> <p> Figure 5-3 \u00a0 Implementing Stack with Array for Push and Pop Operations </p> <p>Since the elements to be pushed onto the stack may continuously increase, we can use a dynamic array, thus avoiding the need to handle array expansion ourselves. Here is an example code:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_stack.py<pre><code>class ArrayStack:\n    \"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._stack: list[int] = []\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\n        return len(self._stack)\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self._stack == []\n\n    def push(self, item: int):\n        \"\"\"\u5165\u6808\"\"\"\n        self._stack.append(item)\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u6808\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u6808\u4e3a\u7a7a\")\n        return self._stack.pop()\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u6808\u4e3a\u7a7a\")\n        return self._stack[-1]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        return self._stack\n</code></pre> array_stack.cpp<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n  private:\n    vector&lt;int&gt; stack;\n\n  public:\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    int size() {\n        return stack.size();\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return stack.size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    void push(int num) {\n        stack.push_back(num);\n    }\n\n    /* \u51fa\u6808 */\n    int pop() {\n        int num = top();\n        stack.pop_back();\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"\u6808\u4e3a\u7a7a\");\n        return stack.back();\n    }\n\n    /* \u8fd4\u56de Vector */\n    vector&lt;int&gt; toVector() {\n        return stack;\n    }\n};\n</code></pre> array_stack.java<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    private ArrayList&lt;Integer&gt; stack;\n\n    public ArrayStack() {\n        // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n        stack = new ArrayList&lt;&gt;();\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int size() {\n        return stack.size();\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void push(int num) {\n        stack.add(num);\n    }\n\n    /* \u51fa\u6808 */\n    public int pop() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.remove(size() - 1);\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.get(size() - 1);\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public Object[] toArray() {\n        return stack.toArray();\n    }\n}\n</code></pre> array_stack.cs<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    List&lt;int&gt; stack;\n    public ArrayStack() {\n        // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n        stack = [];\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int Size() {\n        return stack.Count;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void Push(int num) {\n        stack.Add(num);\n    }\n\n    /* \u51fa\u6808 */\n    public int Pop() {\n        if (IsEmpty())\n            throw new Exception();\n        var val = Peek();\n        stack.RemoveAt(Size() - 1);\n        return val;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stack[Size() - 1];\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] ToArray() {\n        return [.. stack];\n    }\n}\n</code></pre> array_stack.go<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntype arrayStack struct {\n    data []int // \u6570\u636e\n}\n\n/* \u521d\u59cb\u5316\u6808 */\nfunc newArrayStack() *arrayStack {\n    return &amp;arrayStack{\n        // \u8bbe\u7f6e\u6808\u7684\u957f\u5ea6\u4e3a 0\uff0c\u5bb9\u91cf\u4e3a 16\n        data: make([]int, 0, 16),\n    }\n}\n\n/* \u6808\u7684\u957f\u5ea6 */\nfunc (s *arrayStack) size() int {\n    return len(s.data)\n}\n\n/* \u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *arrayStack) isEmpty() bool {\n    return s.size() == 0\n}\n\n/* \u5165\u6808 */\nfunc (s *arrayStack) push(v int) {\n    // \u5207\u7247\u4f1a\u81ea\u52a8\u6269\u5bb9\n    s.data = append(s.data, v)\n}\n\n/* \u51fa\u6808 */\nfunc (s *arrayStack) pop() any {\n    val := s.peek()\n    s.data = s.data[:len(s.data)-1]\n    return val\n}\n\n/* \u83b7\u53d6\u6808\u9876\u5143\u7d20 */\nfunc (s *arrayStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    val := s.data[len(s.data)-1]\n    return val\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (s *arrayStack) toSlice() []int {\n    return s.data\n}\n</code></pre> array_stack.swift<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    private var stack: [Int]\n\n    init() {\n        // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n        stack = []\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        stack.count\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        stack.isEmpty\n    }\n\n    /* \u5165\u6808 */\n    func push(num: Int) {\n        stack.append(num)\n    }\n\n    /* \u51fa\u6808 */\n    @discardableResult\n    func pop() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u6808\u4e3a\u7a7a\")\n        }\n        return stack.removeLast()\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u6808\u4e3a\u7a7a\")\n        }\n        return stack.last!\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        stack\n    }\n}\n</code></pre> array_stack.js<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    #stack;\n    constructor() {\n        this.#stack = [];\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#stack.length;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#stack.length === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num) {\n        this.#stack.push(num);\n    }\n\n    /* \u51fa\u6808 */\n    pop() {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.#stack.pop();\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    top() {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.#stack[this.#stack.length - 1];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        return this.#stack;\n    }\n}\n</code></pre> array_stack.ts<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    private stack: number[];\n    constructor() {\n        this.stack = [];\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.stack.length;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.stack.length === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num: number): void {\n        this.stack.push(num);\n    }\n\n    /* \u51fa\u6808 */\n    pop(): number | undefined {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.stack.pop();\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    top(): number | undefined {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.stack[this.stack.length - 1];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        return this.stack;\n    }\n}\n</code></pre> array_stack.dart<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n  late List&lt;int&gt; _stack;\n  ArrayStack() {\n    _stack = [];\n  }\n\n  /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n  int size() {\n    return _stack.length;\n  }\n\n  /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _stack.isEmpty;\n  }\n\n  /* \u5165\u6808 */\n  void push(int _num) {\n    _stack.add(_num);\n  }\n\n  /* \u51fa\u6808 */\n  int pop() {\n    if (isEmpty()) {\n      throw Exception(\"\u6808\u4e3a\u7a7a\");\n    }\n    return _stack.removeLast();\n  }\n\n  /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"\u6808\u4e3a\u7a7a\");\n    }\n    return _stack.last;\n  }\n\n  /* \u5c06\u6808\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n  List&lt;int&gt; toArray() =&gt; _stack;\n}\n</code></pre> array_stack.rs<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nstruct ArrayStack&lt;T&gt; {\n    stack: Vec&lt;T&gt;,\n}\n\nimpl&lt;T&gt; ArrayStack&lt;T&gt; {\n    /* \u521d\u59cb\u5316\u6808 */\n    fn new() -&gt; ArrayStack&lt;T&gt; {\n        ArrayStack::&lt;T&gt; {\n            stack: Vec::&lt;T&gt;::new(),\n        }\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    fn size(&amp;self) -&gt; usize {\n        self.stack.len()\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.size() == 0\n    }\n\n    /* \u5165\u6808 */\n    fn push(&amp;mut self, num: T) {\n        self.stack.push(num);\n    }\n\n    /* \u51fa\u6808 */\n    fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack.pop()\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    fn peek(&amp;self) -&gt; Option&lt;&amp;T&gt; {\n        if self.is_empty() {\n            panic!(\"\u6808\u4e3a\u7a7a\")\n        };\n        self.stack.last()\n    }\n\n    /* \u8fd4\u56de &amp;Vec */\n    fn to_array(&amp;self) -&gt; &amp;Vec&lt;T&gt; {\n        &amp;self.stack\n    }\n}\n</code></pre> array_stack.c<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\n    int *data;\n    int size;\n} ArrayStack;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayStack *newArrayStack() {\n    ArrayStack *stack = malloc(sizeof(ArrayStack));\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5927\u5bb9\u91cf\uff0c\u907f\u514d\u6269\u5bb9\n    stack-&gt;data = malloc(sizeof(int) * MAX_SIZE);\n    stack-&gt;size = 0;\n    return stack;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayStack(ArrayStack *stack) {\n    free(stack-&gt;data);\n    free(stack);\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(ArrayStack *stack) {\n    return stack-&gt;size;\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(ArrayStack *stack) {\n    return stack-&gt;size == 0;\n}\n\n/* \u5165\u6808 */\nvoid push(ArrayStack *stack, int num) {\n    if (stack-&gt;size == MAX_SIZE) {\n        printf(\"\u6808\u5df2\u6ee1\\n\");\n        return;\n    }\n    stack-&gt;data[stack-&gt;size] = num;\n    stack-&gt;size++;\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(ArrayStack *stack) {\n    if (stack-&gt;size == 0) {\n        printf(\"\u6808\u4e3a\u7a7a\\n\");\n        return INT_MAX;\n    }\n    return stack-&gt;data[stack-&gt;size - 1];\n}\n\n/* \u51fa\u6808 */\nint pop(ArrayStack *stack) {\n    int val = peek(stack);\n    stack-&gt;size--;\n    return val;\n}\n</code></pre> array_stack.kt<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n    private val stack = ArrayList&lt;Int&gt;()\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return stack.size\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u6808 */\n    fun push(num: Int) {\n        stack.add(num)\n    }\n\n    /* \u51fa\u6808 */\n    fun pop(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack.removeAt(size() - 1)\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack[size() - 1]\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    fun toArray(): Array&lt;Any&gt; {\n        return stack.toArray()\n    }\n}\n</code></pre> array_stack.rb<pre><code>[class]{ArrayStack}-[func]{}\n</code></pre> array_stack.zig<pre><code>// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\nfn ArrayStack(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        stack: ?std.ArrayList(T) = null,     \n\n        // \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) void {\n            if (self.stack == null) {\n                self.stack = std.ArrayList(T).init(allocator);\n            }\n        }\n\n        // \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.stack == null) return;\n            self.stack.?.deinit();\n        }\n\n        // \u83b7\u53d6\u6808\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.stack.?.items.len;\n        }\n\n        // \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8bbf\u95ee\u6808\u9876\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u6808\u4e3a\u7a7a\");\n            return self.stack.?.items[self.size() - 1];\n        }  \n\n        // \u5165\u6808\n        pub fn push(self: *Self, num: T) !void {\n            try self.stack.?.append(num);\n        } \n\n        // \u51fa\u6808\n        pub fn pop(self: *Self) T {\n            var num = self.stack.?.pop();\n            return num;\n        } \n\n        // \u8fd4\u56de ArrayList\n        pub fn toList(self: *Self) std.ArrayList(T) {\n            return self.stack.?;\n        }\n    };\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#513-comparison-of-the-two-implementations","title":"5.1.3 \u00a0 Comparison of the Two Implementations","text":"<p>Supported Operations</p> <p>Both implementations support all the operations defined in a stack. The array implementation additionally supports random access, but this is beyond the scope of a stack definition and is generally not used.</p> <p>Time Efficiency</p> <p>In the array-based implementation, both push and pop operations occur in pre-allocated contiguous memory, which has good cache locality and therefore higher efficiency. However, if the push operation exceeds the array capacity, it triggers a resizing mechanism, making the time complexity of that push operation \\(O(n)\\).</p> <p>In the linked list implementation, list expansion is very flexible, and there is no efficiency decrease issue as in array expansion. However, the push operation requires initializing a node object and modifying pointers, so its efficiency is relatively lower. If the elements being pushed are already node objects, then the initialization step can be skipped, improving efficiency.</p> <p>Thus, when the elements for push and pop operations are basic data types like <code>int</code> or <code>double</code>, we can draw the following conclusions:</p> <ul> <li>The array-based stack implementation's efficiency decreases during expansion, but since expansion is a low-frequency operation, its average efficiency is higher.</li> <li>The linked list-based stack implementation provides more stable efficiency performance.</li> </ul> <p>Space Efficiency</p> <p>When initializing a list, the system allocates an \"initial capacity,\" which might exceed the actual need; moreover, the expansion mechanism usually increases capacity by a specific factor (like doubling), which may also exceed the actual need. Therefore, the array-based stack might waste some space.</p> <p>However, since linked list nodes require extra space for storing pointers, the space occupied by linked list nodes is relatively larger.</p> <p>In summary, we cannot simply determine which implementation is more memory-efficient. It requires analysis based on specific circumstances.</p>"},{"location":"chapter_stack_and_queue/stack/#514-typical-applications-of-stack","title":"5.1.4 \u00a0 Typical Applications of Stack","text":"<ul> <li>Back and forward in browsers, undo and redo in software. Every time we open a new webpage, the browser pushes the previous page onto the stack, allowing us to go back to the previous page through the back operation, which is essentially a pop operation. To support both back and forward, two stacks are needed to work together.</li> <li>Memory management in programs. Each time a function is called, the system adds a stack frame at the top of the stack to record the function's context information. In recursive functions, the downward recursion phase keeps pushing onto the stack, while the upward backtracking phase keeps popping from the stack.</li> </ul>"},{"location":"chapter_stack_and_queue/summary/","title":"5.4 \u00a0 Summary","text":""},{"location":"chapter_stack_and_queue/summary/#1-key-review","title":"1. \u00a0 Key Review","text":"<ul> <li>Stack is a data structure that follows the Last-In-First-Out (LIFO) principle and can be implemented using arrays or linked lists.</li> <li>In terms of time efficiency, the array implementation of the stack has a higher average efficiency. However, during expansion, the time complexity for a single push operation can degrade to \\(O(n)\\). In contrast, the linked list implementation of a stack offers more stable efficiency.</li> <li>Regarding space efficiency, the array implementation of the stack may lead to a certain degree of space wastage. However, it's important to note that the memory space occupied by nodes in a linked list is generally larger than that for elements in an array.</li> <li>A queue is a data structure that follows the First-In-First-Out (FIFO) principle, and it can also be implemented using arrays or linked lists. The conclusions regarding time and space efficiency for queues are similar to those for stacks.</li> <li>A double-ended queue (deque) is a more flexible type of queue that allows adding and removing elements at both ends.</li> </ul>"},{"location":"chapter_stack_and_queue/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: Is the browser's forward and backward functionality implemented with a doubly linked list?</p> <p>A browser's forward and backward navigation is essentially a manifestation of the \"stack\" concept. When a user visits a new page, the page is added to the top of the stack; when they click the back button, the page is popped from the top of the stack. A double-ended queue (deque) can conveniently implement some additional operations, as mentioned in the \"Double-Ended Queue\" section.</p> <p>Q: After popping from a stack, is it necessary to free the memory of the popped node?</p> <p>If the popped node will still be used later, it's not necessary to free its memory. In languages like Java and Python that have automatic garbage collection, manual memory release is not necessary; in C and C++, manual memory release is required.</p> <p>Q: A double-ended queue seems like two stacks joined together. What are its uses?</p> <p>A double-ended queue, which is a combination of a stack and a queue or two stacks joined together, exhibits both stack and queue logic. Thus, it can implement all applications of stacks and queues while offering more flexibility.</p> <p>Q: How exactly are undo and redo implemented?</p> <p>Undo and redo operations are implemented using two stacks: Stack A for undo and Stack B for redo.</p> <ol> <li>Each time a user performs an operation, it is pushed onto Stack A, and Stack B is cleared.</li> <li>When the user executes an \"undo\", the most recent operation is popped from Stack A and pushed onto Stack B.</li> <li>When the user executes a \"redo\", the most recent operation is popped from Stack B and pushed back onto Stack A.</li> </ol>"},{"location":"chapter_tree/","title":"Chapter 7. \u00a0 Tree","text":"<p>Abstract</p> <p>The towering tree, full of vitality with its roots deep and leaves lush, branches spreading wide.</p> <p>It vividly demonstrates the form of data divide-and-conquer.</p>"},{"location":"chapter_tree/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>7.1 \u00a0 Binary Tree</li> <li>7.2 \u00a0 Binary Tree Traversal</li> <li>7.3 \u00a0 Array Representation of Tree</li> <li>7.4 \u00a0 Binary Search Tree</li> <li>7.5 \u00a0 AVL Tree *</li> <li>7.6 \u00a0 Summary</li> </ul>"},{"location":"chapter_tree/array_representation_of_tree/","title":"7.3 \u00a0 Array representation of binary trees","text":"<p>Under the linked list representation, the storage unit of a binary tree is a node <code>TreeNode</code>, with nodes connected by pointers. The basic operations of binary trees under the linked list representation were introduced in the previous section.</p> <p>So, can we use an array to represent a binary tree? The answer is yes.</p>"},{"location":"chapter_tree/array_representation_of_tree/#731-representing-perfect-binary-trees","title":"7.3.1 \u00a0 Representing perfect binary trees","text":"<p>Let's analyze a simple case first. Given a perfect binary tree, we store all nodes in an array according to the order of level-order traversal, where each node corresponds to a unique array index.</p> <p>Based on the characteristics of level-order traversal, we can deduce a \"mapping formula\" between the index of a parent node and its children: If a node's index is \\(i\\), then the index of its left child is \\(2i + 1\\) and the right child is \\(2i + 2\\). The Figure 7-12  shows the mapping relationship between the indices of various nodes.</p> <p></p> <p> Figure 7-12 \u00a0 Array representation of a perfect binary tree </p> <p>The mapping formula plays a role similar to the node references (pointers) in linked lists. Given any node in the array, we can access its left (right) child node using the mapping formula.</p>"},{"location":"chapter_tree/array_representation_of_tree/#732-representing-any-binary-tree","title":"7.3.2 \u00a0 Representing any binary tree","text":"<p>Perfect binary trees are a special case; there are often many <code>None</code> values in the middle levels of a binary tree. Since the sequence of level-order traversal does not include these <code>None</code> values, we cannot solely rely on this sequence to deduce the number and distribution of <code>None</code> values. This means that multiple binary tree structures can match the same level-order traversal sequence.</p> <p>As shown in the Figure 7-13 , given a non-perfect binary tree, the above method of array representation fails.</p> <p></p> <p> Figure 7-13 \u00a0 Level-order traversal sequence corresponds to multiple binary tree possibilities </p> <p>To solve this problem, we can consider explicitly writing out all <code>None</code> values in the level-order traversal sequence. As shown in the following figure, after this treatment, the level-order traversal sequence can uniquely represent a binary tree. Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># Array representation of a binary tree\n# Using None to represent empty slots\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using the maximum integer value INT_MAX to mark empty slots\nvector&lt;int&gt; tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using the Integer wrapper class allows for using null to mark empty slots\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using nullable int (int?) allows for using null to mark empty slots\nint?[] tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using an any type slice, allowing for nil to mark empty slots\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using optional Int (Int?) allows for using nil to mark empty slots\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using null to represent empty slots\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using null to represent empty slots\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using nullable int (int?) allows for using null to mark empty slots\nList&lt;int?&gt; tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using None to mark empty slots\nlet tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using the maximum int value to mark empty slots, therefore, node values must not be INT_MAX\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using null to represent empty slots\nval tree = mutableListOf( 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 )\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p></p> <p> Figure 7-14 \u00a0 Array representation of any type of binary tree </p> <p>It's worth noting that complete binary trees are very suitable for array representation. Recalling the definition of a complete binary tree, <code>None</code> appears only at the bottom level and towards the right, meaning all <code>None</code> values definitely appear at the end of the level-order traversal sequence.</p> <p>This means that when using an array to represent a complete binary tree, it's possible to omit storing all <code>None</code> values, which is very convenient. The Figure 7-15  gives an example.</p> <p></p> <p> Figure 7-15 \u00a0 Array representation of a complete binary tree </p> <p>The following code implements a binary tree based on array representation, including the following operations:</p> <ul> <li>Given a node, obtain its value, left (right) child node, and parent node.</li> <li>Obtain the preorder, inorder, postorder, and level-order traversal sequences.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_binary_tree.py<pre><code>class ArrayBinaryTree:\n    \"\"\"\u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b\"\"\"\n\n    def __init__(self, arr: list[int | None]):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._tree = list(arr)\n\n    def size(self):\n        \"\"\"\u5217\u8868\u5bb9\u91cf\"\"\"\n        return len(self._tree)\n\n    def val(self, i: int) -&gt; int:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c\"\"\"\n        # \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 or i &gt;= self.size():\n            return None\n        return self._tree[i]\n\n    def left(self, i: int) -&gt; int | None:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n        return 2 * i + 1\n\n    def right(self, i: int) -&gt; int | None:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n        return 2 * i + 2\n\n    def parent(self, i: int) -&gt; int | None:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n        return (i - 1) // 2\n\n    def level_order(self) -&gt; list[int]:\n        \"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for i in range(self.size()):\n            if self.val(i) is not None:\n                self.res.append(self.val(i))\n        return self.res\n\n    def dfs(self, i: int, order: str):\n        \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\n        if self.val(i) is None:\n            return\n        # \u524d\u5e8f\u904d\u5386\n        if order == \"pre\":\n            self.res.append(self.val(i))\n        self.dfs(self.left(i), order)\n        # \u4e2d\u5e8f\u904d\u5386\n        if order == \"in\":\n            self.res.append(self.val(i))\n        self.dfs(self.right(i), order)\n        # \u540e\u5e8f\u904d\u5386\n        if order == \"post\":\n            self.res.append(self.val(i))\n\n    def pre_order(self) -&gt; list[int]:\n        \"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        self.dfs(0, order=\"pre\")\n        return self.res\n\n    def in_order(self) -&gt; list[int]:\n        \"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        self.dfs(0, order=\"in\")\n        return self.res\n\n    def post_order(self) -&gt; list[int]:\n        \"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        self.dfs(0, order=\"post\")\n        return self.res\n</code></pre> array_binary_tree.cpp<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    ArrayBinaryTree(vector&lt;int&gt; arr) {\n        tree = arr;\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    int size() {\n        return tree.size();\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    int val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size())\n            return INT_MAX;\n        return tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    int left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    int right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    int parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; levelOrder() {\n        vector&lt;int&gt; res;\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != INT_MAX)\n                res.push_back(val(i));\n        }\n        return res;\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; preOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; inOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; postOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"post\", res);\n        return res;\n    }\n\n  private:\n    vector&lt;int&gt; tree;\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    void dfs(int i, string order, vector&lt;int&gt; &amp;res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (val(i) == INT_MAX)\n            return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order == \"pre\")\n            res.push_back(val(i));\n        dfs(left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order == \"in\")\n            res.push_back(val(i));\n        dfs(right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order == \"post\")\n            res.push_back(val(i));\n    }\n};\n</code></pre> array_binary_tree.java<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    private List&lt;Integer&gt; tree;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public ArrayBinaryTree(List&lt;Integer&gt; arr) {\n        tree = new ArrayList&lt;&gt;(arr);\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    public int size() {\n        return tree.size();\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    public Integer val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size())\n            return null;\n        return tree.get(i);\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public Integer left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public Integer right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    public Integer parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; levelOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != null)\n                res.add(val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    private void dfs(Integer i, String order, List&lt;Integer&gt; res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (val(i) == null)\n            return;\n        // \u524d\u5e8f\u904d\u5386\n        if (\"pre\".equals(order))\n            res.add(val(i));\n        dfs(left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (\"in\".equals(order))\n            res.add(val(i));\n        dfs(right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (\"post\".equals(order))\n            res.add(val(i));\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; preOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; inOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; postOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.cs<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree(List&lt;int?&gt; arr) {\n    List&lt;int?&gt; tree = new(arr);\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    public int Size() {\n        return tree.Count;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    public int? Val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= Size())\n            return null;\n        return tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public int Left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public int Right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    public int Parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; LevelOrder() {\n        List&lt;int&gt; res = [];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (int i = 0; i &lt; Size(); i++) {\n            if (Val(i).HasValue)\n                res.Add(Val(i)!.Value);\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    void DFS(int i, string order, List&lt;int&gt; res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (!Val(i).HasValue)\n            return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order == \"pre\")\n            res.Add(Val(i)!.Value);\n        DFS(Left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order == \"in\")\n            res.Add(Val(i)!.Value);\n        DFS(Right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order == \"post\")\n            res.Add(Val(i)!.Value);\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; PreOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; InOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"in\", res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; PostOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.go<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\ntype arrayBinaryTree struct {\n    tree []any\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newArrayBinaryTree(arr []any) *arrayBinaryTree {\n    return &amp;arrayBinaryTree{\n        tree: arr,\n    }\n}\n\n/* \u5217\u8868\u5bb9\u91cf */\nfunc (abt *arrayBinaryTree) size() int {\n    return len(abt.tree)\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nfunc (abt *arrayBinaryTree) val(i int) any {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if i &lt; 0 || i &gt;= abt.size() {\n        return nil\n    }\n    return abt.tree[i]\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) left(i int) int {\n    return 2*i + 1\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) right(i int) int {\n    return 2*i + 2\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) parent(i int) int {\n    return (i - 1) / 2\n}\n\n/* \u5c42\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) levelOrder() []any {\n    var res []any\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n    for i := 0; i &lt; abt.size(); i++ {\n        if abt.val(i) != nil {\n            res = append(res, abt.val(i))\n        }\n    }\n    return res\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nfunc (abt *arrayBinaryTree) dfs(i int, order string, res *[]any) {\n    // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n    if abt.val(i) == nil {\n        return\n    }\n    // \u524d\u5e8f\u904d\u5386\n    if order == \"pre\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.left(i), order, res)\n    // \u4e2d\u5e8f\u904d\u5386\n    if order == \"in\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.right(i), order, res)\n    // \u540e\u5e8f\u904d\u5386\n    if order == \"post\" {\n        *res = append(*res, abt.val(i))\n    }\n}\n\n/* \u524d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) preOrder() []any {\n    var res []any\n    abt.dfs(0, \"pre\", &amp;res)\n    return res\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) inOrder() []any {\n    var res []any\n    abt.dfs(0, \"in\", &amp;res)\n    return res\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) postOrder() []any {\n    var res []any\n    abt.dfs(0, \"post\", &amp;res)\n    return res\n}\n</code></pre> array_binary_tree.swift<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    private var tree: [Int?]\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init(arr: [Int?]) {\n        tree = arr\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    func size() -&gt; Int {\n        tree.count\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    func val(i: Int) -&gt; Int? {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 || i &gt;= size() {\n            return nil\n        }\n        return tree[i]\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    func left(i: Int) -&gt; Int {\n        2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    func right(i: Int) -&gt; Int {\n        2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    func parent(i: Int) -&gt; Int {\n        (i - 1) / 2\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    func levelOrder() -&gt; [Int] {\n        var res: [Int] = []\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for i in 0 ..&lt; size() {\n            if let val = val(i: i) {\n                res.append(val)\n            }\n        }\n        return res\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    private func dfs(i: Int, order: String, res: inout [Int]) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        guard let val = val(i: i) else {\n            return\n        }\n        // \u524d\u5e8f\u904d\u5386\n        if order == \"pre\" {\n            res.append(val)\n        }\n        dfs(i: left(i: i), order: order, res: &amp;res)\n        // \u4e2d\u5e8f\u904d\u5386\n        if order == \"in\" {\n            res.append(val)\n        }\n        dfs(i: right(i: i), order: order, res: &amp;res)\n        // \u540e\u5e8f\u904d\u5386\n        if order == \"post\" {\n            res.append(val)\n        }\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    func preOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"pre\", res: &amp;res)\n        return res\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    func inOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"in\", res: &amp;res)\n        return res\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    func postOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"post\", res: &amp;res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.js<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    #tree;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(arr) {\n        this.#tree = arr;\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    size() {\n        return this.#tree.length;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    val(i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    left(i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    right(i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    parent(i) {\n        return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    levelOrder() {\n        let res = [];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    #dfs(i, order, res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (this.val(i) === null) return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    preOrder() {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    inOrder() {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    postOrder() {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.ts<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    #tree: (number | null)[];\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(arr: (number | null)[]) {\n        this.#tree = arr;\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    size(): number {\n        return this.#tree.length;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    val(i: number): number | null {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    left(i: number): number {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    right(i: number): number {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    parent(i: number): number {\n        return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    levelOrder(): number[] {\n        let res = [];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    #dfs(i: number, order: Order, res: (number | null)[]): void {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (this.val(i) === null) return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    preOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    inOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    postOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.dart<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n  late List&lt;int?&gt; _tree;\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  ArrayBinaryTree(this._tree);\n\n  /* \u5217\u8868\u5bb9\u91cf */\n  int size() {\n    return _tree.length;\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n  int? val(int i) {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if (i &lt; 0 || i &gt;= size()) {\n      return null;\n    }\n    return _tree[i];\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n  int? left(int i) {\n    return 2 * i + 1;\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n  int? right(int i) {\n    return 2 * i + 2;\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n  int? parent(int i) {\n    return (i - 1) ~/ 2;\n  }\n\n  /* \u5c42\u5e8f\u904d\u5386 */\n  List&lt;int&gt; levelOrder() {\n    List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; size(); i++) {\n      if (val(i) != null) {\n        res.add(val(i)!);\n      }\n    }\n    return res;\n  }\n\n  /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n  void dfs(int i, String order, List&lt;int?&gt; res) {\n    // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n    if (val(i) == null) {\n      return;\n    }\n    // \u524d\u5e8f\u904d\u5386\n    if (order == 'pre') {\n      res.add(val(i));\n    }\n    dfs(left(i)!, order, res);\n    // \u4e2d\u5e8f\u904d\u5386\n    if (order == 'in') {\n      res.add(val(i));\n    }\n    dfs(right(i)!, order, res);\n    // \u540e\u5e8f\u904d\u5386\n    if (order == 'post') {\n      res.add(val(i));\n    }\n  }\n\n  /* \u524d\u5e8f\u904d\u5386 */\n  List&lt;int?&gt; preOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'pre', res);\n    return res;\n  }\n\n  /* \u4e2d\u5e8f\u904d\u5386 */\n  List&lt;int?&gt; inOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'in', res);\n    return res;\n  }\n\n  /* \u540e\u5e8f\u904d\u5386 */\n  List&lt;int?&gt; postOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'post', res);\n    return res;\n  }\n}\n</code></pre> array_binary_tree.rs<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nstruct ArrayBinaryTree {\n    tree: Vec&lt;Option&lt;i32&gt;&gt;,\n}\n\nimpl ArrayBinaryTree {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(arr: Vec&lt;Option&lt;i32&gt;&gt;) -&gt; Self {\n        Self { tree: arr }\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    fn size(&amp;self) -&gt; i32 {\n        self.tree.len() as i32\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    fn val(&amp;self, i: i32) -&gt; Option&lt;i32&gt; {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 || i &gt;= self.size() {\n            None\n        } else {\n            self.tree[i as usize]\n        }\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fn left(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fn right(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    fn parent(&amp;self, i: i32) -&gt; i32 {\n        (i - 1) / 2\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    fn level_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for i in 0..self.size() {\n            if let Some(val) = self.val(i) {\n                res.push(val)\n            }\n        }\n        res\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    fn dfs(&amp;self, i: i32, order: &amp;str, res: &amp;mut Vec&lt;i32&gt;) {\n        if self.val(i).is_none() {\n            return;\n        }\n        let val = self.val(i).unwrap();\n        // \u524d\u5e8f\u904d\u5386\n        if order == \"pre\" {\n            res.push(val);\n        }\n        self.dfs(self.left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if order == \"in\" {\n            res.push(val);\n        }\n        self.dfs(self.right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if order == \"post\" {\n            res.push(val);\n        }\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    fn pre_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"pre\", &amp;mut res);\n        res\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    fn in_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"in\", &amp;mut res);\n        res\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    fn post_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"post\", &amp;mut res);\n        res\n    }\n}\n</code></pre> array_binary_tree.c<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7ed3\u6784\u4f53 */\ntypedef struct {\n    int *tree;\n    int size;\n} ArrayBinaryTree;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayBinaryTree *newArrayBinaryTree(int *arr, int arrSize) {\n    ArrayBinaryTree *abt = (ArrayBinaryTree *)malloc(sizeof(ArrayBinaryTree));\n    abt-&gt;tree = malloc(sizeof(int) * arrSize);\n    memcpy(abt-&gt;tree, arr, sizeof(int) * arrSize);\n    abt-&gt;size = arrSize;\n    return abt;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayBinaryTree(ArrayBinaryTree *abt) {\n    free(abt-&gt;tree);\n    free(abt);\n}\n\n/* \u5217\u8868\u5bb9\u91cf */\nint size(ArrayBinaryTree *abt) {\n    return abt-&gt;size;\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nint val(ArrayBinaryTree *abt, int i) {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if (i &lt; 0 || i &gt;= size(abt))\n        return INT_MAX;\n    return abt-&gt;tree[i];\n}\n\n/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; size(abt); i++) {\n        if (val(abt, i) != INT_MAX)\n            res[index++] = val(abt, i);\n    }\n    *returnSize = index;\n    return res;\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nvoid dfs(ArrayBinaryTree *abt, int i, char *order, int *res, int *index) {\n    // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n    if (val(abt, i) == INT_MAX)\n        return;\n    // \u524d\u5e8f\u904d\u5386\n    if (strcmp(order, \"pre\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, left(i), order, res, index);\n    // \u4e2d\u5e8f\u904d\u5386\n    if (strcmp(order, \"in\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, right(i), order, res, index);\n    // \u540e\u5e8f\u904d\u5386\n    if (strcmp(order, \"post\") == 0)\n        res[(*index)++] = val(abt, i);\n}\n\n/* \u524d\u5e8f\u904d\u5386 */\nint *preOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"pre\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nint *inOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"in\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nint *postOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"post\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n</code></pre> array_binary_tree.kt<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree(val tree: List&lt;Int?&gt;) {\n    /* \u5217\u8868\u5bb9\u91cf */\n    fun size(): Int {\n        return tree.size\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    fun value(i: Int): Int? {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size()) return null\n        return tree[i]\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    fun parent(i: Int): Int {\n        return (i - 1) / 2\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    fun levelOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (i in 0..&lt;size()) {\n            if (value(i) != null) res.add(value(i))\n        }\n        return res\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    fun dfs(i: Int, order: String, res: MutableList&lt;Int?&gt;) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (value(i) == null) return\n        // \u524d\u5e8f\u904d\u5386\n        if (\"pre\" == order) res.add(value(i))\n        dfs(left(i), order, res)\n        // \u4e2d\u5e8f\u904d\u5386\n        if (\"in\" == order) res.add(value(i))\n        dfs(right(i), order, res)\n        // \u540e\u5e8f\u904d\u5386\n        if (\"post\" == order) res.add(value(i))\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    fun preOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        dfs(0, \"pre\", res)\n        return res\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    fun inOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        dfs(0, \"in\", res)\n        return res\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    fun postOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        dfs(0, \"post\", res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.rb<pre><code>[class]{ArrayBinaryTree}-[func]{}\n</code></pre> array_binary_tree.zig<pre><code>[class]{ArrayBinaryTree}-[func]{}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_tree/array_representation_of_tree/#733-advantages-and-limitations","title":"7.3.3 \u00a0 Advantages and limitations","text":"<p>The array representation of binary trees has the following advantages:</p> <ul> <li>Arrays are stored in contiguous memory spaces, which is cache-friendly and allows for faster access and traversal.</li> <li>It does not require storing pointers, which saves space.</li> <li>It allows random access to nodes.</li> </ul> <p>However, the array representation also has some limitations:</p> <ul> <li>Array storage requires contiguous memory space, so it is not suitable for storing trees with a large amount of data.</li> <li>Adding or deleting nodes requires array insertion and deletion operations, which are less efficient.</li> <li>When there are many <code>None</code> values in the binary tree, the proportion of node data contained in the array is low, leading to lower space utilization.</li> </ul>"},{"location":"chapter_tree/avl_tree/","title":"7.5 \u00a0 AVL tree *","text":"<p>In the \"Binary Search Tree\" section, we mentioned that after multiple insertions and removals, a binary search tree might degrade to a linked list. In such cases, the time complexity of all operations degrades from \\(O(\\log n)\\) to \\(O(n)\\).</p> <p>As shown in the Figure 7-24 , after two node removal operations, this binary search tree will degrade into a linked list.</p> <p></p> <p> Figure 7-24 \u00a0 Degradation of an AVL tree after removing nodes </p> <p>For example, in the perfect binary tree shown in the Figure 7-25 , after inserting two nodes, the tree will lean heavily to the left, and the time complexity of search operations will also degrade.</p> <p></p> <p> Figure 7-25 \u00a0 Degradation of an AVL tree after inserting nodes </p> <p>In 1962, G. M. Adelson-Velsky and E. M. Landis proposed the \"AVL Tree\" in their paper \"An algorithm for the organization of information\". The paper detailed a series of operations to ensure that after continuously adding and removing nodes, the AVL tree would not degrade, thus maintaining the time complexity of various operations at \\(O(\\log n)\\) level. In other words, in scenarios where frequent additions, removals, searches, and modifications are needed, the AVL tree can always maintain efficient data operation performance, which has great application value.</p>"},{"location":"chapter_tree/avl_tree/#751-common-terminology-in-avl-trees","title":"7.5.1 \u00a0 Common terminology in AVL trees","text":"<p>An AVL tree is both a binary search tree and a balanced binary tree, satisfying all properties of these two types of binary trees, hence it is a \"balanced binary search tree\".</p>"},{"location":"chapter_tree/avl_tree/#1-node-height","title":"1. \u00a0 Node height","text":"<p>Since the operations related to AVL trees require obtaining node heights, we need to add a <code>height</code> variable to the node class:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class TreeNode:\n    \"\"\"AVL tree node\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                 # Node value\n        self.height: int = 0                # Node height\n        self.left: TreeNode | None = None   # Left child reference\n        self.right: TreeNode | None = None  # Right child reference\n</code></pre> <pre><code>/* AVL tree node */\nstruct TreeNode {\n    int val{};          // Node value\n    int height = 0;     // Node height\n    TreeNode *left{};   // Left child\n    TreeNode *right{};  // Right child\n    TreeNode() = default;\n    explicit TreeNode(int x) : val(x){}\n};\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode {\n    public int val;        // Node value\n    public int height;     // Node height\n    public TreeNode left;  // Left child\n    public TreeNode right; // Right child\n    public TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode(int? x) {\n    public int? val = x;    // Node value\n    public int height;      // Node height\n    public TreeNode? left;  // Left child reference\n    public TreeNode? right; // Right child reference\n}\n</code></pre> <pre><code>/* AVL tree node */\ntype TreeNode struct {\n    Val    int       // Node value\n    Height int       // Node height\n    Left   *TreeNode // Left child reference\n    Right  *TreeNode // Right child reference\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode {\n    var val: Int // Node value\n    var height: Int // Node height\n    var left: TreeNode? // Left child\n    var right: TreeNode? // Right child\n\n    init(x: Int) {\n        val = x\n        height = 0\n    }\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode {\n    val; // Node value\n    height; // Node height\n    left; // Left child pointer\n    right; // Right child pointer\n    constructor(val, left, right, height) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode {\n    val: number;            // Node value\n    height: number;         // Node height\n    left: TreeNode | null;  // Left child pointer\n    right: TreeNode | null; // Right child pointer\n    constructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height; \n        this.left = left === undefined ? null : left; \n        this.right = right === undefined ? null : right; \n    }\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode {\n  int val;         // Node value\n  int height;      // Node height\n  TreeNode? left;  // Left child\n  TreeNode? right; // Right child\n  TreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* AVL tree node */\nstruct TreeNode {\n    val: i32,                               // Node value\n    height: i32,                            // Node height\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // Left child\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // Right child\n}\n\nimpl TreeNode {\n    /* Constructor */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            height: 0,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* AVL tree node */\nTreeNode struct TreeNode {\n    int val;\n    int height;\n    struct TreeNode *left;\n    struct TreeNode *right;\n} TreeNode;\n\n/* Constructor */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode(val _val: Int) {  // Node value\n    val height: Int = 0          // Node height\n    val left: TreeNode? = null   // Left child\n    val right: TreeNode? = null  // Right child\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>The \"node height\" refers to the distance from that node to its farthest leaf node, i.e., the number of \"edges\" passed. It is important to note that the height of a leaf node is \\(0\\), and the height of a null node is \\(-1\\). We will create two utility functions for getting and updating the height of a node:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def height(self, node: TreeNode | None) -&gt; int:\n    \"\"\"\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\"\"\"\n    # \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    if node is not None:\n        return node.height\n    return -1\n\ndef update_height(self, node: TreeNode | None):\n    \"\"\"\u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\"\"\"\n    # \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height = max([self.height(node.left), self.height(node.right)]) + 1\n</code></pre> avl_tree.cpp<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node == nullptr ? -1 : node-&gt;height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node-&gt;height = max(height(node-&gt;left), height(node-&gt;right)) + 1;\n}\n</code></pre> avl_tree.java<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint Height(TreeNode? node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid UpdateHeight(TreeNode node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height = Math.Max(Height(node.left), Height(node.right)) + 1;\n}\n</code></pre> avl_tree.go<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    if node != nil {\n        return node.Height\n    }\n    return -1\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\n    lh := t.height(node.Left)\n    rh := t.height(node.Right)\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    if lh &gt; rh {\n        node.Height = lh + 1\n    } else {\n        node.Height = rh + 1\n    }\n}\n</code></pre> avl_tree.swift<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc height(node: TreeNode?) -&gt; Int {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    node?.height ?? -1\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc updateHeight(node: TreeNode?) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n</code></pre> avl_tree.js<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\n#updateHeight(node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node: TreeNode): number {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nupdateHeight(node: TreeNode): void {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode? node) {\n  // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n  return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode? node) {\n  // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n  node!.height = max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfn height(node: OptionTreeNodeRc) -&gt; i32 {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    match node {\n        Some(node) =&gt; node.borrow().height,\n        None =&gt; -1,\n    }\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfn update_height(node: OptionTreeNodeRc) {\n    if let Some(node) = node {\n        let left = node.borrow().left.clone();\n        let right = node.borrow().right.clone();\n        // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n        node.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    if (node != NULL) {\n        return node-&gt;height;\n    }\n    return -1;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n    int lh = height(node-&gt;left);\n    int rh = height(node-&gt;right);\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    if (lh &gt; rh) {\n        node-&gt;height = lh + 1;\n    } else {\n        node-&gt;height = rh + 1;\n    }\n}\n</code></pre> avl_tree.kt<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfun height(node: TreeNode?): Int {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node?.height ?: -1\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfun updateHeight(node: TreeNode?) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node?.height = (max(height(node?.left).toDouble(), height(node?.right).toDouble()) + 1).toInt()\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{height}\n\n[class]{AVLTree}-[func]{update_height}\n</code></pre> avl_tree.zig<pre><code>// \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\nfn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n    _ = self;\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return if (node == null) -1 else node.?.height;\n}\n\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nfn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.?.height = @max(self.height(node.?.left), self.height(node.?.right)) + 1;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2-node-balance-factor","title":"2. \u00a0 Node balance factor","text":"<p>The \"balance factor\" of a node is defined as the height of the node's left subtree minus the height of its right subtree, with the balance factor of a null node defined as \\(0\\). We will also encapsulate the functionality of obtaining the node balance factor into a function for easy use later on:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def balance_factor(self, node: TreeNode | None) -&gt; int:\n    \"\"\"\u83b7\u53d6\u5e73\u8861\u56e0\u5b50\"\"\"\n    # \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if node is None:\n        return 0\n    # \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return self.height(node.left) - self.height(node.right)\n</code></pre> avl_tree.cpp<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == nullptr)\n        return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.java<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null)\n        return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.cs<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint BalanceFactor(TreeNode? node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return Height(node.left) - Height(node.right);\n}\n</code></pre> avl_tree.go<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if node == nil {\n        return 0\n    }\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return t.height(node.Left) - t.height(node.Right)\n}\n</code></pre> avl_tree.swift<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc balanceFactor(node: TreeNode?) -&gt; Int {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    guard let node = node else { return 0 }\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node: node.left) - height(node: node.right)\n}\n</code></pre> avl_tree.js<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node === null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.ts<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node: TreeNode): number {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node === null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.dart<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node) {\n  // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n  if (node == null) return 0;\n  // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n  return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.rs<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfn balance_factor(node: OptionTreeNodeRc) -&gt; i32 {\n    match node {\n        // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n        None =&gt; 0,\n        // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n        Some(node) =&gt; {\n            Self::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())\n        }\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == NULL) {\n        return 0;\n    }\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.kt<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfun balanceFactor(node: TreeNode?): Int {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null) return 0\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node.left) - height(node.right)\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{balance_factor}\n</code></pre> avl_tree.zig<pre><code>// \u83b7\u53d6\u5e73\u8861\u56e0\u5b50\nfn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return self.height(node.?.left) - self.height(node.?.right);\n}\n</code></pre> <p>Note</p> <p>Let the balance factor be \\(f\\), then the balance factor of any node in an AVL tree satisfies \\(-1 \\le f \\le 1\\).</p>"},{"location":"chapter_tree/avl_tree/#752-rotations-in-avl-trees","title":"7.5.2 \u00a0 Rotations in AVL trees","text":"<p>The characteristic feature of an AVL tree is the \"rotation\" operation, which can restore balance to an unbalanced node without affecting the in-order traversal sequence of the binary tree. In other words, the rotation operation can maintain the property of a \"binary search tree\" while also turning the tree back into a \"balanced binary tree\".</p> <p>We call nodes with an absolute balance factor \\(&gt; 1\\) \"unbalanced nodes\". Depending on the type of imbalance, there are four kinds of rotations: right rotation, left rotation, right-left rotation, and left-right rotation. Below, we detail these rotation operations.</p>"},{"location":"chapter_tree/avl_tree/#1-right-rotation","title":"1. \u00a0 Right rotation","text":"<p>As shown in the Figure 7-26 , the first unbalanced node from the bottom up in the binary tree is \"node 3\". Focusing on the subtree with this unbalanced node as the root, denoted as <code>node</code>, and its left child as <code>child</code>, perform a \"right rotation\". After the right rotation, the subtree is balanced again while still maintaining the properties of a binary search tree.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 7-26 \u00a0 Steps of right rotation </p> <p>As shown in the Figure 7-27 , when the <code>child</code> node has a right child (denoted as <code>grand_child</code>), a step needs to be added in the right rotation: set <code>grand_child</code> as the left child of <code>node</code>.</p> <p></p> <p> Figure 7-27 \u00a0 Right rotation with grand_child </p> <p>\"Right rotation\" is a figurative term; in practice, it is achieved by modifying node pointers, as shown in the following code:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def right_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u53f3\u65cb\u64cd\u4f5c\"\"\"\n    child = node.left\n    grand_child = child.right\n    # \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node\n    node.left = grand_child\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    self.update_height(child)\n    # \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;left;\n    TreeNode *grandChild = child-&gt;right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode rightRotate(TreeNode node) {\n    TreeNode child = node.left;\n    TreeNode grandChild = child.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? RightRotate(TreeNode? node) {\n    TreeNode? child = node?.left;\n    TreeNode? grandChild = child?.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\n    child := node.Left\n    grandChild := child.Right\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.Right = node\n    node.Left = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc rightRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.left\n    let grandChild = child?.right\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child?.right = node\n    node?.left = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\n#rightRotate(node) {\n    const child = node.left;\n    const grandChild = child.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nrightRotate(node: TreeNode): TreeNode {\n    const child = node.left;\n    const grandChild = child.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node) {\n  TreeNode? child = node!.left;\n  TreeNode? grandChild = child!.right;\n  // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n  child.right = node;\n  node.left = grandChild;\n  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  updateHeight(node);\n  updateHeight(child);\n  // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfn right_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().left.clone().unwrap();\n            let grand_child = child.borrow().right.clone();\n            // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n            child.borrow_mut().right = Some(node.clone());\n            node.borrow_mut().left = grand_child;\n            // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;left;\n    grandChild = child-&gt;right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfun rightRotate(node: TreeNode?): TreeNode {\n    val child = node!!.left\n    val grandChild = child!!.right\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node\n    node.left = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node)\n    updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{right_rotate}\n</code></pre> avl_tree.zig<pre><code>// \u53f3\u65cb\u64cd\u4f5c\nfn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    var child = node.?.left;\n    var grandChild = child.?.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.?.right = node;\n    node.?.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.updateHeight(node);\n    self.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2-left-rotation","title":"2. \u00a0 Left rotation","text":"<p>Correspondingly, if considering the \"mirror\" of the above unbalanced binary tree, the \"left rotation\" operation shown in the Figure 7-28  needs to be performed.</p> <p></p> <p> Figure 7-28 \u00a0 Left rotation operation </p> <p>Similarly, as shown in the Figure 7-29 , when the <code>child</code> node has a left child (denoted as <code>grand_child</code>), a step needs to be added in the left rotation: set <code>grand_child</code> as the right child of <code>node</code>.</p> <p></p> <p> Figure 7-29 \u00a0 Left rotation with grand_child </p> <p>It can be observed that the right and left rotation operations are logically symmetrical, and they solve two symmetrical types of imbalance. Based on symmetry, by replacing all <code>left</code> with <code>right</code>, and all <code>right</code> with <code>left</code> in the implementation code of right rotation, we can get the implementation code for left rotation:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def left_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u5de6\u65cb\u64cd\u4f5c\"\"\"\n    child = node.right\n    grand_child = child.left\n    # \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node\n    node.right = grand_child\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    self.update_height(child)\n    # \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;right;\n    TreeNode *grandChild = child-&gt;left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode leftRotate(TreeNode node) {\n    TreeNode child = node.right;\n    TreeNode grandChild = child.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? LeftRotate(TreeNode? node) {\n    TreeNode? child = node?.right;\n    TreeNode? grandChild = child?.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\n    child := node.Right\n    grandChild := child.Left\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.Left = node\n    node.Right = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc leftRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.right\n    let grandChild = child?.left\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child?.left = node\n    node?.right = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\n#leftRotate(node) {\n    const child = node.right;\n    const grandChild = child.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nleftRotate(node: TreeNode): TreeNode {\n    const child = node.right;\n    const grandChild = child.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node) {\n  TreeNode? child = node!.right;\n  TreeNode? grandChild = child!.left;\n  // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n  child.left = node;\n  node.right = grandChild;\n  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  updateHeight(node);\n  updateHeight(child);\n  // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfn left_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().right.clone().unwrap();\n            let grand_child = child.borrow().left.clone();\n            // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n            child.borrow_mut().left = Some(node.clone());\n            node.borrow_mut().right = grand_child;\n            // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;right;\n    grandChild = child-&gt;left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfun leftRotate(node: TreeNode?): TreeNode {\n    val child = node!!.right\n    val grandChild = child!!.left\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node\n    node.right = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node)\n    updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{left_rotate}\n</code></pre> avl_tree.zig<pre><code>// \u5de6\u65cb\u64cd\u4f5c\nfn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    var child = node.?.right;\n    var grandChild = child.?.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.?.left = node;\n    node.?.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.updateHeight(node);\n    self.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3-right-left-rotation","title":"3. \u00a0 Right-left rotation","text":"<p>For the unbalanced node 3 shown in the Figure 7-30 , using either left or right rotation alone cannot restore balance to the subtree. In this case, a \"left rotation\" needs to be performed on <code>child</code> first, followed by a \"right rotation\" on <code>node</code>.</p> <p></p> <p> Figure 7-30 \u00a0 Right-left rotation </p>"},{"location":"chapter_tree/avl_tree/#4-left-right-rotation","title":"4. \u00a0 Left-right rotation","text":"<p>As shown in the Figure 7-31 , for the mirror case of the above unbalanced binary tree, a \"right rotation\" needs to be performed on <code>child</code> first, followed by a \"left rotation\" on <code>node</code>.</p> <p></p> <p> Figure 7-31 \u00a0 Left-right rotation </p>"},{"location":"chapter_tree/avl_tree/#5-choice-of-rotation","title":"5. \u00a0 Choice of rotation","text":"<p>The four kinds of imbalances shown in the Figure 7-32  correspond to the cases described above, respectively requiring right rotation, left-right rotation, right-left rotation, and left rotation.</p> <p></p> <p> Figure 7-32 \u00a0 The four rotation cases of AVL tree </p> <p>As shown in the Table 7-3 , we determine which of the above cases an unbalanced node belongs to by judging the sign of the balance factor of the unbalanced node and its higher-side child's balance factor.</p> <p> Table 7-3 \u00a0 Conditions for Choosing Among the Four Rotation Cases </p> Balance factor of unbalanced node Balance factor of child node Rotation method to use \\(&gt; 1\\) (Left-leaning tree) \\(\\geq 0\\) Right rotation \\(&gt; 1\\) (Left-leaning tree) \\(&lt;0\\) Left rotation then right rotation \\(&lt; -1\\) (Right-leaning tree) \\(\\leq 0\\) Left rotation \\(&lt; -1\\) (Right-leaning tree) \\(&gt;0\\) Right rotation then left rotation <p>For convenience, we encapsulate the rotation operations into a function. With this function, we can perform rotations on various kinds of imbalances, restoring balance to unbalanced nodes. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\"\"\"\n    # \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    balance_factor = self.balance_factor(node)\n    # \u5de6\u504f\u6811\n    if balance_factor &gt; 1:\n        if self.balance_factor(node.left) &gt;= 0:\n            # \u53f3\u65cb\n            return self.right_rotate(node)\n        else:\n            # \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = self.left_rotate(node.left)\n            return self.right_rotate(node)\n    # \u53f3\u504f\u6811\n    elif balance_factor &lt; -1:\n        if self.balance_factor(node.right) &lt;= 0:\n            # \u5de6\u65cb\n            return self.left_rotate(node)\n        else:\n            # \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = self.right_rotate(node.right)\n            return self.left_rotate(node)\n    # \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n</code></pre> avl_tree.cpp<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int _balanceFactor = balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (_balanceFactor &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (_balanceFactor &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode rotate(TreeNode node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int balanceFactor = balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = leftRotate(node.left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = rightRotate(node.right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? Rotate(TreeNode? node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int balanceFactorInt = BalanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactorInt &gt; 1) {\n        if (BalanceFactor(node?.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return RightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node!.left = LeftRotate(node!.left);\n            return RightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactorInt &lt; -1) {\n        if (BalanceFactor(node?.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return LeftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node!.right = RightRotate(node!.right);\n            return LeftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    // Go \u63a8\u8350\u77ed\u53d8\u91cf\uff0c\u8fd9\u91cc bf \u6307\u4ee3 t.balanceFactor\n    bf := t.balanceFactor(node)\n    // \u5de6\u504f\u6811\n    if bf &gt; 1 {\n        if t.balanceFactor(node.Left) &gt;= 0 {\n            // \u53f3\u65cb\n            return t.rightRotate(node)\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.Left = t.leftRotate(node.Left)\n            return t.rightRotate(node)\n        }\n    }\n    // \u53f3\u504f\u6811\n    if bf &lt; -1 {\n        if t.balanceFactor(node.Right) &lt;= 0 {\n            // \u5de6\u65cb\n            return t.leftRotate(node)\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.Right = t.rightRotate(node.Right)\n            return t.leftRotate(node)\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc rotate(node: TreeNode?) -&gt; TreeNode? {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    let balanceFactor = balanceFactor(node: node)\n    // \u5de6\u504f\u6811\n    if balanceFactor &gt; 1 {\n        if self.balanceFactor(node: node?.left) &gt;= 0 {\n            // \u53f3\u65cb\n            return rightRotate(node: node)\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node?.left = leftRotate(node: node?.left)\n            return rightRotate(node: node)\n        }\n    }\n    // \u53f3\u504f\u6811\n    if balanceFactor &lt; -1 {\n        if self.balanceFactor(node: node?.right) &lt;= 0 {\n            // \u5de6\u65cb\n            return leftRotate(node: node)\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node?.right = rightRotate(node: node?.right)\n            return leftRotate(node: node)\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n#rotate(node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    const balanceFactor = this.balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return this.#rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = this.#leftRotate(node.left);\n            return this.#rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return this.#leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = this.#rightRotate(node.right);\n            return this.#leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nrotate(node: TreeNode): TreeNode {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    const balanceFactor = this.balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return this.rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = this.leftRotate(node.left);\n            return this.rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return this.leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = this.rightRotate(node.right);\n            return this.leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node) {\n  // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n  int factor = balanceFactor(node);\n  // \u5de6\u504f\u6811\n  if (factor &gt; 1) {\n    if (balanceFactor(node!.left) &gt;= 0) {\n      // \u53f3\u65cb\n      return rightRotate(node);\n    } else {\n      // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n      node.left = leftRotate(node.left);\n      return rightRotate(node);\n    }\n  }\n  // \u53f3\u504f\u6811\n  if (factor &lt; -1) {\n    if (balanceFactor(node!.right) &lt;= 0) {\n      // \u5de6\u65cb\n      return leftRotate(node);\n    } else {\n      // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n      node.right = rightRotate(node.right);\n      return leftRotate(node);\n    }\n  }\n  // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfn rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    let balance_factor = Self::balance_factor(node.clone());\n    // \u5de6\u504f\u6811\n    if balance_factor &gt; 1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().left.clone()) &gt;= 0 {\n            // \u53f3\u65cb\n            Self::right_rotate(Some(node))\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            let left = node.borrow().left.clone();\n            node.borrow_mut().left = Self::left_rotate(left);\n            Self::right_rotate(Some(node))\n        }\n    }\n    // \u53f3\u504f\u6811\n    else if balance_factor &lt; -1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().right.clone()) &lt;= 0 {\n            // \u5de6\u65cb\n            Self::left_rotate(Some(node))\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            let right = node.borrow().right.clone();\n            node.borrow_mut().right = Self::right_rotate(right);\n            Self::left_rotate(Some(node))\n        }\n    } else {\n        // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n        node\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int bf = balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (bf &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (bf &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfun rotate(node: TreeNode): TreeNode {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    val balanceFactor = balanceFactor(node)\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node)\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = leftRotate(node.left)\n            return rightRotate(node)\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node)\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = rightRotate(node.right)\n            return leftRotate(node)\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{rotate}\n</code></pre> avl_tree.zig<pre><code>// \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nfn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    var balance_factor = self.balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balance_factor &gt; 1) {\n        if (self.balanceFactor(node.?.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return self.rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.?.left = self.leftRotate(node.?.left);\n            return self.rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balance_factor &lt; -1) {\n        if (self.balanceFactor(node.?.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return self.leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.?.right = self.rightRotate(node.?.right);\n            return self.leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#753-common-operations-in-avl-trees","title":"7.5.3 \u00a0 Common operations in AVL trees","text":""},{"location":"chapter_tree/avl_tree/#1-node-insertion","title":"1. \u00a0 Node insertion","text":"<p>The node insertion operation in AVL trees is similar to that in binary search trees. The only difference is that after inserting a node in an AVL tree, a series of unbalanced nodes may appear along the path from that node to the root node. Therefore, we need to start from this node and perform rotation operations upwards to restore balance to all unbalanced nodes. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def insert(self, val):\n    \"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n    self._root = self.insert_helper(self._root, val)\n\ndef insert_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode:\n    \"\"\"\u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\n    if node is None:\n        return TreeNode(val)\n    # 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9\n    if val &lt; node.val:\n        node.left = self.insert_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.insert_helper(node.right, val)\n    else:\n        # \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    # 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    else\n        return node;    // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode insertHelper(TreeNode node, int val) {\n    if (node == null)\n        return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val)\n        node.left = insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = insertHelper(node.right, val);\n    else\n        return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int val) {\n    root = InsertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? InsertHelper(TreeNode? node, int val) {\n    if (node == null) return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val)\n        node.left = InsertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = InsertHelper(node.right, val);\n    else\n        return node;     // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    UpdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = Rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc (t *aVLTree) insert(val int) {\n    t.root = t.insertHelper(t.root, val)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return NewTreeNode(val)\n    }\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if val &lt; node.Val.(int) {\n        node.Left = t.insertHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.insertHelper(node.Right, val)\n    } else {\n        // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = t.rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(val: Int) {\n    root = insertHelper(node: root, val: val)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc insertHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return TreeNode(x: val)\n    }\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if val &lt; node!.val {\n        node?.left = insertHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = insertHelper(node: node?.right, val: val)\n    } else {\n        return node // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    updateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node: node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(val) {\n    this.root = this.#insertHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#insertHelper(node, val) {\n    if (node === null) return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val) node.left = this.#insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#insertHelper(node.right, val);\n    else return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    this.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.#rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(val: number): void {\n    this.root = this.insertHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\ninsertHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val) {\n        node.left = this.insertHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.insertHelper(node.right, val);\n    } else {\n        return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    this.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n  root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val) {\n  if (node == null) return TreeNode(val);\n  /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n  if (val &lt; node.val)\n    node.left = insertHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = insertHelper(node.right, val);\n  else\n    return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n  updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n  node = rotate(node);\n  // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfn insert(&amp;mut self, val: i32) {\n    self.root = Self::insert_helper(self.root.clone(), val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn insert_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n            match {\n                let node_val = node.borrow().val;\n                node_val\n            }\n            .cmp(&amp;val)\n            {\n                Ordering::Greater =&gt; {\n                    let left = node.borrow().left.clone();\n                    node.borrow_mut().left = Self::insert_helper(left, val);\n                }\n                Ordering::Less =&gt; {\n                    let right = node.borrow().right.clone();\n                    node.borrow_mut().right = Self::insert_helper(right, val);\n                }\n                Ordering::Equal =&gt; {\n                    return Some(node); // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n                }\n            }\n            Self::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n\n            /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n            node = Self::rotate(Some(node)).unwrap();\n            // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(node)\n        }\n        None =&gt; Some(TreeNode::new(val)),\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(AVLTree *tree, int val) {\n    tree-&gt;root = insertHelper(tree-&gt;root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == NULL) {\n        return newTreeNode(val);\n    }\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    } else {\n        // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node;\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfun insert(value: Int) {\n    root = insertHelper(root, value)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfun insertHelper(n: TreeNode?, value: Int): TreeNode {\n    if (n == null)\n        return TreeNode(value)\n    var node = n\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (value &lt; node.value) node.left = insertHelper(node.left, value)\n    else if (value &gt; node.value) node.right = insertHelper(node.right, value)\n    else return node // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n\n    updateHeight(node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{insert}\n\n[class]{AVLTree}-[func]{insert_helper}\n</code></pre> avl_tree.zig<pre><code>// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, val: T) !void {\n    self.root = (try self.insertHelper(self.root, val)).?;\n}\n\n// \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {\n    var node = node_;\n    if (node == null) {\n        var tmp_node = try self.mem_allocator.create(inc.TreeNode(T));\n        tmp_node.init(val);\n        return tmp_node;\n    }\n    // 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9\n    if (val &lt; node.?.val) {\n        node.?.left = try self.insertHelper(node.?.left, val);\n    } else if (val &gt; node.?.val) {\n        node.?.right = try self.insertHelper(node.?.right, val);\n    } else {\n        return node;            // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    self.updateHeight(node);    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    // 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    node = self.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2-node-removal","title":"2. \u00a0 Node removal","text":"<p>Similarly, based on the method of removing nodes in binary search trees, rotation operations need to be performed from the bottom up to restore balance to all unbalanced nodes. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def remove(self, val: int):\n    \"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n    self._root = self.remove_helper(self._root, val)\n\ndef remove_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode | None:\n    \"\"\"\u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\n    if node is None:\n        return None\n    # 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664\n    if val &lt; node.val:\n        node.left = self.remove_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.remove_helper(node.right, val)\n    else:\n        if node.left is None or node.right is None:\n            child = node.left or node.right\n            # \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if child is None:\n                return None\n            # \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else:\n                node = child\n        else:\n            # \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            temp = node.right\n            while temp.left is not None:\n                temp = temp.left\n            node.right = self.remove_helper(node.right, temp.val)\n            node.val = temp.val\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    # 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return nullptr;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    else {\n        if (node-&gt;left == nullptr || node-&gt;right == nullptr) {\n            TreeNode *child = node-&gt;left != nullptr ? node-&gt;left : node-&gt;right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == nullptr) {\n                delete node;\n                return nullptr;\n            }\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else {\n                delete node;\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != nullptr) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode removeHelper(TreeNode node, int val) {\n    if (node == null)\n        return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val)\n        node.left = removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = removeHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode child = node.left != null ? node.left : node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null)\n                return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else\n                node = child;\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int val) {\n    root = RemoveHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? RemoveHelper(TreeNode? node, int val) {\n    if (node == null) return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val)\n        node.left = RemoveHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = RemoveHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode? child = node.left ?? node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null)\n                return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else\n                node = child;\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode? temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = RemoveHelper(node.right, temp.val!.Value);\n            node.val = temp.val;\n        }\n    }\n    UpdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = Rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc (t *aVLTree) remove(val int) {\n    t.root = t.removeHelper(t.root, val)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return nil\n    }\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if val &lt; node.Val.(int) {\n        node.Left = t.removeHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.removeHelper(node.Right, val)\n    } else {\n        if node.Left == nil || node.Right == nil {\n            child := node.Left\n            if node.Right != nil {\n                child = node.Right\n            }\n            if child == nil {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n                return nil\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                node = child\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            temp := node.Right\n            for temp.Left != nil {\n                temp = temp.Left\n            }\n            node.Right = t.removeHelper(node.Right, temp.Val.(int))\n            node.Val = temp.Val\n        }\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = t.rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc remove(val: Int) {\n    root = removeHelper(node: root, val: val)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc removeHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return nil\n    }\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if val &lt; node!.val {\n        node?.left = removeHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = removeHelper(node: node?.right, val: val)\n    } else {\n        if node?.left == nil || node?.right == nil {\n            let child = node?.left ?? node?.right\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if child == nil {\n                return nil\n            }\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else {\n                node = child\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            var temp = node?.right\n            while temp?.left != nil {\n                temp = temp?.left\n            }\n            node?.right = removeHelper(node: node?.right, val: temp!.val)\n            node?.val = temp!.val\n        }\n    }\n    updateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node: node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(val) {\n    this.root = this.#removeHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#removeHelper(node, val) {\n    if (node === null) return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val) node.left = this.#removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#removeHelper(node.right, val);\n    else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child === null) return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else node = child;\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.#removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.#rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(val: number): void {\n    this.root = this.removeHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nremoveHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val) {\n        node.left = this.removeHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.removeHelper(node.right, val);\n    } else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child === null) {\n                return null;\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n  root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val) {\n  if (node == null) return null;\n  /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n  if (val &lt; node.val)\n    node.left = removeHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = removeHelper(node.right, val);\n  else {\n    if (node.left == null || node.right == null) {\n      TreeNode? child = node.left ?? node.right;\n      // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n      if (child == null)\n        return null;\n      // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n      else\n        node = child;\n    } else {\n      // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n      TreeNode? temp = node.right;\n      while (temp!.left != null) {\n        temp = temp.left;\n      }\n      node.right = removeHelper(node.right, temp.val);\n      node.val = temp.val;\n    }\n  }\n  updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n  node = rotate(node);\n  // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfn remove(&amp;self, val: i32) {\n    Self::remove_helper(self.root.clone(), val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn remove_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n            if val &lt; node.borrow().val {\n                let left = node.borrow().left.clone();\n                node.borrow_mut().left = Self::remove_helper(left, val);\n            } else if val &gt; node.borrow().val {\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, val);\n            } else if node.borrow().left.is_none() || node.borrow().right.is_none() {\n                let child = if node.borrow().left.is_some() {\n                    node.borrow().left.clone()\n                } else {\n                    node.borrow().right.clone()\n                };\n                match child {\n                    // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n                    None =&gt; {\n                        return None;\n                    }\n                    // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                    Some(child) =&gt; node = child,\n                }\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n                let mut temp = node.borrow().right.clone().unwrap();\n                loop {\n                    let temp_left = temp.borrow().left.clone();\n                    if temp_left.is_none() {\n                        break;\n                    }\n                    temp = temp_left.unwrap();\n                }\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);\n                node.borrow_mut().val = temp.borrow().val;\n            }\n            Self::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n\n            /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n            node = Self::rotate(Some(node)).unwrap();\n            // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(node)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(AVLTree *tree, int val) {\n    TreeNode *root = removeHelper(tree-&gt;root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    TreeNode *child, *grandChild;\n    if (node == NULL) {\n        return NULL;\n    }\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    } else {\n        if (node-&gt;left == NULL || node-&gt;right == NULL) {\n            child = node-&gt;left;\n            if (node-&gt;right != NULL) {\n                child = node-&gt;right;\n            }\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == NULL) {\n                return NULL;\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != NULL) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfun remove(value: Int) {\n    root = removeHelper(root, value)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfun removeHelper(n: TreeNode?, value: Int): TreeNode? {\n    var node = n ?: return null\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (value &lt; node.value) node.left = removeHelper(node.left, value)\n    else if (value &gt; node.value) node.right = removeHelper(node.right, value)\n    else {\n        if (node.left == null || node.right == null) {\n            val child = if (node.left != null) node.left else node.right\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null) return null\n            else node = child\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            var temp = node.right\n            while (temp!!.left != null) {\n                temp = temp.left\n            }\n            node.right = removeHelper(node.right, temp.value)\n            node.value = temp.value\n        }\n    }\n    updateHeight(node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{remove}\n\n[class]{AVLTree}-[func]{remove_helper}\n</code></pre> avl_tree.zig<pre><code>// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, val: T) void {\n   self.root = self.removeHelper(self.root, val).?;\n}\n\n// \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {\n    var node = node_;\n    if (node == null) return null;\n    // 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664\n    if (val &lt; node.?.val) {\n        node.?.left = self.removeHelper(node.?.left, val);\n    } else if (val &gt; node.?.val) {\n        node.?.right = self.removeHelper(node.?.right, val);\n    } else {\n        if (node.?.left == null or node.?.right == null) {\n            var child = if (node.?.left != null) node.?.left else node.?.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null) {\n                return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            } else {\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            var temp = node.?.right;\n            while (temp.?.left != null) {\n                temp = temp.?.left;\n            }\n            node.?.right = self.removeHelper(node.?.right, temp.?.val);\n            node.?.val = temp.?.val;\n        }\n    }\n    self.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    // 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    node = self.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3-node-search","title":"3. \u00a0 Node search","text":"<p>The node search operation in AVL trees is consistent with that in binary search trees and will not be detailed here.</p>"},{"location":"chapter_tree/avl_tree/#754-typical-applications-of-avl-trees","title":"7.5.4 \u00a0 Typical applications of AVL trees","text":"<ul> <li>Organizing and storing large amounts of data, suitable for scenarios with high-frequency searches and low-frequency intertions and removals.</li> <li>Used to build index systems in databases.</li> <li>Red-black trees are also a common type of balanced binary search tree. Compared to AVL trees, red-black trees have more relaxed balancing conditions, require fewer rotations for node insertion and removal, and have a higher average efficiency for node addition and removal operations.</li> </ul>"},{"location":"chapter_tree/binary_search_tree/","title":"7.4 \u00a0 Binary search tree","text":"<p>As shown in the Figure 7-16 , a \"binary search tree\" satisfies the following conditions.</p> <ol> <li>For the root node, the value of all nodes in the left subtree &lt; the value of the root node &lt; the value of all nodes in the right subtree.</li> <li>The left and right subtrees of any node are also binary search trees, i.e., they satisfy condition <code>1.</code> as well.</li> </ol> <p></p> <p> Figure 7-16 \u00a0 Binary search tree </p>"},{"location":"chapter_tree/binary_search_tree/#741-operations-on-a-binary-search-tree","title":"7.4.1 \u00a0 Operations on a binary search tree","text":"<p>We encapsulate the binary search tree as a class <code>BinarySearchTree</code> and declare a member variable <code>root</code>, pointing to the tree's root node.</p>"},{"location":"chapter_tree/binary_search_tree/#1-searching-for-a-node","title":"1. \u00a0 Searching for a node","text":"<p>Given a target node value <code>num</code>, one can search according to the properties of the binary search tree. As shown in the Figure 7-17 , we declare a node <code>cur</code> and start from the binary tree's root node <code>root</code>, looping to compare the size relationship between the node value <code>cur.val</code> and <code>num</code>.</p> <ul> <li>If <code>cur.val &lt; num</code>, it means the target node is in <code>cur</code>'s right subtree, thus execute <code>cur = cur.right</code>.</li> <li>If <code>cur.val &gt; num</code>, it means the target node is in <code>cur</code>'s left subtree, thus execute <code>cur = cur.left</code>.</li> <li>If <code>cur.val = num</code>, it means the target node is found, exit the loop and return the node.</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 7-17 \u00a0 Example of searching for a node in a binary search tree </p> <p>The search operation in a binary search tree works on the same principle as the binary search algorithm, eliminating half of the possibilities in each round. The number of loops is at most the height of the binary tree. When the binary tree is balanced, it uses \\(O(\\log n)\\) time. Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def search(self, num: int) -&gt; TreeNode | None:\n    \"\"\"\u67e5\u627e\u8282\u70b9\"\"\"\n    cur = self._root\n    # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur is not None:\n        # \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        elif cur.val &gt; num:\n            cur = cur.left\n        # \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else:\n            break\n    return cur\n</code></pre> binary_search_tree.cpp<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(int num) {\n    TreeNode *cur = root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur-&gt;val &gt; num)\n            cur = cur-&gt;left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode search(int num) {\n    TreeNode cur = root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? Search(int num) {\n    TreeNode? cur = root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur =\n            cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\n    node := bst.root\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    for node != nil {\n        if node.Val.(int) &lt; num {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            node = node.Right\n        } else if node.Val.(int) &gt; num {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            node = node.Left\n        } else {\n            // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            break\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return node\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfunc search(num: Int) -&gt; TreeNode? {\n    var cur = root\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur != nil {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if cur!.val &gt; num {\n            cur = cur?.left\n        }\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else {\n            break\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nsearch(num) {\n    let cur = this.root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num) cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nsearch(num: number): TreeNode | null {\n    let cur = this.root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num) cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? search(int _num) {\n  TreeNode? cur = _root;\n  // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n  while (cur != null) {\n    // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n    else if (cur.val &gt; _num)\n      cur = cur.left;\n    // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n    else\n      break;\n  }\n  // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n  return cur;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u67e5\u627e\u8282\u70b9 */\npub fn search(&amp;self, num: i32) -&gt; OptionTreeNodeRc {\n    let mut cur = self.root.clone();\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            Ordering::Greater =&gt; cur = node.borrow().right.clone(),\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            Ordering::Less =&gt; cur = node.borrow().left.clone(),\n            // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            Ordering::Equal =&gt; break,\n        }\n    }\n\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    cur\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(BinarySearchTree *bst, int num) {\n    TreeNode *cur = bst-&gt;root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != NULL) {\n        if (cur-&gt;val &lt; num) {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            cur = cur-&gt;right;\n        } else if (cur-&gt;val &gt; num) {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            cur = cur-&gt;left;\n        } else {\n            // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfun search(num: Int): TreeNode? {\n    var cur = root\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        cur = if (cur.value &lt; num) cur.right\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.value &gt; num) cur.left\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else break\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur\n}\n</code></pre> binary_search_tree.rb<pre><code>[class]{BinarySearchTree}-[func]{search}\n</code></pre> binary_search_tree.zig<pre><code>// \u67e5\u627e\u8282\u70b9\nfn search(self: *Self, num: T) ?*inc.TreeNode(T) {\n    var cur = self.root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        } else if (cur.?.val &gt; num) {\n            cur = cur.?.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        } else {\n            break;\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_tree/binary_search_tree/#2-inserting-a-node","title":"2. \u00a0 Inserting a node","text":"<p>Given an element <code>num</code> to be inserted, to maintain the property of the binary search tree \"left subtree &lt; root node &lt; right subtree,\" the insertion operation proceeds as shown in the Figure 7-18 .</p> <ol> <li>Finding the insertion position: Similar to the search operation, start from the root node and loop downwards according to the size relationship between the current node value and <code>num</code> until passing through the leaf node (traversing to <code>None</code>) then exit the loop.</li> <li>Insert the node at that position: Initialize the node <code>num</code> and place it where <code>None</code> was.</li> </ol> <p></p> <p> Figure 7-18 \u00a0 Inserting a node into a binary search tree </p> <p>In the code implementation, note the following two points.</p> <ul> <li>The binary search tree does not allow duplicate nodes; otherwise, it will violate its definition. Therefore, if the node to be inserted already exists in the tree, the insertion is not performed, and it directly returns.</li> <li>To perform the insertion operation, we need to use the node <code>pre</code> to save the node from the last loop. This way, when traversing to <code>None</code>, we can get its parent node, thus completing the node insertion operation.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def insert(self, num: int):\n    \"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n    # \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if self._root is None:\n        self._root = TreeNode(num)\n        return\n    # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    cur, pre = self._root, None\n    while cur is not None:\n        # \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if cur.val == num:\n            return\n        pre = cur\n        # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else:\n            cur = cur.left\n    # \u63d2\u5165\u8282\u70b9\n    node = TreeNode(num)\n    if pre.val &lt; num:\n        pre.right = node\n    else:\n        pre.left = node\n</code></pre> binary_search_tree.cpp<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == nullptr) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode *cur = root, *pre = nullptr;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur-&gt;val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur-&gt;left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode *node = new TreeNode(num);\n    if (pre-&gt;val &lt; num)\n        pre-&gt;right = node;\n    else\n        pre-&gt;left = node;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode node = new TreeNode(num);\n    if (pre.val &lt; num)\n        pre.right = node;\n    else\n        pre.left = node;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode? cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode node = new(num);\n    if (pre != null) {\n        if (pre.val &lt; num)\n            pre.right = node;\n        else\n            pre.left = node;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc (bst *binarySearchTree) insert(num int) {\n    cur := bst.root\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if cur == nil {\n        bst.root = NewTreeNode(num)\n        return\n    }\n    // \u5f85\u63d2\u5165\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\n    var pre *TreeNode = nil\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    for cur != nil {\n        if cur.Val == num {\n            return\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            cur = cur.Right\n        } else {\n            cur = cur.Left\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    node := NewTreeNode(num)\n    if pre.Val.(int) &lt; num {\n        pre.Right = node\n    } else {\n        pre.Left = node\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if root == nil {\n        root = TreeNode(x: num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur != nil {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if cur!.val == num {\n            return\n        }\n        pre = cur\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else {\n            cur = cur?.left\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    let node = TreeNode(x: num)\n    if pre!.val &lt; num {\n        pre?.right = node\n    } else {\n        pre?.left = node\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur = this.root,\n        pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val === num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    const node = new TreeNode(num);\n    if (pre.val &lt; num) pre.right = node;\n    else pre.left = node;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(num: number): void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val === num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    const node = new TreeNode(num);\n    if (pre!.val &lt; num) pre!.right = node;\n    else pre!.left = node;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int _num) {\n  // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n  if (_root == null) {\n    _root = TreeNode(_num);\n    return;\n  }\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n  while (cur != null) {\n    // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n    if (cur.val == _num) return;\n    pre = cur;\n    // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n    else\n      cur = cur.left;\n  }\n  // \u63d2\u5165\u8282\u70b9\n  TreeNode? node = TreeNode(_num);\n  if (pre!.val &lt; _num)\n    pre.right = node;\n  else\n    pre.left = node;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\npub fn insert(&amp;mut self, num: i32) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if self.root.is_none() {\n        self.root = Some(TreeNode::new(num));\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n            Ordering::Equal =&gt; return,\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    let pre = pre.unwrap();\n    let node = Some(TreeNode::new(num));\n    if num &gt; pre.borrow().val {\n        pre.borrow_mut().right = node;\n    } else {\n        pre.borrow_mut().left = node;\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(BinarySearchTree *bst, int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (bst-&gt;root == NULL) {\n        bst-&gt;root = newTreeNode(num);\n        return;\n    }\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != NULL) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur-&gt;val == num) {\n            return;\n        }\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            cur = cur-&gt;right;\n        } else {\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            cur = cur-&gt;left;\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode *node = newTreeNode(num);\n    if (pre-&gt;val &lt; num) {\n        pre-&gt;right = node;\n    } else {\n        pre-&gt;left = node;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfun insert(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == null) {\n        root = TreeNode(num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode? = null\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.value == num) return\n        pre = cur\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        cur = if (cur.value &lt; num) cur.right\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur.left\n    }\n    // \u63d2\u5165\u8282\u70b9\n    val node = TreeNode(num)\n    if (pre?.value!! &lt; num) pre.right = node\n    else pre.left = node\n}\n</code></pre> binary_search_tree.rb<pre><code>[class]{BinarySearchTree}-[func]{insert}\n</code></pre> binary_search_tree.zig<pre><code>// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, num: T) !void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (self.root == null) {\n        self.root = try self.mem_allocator.create(inc.TreeNode(T));\n        return;\n    }\n    var cur = self.root;\n    var pre: ?*inc.TreeNode(T) = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.?.val == num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        } else {\n            cur = cur.?.left;\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    var node = try self.mem_allocator.create(inc.TreeNode(T));\n    node.init(num);\n    if (pre.?.val &lt; num) {\n        pre.?.right = node;\n    } else {\n        pre.?.left = node;\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Similar to searching for a node, inserting a node uses \\(O(\\log n)\\) time.</p>"},{"location":"chapter_tree/binary_search_tree/#3-removing-a-node","title":"3. \u00a0 Removing a node","text":"<p>First, find the target node in the binary tree, then remove it. Similar to inserting a node, we need to ensure that after the removal operation is completed, the property of the binary search tree \"left subtree &lt; root node &lt; right subtree\" is still satisfied. Therefore, based on the number of child nodes of the target node, we divide it into 0, 1, and 2 cases, performing the corresponding node removal operations.</p> <p>As shown in the Figure 7-19 , when the degree of the node to be removed is \\(0\\), it means the node is a leaf node, and it can be directly removed.</p> <p></p> <p> Figure 7-19 \u00a0 Removing a node in a binary search tree (degree 0) </p> <p>As shown in the Figure 7-20 , when the degree of the node to be removed is \\(1\\), replacing the node to be removed with its child node is sufficient.</p> <p></p> <p> Figure 7-20 \u00a0 Removing a node in a binary search tree (degree 1) </p> <p>When the degree of the node to be removed is \\(2\\), we cannot remove it directly, but need to use a node to replace it. To maintain the property of the binary search tree \"left subtree &lt; root node &lt; right subtree,\" this node can be either the smallest node of the right subtree or the largest node of the left subtree.</p> <p>Assuming we choose the smallest node of the right subtree (the next node in in-order traversal), then the removal operation proceeds as shown in the Figure 7-21 .</p> <ol> <li>Find the next node in the \"in-order traversal sequence\" of the node to be removed, denoted as <code>tmp</code>.</li> <li>Replace the value of the node to be removed with <code>tmp</code>'s value, and recursively remove the node <code>tmp</code> in the tree.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 7-21 \u00a0 Removing a node in a binary search tree (degree 2) </p> <p>The operation of removing a node also uses \\(O(\\log n)\\) time, where finding the node to be removed requires \\(O(\\log n)\\) time, and obtaining the in-order traversal successor node requires \\(O(\\log n)\\) time. Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def remove(self, num: int):\n    \"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n    # \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if self._root is None:\n        return\n    # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    cur, pre = self._root, None\n    while cur is not None:\n        # \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if cur.val == num:\n            break\n        pre = cur\n        # \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else:\n            cur = cur.left\n    # \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur is None:\n        return\n\n    # \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if cur.left is None or cur.right is None:\n        # \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        child = cur.left or cur.right\n        # \u5220\u9664\u8282\u70b9 cur\n        if cur != self._root:\n            if pre.left == cur:\n                pre.left = child\n            else:\n                pre.right = child\n        else:\n            # \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            self._root = child\n    # \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else:\n        # \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        tmp: TreeNode = cur.right\n        while tmp.left is not None:\n            tmp = tmp.left\n        # \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        self.remove(tmp.val)\n        # \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val\n</code></pre> binary_search_tree.cpp<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == nullptr)\n        return;\n    TreeNode *cur = root, *pre = nullptr;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur-&gt;left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == nullptr)\n        return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur-&gt;left == nullptr || cur-&gt;right == nullptr) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n        TreeNode *child = cur-&gt;left != nullptr ? cur-&gt;left : cur-&gt;right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre-&gt;left == cur)\n                pre-&gt;left = child;\n            else\n                pre-&gt;right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child;\n        }\n        // \u91ca\u653e\u5185\u5b58\n        delete cur;\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != nullptr) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(tmp-&gt;val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return;\n    TreeNode cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        TreeNode child = cur.left != null ? cur.left : cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(tmp.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return;\n    TreeNode? cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        TreeNode? child = cur.left ?? cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre!.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode? tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        Remove(tmp.val!.Value);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc (bst *binarySearchTree) remove(num int) {\n    cur := bst.root\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5f85\u5220\u9664\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\n    var pre *TreeNode = nil\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    for cur != nil {\n        if cur.Val == num {\n            break\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728\u53f3\u5b50\u6811\u4e2d\n            cur = cur.Right\n        } else {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728\u5de6\u5b50\u6811\u4e2d\n            cur = cur.Left\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5b50\u8282\u70b9\u6570\u4e3a 0 \u6216 1\n    if cur.Left == nil || cur.Right == nil {\n        var child *TreeNode = nil\n        // \u53d6\u51fa\u5f85\u5220\u9664\u8282\u70b9\u7684\u5b50\u8282\u70b9\n        if cur.Left != nil {\n            child = cur.Left\n        } else {\n            child = cur.Right\n        }\n        // \u5220\u9664\u8282\u70b9 cur\n        if cur != bst.root {\n            if pre.Left == cur {\n                pre.Left = child\n            } else {\n                pre.Right = child\n            }\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            bst.root = child\n        }\n        // \u5b50\u8282\u70b9\u6570\u4e3a 2\n    } else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u5f85\u5220\u9664\u8282\u70b9 cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        tmp := cur.Right\n        for tmp.Left != nil {\n            tmp = tmp.Left\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        bst.remove(tmp.Val.(int))\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.Val = tmp.Val\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc remove(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if root == nil {\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur != nil {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if cur!.val == num {\n            break\n        }\n        pre = cur\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else {\n            cur = cur?.left\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if cur?.left == nil || cur?.right == nil {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        let child = cur?.left ?? cur?.right\n        // \u5220\u9664\u8282\u70b9 cur\n        if cur !== root {\n            if pre?.left === cur {\n                pre?.left = child\n            } else {\n                pre?.right = child\n            }\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        var tmp = cur?.right\n        while tmp?.left != nil {\n            tmp = tmp?.left\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(num: tmp!.val)\n        // \u7528 tmp \u8986\u76d6 cur\n        cur?.val = tmp!.val\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (this.root === null) return;\n    let cur = this.root,\n        pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val === num) break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur === null) return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        const child = cur.left !== null ? cur.left : cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur !== this.root) {\n            if (pre.left === cur) pre.left = child;\n            else pre.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            this.root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        let tmp = cur.right;\n        while (tmp.left !== null) {\n            tmp = tmp.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        this.remove(tmp.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(num: number): void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (this.root === null) return;\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val === num) break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur === null) return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        const child: TreeNode | null =\n            cur.left !== null ? cur.left : cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur !== this.root) {\n            if (pre!.left === cur) pre!.left = child;\n            else pre!.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            this.root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        let tmp: TreeNode | null = cur.right;\n        while (tmp!.left !== null) {\n            tmp = tmp!.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        this.remove(tmp!.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp!.val;\n    }\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int _num) {\n  // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n  if (_root == null) return;\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n  while (cur != null) {\n    // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n    if (cur.val == _num) break;\n    pre = cur;\n    // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n    else\n      cur = cur.left;\n  }\n  // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n  if (cur == null) return;\n  // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n  if (cur.left == null || cur.right == null) {\n    // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n    TreeNode? child = cur.left ?? cur.right;\n    // \u5220\u9664\u8282\u70b9 cur\n    if (cur != _root) {\n      if (pre!.left == cur)\n        pre.left = child;\n      else\n        pre.right = child;\n    } else {\n      // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n      _root = child;\n    }\n  } else {\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n    TreeNode? tmp = cur.right;\n    while (tmp!.left != null) {\n      tmp = tmp.left;\n    }\n    // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n    remove(tmp.val);\n    // \u7528 tmp \u8986\u76d6 cur\n    cur.val = tmp.val;\n  }\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u5220\u9664\u8282\u70b9 */\npub fn remove(&amp;mut self, num: i32) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if self.root.is_none() {\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            Ordering::Equal =&gt; break,\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur.is_none() {\n        return;\n    }\n    let cur = cur.unwrap();\n    let (left_child, right_child) = (cur.borrow().left.clone(), cur.borrow().right.clone());\n    match (left_child.clone(), right_child.clone()) {\n        // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n        (None, None) | (Some(_), None) | (None, Some(_)) =&gt; {\n            // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n            let child = left_child.or(right_child);\n            let pre = pre.unwrap();\n            // \u5220\u9664\u8282\u70b9 cur\n            if !Rc::ptr_eq(&amp;cur, self.root.as_ref().unwrap()) {\n                let left = pre.borrow().left.clone();\n                if left.is_some() &amp;&amp; Rc::ptr_eq(&amp;left.as_ref().unwrap(), &amp;cur) {\n                    pre.borrow_mut().left = child;\n                } else {\n                    pre.borrow_mut().right = child;\n                }\n            } else {\n                // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n                self.root = child;\n            }\n        }\n        // \u5b50\u8282\u70b9\u6570\u91cf = 2\n        (Some(_), Some(_)) =&gt; {\n            // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n            let mut tmp = cur.borrow().right.clone();\n            while let Some(node) = tmp.clone() {\n                if node.borrow().left.is_some() {\n                    tmp = node.borrow().left.clone();\n                } else {\n                    break;\n                }\n            }\n            let tmpval = tmp.unwrap().borrow().val;\n            // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n            self.remove(tmpval);\n            // \u7528 tmp \u8986\u76d6 cur\n            cur.borrow_mut().val = tmpval;\n        }\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(BinarySearchTree *bst, int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (bst-&gt;root == NULL)\n        return;\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != NULL) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u53f3\u5b50\u6811\u4e2d\n            cur = cur-&gt;right;\n        } else {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u5de6\u5b50\u6811\u4e2d\n            cur = cur-&gt;left;\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == NULL)\n        return;\n    // \u5224\u65ad\u5f85\u5220\u9664\u8282\u70b9\u662f\u5426\u5b58\u5728\u5b50\u8282\u70b9\n    if (cur-&gt;left == NULL || cur-&gt;right == NULL) {\n        /* \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1 */\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n        TreeNode *child = cur-&gt;left != NULL ? cur-&gt;left : cur-&gt;right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (pre-&gt;left == cur) {\n            pre-&gt;left = child;\n        } else {\n            pre-&gt;right = child;\n        }\n        // \u91ca\u653e\u5185\u5b58\n        free(cur);\n    } else {\n        /* \u5b50\u8282\u70b9\u6570\u91cf = 2 */\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != NULL) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        removeItem(bst, tmp-&gt;val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfun remove(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null) return\n    var cur = root\n    var pre: TreeNode? = null\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.value == num) break\n        pre = cur\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        cur = if (cur.value &lt; num) cur.right\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur.left\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null) return\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        val child = if (cur.left != null) cur.left else cur.right\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre!!.left == cur) pre.left = child\n            else pre.right = child\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child\n        }\n        // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    } else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        var tmp = cur.right\n        while (tmp!!.left != null) {\n            tmp = tmp.left\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(tmp.value)\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.value = tmp.value\n    }\n}\n</code></pre> binary_search_tree.rb<pre><code>[class]{BinarySearchTree}-[func]{remove}\n</code></pre> binary_search_tree.zig<pre><code>// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, num: T) void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (self.root == null) return;\n    var cur = self.root;\n    var pre: ?*inc.TreeNode(T) = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.?.val == num) break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        } else {\n            cur = cur.?.left;\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null) return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.?.left == null or cur.?.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        var child = if (cur.?.left != null) cur.?.left else cur.?.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (pre.?.left == cur) {\n            pre.?.left = child;\n        } else {\n            pre.?.right = child;\n        }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    } else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        var tmp = cur.?.right;\n        while (tmp.?.left != null) {\n            tmp = tmp.?.left;\n        }\n        var tmp_val = tmp.?.val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        self.remove(tmp.?.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.?.val = tmp_val;\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_tree/binary_search_tree/#4-in-order-traversal-is-ordered","title":"4. \u00a0 In-order traversal is ordered","text":"<p>As shown in the Figure 7-22 , the in-order traversal of a binary tree follows the \"left \\(\\rightarrow\\) root \\(\\rightarrow\\) right\" traversal order, and a binary search tree satisfies the size relationship \"left child node &lt; root node &lt; right child node\".</p> <p>This means that in-order traversal in a binary search tree always traverses the next smallest node first, thus deriving an important property: The in-order traversal sequence of a binary search tree is ascending.</p> <p>Using the ascending property of in-order traversal, obtaining ordered data in a binary search tree requires only \\(O(n)\\) time, without the need for additional sorting operations, which is very efficient.</p> <p></p> <p> Figure 7-22 \u00a0 In-order traversal sequence of a binary search tree </p>"},{"location":"chapter_tree/binary_search_tree/#742-efficiency-of-binary-search-trees","title":"7.4.2 \u00a0 Efficiency of binary search trees","text":"<p>Given a set of data, we consider using an array or a binary search tree for storage. Observing the Table 7-2 , the operations on a binary search tree all have logarithmic time complexity, which is stable and efficient. Only in scenarios of high-frequency addition and low-frequency search and removal, arrays are more efficient than binary search trees.</p> <p> Table 7-2 \u00a0 Efficiency comparison between arrays and search trees </p> Unsorted array Binary search tree Search element \\(O(n)\\) \\(O(\\log n)\\) Insert element \\(O(1)\\) \\(O(\\log n)\\) Remove element \\(O(n)\\) \\(O(\\log n)\\) <p>In ideal conditions, the binary search tree is \"balanced,\" thus any node can be found within \\(\\log n\\) loops.</p> <p>However, continuously inserting and removing nodes in a binary search tree may lead to the binary tree degenerating into a chain list as shown in the Figure 7-23 , at which point the time complexity of various operations also degrades to \\(O(n)\\).</p> <p></p> <p> Figure 7-23 \u00a0 Degradation of a binary search tree </p>"},{"location":"chapter_tree/binary_search_tree/#743-common-applications-of-binary-search-trees","title":"7.4.3 \u00a0 Common applications of binary search trees","text":"<ul> <li>Used as multi-level indexes in systems to implement efficient search, insertion, and removal operations.</li> <li>Serves as the underlying data structure for certain search algorithms.</li> <li>Used to store data streams to maintain their ordered state.</li> </ul>"},{"location":"chapter_tree/binary_tree/","title":"7.1 \u00a0 Binary tree","text":"<p>A \"binary tree\" is a non-linear data structure that represents the ancestral and descendent relationships, embodying the \"divide and conquer\" logic. Similar to a linked list, the basic unit of a binary tree is a node, each containing a value, a reference to the left child node, and a reference to the right child node.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class TreeNode:\n    \"\"\"Binary tree node\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # Node value\n        self.left: TreeNode | None = None  # Reference to left child node\n        self.right: TreeNode | None = None # Reference to right child node\n</code></pre> <pre><code>/* Binary tree node */\nstruct TreeNode {\n    int val;          // Node value\n    TreeNode *left;   // Pointer to left child node\n    TreeNode *right;  // Pointer to right child node\n    TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode {\n    int val;         // Node value\n    TreeNode left;   // Reference to left child node\n    TreeNode right;  // Reference to right child node\n    TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode(int? x) {\n    public int? val = x;    // Node value\n    public TreeNode? left;  // Reference to left child node\n    public TreeNode? right; // Reference to right child node\n}\n</code></pre> <pre><code>/* Binary tree node */\ntype TreeNode struct {\n    Val   int\n    Left  *TreeNode\n    Right *TreeNode\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc NewTreeNode(v int) *TreeNode {\n    return &amp;TreeNode{\n        Left:  nil, // Pointer to left child node\n        Right: nil, // Pointer to right child node\n        Val:   v,   // Node value\n    }\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode {\n    var val: Int // Node value\n    var left: TreeNode? // Reference to left child node\n    var right: TreeNode? // Reference to right child node\n\n    init(x: Int) {\n        val = x\n    }\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode {\n    val; // Node value\n    left; // Pointer to left child node\n    right; // Pointer to right child node\n    constructor(val, left, right) {\n        this.val = val === undefined ? 0 : val;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode {\n    val: number;\n    left: TreeNode | null;\n    right: TreeNode | null;\n\n    constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val; // Node value\n        this.left = left === undefined ? null : left; // Reference to left child node\n        this.right = right === undefined ? null : right; // Reference to right child node\n    }\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode {\n  int val;         // Node value\n  TreeNode? left;  // Reference to left child node\n  TreeNode? right; // Reference to right child node\n  TreeNode(this.val, [this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* Binary tree node */\nstruct TreeNode {\n    val: i32,                               // Node value\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // Reference to left child node\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // Reference to right child node\n}\n\nimpl TreeNode {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* Binary tree node */\ntypedef struct TreeNode {\n    int val;                // Node value\n    int height;             // \u8282\u70b9\u9ad8\u5ea6\n    struct TreeNode *left;  // Pointer to left child node\n    struct TreeNode *right; // Pointer to right child node\n} TreeNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode(val _val: Int) {  // Node value\n    val left: TreeNode? = null   // Reference to left child node\n    val right: TreeNode? = null  // Reference to right child node\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>Each node has two references (pointers), pointing to the \"left-child node\" and \"right-child node,\" respectively. This node is called the \"parent node\" of these two child nodes. When given a node of a binary tree, we call the tree formed by this node's left child and all nodes under it the \"left subtree\" of this node. Similarly, the \"right subtree\" can be defined.</p> <p>In a binary tree, except for leaf nodes, all other nodes contain child nodes and non-empty subtrees. As shown in the Figure 7-1 , if \"Node 2\" is considered as the parent node, then its left and right child nodes are \"Node 4\" and \"Node 5,\" respectively. The left subtree is \"the tree formed by Node 4 and all nodes under it,\" and the right subtree is \"the tree formed by Node 5 and all nodes under it.\"</p> <p></p> <p> Figure 7-1 \u00a0 Parent Node, child Node, subtree </p>"},{"location":"chapter_tree/binary_tree/#711-common-terminology-of-binary-trees","title":"7.1.1 \u00a0 Common terminology of binary trees","text":"<p>The commonly used terminology of binary trees is shown in the following figure.</p> <ul> <li>\"Root node\": The node at the top level of the binary tree, which has no parent node.</li> <li>\"Leaf node\": A node with no children, both of its pointers point to <code>None</code>.</li> <li>\"Edge\": The line segment connecting two nodes, i.e., node reference (pointer).</li> <li>The \"level\" of a node: Incrementing from top to bottom, with the root node's level being 1.</li> <li>The \"degree\" of a node: The number of a node's children. In a binary tree, the degree can be 0, 1, or 2.</li> <li>The \"height\" of a binary tree: The number of edges passed from the root node to the farthest leaf node.</li> <li>The \"depth\" of a node: The number of edges passed from the root node to the node.</li> <li>The \"height\" of a node: The number of edges from the farthest leaf node to the node.</li> </ul> <p></p> <p> Figure 7-2 \u00a0 Common Terminology of Binary Trees </p> <p>Tip</p> <p>Please note that we usually define \"height\" and \"depth\" as \"the number of edges passed,\" but some problems or textbooks may define them as \"the number of nodes passed.\" In this case, both height and depth need to be incremented by 1.</p>"},{"location":"chapter_tree/binary_tree/#712-basic-operations-of-binary-trees","title":"7.1.2 \u00a0 Basic operations of binary trees","text":""},{"location":"chapter_tree/binary_tree/#1-initializing-a-binary-tree","title":"1. \u00a0 Initializing a binary tree","text":"<p>Similar to a linked list, initialize nodes first, then construct references (pointers).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree.py<pre><code># Initializing a binary tree\n# Initializing nodes\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# Linking references (pointers) between nodes\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.cpp<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// Linking references (pointers) between nodes\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.java<pre><code>// Initializing nodes\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// Linking references (pointers) between nodes\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.cs<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nTreeNode n1 = new(1);\nTreeNode n2 = new(2);\nTreeNode n3 = new(3);\nTreeNode n4 = new(4);\nTreeNode n5 = new(5);\n// Linking references (pointers) between nodes\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.go<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// Linking references (pointers) between nodes\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n</code></pre> binary_tree.swift<pre><code>// Initializing nodes\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// Linking references (pointers) between nodes\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.js<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// Linking references (pointers) between nodes\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.ts<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// Linking references (pointers) between nodes\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.dart<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// Linking references (pointers) between nodes\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.rs<pre><code>// Initializing nodes\nlet n1 = TreeNode::new(1);\nlet n2 = TreeNode::new(2);\nlet n3 = TreeNode::new(3);\nlet n4 = TreeNode::new(4);\nlet n5 = TreeNode::new(5);\n// Linking references (pointers) between nodes\nn1.borrow_mut().left = Some(n2.clone());\nn1.borrow_mut().right = Some(n3);\nn2.borrow_mut().left = Some(n4);\nn2.borrow_mut().right = Some(n5);\n</code></pre> binary_tree.c<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// Linking references (pointers) between nodes\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.kt<pre><code>// Initializing nodes\nval n1 = TreeNode(1)\nval n2 = TreeNode(2)\nval n3 = TreeNode(3)\nval n4 = TreeNode(4)\nval n5 = TreeNode(5)\n// Linking references (pointers) between nodes\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.rb<pre><code>\n</code></pre> binary_tree.zig<pre><code>\n</code></pre> Code visualization <p>https://pythontutor.com/render.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%8F%89%E6%A0%91%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E8%8A%82%E7%82%B9%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E9%92%88%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5&amp;cumulative=false&amp;curInstr=3&amp;heapPrimitives=nevernest&amp;mode=display&amp;origin=opt-frontend.js&amp;py=311&amp;rawInputLstJSON=%5B%5D&amp;textReferences=false</p>"},{"location":"chapter_tree/binary_tree/#2-inserting-and-removing-nodes","title":"2. \u00a0 Inserting and removing nodes","text":"<p>Similar to a linked list, inserting and removing nodes in a binary tree can be achieved by modifying pointers. The Figure 7-3  provides an example.</p> <p></p> <p> Figure 7-3 \u00a0 Inserting and removing nodes in a binary tree </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree.py<pre><code># Inserting and removing nodes\np = TreeNode(0)\n# Inserting node P between n1 -&gt; n2\nn1.left = p\np.left = n2\n# Removing node P\nn1.left = n2\n</code></pre> binary_tree.cpp<pre><code>/* Inserting and removing nodes */\nTreeNode* P = new TreeNode(0);\n// Inserting node P between n1 and n2\nn1-&gt;left = P;\nP-&gt;left = n2;\n// Removing node P\nn1-&gt;left = n2;\n</code></pre> binary_tree.java<pre><code>TreeNode P = new TreeNode(0);\n// Inserting node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Removing node P\nn1.left = n2;\n</code></pre> binary_tree.cs<pre><code>/* Inserting and removing nodes */\nTreeNode P = new(0);\n// Inserting node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Removing node P\nn1.left = n2;\n</code></pre> binary_tree.go<pre><code>/* Inserting and removing nodes */\n// Inserting node P between n1 and n2\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// Removing node P\nn1.Left = n2\n</code></pre> binary_tree.swift<pre><code>let P = TreeNode(x: 0)\n// Inserting node P between n1 and n2\nn1.left = P\nP.left = n2\n// Removing node P\nn1.left = n2\n</code></pre> binary_tree.js<pre><code>/* Inserting and removing nodes */\nlet P = new TreeNode(0);\n// Inserting node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Removing node P\nn1.left = n2;\n</code></pre> binary_tree.ts<pre><code>/* Inserting and removing nodes */\nconst P = new TreeNode(0);\n// Inserting node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Removing node P\nn1.left = n2;\n</code></pre> binary_tree.dart<pre><code>/* Inserting and removing nodes */\nTreeNode P = new TreeNode(0);\n// Inserting node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Removing node P\nn1.left = n2;\n</code></pre> binary_tree.rs<pre><code>let p = TreeNode::new(0);\n// Inserting node P between n1 and n2\nn1.borrow_mut().left = Some(p.clone());\np.borrow_mut().left = Some(n2.clone());\n// Removing node P\nn1.borrow_mut().left = Some(n2);\n</code></pre> binary_tree.c<pre><code>/* Inserting and removing nodes */\nTreeNode *P = newTreeNode(0);\n// Inserting node P between n1 and n2\nn1-&gt;left = P;\nP-&gt;left = n2;\n// Removing node P\nn1-&gt;left = n2;\n</code></pre> binary_tree.kt<pre><code>val P = TreeNode(0)\n// Inserting node P between n1 and n2\nn1.left = P\nP.left = n2\n// Removing node P\nn1.left = n2\n</code></pre> binary_tree.rb<pre><code>\n</code></pre> binary_tree.zig<pre><code>\n</code></pre> Code visualization <p>https://pythontutor.com/render.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%8F%89%E6%A0%91%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E8%8A%82%E7%82%B9%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E9%92%88%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E4%B8%8E%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%0A%20%20%20%20p%20%3D%20TreeNode%280%29%0A%20%20%20%20%23%20%E5%9C%A8%20n1%20-%3E%20n2%20%E4%B8%AD%E9%97%B4%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20p%0A%20%20%20%20p.left%20%3D%20n2%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20n2&amp;cumulative=false&amp;curInstr=37&amp;heapPrimitives=nevernest&amp;mode=display&amp;origin=opt-frontend.js&amp;py=311&amp;rawInputLstJSON=%5B%5D&amp;textReferences=false</p> <p>Note</p> <p>It's important to note that inserting nodes may change the original logical structure of the binary tree, while removing nodes usually means removing the node and all its subtrees. Therefore, in a binary tree, insertion and removal are usually performed through a set of operations to achieve meaningful actions.</p>"},{"location":"chapter_tree/binary_tree/#713-common-types-of-binary-trees","title":"7.1.3 \u00a0 Common types of binary trees","text":""},{"location":"chapter_tree/binary_tree/#1-perfect-binary-tree","title":"1. \u00a0 Perfect binary tree","text":"<p>As shown in the Figure 7-4 , in a \"perfect binary tree,\" all levels of nodes are fully filled. In a perfect binary tree, the degree of leaf nodes is \\(0\\), and the degree of all other nodes is \\(2\\); if the tree's height is \\(h\\), then the total number of nodes is \\(2^{h+1} - 1\\), showing a standard exponential relationship, reflecting the common phenomenon of cell division in nature.</p> <p>Tip</p> <p>Please note that in the Chinese community, a perfect binary tree is often referred to as a \"full binary tree.\"</p> <p></p> <p> Figure 7-4 \u00a0 Perfect binary tree </p>"},{"location":"chapter_tree/binary_tree/#2-complete-binary-tree","title":"2. \u00a0 Complete binary tree","text":"<p>As shown in the Figure 7-5 , a \"complete binary tree\" has only the bottom level nodes not fully filled, and the bottom level nodes are filled as far left as possible.</p> <p></p> <p> Figure 7-5 \u00a0 Complete binary tree </p>"},{"location":"chapter_tree/binary_tree/#3-full-binary-tree","title":"3. \u00a0 Full binary tree","text":"<p>As shown in the Figure 7-6 , a \"full binary tree\" has all nodes except leaf nodes having two children.</p> <p></p> <p> Figure 7-6 \u00a0 Full binary tree </p>"},{"location":"chapter_tree/binary_tree/#4-balanced-binary-tree","title":"4. \u00a0 Balanced binary tree","text":"<p>As shown in the Figure 7-7 , in a \"balanced binary tree,\" the absolute difference in height between the left and right subtrees of any node does not exceed 1.</p> <p></p> <p> Figure 7-7 \u00a0 Balanced binary tree </p>"},{"location":"chapter_tree/binary_tree/#714-degeneration-of-binary-trees","title":"7.1.4 \u00a0 Degeneration of binary trees","text":"<p>The Figure 7-8  shows the ideal and degenerate structures of binary trees. When every level of a binary tree is filled, it reaches the \"perfect binary tree\"; when all nodes are biased towards one side, the binary tree degenerates into a \"linked list\".</p> <ul> <li>The perfect binary tree is the ideal situation, fully leveraging the \"divide and conquer\" advantage of binary trees.</li> <li>A linked list is another extreme, where operations become linear, degrading the time complexity to \\(O(n)\\).</li> </ul> <p></p> <p> Figure 7-8 \u00a0 The Best and Worst Structures of Binary Trees </p> <p>As shown in the Table 7-1 , in the best and worst structures, the number of leaf nodes, total number of nodes, and height of the binary tree reach their maximum or minimum values.</p> <p> Table 7-1 \u00a0 The Best and Worst Structures of Binary Trees </p> Perfect binary tree Linked list Number of nodes at level \\(i\\) \\(2^{i-1}\\) \\(1\\) Number of leaf nodes in a tree with height \\(h\\) \\(2^h\\) \\(1\\) Total number of nodes in a tree with height \\(h\\) \\(2^{h+1} - 1\\) \\(h + 1\\) Height of a tree with \\(n\\) total nodes \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)"},{"location":"chapter_tree/binary_tree_traversal/","title":"7.2 \u00a0 Binary tree traversal","text":"<p>From the perspective of physical structure, a tree is a data structure based on linked lists, hence its traversal method involves accessing nodes one by one through pointers. However, a tree is a non-linear data structure, which makes traversing a tree more complex than traversing a linked list, requiring the assistance of search algorithms to achieve.</p> <p>Common traversal methods for binary trees include level-order traversal, preorder traversal, inorder traversal, and postorder traversal, among others.</p>"},{"location":"chapter_tree/binary_tree_traversal/#721-level-order-traversal","title":"7.2.1 \u00a0 Level-order traversal","text":"<p>As shown in the Figure 7-9 , \"level-order traversal\" traverses the binary tree from top to bottom, layer by layer, and accesses nodes in each layer in a left-to-right order.</p> <p>Level-order traversal essentially belongs to \"breadth-first traversal\", also known as \"breadth-first search (BFS)\", which embodies a \"circumferentially outward expanding\" layer-by-layer traversal method.</p> <p></p> <p> Figure 7-9 \u00a0 Level-order traversal of a binary tree </p>"},{"location":"chapter_tree/binary_tree_traversal/#1-code-implementation","title":"1. \u00a0 Code implementation","text":"<p>Breadth-first traversal is usually implemented with the help of a \"queue\". The queue follows the \"first in, first out\" rule, while breadth-first traversal follows the \"layer-by-layer progression\" rule, the underlying ideas of the two are consistent. The implementation code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree_bfs.py<pre><code>def level_order(root: TreeNode | None) -&gt; list[int]:\n    \"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n    # \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    queue: deque[TreeNode] = deque()\n    queue.append(root)\n    # \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    res = []\n    while queue:\n        node: TreeNode = queue.popleft()  # \u961f\u5217\u51fa\u961f\n        res.append(node.val)  # \u4fdd\u5b58\u8282\u70b9\u503c\n        if node.left is not None:\n            queue.append(node.left)  # \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if node.right is not None:\n            queue.append(node.right)  # \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    return res\n</code></pre> binary_tree_bfs.cpp<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nvector&lt;int&gt; levelOrder(TreeNode *root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    queue&lt;TreeNode *&gt; queue;\n    queue.push(root);\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    vector&lt;int&gt; vec;\n    while (!queue.empty()) {\n        TreeNode *node = queue.front();\n        queue.pop();              // \u961f\u5217\u51fa\u961f\n        vec.push_back(node-&gt;val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node-&gt;left != nullptr)\n            queue.push(node-&gt;left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node-&gt;right != nullptr)\n            queue.push(node-&gt;right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return vec;\n}\n</code></pre> binary_tree_bfs.java<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;Integer&gt; levelOrder(TreeNode root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    Queue&lt;TreeNode&gt; queue = new LinkedList&lt;&gt;();\n    queue.add(root);\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    List&lt;Integer&gt; list = new ArrayList&lt;&gt;();\n    while (!queue.isEmpty()) {\n        TreeNode node = queue.poll(); // \u961f\u5217\u51fa\u961f\n        list.add(node.val);           // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null)\n            queue.offer(node.left);   // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node.right != null)\n            queue.offer(node.right);  // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.cs<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;int&gt; LevelOrder(TreeNode root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    Queue&lt;TreeNode&gt; queue = new();\n    queue.Enqueue(root);\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    List&lt;int&gt; list = [];\n    while (queue.Count != 0) {\n        TreeNode node = queue.Dequeue(); // \u961f\u5217\u51fa\u961f\n        list.Add(node.val!.Value);       // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null)\n            queue.Enqueue(node.left);    // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node.right != null)\n            queue.Enqueue(node.right);   // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.go<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root *TreeNode) []any {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    queue := list.New()\n    queue.PushBack(root)\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5207\u7247\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    nums := make([]any, 0)\n    for queue.Len() &gt; 0 {\n        // \u961f\u5217\u51fa\u961f\n        node := queue.Remove(queue.Front()).(*TreeNode)\n        // \u4fdd\u5b58\u8282\u70b9\u503c\n        nums = append(nums, node.Val)\n        if node.Left != nil {\n            // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n            queue.PushBack(node.Left)\n        }\n        if node.Right != nil {\n            // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n            queue.PushBack(node.Right)\n        }\n    }\n    return nums\n}\n</code></pre> binary_tree_bfs.swift<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root: TreeNode) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    var queue: [TreeNode] = [root]\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    var list: [Int] = []\n    while !queue.isEmpty {\n        let node = queue.removeFirst() // \u961f\u5217\u51fa\u961f\n        list.append(node.val) // \u4fdd\u5b58\u8282\u70b9\u503c\n        if let left = node.left {\n            queue.append(left) // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if let right = node.right {\n            queue.append(right) // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        }\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.js<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    const queue = [root];\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    const list = [];\n    while (queue.length) {\n        let node = queue.shift(); // \u961f\u5217\u51fa\u961f\n        list.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left) queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node.right) queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.ts<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root: TreeNode | null): number[] {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    const queue = [root];\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    const list: number[] = [];\n    while (queue.length) {\n        let node = queue.shift() as TreeNode; // \u961f\u5217\u51fa\u961f\n        list.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left) {\n            queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if (node.right) {\n            queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        }\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.dart<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;int&gt; levelOrder(TreeNode? root) {\n  // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n  Queue&lt;TreeNode?&gt; queue = Queue();\n  queue.add(root);\n  // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n  List&lt;int&gt; res = [];\n  while (queue.isNotEmpty) {\n    TreeNode? node = queue.removeFirst(); // \u961f\u5217\u51fa\u961f\n    res.add(node!.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n    if (node.left != null) queue.add(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n    if (node.right != null) queue.add(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n  }\n  return res;\n}\n</code></pre> binary_tree_bfs.rs<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfn level_order(root: &amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    let mut que = VecDeque::new();\n    que.push_back(Rc::clone(&amp;root));\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    let mut vec = Vec::new();\n\n    while let Some(node) = que.pop_front() {\n        // \u961f\u5217\u51fa\u961f\n        vec.push(node.borrow().val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if let Some(left) = node.borrow().left.as_ref() {\n            que.push_back(Rc::clone(left)); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if let Some(right) = node.borrow().right.as_ref() {\n            que.push_back(Rc::clone(right)); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        };\n    }\n    vec\n}\n</code></pre> binary_tree_bfs.c<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(TreeNode *root, int *size) {\n    /* \u8f85\u52a9\u961f\u5217 */\n    int front, rear;\n    int index, *arr;\n    TreeNode *node;\n    TreeNode **queue;\n\n    /* \u8f85\u52a9\u961f\u5217 */\n    queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);\n    // \u961f\u5217\u6307\u9488\n    front = 0, rear = 0;\n    // \u52a0\u5165\u6839\u8282\u70b9\n    queue[rear++] = root;\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    /* \u8f85\u52a9\u6570\u7ec4 */\n    arr = (int *)malloc(sizeof(int) * MAX_SIZE);\n    // \u6570\u7ec4\u6307\u9488\n    index = 0;\n    while (front &lt; rear) {\n        // \u961f\u5217\u51fa\u961f\n        node = queue[front++];\n        // \u4fdd\u5b58\u8282\u70b9\u503c\n        arr[index++] = node-&gt;val;\n        if (node-&gt;left != NULL) {\n            // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n            queue[rear++] = node-&gt;left;\n        }\n        if (node-&gt;right != NULL) {\n            // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n            queue[rear++] = node-&gt;right;\n        }\n    }\n    // \u66f4\u65b0\u6570\u7ec4\u957f\u5ea6\u7684\u503c\n    *size = index;\n    arr = realloc(arr, sizeof(int) * (*size));\n\n    // \u91ca\u653e\u8f85\u52a9\u6570\u7ec4\u7a7a\u95f4\n    free(queue);\n    return arr;\n}\n</code></pre> binary_tree_bfs.kt<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfun levelOrder(root: TreeNode?): MutableList&lt;Int&gt; {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    val queue = LinkedList&lt;TreeNode?&gt;()\n    queue.add(root)\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    val list = ArrayList&lt;Int&gt;()\n    while (!queue.isEmpty()) {\n        val node = queue.poll() // \u961f\u5217\u51fa\u961f\n        list.add(node?.value!!) // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null) queue.offer(node.left) // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n\n        if (node.right != null) queue.offer(node.right) // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.rb<pre><code>[class]{}-[func]{level_order}\n</code></pre> binary_tree_bfs.zig<pre><code>// \u5c42\u5e8f\u904d\u5386\nfn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    const L = std.TailQueue(*inc.TreeNode(T));\n    var queue = L{};\n    var root_node = try mem_allocator.create(L.Node);\n    root_node.data = root;\n    queue.append(root_node); \n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    var list = std.ArrayList(T).init(std.heap.page_allocator);\n    while (queue.len &gt; 0) {\n        var queue_node = queue.popFirst().?;    // \u961f\u5217\u51fa\u961f\n        var node = queue_node.data;\n        try list.append(node.val);              // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null) {\n            var tmp_node = try mem_allocator.create(L.Node);\n            tmp_node.data = node.left.?;\n            queue.append(tmp_node);             // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if (node.right != null) {\n            var tmp_node = try mem_allocator.create(L.Node);\n            tmp_node.data = node.right.?;\n            queue.append(tmp_node);             // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        }        \n    }\n    return list;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_tree/binary_tree_traversal/#2-complexity-analysis","title":"2. \u00a0 Complexity analysis","text":"<ul> <li>Time complexity is \\(O(n)\\): All nodes are visited once, using \\(O(n)\\) time, where \\(n\\) is the number of nodes.</li> <li>Space complexity is \\(O(n)\\): In the worst case, i.e., a full binary tree, before traversing to the lowest level, the queue can contain at most \\((n + 1) / 2\\) nodes at the same time, occupying \\(O(n)\\) space.</li> </ul>"},{"location":"chapter_tree/binary_tree_traversal/#722-preorder-inorder-and-postorder-traversal","title":"7.2.2 \u00a0 Preorder, inorder, and postorder traversal","text":"<p>Correspondingly, preorder, inorder, and postorder traversal all belong to \"depth-first traversal\", also known as \"depth-first search (DFS)\", which embodies a \"proceed to the end first, then backtrack and continue\" traversal method.</p> <p>The Figure 7-10  shows the working principle of performing a depth-first traversal on a binary tree. Depth-first traversal is like walking around the perimeter of the entire binary tree, encountering three positions at each node, corresponding to preorder traversal, inorder traversal, and postorder traversal.</p> <p></p> <p> Figure 7-10 \u00a0 Preorder, inorder, and postorder traversal of a binary search tree </p>"},{"location":"chapter_tree/binary_tree_traversal/#1-code-implementation_1","title":"1. \u00a0 Code implementation","text":"<p>Depth-first search is usually implemented based on recursion:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree_dfs.py<pre><code>def pre_order(root: TreeNode | None):\n    \"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\n    if root is None:\n        return\n    # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    res.append(root.val)\n    pre_order(root=root.left)\n    pre_order(root=root.right)\n\ndef in_order(root: TreeNode | None):\n    \"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\n    if root is None:\n        return\n    # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    in_order(root=root.left)\n    res.append(root.val)\n    in_order(root=root.right)\n\ndef post_order(root: TreeNode | None):\n    \"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\n    if root is None:\n        return\n    # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    post_order(root=root.left)\n    post_order(root=root.right)\n    res.append(root.val)\n</code></pre> binary_tree_dfs.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    vec.push_back(root-&gt;val);\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root-&gt;left);\n    vec.push_back(root-&gt;val);\n    inOrder(root-&gt;right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root-&gt;left);\n    postOrder(root-&gt;right);\n    vec.push_back(root-&gt;val);\n}\n</code></pre> binary_tree_dfs.java<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.add(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left);\n    list.add(root.val);\n    inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left);\n    postOrder(root.right);\n    list.add(root.val);\n}\n</code></pre> binary_tree_dfs.cs<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.Add(root.val!.Value);\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid InOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    InOrder(root.left);\n    list.Add(root.val!.Value);\n    InOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid PostOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    PostOrder(root.left);\n    PostOrder(root.right);\n    list.Add(root.val!.Value);\n}\n</code></pre> binary_tree_dfs.go<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    nums = append(nums, node.Val)\n    preOrder(node.Left)\n    preOrder(node.Right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(node.Left)\n    nums = append(nums, node.Val)\n    inOrder(node.Right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(node.Left)\n    postOrder(node.Right)\n    nums = append(nums, node.Val)\n}\n</code></pre> binary_tree_dfs.swift<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.append(root.val)\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root: root.left)\n    list.append(root.val)\n    inOrder(root: root.right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root: root.left)\n    postOrder(root: root.right)\n    list.append(root.val)\n}\n</code></pre> binary_tree_dfs.js<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root) {\n    if (root === null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root) {\n    if (root === null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root) {\n    if (root === null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.ts<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.dart<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n  list.add(node.val);\n  preOrder(node.left);\n  preOrder(node.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n  inOrder(node.left);\n  list.add(node.val);\n  inOrder(node.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n  postOrder(node.left);\n  postOrder(node.right);\n  list.add(node.val);\n}\n</code></pre> binary_tree_dfs.rs<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfn pre_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    if let Some(node) = root {\n        // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n        result.push(node.borrow().val);\n        result.append(&amp;mut pre_order(node.borrow().left.as_ref()));\n        result.append(&amp;mut pre_order(node.borrow().right.as_ref()));\n    }\n    result\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfn in_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    if let Some(node) = root {\n        // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n        result.append(&amp;mut in_order(node.borrow().left.as_ref()));\n        result.push(node.borrow().val);\n        result.append(&amp;mut in_order(node.borrow().right.as_ref()));\n    }\n    result\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfn post_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    if let Some(node) = root {\n        // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n        result.append(&amp;mut post_order(node.borrow().left.as_ref()));\n        result.append(&amp;mut post_order(node.borrow().right.as_ref()));\n        result.push(node.borrow().val);\n    }\n    result\n}\n</code></pre> binary_tree_dfs.c<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    arr[(*size)++] = root-&gt;val;\n    preOrder(root-&gt;left, size);\n    preOrder(root-&gt;right, size);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root-&gt;left, size);\n    arr[(*size)++] = root-&gt;val;\n    inOrder(root-&gt;right, size);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root-&gt;left, size);\n    postOrder(root-&gt;right, size);\n    arr[(*size)++] = root-&gt;val;\n}\n</code></pre> binary_tree_dfs.kt<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.add(root.value)\n    preOrder(root.left)\n    preOrder(root.right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfun inOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left)\n    list.add(root.value)\n    inOrder(root.right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfun postOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left)\n    postOrder(root.right)\n    list.add(root.value)\n}\n</code></pre> binary_tree_dfs.rb<pre><code>[class]{}-[func]{pre_order}\n\n[class]{}-[func]{in_order}\n\n[class]{}-[func]{post_order}\n</code></pre> binary_tree_dfs.zig<pre><code>// \u524d\u5e8f\u904d\u5386\nfn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    try list.append(root.?.val);\n    try preOrder(T, root.?.left);\n    try preOrder(T, root.?.right);\n}\n\n// \u4e2d\u5e8f\u904d\u5386\nfn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    try inOrder(T, root.?.left);\n    try list.append(root.?.val);\n    try inOrder(T, root.?.right);\n}\n\n// \u540e\u5e8f\u904d\u5386\nfn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    try postOrder(T, root.?.left);\n    try postOrder(T, root.?.right);\n    try list.append(root.?.val);\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Tip</p> <p>Depth-first search can also be implemented based on iteration, interested readers can study this on their own.</p> <p>The Figure 7-11  shows the recursive process of preorder traversal of a binary tree, which can be divided into two opposite parts: \"recursion\" and \"return\".</p> <ol> <li>\"Recursion\" means starting a new method, the program accesses the next node in this process.</li> <li>\"Return\" means the function returns, indicating the current node has been fully accessed.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 7-11 \u00a0 The recursive process of preorder traversal </p>"},{"location":"chapter_tree/binary_tree_traversal/#2-complexity-analysis_1","title":"2. \u00a0 Complexity analysis","text":"<ul> <li>Time complexity is \\(O(n)\\): All nodes are visited once, using \\(O(n)\\) time.</li> <li>Space complexity is \\(O(n)\\): In the worst case, i.e., the tree degrades into a linked list, the recursion depth reaches \\(n\\), the system occupies \\(O(n)\\) stack frame space.</li> </ul>"},{"location":"chapter_tree/summary/","title":"7.6 \u00a0 Summary","text":""},{"location":"chapter_tree/summary/#1-key-review","title":"1. \u00a0 Key review","text":"<ul> <li>A binary tree is a non-linear data structure that reflects the \"divide and conquer\" logic of splitting one into two. Each binary tree node contains a value and two pointers, which point to its left and right child nodes, respectively.</li> <li>For a node in a binary tree, the tree formed by its left (right) child node and all nodes under it is called the node's left (right) subtree.</li> <li>Related terminology of binary trees includes root node, leaf node, level, degree, edge, height, and depth, among others.</li> <li>The operations of initializing a binary tree, inserting nodes, and removing nodes are similar to those of linked list operations.</li> <li>Common types of binary trees include perfect binary trees, complete binary trees, full binary trees, and balanced binary trees. The perfect binary tree represents the ideal state, while the linked list is the worst state after degradation.</li> <li>A binary tree can be represented using an array by arranging the node values and empty slots in a level-order traversal sequence and implementing pointers based on the index mapping relationship between parent nodes and child nodes.</li> <li>The level-order traversal of a binary tree is a breadth-first search method, which reflects a layer-by-layer traversal manner of \"expanding circle by circle.\" It is usually implemented using a queue.</li> <li>Pre-order, in-order, and post-order traversals are all depth-first search methods, reflecting the traversal manner of \"going to the end first, then backtracking to continue.\" They are usually implemented using recursion.</li> <li>A binary search tree is an efficient data structure for element searching, with the time complexity of search, insert, and remove operations all being \\(O(\\log n)\\). When a binary search tree degrades into a linked list, these time complexities deteriorate to \\(O(n)\\).</li> <li>An AVL tree, also known as a balanced binary search tree, ensures that the tree remains balanced after continuous node insertions and removals through rotation operations.</li> <li>Rotation operations in an AVL tree include right rotation, left rotation, right-then-left rotation, and left-then-right rotation. After inserting or removing nodes, an AVL tree performs rotation operations from bottom to top to rebalance the tree.</li> </ul>"},{"location":"chapter_tree/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: For a binary tree with only one node, are both the height of the tree and the depth of the root node \\(0\\)?</p> <p>Yes, because height and depth are typically defined as \"the number of edges passed.\"</p> <p>Q: The insertion and removal in a binary tree are generally completed by a set of operations. What does \"a set of operations\" refer to here? Can it be understood as the release of resources of the child nodes?</p> <p>Taking the binary search tree as an example, the operation of removing a node needs to be handled in three different scenarios, each requiring multiple steps of node operations.</p> <p>Q: Why are there three sequences: pre-order, in-order, and post-order for DFS traversal of a binary tree, and what are their uses?</p> <p>Similar to sequential and reverse traversal of arrays, pre-order, in-order, and post-order traversals are three methods of traversing a binary tree, allowing us to obtain a traversal result in a specific order. For example, in a binary search tree, since the node sizes satisfy <code>left child node value &lt; root node value &lt; right child node value</code>, we can obtain an ordered node sequence by traversing the tree in the \"left \u2192 root \u2192 right\" priority.</p> <p>Q: In a right rotation operation that deals with the relationship between the imbalance nodes <code>node</code>, <code>child</code>, <code>grand_child</code>, isn't the connection between <code>node</code> and its parent node and the original link of <code>node</code> lost after the right rotation?</p> <p>We need to view this problem from a recursive perspective. The <code>right_rotate(root)</code> operation passes the root node of the subtree and eventually returns the root node of the rotated subtree with <code>return child</code>. The connection between the subtree's root node and its parent node is established after this function returns, which is outside the scope of the right rotation operation's maintenance.</p> <p>Q: In C++, functions are divided into <code>private</code> and <code>public</code> sections. What considerations are there for this? Why are the <code>height()</code> function and the <code>updateHeight()</code> function placed in <code>public</code> and <code>private</code>, respectively?</p> <p>It depends on the scope of the method's use. If a method is only used within the class, then it is designed to be <code>private</code>. For example, it makes no sense for users to call <code>updateHeight()</code> on their own, as it is just a step in the insertion or removal operations. However, <code>height()</code> is for accessing node height, similar to <code>vector.size()</code>, thus it is set to <code>public</code> for use.</p> <p>Q: How do you build a binary search tree from a set of input data? Is the choice of root node very important?</p> <p>Yes, the method for building the tree is provided in the <code>build_tree()</code> method in the binary search tree code. As for the choice of the root node, we usually sort the input data and then select the middle element as the root node, recursively building the left and right subtrees. This approach maximizes the balance of the tree.</p> <p>Q: In Java, do you always have to use the <code>equals()</code> method for string comparison?</p> <p>In Java, for primitive data types, <code>==</code> is used to compare whether the values of two variables are equal. For reference types, the working principles of the two symbols are different.</p> <ul> <li><code>==</code>: Used to compare whether two variables point to the same object, i.e., whether their positions in memory are the same.</li> <li><code>equals()</code>: Used to compare whether the values of two objects are equal.</li> </ul> <p>Therefore, to compare values, we should use <code>equals()</code>. However, strings initialized with <code>String a = \"hi\"; String b = \"hi\";</code> are stored in the string constant pool and point to the same object, so <code>a == b</code> can also be used to compare the contents of two strings.</p> <p>Q: Before reaching the bottom level, is the number of nodes in the queue \\(2^h\\) in breadth-first traversal?</p> <p>Yes, for example, a full binary tree with height \\(h = 2\\) has a total of \\(n = 7\\) nodes, then the bottom level has \\(4 = 2^h = (n + 1) / 2\\) nodes.</p>"}]}
\ No newline at end of file
+{"config":{"lang":["en"],"separator":"[\\s\\-]+","pipeline":["stopWordFilter"]},"docs":[{"location":"","title":"Home","text":"Hello Algo <p>Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code</p> <p> </p> <p>        Dive In           Clone Repo           Get PDF    </p> <p>The English edition is brewing...</p> <p>Feel free to engage in Chinese-to-English translation and pull request review! For guidelines, please see #914.</p> <p>Quote</p> <p>\"An easy-to-understand book on data structures and algorithms, which guides readers to learn by minds-on and hands-on. Strongly recommended for algorithm beginners!\"</p> <p>\u2014\u2014 Junhui Deng, Professor of Computer Science, Tsinghua University</p> <p>Quote</p> <p>\"If I had 'Hello Algo' when I was learning data structures and algorithms, it would have been 10 times easier!\"</p> <p>\u2014\u2014 Mu Li, Senior Principal Scientist, Amazon</p> Animated illustrations <p>Easy to understandSmooth learning curve</p> <p>\"A picture is worth a thousand words.\"</p> Off-the-Shelf Code <p>Multi programming languagesRun with one click</p> <p>\"Talk is cheap. Show me the code.\"</p> Learning Together <p>Discussion and questions welcomeReaders progress together</p> <p>\"Learning by teaching.\"</p>  Preface  <p>Two years ago, I shared the \"Sword Offer\" series of problem solutions on LeetCode, which received much love and support from many students. During my interactions with readers, the most common question I encountered was \"How to get started with algorithms.\" Gradually, I developed a deep interest in this question.</p> <p>Blindly solving problems seems to be the most popular method, being simple, direct, and effective. However, problem-solving is like playing a \"Minesweeper\" game, where students with strong self-learning abilities can successfully clear the mines one by one, but those with insufficient foundations may end up bruised from explosions, retreating step by step in frustration. Thoroughly reading textbooks is also common, but for students aiming for job applications, the energy consumed by graduation, resume submissions, and preparing for written tests and interviews makes tackling thick books a daunting challenge.</p> <p>If you are facing similar troubles, then you are lucky to have found this book. This book is my answer to this question, not necessarily the best solution, but at least an active attempt. Although this book won't directly land you an Offer, it will guide you through the \"knowledge map\" of data structures and algorithms, help you understand the shape, size, and distribution of different \"mines,\" and equip you with various \"demining methods.\" With these skills, I believe you can more comfortably solve problems and read literature, gradually building a complete knowledge system.</p> <p>I deeply agree with Professor Feynman's saying: \"Knowledge isn't free. You have to pay attention.\" In this sense, this book is not entirely \"free.\" To not disappoint the precious \"attention\" you pay to this book, I will do my utmost, investing the greatest \"attention\" to complete the creation of this book.</p>  Author  <p>Yudong Jin(krahets), Senior Algorithm Engineer in a top tech company, Master's degree from Shanghai Jiao Tong University. The highest-read blogger across the entire LeetCode, his published \"Illustration of Algorithm Data Structures\" has been subscribed to by over 300k.</p>  Contribution  <p>This book is continuously improved with the joint efforts of many contributors from the open-source community. Thanks to each writer who invested their time and energy, listed in the order generated by GitHub:</p> <p> </p> <p>The code review work for this book was completed by codingonion, Gonglja, gvenusleo, hpstory, justin\u2010tse, krahets, night-cruise, nuomi1, and Reanon (listed in alphabetical order). Thanks to them for their time and effort, ensuring the standardization and uniformity of the code in various languages.</p> <sub>codingonion</sub><sub>Rust, Zig</sub> <sub>Gonglja</sub><sub>C, C++</sub> <sub>gvenusleo</sub><sub>Dart</sub> <sub>hpstory</sub><sub>C#</sub> <sub>justin-tse</sub><sub>JS, TS</sub> <sub>krahets</sub><sub>Java, Python</sub> <sub>night-cruise</sub><sub>Rust</sub> <sub>nuomi1</sub><sub>Swift</sub> <sub>Reanon</sub><sub>Go, C</sub>"},{"location":"chapter_array_and_linkedlist/","title":"Chapter 4. \u00a0 Arrays and Linked Lists","text":"<p>Abstract</p> <p>The world of data structures resembles a sturdy brick wall.</p> <p>In arrays, envision bricks snugly aligned, each resting seamlessly beside the next, creating a unified formation. Meanwhile, in linked lists, these bricks disperse freely, embraced by vines gracefully knitting connections between them.</p>"},{"location":"chapter_array_and_linkedlist/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>4.1 \u00a0 Array</li> <li>4.2 \u00a0 Linked List</li> <li>4.3 \u00a0 List</li> <li>4.4 \u00a0 Memory and Cache</li> <li>4.5 \u00a0 Summary</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/","title":"4.1 \u00a0 Arrays","text":"<p>An \"array\" is a linear data structure that operates as a lineup of similar items, stored together in a computer's memory in contiguous spaces. It's like a sequence that maintains organized storage. Each item in this lineup has its unique 'spot' known as an \"index\". Please refer to the Figure 4-1  to observe how arrays work and grasp these key terms.</p> <p></p> <p> Figure 4-1 \u00a0 Array Definition and Storage Method </p>"},{"location":"chapter_array_and_linkedlist/array/#411-common-operations-on-arrays","title":"4.1.1 \u00a0 Common Operations on Arrays","text":""},{"location":"chapter_array_and_linkedlist/array/#1-initializing-arrays","title":"1. \u00a0 Initializing Arrays","text":"<p>Arrays can be initialized in two ways depending on the needs: either without initial values or with specified initial values. When initial values are not specified, most programming languages will set the array elements to \\(0\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig array.py<pre><code># Initialize array\narr: list[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> array.cpp<pre><code>/* Initialize array */\n// Stored on stack\nint arr[5];\nint nums[5] = { 1, 3, 2, 5, 4 };\n// Stored on heap (manual memory release needed)\nint* arr1 = new int[5];\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n</code></pre> array.java<pre><code>/* Initialize array */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n</code></pre> array.cs<pre><code>/* Initialize array */\nint[] arr = new int[5]; // [ 0, 0, 0, 0, 0 ]\nint[] nums = [1, 3, 2, 5, 4];\n</code></pre> array.go<pre><code>/* Initialize array */\nvar arr [5]int\n// In Go, specifying the length ([5]int) denotes an array, while not specifying it ([]int) denotes a slice.\n// Since Go's arrays are designed to have compile-time fixed length, only constants can be used to specify the length.\n// For convenience in implementing the extend() method, the Slice will be considered as an Array here.\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> array.swift<pre><code>/* Initialize array */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n</code></pre> array.js<pre><code>/* Initialize array */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n</code></pre> array.ts<pre><code>/* Initialize array */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> array.dart<pre><code>/* Initialize array */\nList&lt;int&gt; arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> array.rs<pre><code>/* Initialize array */\nlet arr: Vec&lt;i32&gt; = vec![0; 5]; // [0, 0, 0, 0, 0]\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> array.c<pre><code>/* Initialize array */\nint arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n</code></pre> array.kt<pre><code>\n</code></pre> array.zig<pre><code>// Initialize array\nvar arr = [_]i32{0} ** 5; // { 0, 0, 0, 0, 0 }\nvar nums = [_]i32{ 1, 3, 2, 5, 4 };\n</code></pre>"},{"location":"chapter_array_and_linkedlist/array/#2-accessing-elements","title":"2. \u00a0 Accessing Elements","text":"<p>Elements in an array are stored in contiguous memory spaces, making it simpler to compute each element's memory address. The formula shown in the Figure below aids in determining an element's memory address, utilizing the array's memory address (specifically, the first element's address) and the element's index. This computation streamlines direct access to the desired element.</p> <p></p> <p> Figure 4-2 \u00a0 Memory Address Calculation for Array Elements </p> <p>As observed in the above illustration, array indexing conventionally begins at \\(0\\). While this might appear counterintuitive, considering counting usually starts at \\(1\\), within the address calculation formula, an index is essentially an offset from the memory address. For the first element's address, this offset is \\(0\\), validating its index as \\(0\\).</p> <p>Accessing elements in an array is highly efficient, allowing us to randomly access any element in \\(O(1)\\) time.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def random_access(nums: list[int]) -&gt; int:\n    \"\"\"\u968f\u673a\u8bbf\u95ee\u5143\u7d20\"\"\"\n    # \u5728\u533a\u95f4 [0, len(nums)-1] \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    random_index = random.randint(0, len(nums) - 1)\n    # \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    random_num = nums[random_index]\n    return random_num\n</code></pre> array.cpp<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n    // \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = rand() % size;\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.java<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int[] nums) {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.cs<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint RandomAccess(int[] nums) {\n    Random random = new();\n    // \u5728\u533a\u95f4 [0, nums.Length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = random.Next(nums.Length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.go<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums []int) (randomNum int) {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    randomIndex := rand.Intn(len(nums))\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    randomNum = nums[randomIndex]\n    return\n}\n</code></pre> array.swift<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums: [Int]) -&gt; Int {\n    // \u5728\u533a\u95f4 [0, nums.count) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    let randomIndex = nums.indices.randomElement()!\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    let randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.js<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums) {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    const random_index = Math.floor(Math.random() * nums.length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.ts<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums: number[]): number {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    const random_index = Math.floor(Math.random() * nums.length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.dart<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(List&lt;int&gt; nums) {\n  // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n  int randomIndex = Random().nextInt(nums.length);\n  // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n  int randomNum = nums[randomIndex];\n  return randomNum;\n}\n</code></pre> array.rs<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfn random_access(nums: &amp;[i32]) -&gt; i32 {\n    // \u5728\u533a\u95f4 [0, nums.len()) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    let random_index = rand::thread_rng().gen_range(0..nums.len());\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    let random_num = nums[random_index];\n    random_num\n}\n</code></pre> array.c<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n    // \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = rand() % size;\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.kt<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfun randomAccess(nums: IntArray): Int {\n    // \u5728\u533a\u95f4 [0, nums.size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    val randomIndex = ThreadLocalRandom.current().nextInt(0, nums.size)\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    val randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.rb<pre><code>### \u968f\u673a\u8bbf\u95ee\u5143\u7d20 ###\ndef random_access(nums)\n  # \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n  random_index = Random.rand 0...(nums.length - 1)\n\n  # \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n  nums[random_index]\nend\n</code></pre> array.zig<pre><code>// \u968f\u673a\u8bbf\u95ee\u5143\u7d20\nfn randomAccess(nums: []i32) i32 {\n    // \u5728\u533a\u95f4 [0, nums.len) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6574\u6570\n    var randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    var randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#3-inserting-elements","title":"3. \u00a0 Inserting Elements","text":"<p>Array elements are tightly packed in memory, with no space available to accommodate additional data between them. Illustrated in Figure below, inserting an element in the middle of an array requires shifting all subsequent elements back by one position to create room for the new element.</p> <p></p> <p> Figure 4-3 \u00a0 Array Element Insertion Example </p> <p>It's important to note that due to the fixed length of an array, inserting an element will unavoidably result in the loss of the last element in the array. Solutions to address this issue will be explored in the \"List\" chapter.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def insert(nums: list[int], num: int, index: int):\n    \"\"\"\u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\"\"\"\n    # \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i in range(len(nums) - 1, index, -1):\n        nums[i] = nums[i - 1]\n    # \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n</code></pre> array.cpp<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.java<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid Insert(int[] nums, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = nums.Length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.go<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums []int, num int, index int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i := len(nums) - 1; i &gt; index; i-- {\n        nums[i] = nums[i-1]\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.swift<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i in nums.indices.dropFirst(index).reversed() {\n        nums[i] = nums[i - 1]\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.js<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums, num, index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums: number[], num: number, index: number): void {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 _num */\nvoid insert(List&lt;int&gt; nums, int _num, int index) {\n  // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n  for (var i = nums.length - 1; i &gt; index; i--) {\n    nums[i] = nums[i - 1];\n  }\n  // \u5c06 _num \u8d4b\u7ed9 index \u5904\u5143\u7d20\n  nums[index] = _num;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfn insert(nums: &amp;mut Vec&lt;i32&gt;, num: i32, index: usize) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i in (index + 1..nums.len()).rev() {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.c<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.kt<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfun insert(nums: IntArray, num: Int, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (i in nums.size - 1 downTo index + 1) {\n        nums[i] = nums[i - 1]\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.rb<pre><code>### \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num ###\ndef insert(nums, num, index)\n  # \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n  for i in (nums.length - 1).downto(index + 1)\n    nums[i] = nums[i - 1]\n  end\n\n  # \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n  nums[index] = num\nend\n</code></pre> array.zig<pre><code>// \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\nfn insert(nums: []i32, num: i32, index: usize) void {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    var i = nums.len - 1;\n    while (i &gt; index) : (i -= 1) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#4-deleting-elements","title":"4. \u00a0 Deleting Elements","text":"<p>Similarly, as depicted in the Figure 4-4 , to delete an element at index \\(i\\), all elements following index \\(i\\) must be moved forward by one position.</p> <p></p> <p> Figure 4-4 \u00a0 Array Element Deletion Example </p> <p>Please note that after deletion, the former last element becomes \"meaningless,\" hence requiring no specific modification.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def remove(nums: list[int], index: int):\n    \"\"\"\u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\"\"\"\n    # \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i in range(index, len(nums) - 1):\n        nums[i] = nums[i + 1]\n</code></pre> array.cpp<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int *nums, int size, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.java<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.cs<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid Remove(int[] nums, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; nums.Length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.go<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums []int, index int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i := index; i &lt; len(nums)-1; i++ {\n        nums[i] = nums[i+1]\n    }\n}\n</code></pre> array.swift<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums: inout [Int], index: Int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i in nums.indices.dropFirst(index).dropLast() {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.js<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums, index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.ts<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums: number[], index: number): void {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.dart<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(List&lt;int&gt; nums, int index) {\n  // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n  for (var i = index; i &lt; nums.length - 1; i++) {\n    nums[i] = nums[i + 1];\n  }\n}\n</code></pre> array.rs<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfn remove(nums: &amp;mut Vec&lt;i32&gt;, index: usize) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i in index..nums.len() - 1 {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.c<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(int *nums, int size, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.kt<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfun remove(nums: IntArray, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (i in index..&lt;nums.size - 1) {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.rb<pre><code>### \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 ###\ndef remove(nums, index)\n  # \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n  for i in index...nums.length\n    nums[i] = nums[i + 1] || 0\n  end\nend\n</code></pre> array.zig<pre><code>// \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\nfn remove(nums: []i32, index: usize) void {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    var i = index;\n    while (i &lt; nums.len - 1) : (i += 1) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>In summary, the insertion and deletion operations in arrays present the following disadvantages:</p> <ul> <li>High Time Complexity: Both insertion and deletion in an array have an average time complexity of \\(O(n)\\), where \\(n\\) is the length of the array.</li> <li>Loss of Elements: Due to the fixed length of arrays, elements that exceed the array's capacity are lost during insertion.</li> <li>Waste of Memory: Initializing a longer array and utilizing only the front part results in \"meaningless\" end elements during insertion, leading to some wasted memory space.</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#5-traversing-arrays","title":"5. \u00a0 Traversing Arrays","text":"<p>In most programming languages, we can traverse an array either by using indices or by directly iterating over each element:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def traverse(nums: list[int]):\n    \"\"\"\u904d\u5386\u6570\u7ec4\"\"\"\n    count = 0\n    # \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i in range(len(nums)):\n        count += nums[i]\n    # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for num in nums:\n        count += num\n    # \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n    for i, num in enumerate(nums):\n        count += nums[i]\n        count += num\n</code></pre> array.cpp<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.java<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (int num : nums) {\n        count += num;\n    }\n}\n</code></pre> array.cs<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid Traverse(int[] nums) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; nums.Length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    foreach (int num in nums) {\n        count += num;\n    }\n}\n</code></pre> array.go<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums []int) {\n    count := 0\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i := 0; i &lt; len(nums); i++ {\n        count += nums[i]\n    }\n    count = 0\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for _, num := range nums {\n        count += num\n    }\n    // \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n    for i, num := range nums {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.swift<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums: [Int]) {\n    var count = 0\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i in nums.indices {\n        count += nums[i]\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for num in nums {\n        count += num\n    }\n    // \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n    for (i, num) in nums.enumerated() {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.js<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums) {\n    let count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.ts<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums: number[]): void {\n    let count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.dart<pre><code>/* \u904d\u5386\u6570\u7ec4\u5143\u7d20 */\nvoid traverse(List&lt;int&gt; nums) {\n  int count = 0;\n  // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n  for (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n  }\n  // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n  for (int _num in nums) {\n    count += _num;\n  }\n  // \u901a\u8fc7 forEach \u65b9\u6cd5\u904d\u5386\u6570\u7ec4\n  nums.forEach((_num) {\n    count += _num;\n  });\n}\n</code></pre> array.rs<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfn traverse(nums: &amp;[i32]) {\n    let mut _count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i in 0..nums.len() {\n        _count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for num in nums {\n        _count += num;\n    }\n}\n</code></pre> array.c<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.kt<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfun traverse(nums: IntArray) {\n    var count = 0\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (i in nums.indices) {\n        count += nums[i]\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (j: Int in nums) {\n        count += j\n    }\n}\n</code></pre> array.rb<pre><code>### \u904d\u5386\u6570\u7ec4 ###\ndef traverse(nums)\n  count = 0\n\n  # \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n  for i in 0...nums.length\n    count += nums[i]\n  end\n\n  # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n  for num in nums\n    count += num\n  end\nend\n</code></pre> array.zig<pre><code>// \u904d\u5386\u6570\u7ec4\nfn traverse(nums: []i32) void {\n    var count: i32 = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    var i: i32 = 0;\n    while (i &lt; nums.len) : (i += 1) {\n        count += nums[i];\n    }\n    count = 0;\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (nums) |num| {\n        count += num;\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#6-finding-elements","title":"6. \u00a0 Finding Elements","text":"<p>Locating a specific element within an array involves iterating through the array, checking each element to determine if it matches the desired value.</p> <p>Because arrays are linear data structures, this operation is commonly referred to as \"linear search.\"</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def find(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\"\"\"\n    for i in range(len(nums)):\n        if nums[i] == target:\n            return i\n    return -1\n</code></pre> array.cpp<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.java<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint Find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.go<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums []int, target int) (index int) {\n    index = -1\n    for i := 0; i &lt; len(nums); i++ {\n        if nums[i] == target {\n            index = i\n            break\n        }\n    }\n    return\n}\n</code></pre> array.swift<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums: [Int], target: Int) -&gt; Int {\n    for i in nums.indices {\n        if nums[i] == target {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> array.js<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums, target) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) return i;\n    }\n    return -1;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums: number[], target: number): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(List&lt;int&gt; nums, int target) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    if (nums[i] == target) return i;\n  }\n  return -1;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfn find(nums: &amp;[i32], target: i32) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        if nums[i] == target {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> array.c<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.kt<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfun find(nums: IntArray, target: Int): Int {\n    for (i in nums.indices) {\n        if (nums[i] == target) return i\n    }\n    return -1\n}\n</code></pre> array.rb<pre><code>### \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 ###\ndef find(nums, target)\n  for i in 0...nums.length\n    return i if nums[i] == target\n  end\n\n  -1\nend\n</code></pre> array.zig<pre><code>// \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\nfn find(nums: []i32, target: i32) i32 {\n    for (nums, 0..) |num, i| {\n        if (num == target) return @intCast(i);\n    }\n    return -1;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#7-expanding-arrays","title":"7. \u00a0 Expanding Arrays","text":"<p>In complex system environments, ensuring the availability of memory space after an array for safe capacity extension becomes challenging. Consequently, in most programming languages, the length of an array is immutable.</p> <p>To expand an array,  it's necessary to create a larger array and then copy the elements from the original array. This operation has a time complexity of \\(O(n)\\) and can be time-consuming for large arrays. The code are as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def extend(nums: list[int], enlarge: int) -&gt; list[int]:\n    \"\"\"\u6269\u5c55\u6570\u7ec4\u957f\u5ea6\"\"\"\n    # \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    res = [0] * (len(nums) + enlarge)\n    # \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for i in range(len(nums)):\n        res[i] = nums[i]\n    # \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n</code></pre> array.cpp<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int *res = new int[size + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // \u91ca\u653e\u5185\u5b58\n    delete[] nums;\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.java<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int[] res = new int[nums.length + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.cs<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] Extend(int[] nums, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int[] res = new int[nums.Length + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; nums.Length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.go<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums []int, enlarge int) []int {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    res := make([]int, len(nums)+enlarge)\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for i, num := range nums {\n        res[i] = num\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n}\n</code></pre> array.swift<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums: [Int], enlarge: Int) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    var res = Array(repeating: 0, count: nums.count + enlarge)\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for i in nums.indices {\n        res[i] = nums[i]\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n}\n</code></pre> array.js<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cJavaScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums, enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    const res = new Array(nums.length + enlarge).fill(0);\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.ts<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cTypeScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums: number[], enlarge: number): number[] {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    const res = new Array(nums.length + enlarge).fill(0);\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.dart<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nList&lt;int&gt; extend(List&lt;int&gt; nums, int enlarge) {\n  // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n  List&lt;int&gt; res = List.filled(nums.length + enlarge, 0);\n  // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n  for (var i = 0; i &lt; nums.length; i++) {\n    res[i] = nums[i];\n  }\n  // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n  return res;\n}\n</code></pre> array.rs<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfn extend(nums: Vec&lt;i32&gt;, enlarge: usize) -&gt; Vec&lt;i32&gt; {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    let mut res: Vec&lt;i32&gt; = vec![0; nums.len() + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\n    for i in 0..nums.len() {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    res\n}\n</code></pre> array.c<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int *res = (int *)malloc(sizeof(int) * (size + enlarge));\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // \u521d\u59cb\u5316\u6269\u5c55\u540e\u7684\u7a7a\u95f4\n    for (int i = size; i &lt; size + enlarge; i++) {\n        res[i] = 0;\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.kt<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfun extend(nums: IntArray, enlarge: Int): IntArray {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    val res = IntArray(nums.size + enlarge)\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (i in nums.indices) {\n        res[i] = nums[i]\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n}\n</code></pre> array.rb<pre><code>### \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 ###\n# \u8bf7\u6ce8\u610f\uff0cRuby \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n# \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\ndef extend(nums, enlarge)\n  # \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n  res = Array.new(nums.length + enlarge, 0)\n\n  # \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n  for i in 0...nums.length\n    res[i] = nums[i]\n  end\n\n  # \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n  res\nend\n</code></pre> array.zig<pre><code>// \u6269\u5c55\u6570\u7ec4\u957f\u5ea6\nfn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    var res = try mem_allocator.alloc(i32, nums.len + enlarge);\n    @memset(res, 0);\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    std.mem.copy(i32, res, nums);\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#412-advantages-and-limitations-of-arrays","title":"4.1.2 \u00a0 Advantages and Limitations of Arrays","text":"<p>Arrays are stored in contiguous memory spaces and consist of elements of the same type. This approach provides substantial prior information that systems can leverage to optimize the efficiency of data structure operations.</p> <ul> <li>High Space Efficiency: Arrays allocate a contiguous block of memory for data, eliminating the need for additional structural overhead.</li> <li>Support for Random Access: Arrays allow \\(O(1)\\) time access to any element.</li> <li>Cache Locality: When accessing array elements, the computer not only loads them but also caches the surrounding data, utilizing high-speed cache to enchance subsequent operation speeds.</li> </ul> <p>However, continuous space storage is a double-edged sword, with the following limitations:</p> <ul> <li>Low Efficiency in Insertion and Deletion: As arrays accumulate many elements, inserting or deleting elements requires shifting a large number of elements.</li> <li>Fixed Length: The length of an array is fixed after initialization. Expanding an array requires copying all data to a new array, incurring significant costs.</li> <li>Space Wastage: If the allocated array size exceeds the what is necessary, the extra space is wasted.</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#413-typical-applications-of-arrays","title":"4.1.3 \u00a0 Typical Applications of Arrays","text":"<p>Arrays are fundamental and widely used data structures. They find frequent application in various algorithms and serve in the implementation of complex data structures.</p> <ul> <li>Random Access: Arrays are ideal for storing data when random sampling is required. By generating a random sequence based on indices, we can achieve random sampling efficiently.</li> <li>Sorting and Searching: Arrays are the most commonly used data structure for sorting and searching algorithms.  Techniques like quick sort, merge sort, binary search, etc., are primarily operate on arrays.</li> <li>Lookup Tables: Arrays serve as efficient lookup tables for quick element or relationship retrieval. For instance, mapping characters to ASCII codes becomes seamless by using the ASCII code values as indices and storing corresponding elements in the array.</li> <li>Machine Learning: Within the domain of neural networks, arrays play a pivotal role in executing crucial linear algebra operations involving vectors, matrices, and tensors. Arrays serve as the primary and most extensively used data structure in neural network programming.</li> <li>Data Structure Implementation:  Arrays serve as the building blocks for implementing various data structures like stacks, queues, hash tables, heaps, graphs, etc. For instance, the adjacency matrix representation of a graph is essentially a two-dimensional array.</li> </ul>"},{"location":"chapter_array_and_linkedlist/linked_list/","title":"4.2 \u00a0 Linked Lists","text":"<p>Memory space is a shared resource among all programs. In a complex system environment, available memory can be dispersed throughout the memory space. We understand that the memory allocated for an array must be continuous. However, for very large arrays, finding a sufficiently large contiguous memory space might be challenging. This is where the flexible advantage of linked lists becomes evident.</p> <p>A \"linked list\" is a linear data structure in which each element is a node object, and the nodes are interconnected through \"references\". These references hold the memory addresses of subsequent nodes, enabling navigation from one node to the next.</p> <p>The design of linked lists allows for their nodes to be distributed across memory locations without requiring contiguous memory addresses.</p> <p></p> <p> Figure 4-5 \u00a0 Linked List Definition and Storage Method </p> <p>As shown in the figure, we see that the basic building block of a linked list is the \"node\" object. Each node comprises two key components: the node's \"value\" and a \"reference\" to the next node.</p> <ul> <li>The first node in a linked list is the \"head node\", and the final one is the \"tail node\".</li> <li>The tail node points to \"null\", designated as <code>null</code> in Java, <code>nullptr</code> in C++, and <code>None</code> in Python.</li> <li>In languages that support pointers, like C, C++, Go, and Rust, this \"reference\" is typically implemented as a \"pointer\".</li> </ul> <p>As the code below illustrates, a <code>ListNode</code> in a linked list, besides holding a value, must also maintain an additional reference (or pointer). Therefore, a linked list occupies more memory space than an array when storing the same quantity of data..</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>class ListNode:\n    \"\"\"Linked List Node Class\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val               # Node value\n        self.next: ListNode | None = None # Reference to the next node\n</code></pre> <pre><code>/* Linked List Node Structure */\nstruct ListNode {\n    int val;         // Node value\n    ListNode *next;  // Pointer to the next node\n    ListNode(int x) : val(x), next(nullptr) {}  // Constructor\n};\n</code></pre> <pre><code>/* Linked List Node Class */\nclass ListNode {\n    int val;        // Node value\n    ListNode next;  // Reference to the next node\n    ListNode(int x) { val = x; }  // Constructor\n}\n</code></pre> <pre><code>/* Linked List Node Class */\nclass ListNode(int x) {  // Constructor\n    int val = x;         // Node value\n    ListNode? next;      // Reference to the next node\n}\n</code></pre> <pre><code>/* Linked List Node Structure */\ntype ListNode struct {\n    Val  int       // Node value\n    Next *ListNode // Pointer to the next node\n}\n\n// NewListNode Constructor, creates a new linked list\nfunc NewListNode(val int) *ListNode {\n    return &amp;ListNode{\n        Val:  val,\n        Next: nil,\n    }\n}\n</code></pre> <pre><code>/* Linked List Node Class */\nclass ListNode {\n    var val: Int // Node value\n    var next: ListNode? // Reference to the next node\n\n    init(x: Int) { // Constructor\n        val = x\n    }\n}\n</code></pre> <pre><code>/* Linked List Node Class */\nclass ListNode {\n    constructor(val, next) {\n        this.val = (val === undefined ? 0 : val);       // Node value\n        this.next = (next === undefined ? null : next); // Reference to the next node\n    }\n}\n</code></pre> <pre><code>/* Linked List Node Class */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    constructor(val?: number, next?: ListNode | null) {\n        this.val = val === undefined ? 0 : val;        // Node value\n        this.next = next === undefined ? null : next;  // Reference to the next node\n    }\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n  int val; // Node value\n  ListNode? next; // Reference to the next node\n  ListNode(this.val, [this.next]); // Constructor\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* Linked List Node Class */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // Node value\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // Pointer to the next node\n}\n</code></pre> <pre><code>/* Linked List Node Structure */\ntypedef struct ListNode {\n    int val;               // Node value\n    struct ListNode *next; // Pointer to the next node\n} ListNode;\n\n/* Constructor */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    return node;\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>// Linked List Node Class\npub fn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = 0, // Node value\n        next: ?*Self = null, // Pointer to the next node\n\n        // Constructor\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#421-common-operations-on-linked-lists","title":"4.2.1 \u00a0 Common Operations on Linked Lists","text":""},{"location":"chapter_array_and_linkedlist/linked_list/#1-initializing-a-linked-list","title":"1. \u00a0 Initializing a Linked List","text":"<p>Constructing a linked list is a two-step process: first, initializing each node object, and second, forming the reference links between the nodes. After initialization, we can traverse all nodes sequentially from the head node by following the <code>next</code> reference.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig linked_list.py<pre><code># Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# Initialize each node\nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# Build references between nodes\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.cpp<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// Build references between nodes\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.java<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.cs<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nListNode n0 = new(1);\nListNode n1 = new(3);\nListNode n2 = new(2);\nListNode n3 = new(5);\nListNode n4 = new(4);\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.go<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// Build references between nodes\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n</code></pre> linked_list.swift<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// Build references between nodes\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.js<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.ts<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.dart<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rs<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nlet n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None }));\nlet n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None }));\nlet n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None }));\nlet n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None }));\nlet n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None }));\n\n// Build references between nodes\nn0.borrow_mut().next = Some(n1.clone());\nn1.borrow_mut().next = Some(n2.clone());\nn2.borrow_mut().next = Some(n3.clone());\nn3.borrow_mut().next = Some(n4.clone());\n</code></pre> linked_list.c<pre><code>/* Initialize linked list: 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// Build references between nodes\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.kt<pre><code>\n</code></pre> linked_list.zig<pre><code>// Initialize linked list\n// Initialize each node\nvar n0 = inc.ListNode(i32){.val = 1};\nvar n1 = inc.ListNode(i32){.val = 3};\nvar n2 = inc.ListNode(i32){.val = 2};\nvar n3 = inc.ListNode(i32){.val = 5};\nvar n4 = inc.ListNode(i32){.val = 4};\n// Build references between nodes\nn0.next = &amp;n1;\nn1.next = &amp;n2;\nn2.next = &amp;n3;\nn3.next = &amp;n4;\n</code></pre> <p>The array as a whole is a variable, for instance, the array <code>nums</code> includes elements like <code>nums[0]</code>, <code>nums[1]</code>, and so on, whereas a linked list is made up of several distinct node objects. We typically refer to a linked list by its head node, for example, the linked list in the previous code snippet is referred to as <code>n0</code>.</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#2-inserting-a-node","title":"2. \u00a0 Inserting a Node","text":"<p>Inserting a node into a linked list is very easy. As shown in the figure, let's assume we aim to insert a new node <code>P</code> between two adjacent nodes <code>n0</code> and <code>n1</code>. This can be achieved by simply modifying two node references (pointers), with a time complexity of \\(O(1)\\).</p> <p>By comparison, inserting an element into an array has a time complexity of \\(O(n)\\), which becomes less efficient when dealing with large data volumes.</p> <p></p> <p> Figure 4-6 \u00a0 Linked List Node Insertion Example </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def insert(n0: ListNode, P: ListNode):\n    \"\"\"\u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\"\"\"\n    n1 = n0.next\n    P.next = n1\n    n0.next = P\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\n    ListNode n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid Insert(ListNode n0, ListNode P) {\n    ListNode? n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insertNode(n0 *ListNode, P *ListNode) {\n    n1 := n0.Next\n    P.Next = n1\n    n0.Next = P\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insert(n0: ListNode, P: ListNode) {\n    let n1 = n0.next\n    P.next = n1\n    n0.next = P\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0, P) {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0: ListNode, P: ListNode): void {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\n  ListNode? n1 = n0.next;\n  P.next = n1;\n  n0.next = P;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\n#[allow(non_snake_case)]\npub fn insert&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, P: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    let n1 = n0.borrow_mut().next.take();\n    P.borrow_mut().next = n1;\n    n0.borrow_mut().next = Some(P);\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.kt<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9p */\nfun insert(n0: ListNode?, p: ListNode?) {\n    val n1 = n0?.next\n    p?.next = n1\n    n0?.next = p\n}\n</code></pre> linked_list.rb<pre><code>### \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 _p ###\n# Ruby \u7684 `p` \u662f\u4e00\u4e2a\u5185\u7f6e\u51fd\u6570\uff0c `P` \u662f\u4e00\u4e2a\u5e38\u91cf\uff0c\u6240\u4ee5\u53ef\u4ee5\u4f7f\u7528 `_p` \u4ee3\u66ff\ndef insert(n0, _p)\n  n1 = n0.next\n  _p.next = n1\n  n0.next = _p\nend\n</code></pre> linked_list.zig<pre><code>// \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\nfn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {\n    var n1 = n0.?.next;\n    P.?.next = n1;\n    n0.?.next = P;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#3-deleting-a-node","title":"3. \u00a0 Deleting a Node","text":"<p>As shown in the figure, deleting a node from a linked list is also very easy, involving only the modification of a single node's reference (pointer).</p> <p>It's important to note that even though node <code>P</code> continues to point to <code>n1</code> after being deleted, it becomes inaccessible during linked list traversal. This effectively means that <code>P</code> is no longer a part of the linked list.</p> <p></p> <p> Figure 4-7 \u00a0 Linked List Node Deletion </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def remove(n0: ListNode):\n    \"\"\"\u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\"\"\"\n    if not n0.next:\n        return\n    # n0 -&gt; P -&gt; n1\n    P = n0.next\n    n1 = P.next\n    n0.next = n1\n</code></pre> linked_list.cpp<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode *n0) {\n    if (n0-&gt;next == nullptr)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // \u91ca\u653e\u5185\u5b58\n    delete P;\n}\n</code></pre> linked_list.java<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid Remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode? n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.go<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc removeItem(n0 *ListNode) {\n    if n0.Next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    P := n0.Next\n    n1 := P.Next\n    n0.Next = n1\n}\n</code></pre> linked_list.swift<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc remove(n0: ListNode) {\n    if n0.next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    let P = n0.next\n    let n1 = P?.next\n    n0.next = n1\n}\n</code></pre> linked_list.js<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0) {\n    if (!n0.next) return;\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0: ListNode): void {\n    if (!n0.next) {\n        return;\n    }\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\n  if (n0.next == null) return;\n  // n0 -&gt; P -&gt; n1\n  ListNode P = n0.next!;\n  ListNode? n1 = P.next;\n  n0.next = n1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n#[allow(non_snake_case)]\npub fn remove&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    if n0.borrow().next.is_none() {\n        return;\n    };\n    // n0 -&gt; P -&gt; n1\n    let P = n0.borrow_mut().next.take();\n    if let Some(node) = P {\n        let n1 = node.borrow_mut().next.take();\n        n0.borrow_mut().next = n1;\n    }\n}\n</code></pre> linked_list.c<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(ListNode *n0) {\n    if (!n0-&gt;next)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // \u91ca\u653e\u5185\u5b58\n    free(P);\n}\n</code></pre> linked_list.kt<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfun remove(n0: ListNode?) {\n    val p = n0?.next\n    val n1 = p?.next\n    n0?.next = n1\n}\n</code></pre> linked_list.rb<pre><code>### \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 ###\ndef remove(n0)\n  return if n0.next.nil?\n\n  # n0 -&gt; remove_node -&gt; n1\n  remove_node = n0.next\n  n1 = remove_node.next\n  n0.next = n1\nend\n</code></pre> linked_list.zig<pre><code>// \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\nfn remove(n0: ?*inc.ListNode(i32)) void {\n    if (n0.?.next == null) return;\n    // n0 -&gt; P -&gt; n1\n    var P = n0.?.next;\n    var n1 = P.?.next;\n    n0.?.next = n1;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#4-accessing-nodes","title":"4. \u00a0 Accessing Nodes","text":"<p>Accessing nodes in a linked list is less efficient. As previously mentioned, any element in an array can be accessed in \\(O(1)\\) time. In contrast, with a linked list, the program involves starting from the head node and sequentially traversing through the nodes until the desired node is found. In other words, to access the \\(i\\)-th node in a linked list, the program must iterate through \\(i - 1\\) nodes, resulting in a time complexity of \\(O(n)\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def access(head: ListNode, index: int) -&gt; ListNode | None:\n    \"\"\"\u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\"\"\"\n    for _ in range(index):\n        if not head:\n            return None\n        head = head.next\n    return head\n</code></pre> linked_list.cpp<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == nullptr)\n            return nullptr;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.java<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode access(ListNode head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.cs<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? Access(ListNode? head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.go<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head *ListNode, index int) *ListNode {\n    for i := 0; i &lt; index; i++ {\n        if head == nil {\n            return nil\n        }\n        head = head.Next\n    }\n    return head\n}\n</code></pre> linked_list.swift<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head: ListNode, index: Int) -&gt; ListNode? {\n    var head: ListNode? = head\n    for _ in 0 ..&lt; index {\n        if head == nil {\n            return nil\n        }\n        head = head?.next\n    }\n    return head\n}\n</code></pre> linked_list.js<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head, index) {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.ts<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head: ListNode | null, index: number): ListNode | null {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.dart<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? access(ListNode? head, int index) {\n  for (var i = 0; i &lt; index; i++) {\n    if (head == null) return null;\n    head = head.next;\n  }\n  return head;\n}\n</code></pre> linked_list.rs<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\npub fn access&lt;T&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, index: i32) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n    if index &lt;= 0 {\n        return head;\n    };\n    if let Some(node) = &amp;head.borrow().next {\n        return access(node.clone(), index - 1);\n    }\n\n    return head;\n}\n</code></pre> linked_list.c<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == NULL)\n            return NULL;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.kt<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfun access(head: ListNode?, index: Int): ListNode? {\n    var h = head\n    for (i in 0..&lt;index) {\n        h = h?.next\n    }\n    return h\n}\n</code></pre> linked_list.rb<pre><code>### \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 ###\ndef access(head, index)\n  for i in 0...index\n    return nil if head.nil?\n    head = head.next\n  end\n\n  head\nend\n</code></pre> linked_list.zig<pre><code>// \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\nfn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {\n    var head = node;\n    var i: i32 = 0;\n    while (i &lt; index) : (i += 1) {\n        head = head.?.next;\n        if (head == null) return null;\n    }\n    return head;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#5-finding-nodes","title":"5. \u00a0 Finding Nodes","text":"<p>Traverse the linked list to locate a node whose value matches <code>target</code>, and then output the index of that node within the linked list. This procedure is also an example of linear search. The corresponding code is provided below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def find(head: ListNode, target: int) -&gt; int:\n    \"\"\"\u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\"\"\"\n    index = 0\n    while head:\n        if head.val == target:\n            return index\n        head = head.next\n        index += 1\n    return -1\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head != nullptr) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint Find(ListNode? head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc findNode(head *ListNode, target int) int {\n    index := 0\n    for head != nil {\n        if head.Val == target {\n            return index\n        }\n        head = head.Next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc find(head: ListNode, target: Int) -&gt; Int {\n    var head: ListNode? = head\n    var index = 0\n    while head != nil {\n        if head?.val == target {\n            return index\n        }\n        head = head?.next\n        index += 1\n    }\n    return -1\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head, target) {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head: ListNode | null, target: number): number {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode? head, int target) {\n  int index = 0;\n  while (head != null) {\n    if (head.val == target) {\n      return index;\n    }\n    head = head.next;\n    index++;\n  }\n  return -1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\npub fn find&lt;T: PartialEq&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, target: T, index: i32) -&gt; i32 {\n    if head.borrow().val == target {\n        return index;\n    };\n    if let Some(node) = &amp;head.borrow_mut().next {\n        return find(node.clone(), target, index + 1);\n    }\n    return -1;\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.kt<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfun find(head: ListNode?, target: Int): Int {\n    var index = 0\n    var h = head\n    while (h != null) {\n        if (h.value == target) return index\n        h = h.next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.rb<pre><code>### \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 ###\ndef find(head, target)\n  index = 0\n  while head\n    return index if head.val == target\n    head = head.next\n    index += 1\n  end\n\n  -1\nend\n</code></pre> linked_list.zig<pre><code>// \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\nfn find(node: ?*inc.ListNode(i32), target: i32) i32 {\n    var head = node;\n    var index: i32 = 0;\n    while (head != null) {\n        if (head.?.val == target) return index;\n        head = head.?.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#422-arrays-vs-linked-lists","title":"4.2.2 \u00a0 Arrays vs. Linked Lists","text":"<p>The Table 4-1  summarizes the characteristics of arrays and linked lists, and it also compares their efficiencies in various operations. Because they utilize opposing storage strategies, their respective properties and operational efficiencies exhibit distinct contrasts.</p> <p> Table 4-1 \u00a0 Efficiency Comparison of Arrays and Linked Lists </p> Arrays Linked Lists Storage Contiguous Memory Space Dispersed Memory Space Capacity Expansion Fixed Length Flexible Expansion Memory Efficiency Less Memory per Element, Potential Space Wastage More Memory per Element Accessing Elements \\(O(1)\\) \\(O(n)\\) Adding Elements \\(O(n)\\) \\(O(1)\\) Deleting Elements \\(O(n)\\) \\(O(1)\\)"},{"location":"chapter_array_and_linkedlist/linked_list/#423-common-types-of-linked-lists","title":"4.2.3 \u00a0 Common Types of Linked Lists","text":"<p>As shown in the figure, there are three common types of linked lists.</p> <ul> <li>Singly Linked List: This is the standard linked list described earlier. Nodes in a singly linked list include a value and a reference to the next node. The first node is known as the head node, and the last node, which points to null (<code>None</code>), is the tail node.</li> <li>Circular Linked List: This is formed when the tail node of a singly linked list points back to the head node, creating a loop. In a circular linked list, any node can function as the head node.</li> <li>Doubly Linked List: In contrast to a singly linked list, a doubly linked list maintains references in two directions. Each node contains references (pointer) to both its successor (the next node) and predecessor (the previous node). Although doubly linked lists offer more flexibility for traversing in either direction, they also consume more memory space.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>class ListNode:\n    \"\"\"Bidirectional linked list node class\"\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # Node value\n        self.next: ListNode | None = None  # Reference to the successor node\n        self.prev: ListNode | None = None  # Reference to a predecessor node\n</code></pre> <pre><code>/* Bidirectional linked list node structure */\nstruct ListNode {\n    int val;         // Node value\n    ListNode *next;  // Pointer to the successor node\n    ListNode *prev;  // Pointer to the predecessor node\n    ListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // Constructor\n};\n</code></pre> <pre><code>/* Bidirectional linked list node class */\nclass ListNode {\n    int val;        // Node value\n    ListNode next;  // Reference to the next node\n    ListNode prev;  // Reference to the predecessor node\n    ListNode(int x) { val = x; }  // Constructor\n}\n</code></pre> <pre><code>/* Bidirectional linked list node class */\nclass ListNode(int x) {  // Constructor\n    int val = x;    // Node value\n    ListNode next;  // Reference to the next node\n    ListNode prev;  // Reference to the predecessor node\n}\n</code></pre> <pre><code>/* Bidirectional linked list node structure */\ntype DoublyListNode struct {\n    Val  int             // Node value\n    Next *DoublyListNode // Pointer to the successor node\n    Prev *DoublyListNode // Pointer to the predecessor node\n}\n\n// NewDoublyListNode initialization\nfunc NewDoublyListNode(val int) *DoublyListNode {\n    return &amp;DoublyListNode{\n        Val:  val,\n        Next: nil,\n        Prev: nil,\n    }\n}\n</code></pre> <pre><code>/* Bidirectional linked list node class */\nclass ListNode {\n    var val: Int // Node value\n    var next: ListNode? // Reference to the next node\n    var prev: ListNode? // Reference to the predecessor node\n\n    init(x: Int) { // Constructor\n        val = x\n    }\n}\n</code></pre> <pre><code>/* Bidirectional linked list node class */\nclass ListNode {\n    constructor(val, next, prev) {\n        this.val = val  ===  undefined ? 0 : val;        // Node value\n        this.next = next  ===  undefined ? null : next;  // Reference to the successor node\n        this.prev = prev  ===  undefined ? null : prev;  // Reference to the predecessor node\n    }\n}\n</code></pre> <pre><code>/* Bidirectional linked list node class */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    prev: ListNode | null;\n    constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\n        this.val = val  ===  undefined ? 0 : val;        // Node value\n        this.next = next  ===  undefined ? null : next;  // Reference to the successor node\n        this.prev = prev  ===  undefined ? null : prev;  // Reference to the predecessor node\n    }\n}\n</code></pre> <pre><code>/* Bidirectional linked list node class */\nclass ListNode {\n    int val;        // Node value\n    ListNode next;  // Reference to the next node\n    ListNode prev;  // Reference to the predecessor node\n    ListNode(this.val, [this.next, this.prev]);  // Constructor\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* Bidirectional linked list node type */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // Node value\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // Pointer to successor node\n    prev: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // Pointer to predecessor node\n}\n\n/* Constructors */\nimpl ListNode {\n    fn new(val: i32) -&gt; Self {\n        ListNode {\n            val,\n            next: None,\n            prev: None,\n        }\n    }\n}\n</code></pre> <pre><code>/* Bidirectional linked list node structure */\ntypedef struct ListNode {\n    int val;               // Node value\n    struct ListNode *next; // Pointer to the successor node\n    struct ListNode *prev; // Pointer to the predecessor node\n} ListNode;\n\n/* Constructors */\nListNode *newListNode(int val) {\n    ListNode *node, *next;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    node-&gt;prev = NULL;\n    return node;\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>// Bidirectional linked list node class\npub fn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = 0, // Node value\n        next: ?*Self = null, // Pointer to the successor node\n        prev: ?*Self = null, // Pointer to the predecessor node\n\n        // Constructor\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n            self.prev = null;\n        }\n    };\n}\n</code></pre> <p></p> <p> Figure 4-8 \u00a0 Common Types of Linked Lists </p>"},{"location":"chapter_array_and_linkedlist/linked_list/#424-typical-applications-of-linked-lists","title":"4.2.4 \u00a0 Typical Applications of Linked Lists","text":"<p>Singly linked lists are frequently utilized in implementing stacks, queues, hash tables, and graphs.</p> <ul> <li>Stacks and Queues: In singly linked lists, if insertions and deletions occur at the same end, it behaves like a stack (last-in-first-out). Conversely, if insertions are at one end and deletions at the other, it functions like a queue (first-in-first-out).</li> <li>Hash Tables: Linked lists are used in chaining, a popular method for resolving hash collisions. Here, all collided elements are grouped into a linked list.</li> <li>Graphs: Adjacency lists, a standard method for graph representation, associate each graph vertex with a linked list. This list contains elements that represent vertices connected to the corresponding vertex.</li> </ul> <p>Doubly linked lists are ideal for scenarios requiring rapid access to preceding and succeeding elements.</p> <ul> <li>Advanced Data Structures: In structures like red-black trees and B-trees, accessing a node's parent is essential. This is achieved by incorporating a reference to the parent node in each node, akin to a doubly linked list.</li> <li>Browser History: In web browsers, doubly linked lists facilitate navigating the history of visited pages when users click forward or back.</li> <li>LRU Algorithm: Doubly linked lists are apt for Least Recently Used (LRU) cache eviction algorithms, enabling swift identification of the least recently used data and facilitating fast node addition and removal.</li> </ul> <p>Circular linked lists are ideal for applications that require periodic operations, such as resource scheduling in operating systems.</p> <ul> <li>Round-Robin Scheduling Algorithm: In operating systems, the round-robin scheduling algorithm is a common CPU scheduling method, requiring cycling through a group of processes. Each process is assigned a time slice, and upon expiration, the CPU rotates to the next process. This cyclical operation can be efficiently realized using a circular linked list, allowing for a fair and time-shared system among all processes.</li> <li>Data Buffers: Circular linked lists are also used in data buffers, like in audio and video players, where the data stream is divided into multiple buffer blocks arranged in a circular fashion for seamless playback.</li> </ul>"},{"location":"chapter_array_and_linkedlist/list/","title":"4.3 \u00a0 List","text":"<p>A \"list\" is an abstract data structure concept that represents an ordered collection of elements, supporting operations such as element access, modification, addition, deletion, and traversal, without requiring users to consider capacity limitations. Lists can be implemented based on linked lists or arrays.</p> <ul> <li>A linked list inherently serves as a list, supporting operations for adding, deleting, searching, and modifying elements, with the flexibility to dynamically adjust its size.</li> <li>Arrays also support these operations, but due to their immutable length, they can be considered as a list with a length limit.</li> </ul> <p>When implementing lists using arrays, the immutability of length reduces the practicality of the list. This is because predicting the amount of data to be stored in advance is often challenging, making it difficult to choose an appropriate list length. If the length is too small, it may not meet the requirements; if too large, it may waste memory space.</p> <p>To solve this problem, we can implement lists using a \"dynamic array.\" It inherits the advantages of arrays and can dynamically expand during program execution.</p> <p>In fact, many programming languages' standard libraries implement lists using dynamic arrays, such as Python's <code>list</code>, Java's <code>ArrayList</code>, C++'s <code>vector</code>, and C#'s <code>List</code>. In the following discussion, we will consider \"list\" and \"dynamic array\" as synonymous concepts.</p>"},{"location":"chapter_array_and_linkedlist/list/#431-common-list-operations","title":"4.3.1 \u00a0 Common List Operations","text":""},{"location":"chapter_array_and_linkedlist/list/#1-initializing-a-list","title":"1. \u00a0 Initializing a List","text":"<p>We typically use two initialization methods: \"without initial values\" and \"with initial values\".</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig list.py<pre><code># Initialize list\n# Without initial values\nnums1: list[int] = []\n# With initial values\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> list.cpp<pre><code>/* Initialize list */\n// Note, in C++ the vector is the equivalent of nums described here\n// Without initial values\nvector&lt;int&gt; nums1;\n// With initial values\nvector&lt;int&gt; nums = { 1, 3, 2, 5, 4 };\n</code></pre> list.java<pre><code>/* Initialize list */\n// Without initial values\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;();\n// With initial values (note the element type should be the wrapper class Integer[] for int[])\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList&lt;Integer&gt; nums = new ArrayList&lt;&gt;(Arrays.asList(numbers));\n</code></pre> list.cs<pre><code>/* Initialize list */\n// Without initial values\nList&lt;int&gt; nums1 = [];\n// With initial values\nint[] numbers = [1, 3, 2, 5, 4];\nList&lt;int&gt; nums = [.. numbers];\n</code></pre> list_test.go<pre><code>/* Initialize list */\n// Without initial values\nnums1 := []int{}\n// With initial values\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> list.swift<pre><code>/* Initialize list */\n// Without initial values\nlet nums1: [Int] = []\n// With initial values\nvar nums = [1, 3, 2, 5, 4]\n</code></pre> list.js<pre><code>/* Initialize list */\n// Without initial values\nconst nums1 = [];\n// With initial values\nconst nums = [1, 3, 2, 5, 4];\n</code></pre> list.ts<pre><code>/* Initialize list */\n// Without initial values\nconst nums1: number[] = [];\n// With initial values\nconst nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> list.dart<pre><code>/* Initialize list */\n// Without initial values\nList&lt;int&gt; nums1 = [];\n// With initial values\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> list.rs<pre><code>/* Initialize list */\n// Without initial values\nlet nums1: Vec&lt;i32&gt; = Vec::new();\n// With initial values\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>\n</code></pre> list.zig<pre><code>// Initialize list\nvar nums = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums.deinit();\ntry nums.appendSlice(&amp;[_]i32{ 1, 3, 2, 5, 4 });\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#2-accessing-elements","title":"2. \u00a0 Accessing Elements","text":"<p>Lists are essentially arrays, thus they can access and update elements in \\(O(1)\\) time, which is very efficient.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig list.py<pre><code># Access elements\nnum: int = nums[1]  # Access the element at index 1\n\n# Update elements\nnums[1] = 0    # Update the element at index 1 to 0\n</code></pre> list.cpp<pre><code>/* Access elements */\nint num = nums[1];  // Access the element at index 1\n\n/* Update elements */\nnums[1] = 0;  // Update the element at index 1 to 0\n</code></pre> list.java<pre><code>/* Access elements */\nint num = nums.get(1);  // Access the element at index 1\n\n/* Update elements */\nnums.set(1, 0);  // Update the element at index 1 to 0\n</code></pre> list.cs<pre><code>/* Access elements */\nint num = nums[1];  // Access the element at index 1\n\n/* Update elements */\nnums[1] = 0;  // Update the element at index 1 to 0\n</code></pre> list_test.go<pre><code>/* Access elements */\nnum := nums[1]  // Access the element at index 1\n\n/* Update elements */\nnums[1] = 0     // Update the element at index 1 to 0\n</code></pre> list.swift<pre><code>/* Access elements */\nlet num = nums[1] // Access the element at index 1\n\n/* Update elements */\nnums[1] = 0 // Update the element at index 1 to 0\n</code></pre> list.js<pre><code>/* Access elements */\nconst num = nums[1];  // Access the element at index 1\n\n/* Update elements */\nnums[1] = 0;  // Update the element at index 1 to 0\n</code></pre> list.ts<pre><code>/* Access elements */\nconst num: number = nums[1];  // Access the element at index 1\n\n/* Update elements */\nnums[1] = 0;  // Update the element at index 1 to 0\n</code></pre> list.dart<pre><code>/* Access elements */\nint num = nums[1];  // Access the element at index 1\n\n/* Update elements */\nnums[1] = 0;  // Update the element at index 1 to 0\n</code></pre> list.rs<pre><code>/* Access elements */\nlet num: i32 = nums[1];  // Access the element at index 1\n/* Update elements */\nnums[1] = 0;             // Update the element at index 1 to 0\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>\n</code></pre> list.zig<pre><code>// Access elements\nvar num = nums.items[1]; // Access the element at index 1\n\n// Update elements\nnums.items[1] = 0; // Update the element at index 1 to 0  \n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#3-inserting-and-removing-elements","title":"3. \u00a0 Inserting and Removing Elements","text":"<p>Compared to arrays, lists offer more flexibility in adding and removing elements. While adding elements to the end of a list is an \\(O(1)\\) operation, the efficiency of inserting and removing elements elsewhere in the list remains the same as in arrays, with a time complexity of \\(O(n)\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig list.py<pre><code># Clear list\nnums.clear()\n\n# Append elements at the end\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n# Insert element in the middle\nnums.insert(3, 6)  # Insert number 6 at index 3\n\n# Remove elements\nnums.pop(3)        # Remove the element at index 3\n</code></pre> list.cpp<pre><code>/* Clear list */\nnums.clear();\n\n/* Append elements at the end */\nnums.push_back(1);\nnums.push_back(3);\nnums.push_back(2);\nnums.push_back(5);\nnums.push_back(4);\n\n/* Insert element in the middle */\nnums.insert(nums.begin() + 3, 6);  // Insert number 6 at index 3\n\n/* Remove elements */\nnums.erase(nums.begin() + 3);      // Remove the element at index 3\n</code></pre> list.java<pre><code>/* Clear list */\nnums.clear();\n\n/* Append elements at the end */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* Insert element in the middle */\nnums.add(3, 6);  // Insert number 6 at index 3\n\n/* Remove elements */\nnums.remove(3);  // Remove the element at index 3\n</code></pre> list.cs<pre><code>/* Clear list */\nnums.Clear();\n\n/* Append elements at the end */\nnums.Add(1);\nnums.Add(3);\nnums.Add(2);\nnums.Add(5);\nnums.Add(4);\n\n/* Insert element in the middle */\nnums.Insert(3, 6);\n\n/* Remove elements */\nnums.RemoveAt(3);\n</code></pre> list_test.go<pre><code>/* Clear list */\nnums = nil\n\n/* Append elements at the end */\nnums = append(nums, 1)\nnums = append(nums, 3)\nnums = append(nums, 2)\nnums = append(nums, 5)\nnums = append(nums, 4)\n\n/* Insert element in the middle */\nnums = append(nums[:3], append([]int{6}, nums[3:]...)...) // Insert number 6 at index 3\n\n/* Remove elements */\nnums = append(nums[:3], nums[4:]...) // Remove the element at index 3\n</code></pre> list.swift<pre><code>/* Clear list */\nnums.removeAll()\n\n/* Append elements at the end */\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n/* Insert element in the middle */\nnums.insert(6, at: 3) // Insert number 6 at index 3\n\n/* Remove elements */\nnums.remove(at: 3) // Remove the element at index 3\n</code></pre> list.js<pre><code>/* Clear list */\nnums.length = 0;\n\n/* Append elements at the end */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* Insert element in the middle */\nnums.splice(3, 0, 6);\n\n/* Remove elements */\nnums.splice(3, 1);\n</code></pre> list.ts<pre><code>/* Clear list */\nnums.length = 0;\n\n/* Append elements at the end */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* Insert element in the middle */\nnums.splice(3, 0, 6);\n\n/* Remove elements */\nnums.splice(3, 1);\n</code></pre> list.dart<pre><code>/* Clear list */\nnums.clear();\n\n/* Append elements at the end */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* Insert element in the middle */\nnums.insert(3, 6); // Insert number 6 at index 3\n\n/* Remove elements */\nnums.removeAt(3); // Remove the element at index 3\n</code></pre> list.rs<pre><code>/* Clear list */\nnums.clear();\n\n/* Append elements at the end */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* Insert element in the middle */\nnums.insert(3, 6);  // Insert number 6 at index 3\n\n/* Remove elements */\nnums.remove(3);    // Remove the element at index 3\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>\n</code></pre> list.zig<pre><code>// Clear list\nnums.clearRetainingCapacity();\n\n// Append elements at the end\ntry nums.append(1);\ntry nums.append(3);\ntry nums.append(2);\ntry nums.append(5);\ntry nums.append(4);\n\n// Insert element in the middle\ntry nums.insert(3, 6); // Insert number 6 at index 3\n\n// Remove elements\n_ = nums.orderedRemove(3); // Remove the element at index 3\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#4-iterating-the-list","title":"4. \u00a0 Iterating the List","text":"<p>Similar to arrays, lists can be iterated either by using indices or by directly iterating through each element.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig list.py<pre><code># Iterate through the list by index\ncount = 0\nfor i in range(len(nums)):\n    count += nums[i]\n\n# Iterate directly through list elements\nfor num in nums:\n    count += num\n</code></pre> list.cpp<pre><code>/* Iterate through the list by index */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums[i];\n}\n\n/* Iterate directly through list elements */\ncount = 0;\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.java<pre><code>/* Iterate through the list by index */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums.get(i);\n}\n\n/* Iterate directly through list elements */\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.cs<pre><code>/* Iterate through the list by index */\nint count = 0;\nfor (int i = 0; i &lt; nums.Count; i++) {\n    count += nums[i];\n}\n\n/* Iterate directly through list elements */\ncount = 0;\nforeach (int num in nums) {\n    count += num;\n}\n</code></pre> list_test.go<pre><code>/* Iterate through the list by index */\ncount := 0\nfor i := 0; i &lt; len(nums); i++ {\n    count += nums[i]\n}\n\n/* Iterate directly through list elements */\ncount = 0\nfor _, num := range nums {\n    count += num\n}\n</code></pre> list.swift<pre><code>/* Iterate through the list by index */\nvar count = 0\nfor i in nums.indices {\n    count += nums[i]\n}\n\n/* Iterate directly through list elements */\ncount = 0\nfor num in nums {\n    count += num\n}\n</code></pre> list.js<pre><code>/* Iterate through the list by index */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* Iterate directly through list elements */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.ts<pre><code>/* Iterate through the list by index */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* Iterate directly through list elements */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.dart<pre><code>/* Iterate through the list by index */\nint count = 0;\nfor (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* Iterate directly through list elements */\ncount = 0;\nfor (var num in nums) {\n    count += num;\n}\n</code></pre> list.rs<pre><code>// Iterate through the list by index\nlet mut _count = 0;\nfor i in 0..nums.len() {\n    _count += nums[i];\n}\n\n// Iterate directly through list elements\n_count = 0;\nfor num in &amp;nums {\n    _count += num;\n}\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>\n</code></pre> list.zig<pre><code>// Iterate through the list by index\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i &lt; nums.items.len) : (i += 1) {\n    count += nums[i];\n}\n\n// Iterate directly through list elements\ncount = 0;\nfor (nums.items) |num| {\n    count += num;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#5-concatenating-lists","title":"5. \u00a0 Concatenating Lists","text":"<p>Given a new list <code>nums1</code>, we can append it to the end of the original list.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig list.py<pre><code># Concatenate two lists\nnums1: list[int] = [6, 8, 7, 10, 9]\nnums += nums1  # Concatenate nums1 to the end of nums\n</code></pre> list.cpp<pre><code>/* Concatenate two lists */\nvector&lt;int&gt; nums1 = { 6, 8, 7, 10, 9 };\n// Concatenate nums1 to the end of nums\nnums.insert(nums.end(), nums1.begin(), nums1.end());\n</code></pre> list.java<pre><code>/* Concatenate two lists */\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nnums.addAll(nums1);  // Concatenate nums1 to the end of nums\n</code></pre> list.cs<pre><code>/* Concatenate two lists */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.AddRange(nums1);  // Concatenate nums1 to the end of nums\n</code></pre> list_test.go<pre><code>/* Concatenate two lists */\nnums1 := []int{6, 8, 7, 10, 9}\nnums = append(nums, nums1...)  // Concatenate nums1 to the end of nums\n</code></pre> list.swift<pre><code>/* Concatenate two lists */\nlet nums1 = [6, 8, 7, 10, 9]\nnums.append(contentsOf: nums1) // Concatenate nums1 to the end of nums\n</code></pre> list.js<pre><code>/* Concatenate two lists */\nconst nums1 = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // Concatenate nums1 to the end of nums\n</code></pre> list.ts<pre><code>/* Concatenate two lists */\nconst nums1: number[] = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // Concatenate nums1 to the end of nums\n</code></pre> list.dart<pre><code>/* Concatenate two lists */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.addAll(nums1);  // Concatenate nums1 to the end of nums\n</code></pre> list.rs<pre><code>/* Concatenate two lists */\nlet nums1: Vec&lt;i32&gt; = vec![6, 8, 7, 10, 9];\nnums.extend(nums1);\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>\n</code></pre> list.zig<pre><code>// Concatenate two lists\nvar nums1 = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums1.deinit();\ntry nums1.appendSlice(&amp;[_]i32{ 6, 8, 7, 10, 9 });\ntry nums.insertSlice(nums.items.len, nums1.items); // Concatenate nums1 to the end of nums\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#6-sorting-the-list","title":"6. \u00a0 Sorting the List","text":"<p>Once the list is sorted, we can employ algorithms commonly used in array-related algorithm problems, such as \"binary search\" and \"two-pointer\" algorithms.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig list.py<pre><code># Sort the list\nnums.sort()  # After sorting, the list elements are in ascending order\n</code></pre> list.cpp<pre><code>/* Sort the list */\nsort(nums.begin(), nums.end());  // After sorting, the list elements are in ascending order\n</code></pre> list.java<pre><code>/* Sort the list */\nCollections.sort(nums);  // After sorting, the list elements are in ascending order\n</code></pre> list.cs<pre><code>/* Sort the list */\nnums.Sort(); // After sorting, the list elements are in ascending order\n</code></pre> list_test.go<pre><code>/* Sort the list */\nsort.Ints(nums)  // After sorting, the list elements are in ascending order\n</code></pre> list.swift<pre><code>/* Sort the list */\nnums.sort() // After sorting, the list elements are in ascending order\n</code></pre> list.js<pre><code>/* Sort the list */  \nnums.sort((a, b) =&gt; a - b);  // After sorting, the list elements are in ascending order\n</code></pre> list.ts<pre><code>/* Sort the list */\nnums.sort((a, b) =&gt; a - b);  // After sorting, the list elements are in ascending order\n</code></pre> list.dart<pre><code>/* Sort the list */\nnums.sort(); // After sorting, the list elements are in ascending order\n</code></pre> list.rs<pre><code>/* Sort the list */\nnums.sort(); // After sorting, the list elements are in ascending order\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>\n</code></pre> list.zig<pre><code>// Sort the list\nstd.sort.sort(i32, nums.items, {}, comptime std.sort.asc(i32));\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#432-list-implementation","title":"4.3.2 \u00a0 List Implementation","text":"<p>Many programming languages come with built-in lists, including Java, C++, Python, etc. Their implementations tend to be intricate, featuring carefully considered settings for various parameters, like initial capacity and expansion factors. Readers who are curious can delve into the source code for further learning.</p> <p>To enhance our understanding of how lists work, we will attempt to implement a simplified version of a list, focusing on three crucial design aspects:</p> <ul> <li>Initial Capacity: Choose a reasonable initial capacity for the array. In this example, we choose 10 as the initial capacity.</li> <li>Size Recording: Declare a variable <code>size</code> to record the current number of elements in the list, updating in real-time with element insertion and deletion. With this variable, we can locate the end of the list and determine whether expansion is needed.</li> <li>Expansion Mechanism: If the list reaches full capacity upon an element insertion, an expansion process is required. This involves creating a larger array based on the expansion factor, and then transferring all elements from the current array to the new one. In this example, we stipulate that the array size should double with each expansion.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_list.py<pre><code>class MyList:\n    \"\"\"\u5217\u8868\u7c7b\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._capacity: int = 10  # \u5217\u8868\u5bb9\u91cf\n        self._arr: list[int] = [0] * self._capacity  # \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n        self._size: int = 0  # \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        self._extend_ratio: int = 2  # \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\"\"\"\n        return self._size\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u5217\u8868\u5bb9\u91cf\"\"\"\n        return self._capacity\n\n    def get(self, index: int) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u5143\u7d20\"\"\"\n        # \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        return self._arr[index]\n\n    def set(self, num: int, index: int):\n        \"\"\"\u66f4\u65b0\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        self._arr[index] = num\n\n    def add(self, num: int):\n        \"\"\"\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\"\"\"\n        # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self.size() == self.capacity():\n            self.extend_capacity()\n        self._arr[self._size] = num\n        self._size += 1\n\n    def insert(self, num: int, index: int):\n        \"\"\"\u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self._size == self.capacity():\n            self.extend_capacity()\n        # \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for j in range(self._size - 1, index - 1, -1):\n            self._arr[j + 1] = self._arr[j]\n        self._arr[index] = num\n        # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self._size += 1\n\n    def remove(self, index: int) -&gt; int:\n        \"\"\"\u5220\u9664\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        num = self._arr[index]\n        # \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for j in range(index, self._size - 1):\n            self._arr[j] = self._arr[j + 1]\n        # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self._size -= 1\n        # \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num\n\n    def extend_capacity(self):\n        \"\"\"\u5217\u8868\u6269\u5bb9\"\"\"\n        # \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 _extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        self._arr = self._arr + [0] * self.capacity() * (self._extend_ratio - 1)\n        # \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        self._capacity = len(self._arr)\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868\"\"\"\n        return self._arr[: self._size]\n</code></pre> my_list.cpp<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n  private:\n    int *arr;             // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    int arrCapacity = 10; // \u5217\u8868\u5bb9\u91cf\n    int arrSize = 0;      // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    int extendRatio = 2;   // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~MyList() {\n        delete[] arr;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    int size() {\n        return arrSize;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    int capacity() {\n        return arrCapacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    int get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    void add(int num) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size() == capacity())\n            extendCapacity();\n        arr[size()] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size() == capacity())\n            extendCapacity();\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (int j = size() - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    int remove(int index) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (int j = index; j &lt; size() - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    void extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\n        int newCapacity = capacity() * extendRatio;\n        int *tmp = arr;\n        arr = new int[newCapacity];\n        // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        for (int i = 0; i &lt; size(); i++) {\n            arr[i] = tmp[i];\n        }\n        // \u91ca\u653e\u5185\u5b58\n        delete[] tmp;\n        arrCapacity = newCapacity;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Vector \u7528\u4e8e\u6253\u5370 */\n    vector&lt;int&gt; toVector() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        vector&lt;int&gt; vec(size());\n        for (int i = 0; i &lt; size(); i++) {\n            vec[i] = arr[i];\n        }\n        return vec;\n    }\n};\n</code></pre> my_list.java<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private int[] arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private int capacity = 10; // \u5217\u8868\u5bb9\u91cf\n    private int size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private int extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public MyList() {\n        arr = new int[capacity];\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\n    public int size() {\n        return size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    public int capacity() {\n        return capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    public int get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    public void add(int num) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity();\n        arr[size] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    public void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity();\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (int j = size - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    public int remove(int index) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (int j = index; j &lt; size - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    public void extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        arr = Arrays.copyOf(arr, capacity() * extendRatio);\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        capacity = arr.length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    public int[] toArray() {\n        int size = size();\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] arr = new int[size];\n        for (int i = 0; i &lt; size; i++) {\n            arr[i] = get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.cs<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private int[] arr;           // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private int arrCapacity = 10;    // \u5217\u8868\u5bb9\u91cf\n    private int arrSize = 0;         // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private readonly int extendRatio = 2;  // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    public int Size() {\n        return arrSize;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    public int Capacity() {\n        return arrCapacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    public int Get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public void Set(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    public void Add(int num) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        arr[arrSize] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    public void Insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (int j = arrSize - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    public int Remove(int index) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (int j = index; j &lt; arrSize - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    public void ExtendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a arrCapacity * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        Array.Resize(ref arr, arrCapacity * extendRatio);\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        arrCapacity = arr.Length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    public int[] ToArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] arr = new int[arrSize];\n        for (int i = 0; i &lt; arrSize; i++) {\n            arr[i] = Get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.go<pre><code>/* \u5217\u8868\u7c7b */\ntype myList struct {\n    arrCapacity int\n    arr         []int\n    arrSize     int\n    extendRatio int\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newMyList() *myList {\n    return &amp;myList{\n        arrCapacity: 10,              // \u5217\u8868\u5bb9\u91cf\n        arr:         make([]int, 10), // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n        arrSize:     0,               // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        extendRatio: 2,               // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n    }\n}\n\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\nfunc (l *myList) size() int {\n    return l.arrSize\n}\n\n/*  \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc (l *myList) capacity() int {\n    return l.arrCapacity\n}\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc (l *myList) get(index int) int {\n    // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    return l.arr[index]\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc (l *myList) set(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    l.arr[index] = num\n}\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc (l *myList) add(num int) {\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    l.arr[l.arrSize] = num\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    l.arrSize++\n}\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc (l *myList) insert(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for j := l.arrSize - 1; j &gt;= index; j-- {\n        l.arr[j+1] = l.arr[j]\n    }\n    l.arr[index] = num\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    l.arrSize++\n}\n\n/* \u5220\u9664\u5143\u7d20 */\nfunc (l *myList) remove(index int) int {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    num := l.arr[index]\n    // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for j := index; j &lt; l.arrSize-1; j++ {\n        l.arr[j] = l.arr[j+1]\n    }\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    l.arrSize--\n    // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n    return num\n}\n\n/* \u5217\u8868\u6269\u5bb9 */\nfunc (l *myList) extendCapacity() {\n    // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    l.arr = append(l.arr, make([]int, l.arrCapacity*(l.extendRatio-1))...)\n    // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n    l.arrCapacity = len(l.arr)\n}\n\n/* \u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868 */\nfunc (l *myList) toArray() []int {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    return l.arr[:l.arrSize]\n}\n</code></pre> my_list.swift<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private var arr: [Int] // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private var _capacity: Int // \u5217\u8868\u5bb9\u91cf\n    private var _size: Int // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private let extendRatio: Int // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init() {\n        _capacity = 10\n        _size = 0\n        extendRatio = 2\n        arr = Array(repeating: 0, count: _capacity)\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        _capacity\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    func get(index: Int) -&gt; Int {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u9519\u8bef\uff0c\u4e0b\u540c\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        return arr[index]\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    func set(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        arr[index] = num\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    func add(num: Int) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if size() == capacity() {\n            extendCapacity()\n        }\n        arr[size()] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        _size += 1\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    func insert(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if size() == capacity() {\n            extendCapacity()\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for j in (index ..&lt; size()).reversed() {\n            arr[j + 1] = arr[j]\n        }\n        arr[index] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        _size += 1\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    @discardableResult\n    func remove(index: Int) -&gt; Int {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        let num = arr[index]\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for j in index ..&lt; (size() - 1) {\n            arr[j] = arr[j + 1]\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        _size -= 1\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    func extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        arr = arr + Array(repeating: 0, count: capacity() * (extendRatio - 1))\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        _capacity = arr.count\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    func toArray() -&gt; [Int] {\n        Array(arr.prefix(size()))\n    }\n}\n</code></pre> my_list.js<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    #arr = new Array(); // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    #capacity = 10; // \u5217\u8868\u5bb9\u91cf\n    #size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    #extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#arr = new Array(this.#capacity);\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    size() {\n        return this.#size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    capacity() {\n        return this.#capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    get(index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        return this.#arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    set(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        this.#arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    add(num) {\n        // \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\n        this.#arr[this.#size] = num;\n        this.#size++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    insert(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (let j = this.#size - 1; j &gt;= index; j--) {\n            this.#arr[j + 1] = this.#arr[j];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this.#arr[index] = num;\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    remove(index) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        let num = this.#arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (let j = index; j &lt; this.#size - 1; j++) {\n            this.#arr[j] = this.#arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this.#size--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        this.#arr = this.#arr.concat(\n            new Array(this.capacity() * (this.#extendRatio - 1))\n        );\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        this.#capacity = this.#arr.length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    toArray() {\n        let size = this.size();\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.ts<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private arr: Array&lt;number&gt;; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private _capacity: number = 10; // \u5217\u8868\u5bb9\u91cf\n    private _size: number = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private extendRatio: number = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.arr = new Array(this._capacity);\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    public size(): number {\n        return this._size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    public capacity(): number {\n        return this._capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    public get(index: number): number {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        return this.arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public set(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        this.arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    public add(num: number): void {\n        // \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\n        if (this._size === this._capacity) this.extendCapacity();\n        // \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\n        this.arr[this._size] = num;\n        this._size++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    public insert(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (this._size === this._capacity) {\n            this.extendCapacity();\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (let j = this._size - 1; j &gt;= index; j--) {\n            this.arr[j + 1] = this.arr[j];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this.arr[index] = num;\n        this._size++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    public remove(index: number): number {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        let num = this.arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (let j = index; j &lt; this._size - 1; j++) {\n            this.arr[j] = this.arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this._size--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    public extendCapacity(): void {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        this.arr = this.arr.concat(\n            new Array(this.capacity() * (this.extendRatio - 1))\n        );\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        this._capacity = this.arr.length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    public toArray(): number[] {\n        let size = this.size();\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.dart<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n  late List&lt;int&gt; _arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n  int _capacity = 10; // \u5217\u8868\u5bb9\u91cf\n  int _size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n  int _extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  MyList() {\n    _arr = List.filled(_capacity, 0);\n  }\n\n  /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n  int size() =&gt; _size;\n\n  /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n  int capacity() =&gt; _capacity;\n\n  /* \u8bbf\u95ee\u5143\u7d20 */\n  int get(int index) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    return _arr[index];\n  }\n\n  /* \u66f4\u65b0\u5143\u7d20 */\n  void set(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    _arr[index] = _num;\n  }\n\n  /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n  void add(int _num) {\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if (_size == _capacity) extendCapacity();\n    _arr[_size] = _num;\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    _size++;\n  }\n\n  /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n  void insert(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if (_size == _capacity) extendCapacity();\n    // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (var j = _size - 1; j &gt;= index; j--) {\n      _arr[j + 1] = _arr[j];\n    }\n    _arr[index] = _num;\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    _size++;\n  }\n\n  /* \u5220\u9664\u5143\u7d20 */\n  int remove(int index) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    int _num = _arr[index];\n    // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (var j = index; j &lt; _size - 1; j++) {\n      _arr[j] = _arr[j + 1];\n    }\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    _size--;\n    // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n    return _num;\n  }\n\n  /* \u5217\u8868\u6269\u5bb9 */\n  void extendCapacity() {\n    // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 _extendRatio \u500d\u7684\u65b0\u6570\u7ec4\n    final _newNums = List.filled(_capacity * _extendRatio, 0);\n    // \u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    List.copyRange(_newNums, 0, _arr);\n    // \u66f4\u65b0 _arr \u7684\u5f15\u7528\n    _arr = _newNums;\n    // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n    _capacity = _arr.length;\n  }\n\n  /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n  List&lt;int&gt; toArray() {\n    List&lt;int&gt; arr = [];\n    for (var i = 0; i &lt; _size; i++) {\n      arr.add(get(i));\n    }\n    return arr;\n  }\n}\n</code></pre> my_list.rs<pre><code>/* \u5217\u8868\u7c7b */\n#[allow(dead_code)]\nstruct MyList {\n    arr: Vec&lt;i32&gt;,       // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    capacity: usize,     // \u5217\u8868\u5bb9\u91cf\n    size: usize,         // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    extend_ratio: usize, // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n}\n\n#[allow(unused, unused_comparisons)]\nimpl MyList {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(capacity: usize) -&gt; Self {\n        let mut vec = Vec::new();\n        vec.resize(capacity, 0);\n        Self {\n            arr: vec,\n            capacity,\n            size: 0,\n            extend_ratio: 2,\n        }\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    pub fn size(&amp;self) -&gt; usize {\n        return self.size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    pub fn capacity(&amp;self) -&gt; usize {\n        return self.capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    pub fn get(&amp;self, index: usize) -&gt; i32 {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if index &gt;= self.size {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        return self.arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    pub fn set(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        self.arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    pub fn add(&amp;mut self, num: i32) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        self.arr[self.size] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self.size += 1;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    pub fn insert(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size() {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for j in (index..self.size).rev() {\n            self.arr[j + 1] = self.arr[j];\n        }\n        self.arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self.size += 1;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    pub fn remove(&amp;mut self, index: usize) -&gt; i32 {\n        if index &gt;= self.size() {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        let num = self.arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for j in (index..self.size - 1) {\n            self.arr[j] = self.arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self.size -= 1;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    pub fn extend_capacity(&amp;mut self) {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        let new_capacity = self.capacity * self.extend_ratio;\n        self.arr.resize(new_capacity, 0);\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        self.capacity = new_capacity;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    pub fn to_array(&amp;mut self) -&gt; Vec&lt;i32&gt; {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        let mut arr = Vec::new();\n        for i in 0..self.size {\n            arr.push(self.get(i));\n        }\n        arr\n    }\n}\n</code></pre> my_list.c<pre><code>/* \u5217\u8868\u7c7b */\ntypedef struct {\n    int *arr;        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    int capacity;    // \u5217\u8868\u5bb9\u91cf\n    int size;        // \u5217\u8868\u5927\u5c0f\n    int extendRatio; // \u5217\u8868\u6bcf\u6b21\u6269\u5bb9\u7684\u500d\u6570\n} MyList;\n\n/* \u6784\u9020\u51fd\u6570 */\nMyList *newMyList() {\n    MyList *nums = malloc(sizeof(MyList));\n    nums-&gt;capacity = 10;\n    nums-&gt;arr = malloc(sizeof(int) * nums-&gt;capacity);\n    nums-&gt;size = 0;\n    nums-&gt;extendRatio = 2;\n    return nums;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delMyList(MyList *nums) {\n    free(nums-&gt;arr);\n    free(nums);\n}\n\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6 */\nint size(MyList *nums) {\n    return nums-&gt;size;\n}\n\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity(MyList *nums) {\n    return nums-&gt;capacity;\n}\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    return nums-&gt;arr[index];\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    nums-&gt;arr[index] = num;\n}\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(MyList *nums, int num) {\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // \u6269\u5bb9\n    }\n    nums-&gt;arr[size(nums)] = num;\n    nums-&gt;size++;\n}\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // \u6269\u5bb9\n    }\n    for (int i = size(nums); i &gt; index; --i) {\n        nums-&gt;arr[i] = nums-&gt;arr[i - 1];\n    }\n    nums-&gt;arr[index] = num;\n    nums-&gt;size++;\n}\n\n/* \u5220\u9664\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nint removeItem(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    int num = nums-&gt;arr[index];\n    for (int i = index; i &lt; size(nums) - 1; i++) {\n        nums-&gt;arr[i] = nums-&gt;arr[i + 1];\n    }\n    nums-&gt;size--;\n    return num;\n}\n\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity(MyList *nums) {\n    // \u5148\u5206\u914d\u7a7a\u95f4\n    int newCapacity = capacity(nums) * nums-&gt;extendRatio;\n    int *extend = (int *)malloc(sizeof(int) * newCapacity);\n    int *temp = nums-&gt;arr;\n\n    // \u62f7\u8d1d\u65e7\u6570\u636e\u5230\u65b0\u6570\u636e\n    for (int i = 0; i &lt; size(nums); i++)\n        extend[i] = nums-&gt;arr[i];\n\n    // \u91ca\u653e\u65e7\u6570\u636e\n    free(temp);\n\n    // \u66f4\u65b0\u65b0\u6570\u636e\n    nums-&gt;arr = extend;\n    nums-&gt;capacity = newCapacity;\n}\n\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Array \u7528\u4e8e\u6253\u5370 */\nint *toArray(MyList *nums) {\n    return nums-&gt;arr;\n}\n</code></pre> my_list.kt<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private var arr: IntArray = intArrayOf() // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private var capacity = 10 // \u5217\u8868\u5bb9\u91cf\n    private var size = 0 // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private var extendRatio = 2 // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u51fd\u6570 */\n    init {\n        arr = IntArray(capacity)\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\n    fun size(): Int {\n        return size\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    fun capacity(): Int {\n        return capacity\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    fun get(index: Int): Int {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException()\n        return arr[index]\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    fun set(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        arr[index] = num\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    fun add(num: Int) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity()\n        arr[size] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    fun insert(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity()\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (j in size - 1 downTo index)\n            arr[j + 1] = arr[j]\n        arr[index] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    fun remove(index: Int): Int {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        val num: Int = arr[index]\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (j in index..&lt;size - 1)\n            arr[j] = arr[j + 1]\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size--\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    fun extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        arr = arr.copyOf(capacity() * extendRatio)\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        capacity = arr.size\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    fun toArray(): IntArray {\n        val size = size()\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        val arr = IntArray(size)\n        for (i in 0..&lt;size) {\n            arr[i] = get(i)\n        }\n        return arr\n    }\n}\n</code></pre> my_list.rb<pre><code>### \u5217\u8868\u7c7b ###\nclass MyList\n  attr_reader :size       # \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n  attr_reader :capacity   # \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\n\n  ### \u6784\u9020\u65b9\u6cd5 ###\n  def initialize\n    @capacity = 10\n    @size = 0\n    @extend_ratio = 2\n    @arr = Array.new capacity\n  end\n\n  ### \u8bbf\u95ee\u5143\u7d20 ###\n  def get(index)\n    # \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    @arr[index]\n  end\n\n  ### \u8bbf\u95ee\u5143\u7d20 ###\n  def set(index, num)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    @arr[index] = num\n  end\n\n  ### \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 ###\n  def add(num)\n    # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    extend_capacity if size == capacity\n    @arr[size] = num\n\n    # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    @size += 1\n  end\n\n  ### \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 ###\n  def insert(index, num)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n\n    # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    extend_capacity if size == capacity\n\n    # \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for j in (size - 1).downto(index)\n      @arr[j + 1] = @arr[j] \n    end\n    @arr[index] = num\n\n    # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    @size += 1\n  end\n\n  ### \u5220\u9664\u5143\u7d20 ###\n  def remove(index)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    num = @arr[index]\n\n    # \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for j in index...size\n      @arr[j] = @arr[j + 1]\n    end\n\n    # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    @size -= 1\n\n    # \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n    num\n  end\n\n  ### \u5217\u8868\u6269\u5bb9 ###\n  def extend_capacity\n    # \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    arr = @arr.dup + Array.new(capacity * (@extend_ratio - 1))\n    # \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n    @capacity = arr.length\n  end\n\n  ### \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 ###\n  def to_array\n    sz = size\n    # \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    arr = Array.new sz\n    for i in 0...sz\n      arr[i] = get i\n    end\n    arr\n  end\nend\n</code></pre> my_list.zig<pre><code>// \u5217\u8868\u7c7b\nfn MyList(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        arr: []T = undefined,                        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n        arrCapacity: usize = 10,                     // \u5217\u8868\u5bb9\u91cf\n        numSize: usize = 0,                           // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        extendRatio: usize = 2,                       // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u5217\u8868\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.arr = try self.mem_allocator.alloc(T, self.arrCapacity);\n            @memset(self.arr, @as(T, 0));\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        pub fn size(self: *Self) usize {\n            return self.numSize;\n        }\n\n        // \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\n        pub fn capacity(self: *Self) usize {\n            return self.arrCapacity;\n        }\n\n        // \u8bbf\u95ee\u5143\u7d20\n        pub fn get(self: *Self, index: usize) T {\n            // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            return self.arr[index];\n        }  \n\n        // \u66f4\u65b0\u5143\u7d20\n        pub fn set(self: *Self, index: usize, num: T) void {\n            // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            self.arr[index] = num;\n        }  \n\n        // \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\n        pub fn add(self: *Self, num: T) !void {\n            // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n            if (self.size() == self.capacity()) try self.extendCapacity();\n            self.arr[self.size()] = num;\n            // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n            self.numSize += 1;\n        }  \n\n        // \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\n        pub fn insert(self: *Self, index: usize, num: T) !void {\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n            if (self.size() == self.capacity()) try self.extendCapacity();\n            // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n            var j = self.size() - 1;\n            while (j &gt;= index) : (j -= 1) {\n                self.arr[j + 1] = self.arr[j];\n            }\n            self.arr[index] = num;\n            // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n            self.numSize += 1;\n        }\n\n        // \u5220\u9664\u5143\u7d20\n        pub fn remove(self: *Self, index: usize) T {\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            var num = self.arr[index];\n            // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n            var j = index;\n            while (j &lt; self.size() - 1) : (j += 1) {\n                self.arr[j] = self.arr[j + 1];\n            }\n            // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n            self.numSize -= 1;\n            // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n            return num;\n        }\n\n        // \u5217\u8868\u6269\u5bb9\n        pub fn extendCapacity(self: *Self) !void {\n            // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n            var newCapacity = self.capacity() * self.extendRatio;\n            var extend = try self.mem_allocator.alloc(T, newCapacity);\n            @memset(extend, @as(T, 0));\n            // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n            std.mem.copy(T, extend, self.arr);\n            self.arr = extend;\n            // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n            self.arrCapacity = newCapacity;\n        }\n\n        // \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n            var arr = try self.mem_allocator.alloc(T, self.size());\n           @memset(arr, @as(T, 0));\n            for (arr, 0..) |*num, i| {\n                num.* = self.get(i);\n            }\n            return arr;\n        }\n    };\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/","title":"4.4 \u00a0 Memory and Cache *","text":"<p>In the first two sections of this chapter, we explored arrays and linked lists, two fundamental and important data structures, representing \"continuous storage\" and \"dispersed storage\" respectively.</p> <p>In fact, the physical structure largely determines the efficiency of a program's use of memory and cache, which in turn affects the overall performance of the algorithm.</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#441-computer-storage-devices","title":"4.4.1 \u00a0 Computer Storage Devices","text":"<p>There are three types of storage devices in computers: \"hard disk,\" \"random-access memory (RAM),\" and \"cache memory.\" The following table shows their different roles and performance characteristics in computer systems.</p> <p> Table 4-2 \u00a0 Computer Storage Devices </p> Hard Disk Memory Cache Usage Long-term storage of data, including OS, programs, files, etc. Temporary storage of currently running programs and data being processed Stores frequently accessed data and instructions, reducing the number of CPU accesses to memory Volatility Data is not lost after power off Data is lost after power off Data is lost after power off Capacity Larger, TB level Smaller, GB level Very small, MB level Speed Slower, several hundred to thousands MB/s Faster, several tens of GB/s Very fast, several tens to hundreds of GB/s Price Cheaper, several cents to yuan / GB More expensive, tens to hundreds of yuan / GB Very expensive, priced with CPU <p>We can imagine the computer storage system as a pyramid structure shown in the Figure 4-9 . The storage devices closer to the top of the pyramid are faster, have smaller capacity, and are more costly. This multi-level design is not accidental, but the result of careful consideration by computer scientists and engineers.</p> <ul> <li>Hard disks are difficult to replace with memory. Firstly, data in memory is lost after power off, making it unsuitable for long-term data storage; secondly, the cost of memory is dozens of times that of hard disks, making it difficult to popularize in the consumer market.</li> <li>It is difficult for caches to have both large capacity and high speed. As the capacity of L1, L2, L3 caches gradually increases, their physical size becomes larger, increasing the physical distance from the CPU core, leading to increased data transfer time and higher element access latency. Under current technology, a multi-level cache structure is the best balance between capacity, speed, and cost.</li> </ul> <p></p> <p> Figure 4-9 \u00a0 Computer Storage System </p> <p>Note</p> <p>The storage hierarchy of computers reflects a delicate balance between speed, capacity, and cost. In fact, this kind of trade-off is common in all industrial fields, requiring us to find the best balance between different advantages and limitations.</p> <p>Overall, hard disks are used for long-term storage of large amounts of data, memory is used for temporary storage of data being processed during program execution, and cache is used to store frequently accessed data and instructions to improve program execution efficiency. Together, they ensure the efficient operation of computer systems.</p> <p>As shown in the Figure 4-10 , during program execution, data is read from the hard disk into memory for CPU computation. The cache can be considered a part of the CPU, smartly loading data from memory to provide fast data access to the CPU, significantly enhancing program execution efficiency and reducing reliance on slower memory.</p> <p></p> <p> Figure 4-10 \u00a0 Data Flow Between Hard Disk, Memory, and Cache </p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#442-memory-efficiency-of-data-structures","title":"4.4.2 \u00a0 Memory Efficiency of Data Structures","text":"<p>In terms of memory space utilization, arrays and linked lists have their advantages and limitations.</p> <p>On one hand, memory is limited and cannot be shared by multiple programs, so we hope that data structures can use space as efficiently as possible. The elements of an array are tightly packed without extra space for storing references (pointers) between linked list nodes, making them more space-efficient. However, arrays require allocating sufficient continuous memory space at once, which may lead to memory waste, and array expansion also requires additional time and space costs. In contrast, linked lists allocate and reclaim memory dynamically on a per-node basis, providing greater flexibility.</p> <p>On the other hand, during program execution, as memory is repeatedly allocated and released, the degree of fragmentation of free memory becomes higher, leading to reduced memory utilization efficiency. Arrays, due to their continuous storage method, are relatively less likely to cause memory fragmentation. In contrast, the elements of a linked list are dispersedly stored, and frequent insertion and deletion operations make memory fragmentation more likely.</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#443-cache-efficiency-of-data-structures","title":"4.4.3 \u00a0 Cache Efficiency of Data Structures","text":"<p>Although caches are much smaller in space capacity than memory, they are much faster and play a crucial role in program execution speed. Since the cache's capacity is limited and can only store a small part of frequently accessed data, when the CPU tries to access data not in the cache, a \"cache miss\" occurs, forcing the CPU to load the needed data from slower memory.</p> <p>Clearly, the fewer the cache misses, the higher the CPU's data read-write efficiency, and the better the program performance. The proportion of successful data retrieval from the cache by the CPU is called the \"cache hit rate,\" a metric often used to measure cache efficiency.</p> <p>To achieve higher efficiency, caches adopt the following data loading mechanisms.</p> <ul> <li>Cache Lines: Caches don't store and load data byte by byte but in units of cache lines. Compared to byte-by-byte transfer, the transmission of cache lines is more efficient.</li> <li>Prefetch Mechanism: Processors try to predict data access patterns (such as sequential access, fixed stride jumping access, etc.) and load data into the cache according to specific patterns to improve the hit rate.</li> <li>Spatial Locality: If data is accessed, data nearby is likely to be accessed in the near future. Therefore, when loading certain data, the cache also loads nearby data to improve the hit rate.</li> <li>Temporal Locality: If data is accessed, it's likely to be accessed again in the near future. Caches use this principle to retain recently accessed data to improve the hit rate.</li> </ul> <p>In fact, arrays and linked lists have different cache utilization efficiencies, mainly reflected in the following aspects.</p> <ul> <li>Occupied Space: Linked list elements occupy more space than array elements, resulting in less effective data volume in the cache.</li> <li>Cache Lines: Linked list data is scattered throughout memory, and since caches load \"by line,\" the proportion of loading invalid data is higher.</li> <li>Prefetch Mechanism: The data access pattern of arrays is more \"predictable\" than that of linked lists, meaning the system is more likely to guess which data will be loaded next.</li> <li>Spatial Locality: Arrays are stored in concentrated memory spaces, so the data near the loaded data is more likely to be accessed next.</li> </ul> <p>Overall, arrays have a higher cache hit rate and are generally more efficient in operation than linked lists. This makes data structures based on arrays more popular in solving algorithmic problems.</p> <p>It should be noted that high cache efficiency does not mean that arrays are always better than linked lists. Which data structure to choose in actual applications should be based on specific requirements. For example, both arrays and linked lists can implement the \"stack\" data structure (which will be detailed in the next chapter), but they are suitable for different scenarios.</p> <ul> <li>In algorithm problems, we tend to choose stacks based on arrays because they provide higher operational efficiency and random access capabilities, with the only cost being the need to pre-allocate a certain amount of memory space for the array.</li> <li>If the data volume is very large, highly dynamic, and the expected size of the stack is difficult to estimate, then a stack based on a linked list is more appropriate. Linked lists can disperse a large amount of data in different parts of the memory and avoid the additional overhead of array expansion.</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/","title":"4.5 \u00a0 Summary","text":""},{"location":"chapter_array_and_linkedlist/summary/#1-key-review","title":"1. \u00a0 Key Review","text":"<ul> <li>Arrays and linked lists are two basic data structures, representing two storage methods in computer memory: contiguous space storage and non-contiguous space storage. Their characteristics complement each other.</li> <li>Arrays support random access and use less memory; however, they are inefficient in inserting and deleting elements and have a fixed length after initialization.</li> <li>Linked lists implement efficient node insertion and deletion through changing references (pointers) and can flexibly adjust their length; however, they have lower node access efficiency and consume more memory.</li> <li>Common types of linked lists include singly linked lists, circular linked lists, and doubly linked lists, each with its own application scenarios.</li> <li>Lists are ordered collections of elements that support addition, deletion, and modification, typically implemented based on dynamic arrays, retaining the advantages of arrays while allowing flexible length adjustment.</li> <li>The advent of lists significantly enhanced the practicality of arrays but may lead to some memory space wastage.</li> <li>During program execution, data is mainly stored in memory. Arrays provide higher memory space efficiency, while linked lists are more flexible in memory usage.</li> <li>Caches provide fast data access to CPUs through mechanisms like cache lines, prefetching, spatial locality, and temporal locality, significantly enhancing program execution efficiency.</li> <li>Due to higher cache hit rates, arrays are generally more efficient than linked lists. When choosing a data structure, the appropriate choice should be made based on specific needs and scenarios.</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: Does storing arrays on the stack versus the heap affect time and space efficiency?</p> <p>Arrays stored on both the stack and heap are stored in contiguous memory spaces, and data operation efficiency is essentially the same. However, stacks and heaps have their own characteristics, leading to the following differences.</p> <ol> <li>Allocation and release efficiency: The stack is a smaller memory block, allocated automatically by the compiler; the heap memory is relatively larger and can be dynamically allocated in the code, more prone to fragmentation. Therefore, allocation and release operations on the heap are generally slower than on the stack.</li> <li>Size limitation: Stack memory is relatively small, while the heap size is generally limited by available memory. Therefore, the heap is more suitable for storing large arrays.</li> <li>Flexibility: The size of arrays on the stack needs to be determined at compile-time, while the size of arrays on the heap can be dynamically determined at runtime.</li> </ol> <p>Q: Why do arrays require elements of the same type, while linked lists do not emphasize same-type elements?</p> <p>Linked lists consist of nodes connected by references (pointers), and each node can store data of different types, such as int, double, string, object, etc.</p> <p>In contrast, array elements must be of the same type, allowing the calculation of offsets to access the corresponding element positions. For example, an array containing both int and long types, with single elements occupying 4 bytes and 8 bytes respectively, cannot use the following formula to calculate offsets, as the array contains elements of two different lengths.</p> <pre><code># Element memory address = Array memory address + Element length * Element index\n</code></pre> <p>Q: After deleting a node, is it necessary to set <code>P.next</code> to <code>None</code>?</p> <p>Not modifying <code>P.next</code> is also acceptable. From the perspective of the linked list, traversing from the head node to the tail node will no longer encounter <code>P</code>. This means that node <code>P</code> has been effectively removed from the list, and where <code>P</code> points no longer affects the list.</p> <p>From a garbage collection perspective, for languages with automatic garbage collection mechanisms like Java, Python, and Go, whether node <code>P</code> is collected depends on whether there are still references pointing to it, not on the value of <code>P.next</code>. In languages like C and C++, we need to manually free the node's memory.</p> <p>Q: In linked lists, the time complexity for insertion and deletion operations is <code>O(1)</code>. But searching for the element before insertion or deletion takes <code>O(n)</code> time, so why isn't the time complexity <code>O(n)</code>?</p> <p>If an element is searched first and then deleted, the time complexity is indeed <code>O(n)</code>. However, the <code>O(1)</code> advantage of linked lists in insertion and deletion can be realized in other applications. For example, in the implementation of double-ended queues using linked lists, we maintain pointers always pointing to the head and tail nodes, making each insertion and deletion operation <code>O(1)</code>.</p> <p>Q: In the image \"Linked List Definition and Storage Method\", do the light blue storage nodes occupy a single memory address, or do they share half with the node value?</p> <p>The diagram is just a qualitative representation; quantitative analysis depends on specific situations.</p> <ul> <li>Different types of node values occupy different amounts of space, such as int, long, double, and object instances.</li> <li>The memory space occupied by pointer variables depends on the operating system and compilation environment used, usually 8 bytes or 4 bytes.</li> </ul> <p>Q: Is adding elements to the end of a list always <code>O(1)</code>?</p> <p>If adding an element exceeds the list length, the list needs to be expanded first. The system will request a new memory block and move all elements of the original list over, in which case the time complexity becomes <code>O(n)</code>.</p> <p>Q: The statement \"The emergence of lists greatly improves the practicality of arrays, but may lead to some memory space wastage\" - does this refer to the memory occupied by additional variables like capacity, length, and expansion multiplier?</p> <p>The space wastage here mainly refers to two aspects: on the one hand, lists are set with an initial length, which we may not always need; on the other hand, to prevent frequent expansion, expansion usually multiplies by a coefficient, such as \\(\\times 1.5\\). This results in many empty slots, which we typically cannot fully fill.</p> <p>Q: In Python, after initializing <code>n = [1, 2, 3]</code>, the addresses of these 3 elements are contiguous, but initializing <code>m = [2, 1, 3]</code> shows that each element's <code>id</code> is not consecutive but identical to those in <code>n</code>. If the addresses of these elements are not contiguous, is <code>m</code> still an array?</p> <p>If we replace list elements with linked list nodes <code>n = [n1, n2, n3, n4, n5]</code>, these 5 node objects are also typically dispersed throughout memory. However, given a list index, we can still access the node's memory address in <code>O(1)</code> time, thereby accessing the corresponding node. This is because the array stores references to the nodes, not the nodes themselves.</p> <p>Unlike many languages, in Python, numbers are also wrapped as objects, and lists store references to these numbers, not the numbers themselves. Therefore, we find that the same number in two arrays has the same <code>id</code>, and these numbers' memory addresses need not be contiguous.</p> <p>Q: The <code>std::list</code> in C++ STL has already implemented a doubly linked list, but it seems that some algorithm books don't directly use it. Is there any limitation?</p> <p>On the one hand, we often prefer to use arrays to implement algorithms, only using linked lists when necessary, mainly for two reasons.</p> <ul> <li>Space overhead: Since each element requires two additional pointers (one for the previous element and one for the next), <code>std::list</code> usually occupies more space than <code>std::vector</code>.</li> <li>Cache unfriendly: As the data is not stored continuously, <code>std::list</code> has a lower cache utilization rate. Generally, <code>std::vector</code> performs better.</li> </ul> <p>On the other hand, linked lists are primarily necessary for binary trees and graphs. Stacks and queues are often implemented using the programming language's <code>stack</code> and <code>queue</code> classes, rather than linked lists.</p> <p>Q: Does initializing a list <code>res = [0] * self.size()</code> result in each element of <code>res</code> referencing the same address?</p> <p>No. However, this issue arises with two-dimensional arrays, for example, initializing a two-dimensional list <code>res = [[0] * self.size()]</code> would reference the same list <code>[0]</code> multiple times.</p> <p>Q: In deleting a node, is it necessary to break the reference to its successor node?</p> <p>From the perspective of data structures and algorithms (problem-solving), it's okay not to break the link, as long as the program's logic is correct. From the perspective of standard libraries, breaking the link is safer and more logically clear. If the link is not broken, and the deleted node is not properly recycled, it could affect the recycling of the successor node's memory.</p>"},{"location":"chapter_computational_complexity/","title":"Chapter 2. \u00a0 Complexity Analysis","text":"<p>Abstract</p> <p>Complexity analysis is like a space-time navigator in the vast universe of algorithms.</p> <p>It guides us in exploring deeper within the the dimensions of time and space, seeking more elegant solutions.</p>"},{"location":"chapter_computational_complexity/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>2.1 \u00a0 Algorithm Efficiency Assessment</li> <li>2.2 \u00a0 Iteration and Recursion</li> <li>2.3 \u00a0 Time Complexity</li> <li>2.4 \u00a0 Space Complexity</li> <li>2.5 \u00a0 Summary</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/","title":"2.2 \u00a0 Iteration and Recursion","text":"<p>In algorithms, the repeated execution of a task is quite common and is closely related to the analysis of complexity. Therefore, before delving into the concepts of time complexity and space complexity, let's first explore how to implement repetitive tasks in programming. This involves understanding two fundamental programming control structures: iteration and recursion.</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#221-iteration","title":"2.2.1 \u00a0 Iteration","text":"<p>\"Iteration\" is a control structure for repeatedly performing a task. In iteration, a program repeats a block of code as long as a certain condition is met until this condition is no longer satisfied.</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for-loops","title":"1. \u00a0 For Loops","text":"<p>The <code>for</code> loop is one of the most common forms of iteration, and it's particularly suitable when the number of iterations is known in advance.</p> <p>The following function uses a <code>for</code> loop to perform a summation of \\(1 + 2 + \\dots + n\\), with the sum being stored in the variable <code>res</code>. It's important to note that in Python, <code>range(a, b)</code> creates an interval that is inclusive of <code>a</code> but exclusive of <code>b</code>, meaning it iterates over the range from \\(a\\) up to \\(b\u22121\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def for_loop(n: int) -&gt; int:\n    \"\"\"for \u5faa\u73af\"\"\"\n    res = 0\n    # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        res += i\n    return res\n</code></pre> iteration.cpp<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* for \u5faa\u73af */\nint ForLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* for \u5faa\u73af */\nfunc forLoop(n int) int {\n    res := 0\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* for \u5faa\u73af */\nfunc forLoop(n: Int) -&gt; Int {\n    var res = 0\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* for \u5faa\u73af */\nfunction forLoop(n) {\n    let res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* for \u5faa\u73af */\nfunction forLoop(n: number): number {\n    let res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n  int res = 0;\n  // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    res += i;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* for \u5faa\u73af */\nfn for_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i in 1..=n {\n        res += i;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* for \u5faa\u73af */\nfun forLoop(n: Int): Int {\n    var res = 0\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (i in 1..n) {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{for_loop}\n</code></pre> iteration.zig<pre><code>// for \u5faa\u73af\nfn forLoop(n: usize) i32 {\n    var res: i32 = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (1..n+1) |i| {\n        res = res + @as(i32, @intCast(i));\n    }\n    return res;\n} \n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The flowchart below represents this sum function.</p> <p></p> <p> Figure 2-1 \u00a0 Flowchart of the Sum Function </p> <p>The number of operations in this summation function is proportional to the size of the input data \\(n\\), or in other words, it has a \"linear relationship.\" This \"linear relationship\" is what time complexity describes. This topic will be discussed in more detail in the next section.</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while-loops","title":"2. \u00a0 While Loops","text":"<p>Similar to <code>for</code> loops, <code>while</code> loops are another approach for implementing iteration. In a <code>while</code> loop, the program checks a condition at the beginning of each iteration; if the condition is true, the execution continues, otherwise, the loop ends.</p> <p>Below we use a <code>while</code> loop to implement the sum \\(1 + 2 + \\dots + n\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def while_loop(n: int) -&gt; int:\n    \"\"\"while \u5faa\u73af\"\"\"\n    res = 0\n    i = 1  # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n:\n        res += i\n        i += 1  # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    return res\n</code></pre> iteration.cpp<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u5faa\u73af */\nint WhileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while \u5faa\u73af */\nfunc whileLoop(n int) int {\n    res := 0\n    // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    i := 1\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while \u5faa\u73af */\nfunc whileLoop(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i\n        i += 1 // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while \u5faa\u73af */\nfunction whileLoop(n) {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u5faa\u73af */\nfunction whileLoop(n: number): number {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n  int res = 0;\n  int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n  // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n  while (i &lt;= n) {\n    res += i;\n    i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u5faa\u73af */\nfn while_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i;\n        i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while \u5faa\u73af */\nfun whileLoop(n: Int): Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i\n        i++ // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{while_loop}\n</code></pre> iteration.zig<pre><code>// while \u5faa\u73af\nfn whileLoop(n: i32) i32 {\n    var res: i32 = 0;\n    var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += @intCast(i);\n        i += 1;\n    }\n    return res;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p><code>While</code> loops provide more flexibility than <code>for</code> loops, especially since they allow for custom initialization and modification of the condition variable at each step.</p> <p>For example, in the following code, the condition variable \\(i\\) is updated twice each round, which would be inconvenient to implement with a <code>for</code> loop.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def while_loop_ii(n: int) -&gt; int:\n    \"\"\"while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\"\"\"\n    res = 0\n    i = 1  # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    # \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n:\n        res += i\n        # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1\n        i *= 2\n    return res\n</code></pre> iteration.cpp<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint WhileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1; \n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n int) int {\n    res := 0\n    // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    i := 1\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    for i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n) {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n: number): number {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n  int res = 0;\n  int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n  // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n  while (i &lt;= n) {\n    res += i;\n    // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    i++;\n    i *= 2;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfn while_loop_ii(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1;\n        i *= 2;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfun whileLoopII(n: Int): Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{while_loop_ii}\n</code></pre> iteration.zig<pre><code>//  while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\nfn whileLoopII(n: i32) i32 {\n    var res: i32 = 0;\n    var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += @intCast(i);\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Overall, <code>for</code> loops are more concise, while <code>while</code> loops are more flexible. Both can implement iterative structures. Which one to use should be determined based on the specific requirements of the problem.</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3-nested-loops","title":"3. \u00a0 Nested Loops","text":"<p>We can nest one loop structure within another. Below is an example using <code>for</code> loops:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def nested_for_loop(n: int) -&gt; str:\n    \"\"\"\u53cc\u5c42 for \u5faa\u73af\"\"\"\n    res = \"\"\n    # \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        # \u5faa\u73af j = 1, 2, ..., n-1, n\n        for j in range(1, n + 1):\n            res += f\"({i}, {j}), \"\n    return res\n</code></pre> iteration.cpp<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nstring nestedForLoop(int n) {\n    ostringstream res;\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; ++j) {\n            res &lt;&lt; \"(\" &lt;&lt; i &lt;&lt; \", \" &lt;&lt; j &lt;&lt; \"), \";\n        }\n    }\n    return res.str();\n}\n</code></pre> iteration.java<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n    StringBuilder res = new StringBuilder();\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.append(\"(\" + i + \", \" + j + \"), \");\n        }\n    }\n    return res.toString();\n}\n</code></pre> iteration.cs<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nstring NestedForLoop(int n) {\n    StringBuilder res = new();\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.Append($\"({i}, {j}), \");\n        }\n    }\n    return res.ToString();\n}\n</code></pre> iteration.go<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n int) string {\n    res := \"\"\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= n; j++ {\n            // \u5faa\u73af j = 1, 2, ..., n-1, n\n            res += fmt.Sprintf(\"(%d, %d), \", i, j)\n        }\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n: Int) -&gt; String {\n    var res = \"\"\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for j in 1 ... n {\n            res.append(\"(\\(i), \\(j)), \")\n        }\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n) {\n    let res = '';\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n: number): string {\n    let res = '';\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n  String res = \"\";\n  // \u5faa\u73af i = 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    // \u5faa\u73af j = 1, 2, ..., n-1, n\n    for (int j = 1; j &lt;= n; j++) {\n      res += \"($i, $j), \";\n    }\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfn nested_for_loop(n: i32) -&gt; String {\n    let mut res = vec![];\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i in 1..=n {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for j in 1..=n {\n            res.push(format!(\"({}, {}), \", i, j));\n        }\n    }\n    res.join(\"\")\n}\n</code></pre> iteration.c<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nchar *nestedForLoop(int n) {\n    // n * n \u4e3a\u5bf9\u5e94\u70b9\u6570\u91cf\uff0c\"(i, j), \" \u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u6700\u5927\u4e3a 6+10*2\uff0c\u52a0\u4e0a\u6700\u540e\u4e00\u4e2a\u7a7a\u5b57\u7b26 \\0 \u7684\u989d\u5916\u7a7a\u95f4\n    int size = n * n * 26 + 1;\n    char *res = malloc(size * sizeof(char));\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            char tmp[26];\n            snprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\n            strncat(res, tmp, size - strlen(res) - 1);\n        }\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfun nestedForLoop(n: Int): String {\n    val res = StringBuilder()\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (i in 1..n) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (j in 1..n) {\n            res.append(\" ($i, $j), \")\n        }\n    }\n    return res.toString()\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{nested_for_loop}\n</code></pre> iteration.zig<pre><code>// \u53cc\u5c42 for \u5faa\u73af\nfn nestedForLoop(allocator: Allocator, n: usize) ![]const u8 {\n    var res = std.ArrayList(u8).init(allocator);\n    defer res.deinit();\n    var buffer: [20]u8 = undefined;\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (1..n+1) |i| {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (1..n+1) |j| {\n            var _str = try std.fmt.bufPrint(&amp;buffer, \"({d}, {d}), \", .{i, j});\n            try res.appendSlice(_str);\n        }\n    }\n    return res.toOwnedSlice();\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The flowchart below represents this nested loop.</p> <p></p> <p> Figure 2-2 \u00a0 Flowchart of the Nested Loop </p> <p>In such cases, the number of operations of the function is proportional to \\(n^2\\), meaning the algorithm's runtime and the size of the input data \\(n\\) has a 'quadratic relationship.'</p> <p>We can further increase the complexity by adding more nested loops, each level of nesting effectively \"increasing the dimension,\" which raises the time complexity to \"cubic,\" \"quartic,\" and so on.</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#222-recursion","title":"2.2.2 \u00a0 Recursion","text":"<p>\"Recursion\" is an algorithmic strategy where a function solves a problem by calling itself. It primarily involves two phases:</p> <ol> <li>Calling: This is where the program repeatedly calls itself, often with progressively smaller or simpler arguments, moving towards the \"termination condition.\"</li> <li>Returning: Upon triggering the \"termination condition,\" the program begins to return from the deepest recursive function, aggregating the results of each layer.</li> </ol> <p>From an implementation perspective, recursive code mainly includes three elements.</p> <ol> <li>Termination Condition: Determines when to switch from \"calling\" to \"returning.\"</li> <li>Recursive Call: Corresponds to \"calling,\" where the function calls itself, usually with smaller or more simplified parameters.</li> <li>Return Result: Corresponds to \"returning,\" where the result of the current recursion level is returned to the previous layer.</li> </ol> <p>Observe the following code, where simply calling the function <code>recur(n)</code> can compute the sum of \\(1 + 2 + \\dots + n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def recur(n: int) -&gt; int:\n    \"\"\"\u9012\u5f52\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6\n    if n == 1:\n        return 1\n    # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    res = recur(n - 1)\n    # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res\n</code></pre> recursion.cpp<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.java<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.cs<pre><code>/* \u9012\u5f52 */\nint Recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = Recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.go<pre><code>/* \u9012\u5f52 */\nfunc recur(n int) int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 1 {\n        return 1\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    res := recur(n - 1)\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res\n}\n</code></pre> recursion.swift<pre><code>/* \u9012\u5f52 */\nfunc recur(n: Int) -&gt; Int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 1 {\n        return 1\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    let res = recur(n: n - 1)\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res\n}\n</code></pre> recursion.js<pre><code>/* \u9012\u5f52 */\nfunction recur(n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 1) return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    const res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.ts<pre><code>/* \u9012\u5f52 */\nfunction recur(n: number): number {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 1) return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    const res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.dart<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n  // \u7ec8\u6b62\u6761\u4ef6\n  if (n == 1) return 1;\n  // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n  int res = recur(n - 1);\n  // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n  return n + res;\n}\n</code></pre> recursion.rs<pre><code>/* \u9012\u5f52 */\nfn recur(n: i32) -&gt; i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 1 {\n        return 1;\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    let res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    n + res\n}\n</code></pre> recursion.c<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.kt<pre><code>/* \u9012\u5f52 */\nfun recur(n: Int): Int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1\n    // \u9012: \u9012\u5f52\u8c03\u7528\n    val res = recur(n - 1)\n    // \u5f52: \u8fd4\u56de\u7ed3\u679c\n    return n + res\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{recur}\n</code></pre> recursion.zig<pre><code>// \u9012\u5f52\u51fd\u6570\nfn recur(n: i32) i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1) {\n        return 1;\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    var res: i32 = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The Figure 2-3  shows the recursive process of this function.</p> <p></p> <p> Figure 2-3 \u00a0 Recursive Process of the Sum Function </p> <p>Although iteration and recursion can achieve the same results from a computational standpoint, they represent two entirely different paradigms of thinking and problem-solving.</p> <ul> <li>Iteration: Solves problems \"from the bottom up.\" It starts with the most basic steps, and then repeatedly adds or accumulates these steps until the task is complete.</li> <li>Recursion: Solves problems \"from the top down.\" It breaks down the original problem into smaller sub-problems, each of which has the same form as the original problem. These sub-problems are then further decomposed into even smaller sub-problems, stopping at the base case whose solution is known.</li> </ul> <p>Let's take the earlier example of the summation function, defined as \\(f(n) = 1 + 2 + \\dots + n\\).</p> <ul> <li>Iteration: In this approach, we simulate the summation process within a loop. Starting from \\(1\\) and traversing to \\(n\\), we perform the summation operation in each iteration to eventually compute \\(f(n)\\).</li> <li>Recursion: Here, the problem is broken down into a sub-problem: \\(f(n) = n + f(n-1)\\). This decomposition continues recursively until reaching the base case, \\(f(1) = 1\\), at which point the recursion terminates.</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-call-stack","title":"1. \u00a0 Call Stack","text":"<p>Every time a recursive function calls itself, the system allocates memory for the newly initiated function to store local variables, the return address, and other relevant information. This leads to two primary outcomes.</p> <ul> <li>The function's context data is stored in a memory area called \"stack frame space\" and is only released after the function returns. Therefore, recursion generally consumes more memory space than iteration.</li> <li>Recursive calls introduce additional overhead. Hence, recursion is usually less time-efficient than loops.</li> </ul> <p>As shown in the Figure 2-4 , there are \\(n\\) unreturned recursive functions before triggering the termination condition, indicating a recursion depth of \\(n\\).</p> <p></p> <p> Figure 2-4 \u00a0 Recursion Call Depth </p> <p>In practice, the depth of recursion allowed by programming languages is usually limited, and excessively deep recursion can lead to stack overflow errors.</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-tail-recursion","title":"2. \u00a0 Tail Recursion","text":"<p>Interestingly, if a function performs its recursive call as the very last step before returning, it can be optimized by the compiler or interpreter to be as space-efficient as iteration. This scenario is known as \"tail recursion.\"</p> <ul> <li>Regular Recursion: In standard recursion, when the function returns to the previous level, it continues to execute more code, requiring the system to save the context of the previous call.</li> <li>Tail Recursion: Here, the recursive call is the final operation before the function returns. This means that upon returning to the previous level, no further actions are needed, so the system does not need to save the context of the previous level.</li> </ul> <p>For example, in calculating \\(1 + 2 + \\dots + n\\), we can make the result variable <code>res</code> a parameter of the function, thereby achieving tail recursion:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def tail_recur(n, res):\n    \"\"\"\u5c3e\u9012\u5f52\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6\n    if n == 0:\n        return res\n    # \u5c3e\u9012\u5f52\u8c03\u7528\n    return tail_recur(n - 1, res + n)\n</code></pre> recursion.cpp<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.java<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.cs<pre><code>/* \u5c3e\u9012\u5f52 */\nint TailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return TailRecur(n - 1, res + n);\n}\n</code></pre> recursion.go<pre><code>/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n int, res int) int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 0 {\n        return res\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n-1, res+n)\n}\n</code></pre> recursion.swift<pre><code>/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n: Int, res: Int) -&gt; Int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 0 {\n        return res\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n: n - 1, res: res + n)\n}\n</code></pre> recursion.js<pre><code>/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n, res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 0) return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.ts<pre><code>/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n: number, res: number): number {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 0) return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.dart<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n  // \u7ec8\u6b62\u6761\u4ef6\n  if (n == 0) return res;\n  // \u5c3e\u9012\u5f52\u8c03\u7528\n  return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.rs<pre><code>/* \u5c3e\u9012\u5f52 */\nfn tail_recur(n: i32, res: i32) -&gt; i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 0 {\n        return res;\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    tail_recur(n - 1, res + n)\n}\n</code></pre> recursion.c<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.kt<pre><code>/* \u5c3e\u9012\u5f52 */\ntailrec fun tailRecur(n: Int, res: Int): Int {\n    // \u6dfb\u52a0 tailrec \u5173\u952e\u8bcd\uff0c\u4ee5\u5f00\u542f\u5c3e\u9012\u5f52\u4f18\u5316\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n)\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{tail_recur}\n</code></pre> recursion.zig<pre><code>// \u5c3e\u9012\u5f52\u51fd\u6570\nfn tailRecur(n: i32, res: i32) i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0) {\n        return res;\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The execution process of tail recursion is shown in the following figure. Comparing regular recursion and tail recursion, the point of the summation operation is different.</p> <ul> <li>Regular Recursion: The summation operation occurs during the \"returning\" phase, requiring another summation after each layer returns.</li> <li>Tail Recursion: The summation operation occurs during the \"calling\" phase, and the \"returning\" phase only involves returning through each layer.</li> </ul> <p></p> <p> Figure 2-5 \u00a0 Tail Recursion Process </p> <p>Tip</p> <p>Note that many compilers or interpreters do not support tail recursion optimization. For example, Python does not support tail recursion optimization by default, so even if the function is in the form of tail recursion, it may still encounter stack overflow issues.</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3-recursion-tree","title":"3. \u00a0 Recursion Tree","text":"<p>When dealing with algorithms related to \"divide and conquer\", recursion often offers a more intuitive approach and more readable code than iteration. Take the \"Fibonacci sequence\" as an example.</p> <p>Question</p> <p>Given a Fibonacci sequence \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\), find the \\(n\\)th number in the sequence.</p> <p>Let the \\(n\\)th number of the Fibonacci sequence be \\(f(n)\\), it's easy to deduce two conclusions:</p> <ul> <li>The first two numbers of the sequence are \\(f(1) = 0\\) and \\(f(2) = 1\\).</li> <li>Each number in the sequence is the sum of the two preceding ones, that is, \\(f(n) = f(n - 1) + f(n - 2)\\).</li> </ul> <p>Using the recursive relation, and considering the first two numbers as termination conditions, we can write the recursive code. Calling <code>fib(n)</code> will yield the \\(n\\)th number of the Fibonacci sequence:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def fib(n: int) -&gt; int:\n    \"\"\"\u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 or n == 2:\n        return n - 1\n    # \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    res = fib(n - 1) + fib(n - 2)\n    # \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n</code></pre> recursion.cpp<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint Fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = Fib(n - 1) + Fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n int) int {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    res := fib(n-1) + fib(n-2)\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n: Int) -&gt; Int {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    let res = fib(n: n - 1) + fib(n: n - 2)\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n}\n</code></pre> recursion.js<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n: number): number {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n  // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n  if (n == 1 || n == 2) return n - 1;\n  // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n  int res = fib(n - 1) + fib(n - 2);\n  // \u8fd4\u56de\u7ed3\u679c f(n)\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfn fib(n: i32) -&gt; i32 {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1;\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    let res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c\n    res\n}\n</code></pre> recursion.c<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfun fib(n: Int): Int {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    val res = fib(n - 1) + fib(n - 2)\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{fib}\n</code></pre> recursion.zig<pre><code>// \u6590\u6ce2\u90a3\u5951\u6570\u5217\nfn fib(n: i32) i32 {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 or n == 2) {\n        return n - 1;\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    var res: i32 = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Observing the above code, we see that it recursively calls two functions within itself, meaning that one call generates two branching calls. As illustrated below, this continuous recursive calling eventually creates a \"recursion tree\" with a depth of \\(n\\).</p> <p></p> <p> Figure 2-6 \u00a0 Fibonacci Sequence Recursion Tree </p> <p>Fundamentally, recursion embodies the paradigm of \"breaking down a problem into smaller sub-problems.\" This divide-and-conquer strategy is crucial.</p> <ul> <li>From an algorithmic perspective, many important strategies like searching, sorting, backtracking, divide-and-conquer, and dynamic programming directly or indirectly use this way of thinking.</li> <li>From a data structure perspective, recursion is naturally suited for dealing with linked lists, trees, and graphs, as they are well suited for analysis using the divide-and-conquer approach.</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#223-comparison","title":"2.2.3 \u00a0 Comparison","text":"<p>Summarizing the above content, the following table shows the differences between iteration and recursion in terms of implementation, performance, and applicability.</p> <p> Table: Comparison of Iteration and Recursion Characteristics </p> Iteration Recursion Approach Loop structure Function calls itself Time Efficiency Generally higher efficiency, no function call overhead Each function call generates overhead Memory Usage Typically uses a fixed size of memory space Accumulative function calls can use a substantial amount of stack frame space Suitable Problems Suitable for simple loop tasks, intuitive and readable code Suitable for problem decomposition, like trees, graphs, divide-and-conquer, backtracking, etc., concise and clear code structure <p>Tip</p> <p>If you find the following content difficult to understand, consider revisiting it after reading the \"Stack\" chapter.</p> <p>So, what is the intrinsic connection between iteration and recursion? Taking the above recursive function as an example, the summation operation occurs during the recursion's \"return\" phase. This means that the initially called function is the last to complete its summation operation, mirroring the \"last in, first out\" principle of a stack.</p> <p>Recursive terms like \"call stack\" and \"stack frame space\" hint at the close relationship between recursion and stacks.</p> <ol> <li>Calling: When a function is called, the system allocates a new stack frame on the \"call stack\" for that function, storing local variables, parameters, return addresses, and other data.</li> <li>Returning: When a function completes execution and returns, the corresponding stack frame is removed from the \"call stack,\" restoring the execution environment of the previous function.</li> </ol> <p>Therefore, we can use an explicit stack to simulate the behavior of the call stack, thus transforming recursion into an iterative form:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def for_loop_recur(n: int) -&gt; int:\n    \"\"\"\u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\"\"\"\n    # \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    stack = []\n    res = 0\n    # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i in range(n, 0, -1):\n        # \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.append(i)\n    # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while stack:\n        # \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop()\n    # res = 1+2+3+...+n\n    return res\n</code></pre> recursion.cpp<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    stack&lt;int&gt; stack;\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (!stack.empty()) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.top();\n        stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    Stack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (!stack.isEmpty()) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint ForLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    Stack&lt;int&gt; stack = new();\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.Push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (stack.Count &gt; 0) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.Pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n int) int {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    stack := list.New()\n    res := 0\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i := n; i &gt; 0; i-- {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.PushBack(i)\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    for stack.Len() != 0 {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.Back().Value.(int)\n        stack.Remove(stack.Back())\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n: Int) -&gt; Int {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    var stack: [Int] = []\n    var res = 0\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i in (1 ... n).reversed() {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.append(i)\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while !stack.isEmpty {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.removeLast()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.js<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    const stack = [];\n    let res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (let i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (stack.length) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n: number): number {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808 \n    const stack: number[] = [];\n    let res: number = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (let i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (stack.length) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n  // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n  List&lt;int&gt; stack = [];\n  int res = 0;\n  // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n  for (int i = n; i &gt; 0; i--) {\n    // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n    stack.add(i);\n  }\n  // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n  while (!stack.isEmpty) {\n    // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n    res += stack.removeLast();\n  }\n  // res = 1+2+3+...+n\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfn for_loop_recur(n: i32) -&gt; i32 {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    let mut stack = Vec::new();\n    let mut res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i in (1..=n).rev() {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while !stack.is_empty() {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop().unwrap();\n    }\n    // res = 1+2+3+...+n\n    res\n}\n</code></pre> recursion.c<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n    int stack[1000]; // \u501f\u52a9\u4e00\u4e2a\u5927\u6570\u7ec4\u6765\u6a21\u62df\u6808\n    int top = -1;    // \u6808\u9876\u7d22\u5f15\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack[1 + top++] = i;\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (top &gt;= 0) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack[top--];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfun forLoopRecur(n: Int): Int {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    val stack = Stack&lt;Int&gt;()\n    var res = 0\n    // \u9012: \u9012\u5f52\u8c03\u7528\n    for (i in n downTo 0) {\n        stack.push(i)\n    }\n    // \u5f52: \u8fd4\u56de\u7ed3\u679c\n    while (stack.isNotEmpty()) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{for_loop_recur}\n</code></pre> recursion.zig<pre><code>// \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\nfn forLoopRecur(comptime n: i32) i32 {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    var stack: [n]i32 = undefined;\n    var res: i32 = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    var i: usize = n;\n    while (i &gt; 0) {\n        stack[i - 1] = @intCast(i);\n        i -= 1;\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    var index: usize = n;\n    while (index &gt; 0) {\n        index -= 1;\n        res += stack[index];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Observing the above code, when recursion is transformed into iteration, the code becomes more complex. Although iteration and recursion can often be transformed into each other, it's not always advisable to do so for two reasons:</p> <ul> <li>The transformed code may become more challenging to understand and less readable.</li> <li>For some complex problems, simulating the behavior of the system's call stack can be quite challenging.</li> </ul> <p>In conclusion, whether to choose iteration or recursion depends on the specific nature of the problem. In programming practice, it's crucial to weigh the pros and cons of both and choose the most suitable approach for the situation at hand.</p>"},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1 \u00a0 Algorithm Efficiency Assessment","text":"<p>In algorithm design, we pursue the following two objectives in sequence.</p> <ol> <li>Finding a Solution to the Problem: The algorithm should reliably find the correct solution within the stipulated range of inputs.</li> <li>Seeking the Optimal Solution: For the same problem, multiple solutions might exist, and we aim to find the most efficient algorithm possible.</li> </ol> <p>In other words, under the premise of being able to solve the problem, algorithm efficiency has become the main criterion for evaluating the merits of an algorithm, which includes the following two dimensions.</p> <ul> <li>Time Efficiency: The speed at which an algorithm runs.</li> <li>Space Efficiency: The size of the memory space occupied by an algorithm.</li> </ul> <p>In short, our goal is to design data structures and algorithms that are both fast and memory-efficient. Effectively assessing algorithm efficiency is crucial because only then can we compare various algorithms and guide the process of algorithm design and optimization.</p> <p>There are mainly two methods of efficiency assessment: actual testing and theoretical estimation.</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#211-actual-testing","title":"2.1.1 \u00a0 Actual Testing","text":"<p>Suppose we have algorithms <code>A</code> and <code>B</code>, both capable of solving the same problem, and we need to compare their efficiencies. The most direct method is to use a computer to run these two algorithms and monitor and record their runtime and memory usage. This assessment method reflects the actual situation but has significant limitations.</p> <p>On one hand, it's difficult to eliminate interference from the testing environment. Hardware configurations can affect algorithm performance. For example, algorithm <code>A</code> might run faster than <code>B</code> on one computer, but the opposite result may occur on another computer with different configurations. This means we would need to test on a variety of machines to calculate average efficiency, which is impractical.</p> <p>On the other hand, conducting a full test is very resource-intensive. As the volume of input data changes, the efficiency of the algorithms may vary. For example, with smaller data volumes, algorithm <code>A</code> might run faster than <code>B</code>, but the opposite might be true with larger data volumes. Therefore, to draw convincing conclusions, we need to test a wide range of input data sizes, which requires significant computational resources.</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#212-theoretical-estimation","title":"2.1.2 \u00a0 Theoretical Estimation","text":"<p>Due to the significant limitations of actual testing, we can consider evaluating algorithm efficiency solely through calculations. This estimation method is known as \"asymptotic complexity analysis,\" or simply \"complexity analysis.\"</p> <p>Complexity analysis reflects the relationship between the time and space resources required for algorithm execution and the size of the input data. It describes the trend of growth in the time and space required by the algorithm as the size of the input data increases. This definition might sound complex, but we can break it down into three key points to understand it better.</p> <ul> <li>\"Time and space resources\" correspond to \"time complexity\" and \"space complexity,\" respectively.</li> <li>\"As the size of input data increases\" means that complexity reflects the relationship between algorithm efficiency and the volume of input data.</li> <li>\"The trend of growth in time and space\" indicates that complexity analysis focuses not on the specific values of runtime or space occupied but on the \"rate\" at which time or space grows.</li> </ul> <p>Complexity analysis overcomes the disadvantages of actual testing methods, reflected in the following aspects:</p> <ul> <li>It is independent of the testing environment and applicable to all operating platforms.</li> <li>It can reflect algorithm efficiency under different data volumes, especially in the performance of algorithms with large data volumes.</li> </ul> <p>Tip</p> <p>If you're still confused about the concept of complexity, don't worry. We will introduce it in detail in subsequent chapters.</p> <p>Complexity analysis provides us with a \"ruler\" to measure the time and space resources needed to execute an algorithm and compare the efficiency between different algorithms.</p> <p>Complexity is a mathematical concept and may be abstract and challenging for beginners. From this perspective, complexity analysis might not be the best content to introduce first. However, when discussing the characteristics of a particular data structure or algorithm, it's hard to avoid analyzing its speed and space usage.</p> <p>In summary, it's recommended that you establish a preliminary understanding of complexity analysis before diving deep into data structures and algorithms, so that you can carry out simple complexity analyses of algorithms.</p>"},{"location":"chapter_computational_complexity/space_complexity/","title":"2.4 \u00a0 Space Complexity","text":"<p>\"Space complexity\" is used to measure the growth trend of the memory space occupied by an algorithm as the amount of data increases. This concept is very similar to time complexity, except that \"running time\" is replaced with \"occupied memory space\".</p>"},{"location":"chapter_computational_complexity/space_complexity/#241-space-related-to-algorithms","title":"2.4.1 \u00a0 Space Related to Algorithms","text":"<p>The memory space used by an algorithm during its execution mainly includes the following types.</p> <ul> <li>Input Space: Used to store the input data of the algorithm.</li> <li>Temporary Space: Used to store variables, objects, function contexts, and other data during the algorithm's execution.</li> <li>Output Space: Used to store the output data of the algorithm.</li> </ul> <p>Generally, the scope of space complexity statistics includes both \"Temporary Space\" and \"Output Space\".</p> <p>Temporary space can be further divided into three parts.</p> <ul> <li>Temporary Data: Used to save various constants, variables, objects, etc., during the algorithm's execution.</li> <li>Stack Frame Space: Used to save the context data of the called function. The system creates a stack frame at the top of the stack each time a function is called, and the stack frame space is released after the function returns.</li> <li>Instruction Space: Used to store compiled program instructions, which are usually negligible in actual statistics.</li> </ul> <p>When analyzing the space complexity of a program, we typically count the Temporary Data, Stack Frame Space, and Output Data, as shown in the Figure 2-15 .</p> <p></p> <p> Figure 2-15 \u00a0 Space Types Used in Algorithms </p> <p>The relevant code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>class Node:\n    \"\"\"Classes\"\"\"\"\n    def __init__(self, x: int):\n        self.val: int = x               # node value\n        self.next: Node | None = None   # reference to the next node\n\ndef function() -&gt; int:\n    \"\"\"\"Functions\"\"\"\"\"\n    # Perform certain operations...\n    return 0\n\ndef algorithm(n) -&gt; int:    # input data\n    A = 0                   # temporary data (constant, usually in uppercase)\n    b = 0                   # temporary data (variable)\n    node = Node(0)          # temporary data (object)\n    c = function()          # Stack frame space (call function)\n    return A + b + c        # output data\n</code></pre> <pre><code>/* Structures */\nstruct Node {\n    int val;\n    Node *next;\n    Node(int x) : val(x), next(nullptr) {}\n};\n\n/* Functions */\nint func() {\n    // Perform certain operations...\n    return 0;\n}\n\nint algorithm(int n) {          // input data\n    const int a = 0;            // temporary data (constant)\n    int b = 0;                  // temporary data (variable)\n    Node* node = new Node(0);   // temporary data (object)\n    int c = func();             // stack frame space (call function)\n    return a + b + c;           // output data\n}\n</code></pre> <pre><code>/* Classes */\nclass Node {\n    int val;\n    Node next;\n    Node(int x) { val = x; }\n}\n\n/* Functions */\nint function() {\n    // Perform certain operations...\n    return 0;\n}\n\nint algorithm(int n) {          // input data\n    final int a = 0;            // temporary data (constant)\n    int b = 0;                  // temporary data (variable)\n    Node node = new Node(0);    // temporary data (object)\n    int c = function();         // stack frame space (call function)\n    return a + b + c;           // output data\n}\n</code></pre> <pre><code>/* Classes */\nclass Node {\n    int val;\n    Node next;\n    Node(int x) { val = x; }\n}\n\n/* Functions */\nint Function() {\n    // Perform certain operations...\n    return 0;\n}\n\nint Algorithm(int n) {  // input data\n    const int a = 0;    // temporary data (constant)\n    int b = 0;          // temporary data (variable)\n    Node node = new(0); // temporary data (object)\n    int c = Function(); // stack frame space (call function)\n    return a + b + c;   // output data\n}\n</code></pre> <pre><code>/* Structures */\ntype node struct {\n    val  int\n    next *node\n}\n\n/* Create node structure */\nfunc newNode(val int) *node {\n    return &amp;node{val: val}\n}\n\n/* Functions */\nfunc function() int {\n    // Perform certain operations...\n    return 0\n}\n\nfunc algorithm(n int) int { // input data\n    const a = 0             // temporary data (constant)\n    b := 0                  // temporary storage of data (variable)\n    newNode(0)              // temporary data (object)\n    c := function()         // stack frame space (call function)\n    return a + b + c        // output data\n}\n</code></pre> <pre><code>/* Classes */\nclass Node {\n    var val: Int\n    var next: Node?\n\n    init(x: Int) {\n        val = x\n    }\n}\n\n/* Functions */\nfunc function() -&gt; Int {\n    // Perform certain operations...\n    return 0\n}\n\nfunc algorithm(n: Int) -&gt; Int { // input data\n    let a = 0                   // temporary data (constant)\n    var b = 0                   // temporary data (variable)\n    let node = Node(x: 0)       // temporary data (object)\n    let c = function()          // stack frame space (call function)\n    return a + b + c            // output data\n}\n</code></pre> <pre><code>/* Classes */\nclass Node {\n    val;\n    next;\n    constructor(val) {\n        this.val = val === undefined ? 0 : val; // node value\n        this.next = null;                       // reference to the next node\n    }\n}\n\n/* Functions */\nfunction constFunc() {\n    // Perform certain operations\n    return 0;\n}\n\nfunction algorithm(n) {         // input data\n    const a = 0;                // temporary data (constant)\n    let b = 0;                  // temporary data (variable)\n    const node = new Node(0);   // temporary data (object)\n    const c = constFunc();      // Stack frame space (calling function)\n    return a + b + c;           // output data\n}\n</code></pre> <pre><code>/* Classes */\nclass Node {\n    val: number;\n    next: Node | null;\n    constructor(val?: number) {\n        this.val = val === undefined ? 0 : val; // node value\n        this.next = null;                       // reference to the next node\n    }\n}\n\n/* Functions */\nfunction constFunc(): number {\n    // Perform certain operations\n    return 0;\n}\n\nfunction algorithm(n: number): number { // input data\n    const a = 0;                        // temporary data (constant)\n    let b = 0;                          // temporary data (variable)\n    const node = new Node(0);           // temporary data (object)\n    const c = constFunc();              // Stack frame space (calling function)\n    return a + b + c;                   // output data\n}\n</code></pre> <pre><code>/* Classes */\nclass Node {\n  int val;\n  Node next;\n  Node(this.val, [this.next]);\n}\n\n/* Functions */\nint function() {\n  // Perform certain operations...\n  return 0;\n}\n\nint algorithm(int n) {  // input data\n  const int a = 0;      // temporary data (constant)\n  int b = 0;            // temporary data (variable)\n  Node node = Node(0);  // temporary data (object)\n  int c = function();   // stack frame space (call function)\n  return a + b + c;     // output data\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* Structures */\nstruct Node {\n    val: i32,\n    next: Option&lt;Rc&lt;RefCell&lt;Node&gt;&gt;&gt;,\n}\n\n/* Creating a Node structure */\nimpl Node {\n    fn new(val: i32) -&gt; Self {\n        Self { val: val, next: None }\n    }\n}\n\n/* Functions */\nfn function() -&gt; i32 {     \n    // Perform certain operations...\n    return 0;\n}\n\nfn algorithm(n: i32) -&gt; i32 {   // input data\n    const a: i32 = 0;           // temporary data (constant)\n    let mut b = 0;              // temporary data (variable)\n    let node = Node::new(0);    // temporary data (object)\n    let c = function();         // stack frame space (call function)\n    return a + b + c;           // output data\n}\n</code></pre> <pre><code>/* Functions */\nint func() {\n    // Perform certain operations...\n    return 0;\n}\n\nint algorithm(int n) {  // input data\n    const int a = 0;    // temporary data (constant)\n    int b = 0;          // temporary data (variable)\n    int c = func();     // stack frame space (call function)\n    return a + b + c;   // output data\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre>"},{"location":"chapter_computational_complexity/space_complexity/#242-calculation-method","title":"2.4.2 \u00a0 Calculation Method","text":"<p>The method for calculating space complexity is roughly similar to that of time complexity, with the only change being the shift of the statistical object from \"number of operations\" to \"size of used space\".</p> <p>However, unlike time complexity, we usually only focus on the worst-case space complexity. This is because memory space is a hard requirement, and we must ensure that there is enough memory space reserved under all input data.</p> <p>Consider the following code, the term \"worst-case\" in worst-case space complexity has two meanings.</p> <ol> <li>Based on the worst input data: When \\(n &lt; 10\\), the space complexity is \\(O(1)\\); but when \\(n &gt; 10\\), the initialized array <code>nums</code> occupies \\(O(n)\\) space, thus the worst-case space complexity is \\(O(n)\\).</li> <li>Based on the peak memory used during the algorithm's execution: For example, before executing the last line, the program occupies \\(O(1)\\) space; when initializing the array <code>nums</code>, the program occupies \\(O(n)\\) space, hence the worst-case space complexity is \\(O(n)\\).</li> </ol> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>def algorithm(n: int):\n    a = 0               # O(1)\n    b = [0] * 10000     # O(1)\n    if n &gt; 10:\n        nums = [0] * n  # O(n)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    vector&lt;int&gt; b(10000);    // O(1)\n    if (n &gt; 10)\n        vector&lt;int&gt; nums(n); // O(n)\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10)\n        int[] nums = new int[n]; // O(n)\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10) {\n        int[] nums = new int[n]; // O(n)\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 0                      // O(1)\n    b := make([]int, 10000)     // O(1)\n    var nums []int\n    if n &gt; 10 {\n        nums := make([]int, n)  // O(n)\n    }\n    fmt.Println(a, b, nums)\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    let a = 0 // O(1)\n    let b = Array(repeating: 0, count: 10000) // O(1)\n    if n &gt; 10 {\n        let nums = Array(repeating: 0, count: n) // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 0;                            // O(1)\n  List&lt;int&gt; b = List.filled(10000, 0);  // O(1)\n  if (n &gt; 10) {\n    List&lt;int&gt; nums = List.filled(n, 0); // O(n)\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let a = 0;                              // O(1)\n    let b = [0; 10000];                     // O(1)\n    if n &gt; 10 {\n        let nums = vec![0; n as usize];     // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    int b[10000];            // O(1)\n    if (n &gt; 10)\n        int nums[n] = {0};   // O(n)\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>In recursive functions, stack frame space must be taken into count. Consider the following code:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>def function() -&gt; int:\n    # Perform certain operations\n    return 0\n\ndef loop(n: int):\n    \"\"\"Loop O(1)\"\"\"\"\"\n    for _ in range(n):\n        function()\n\ndef recur(n: int):\n    \"\"\"Recursion O(n)\"\"\"\"\"\n    if n == 1:\n        return\n    return recur(n - 1)\n</code></pre> <pre><code>int func() {\n    // Perform certain operations\n    return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n    // Perform certain operations\n    return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int Function() {\n    // Perform certain operations\n    return 0;\n}\n/* Cycle O(1) */\nvoid Loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n/* Recursion O(n) */\nint Recur(int n) {\n    if (n == 1) return 1;\n    return Recur(n - 1);\n}\n</code></pre> <pre><code>func function() int {\n    // Perform certain operations\n    return 0\n}\n\n/* Cycle O(1) */\nfunc loop(n int) {\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n}\n\n/* Recursion O(n) */\nfunc recur(n int) {\n    if n == 1 {\n        return\n    }\n    recur(n - 1)\n}\n</code></pre> <pre><code>@discardableResult\nfunc function() -&gt; Int {\n    // Perform certain operations\n    return 0\n}\n\n/* Cycle O(1) */\nfunc loop(n: Int) {\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n\n/* Recursion O(n) */\nfunc recur(n: Int) {\n    if n == 1 {\n        return\n    }\n    recur(n: n - 1)\n}\n</code></pre> <pre><code>function constFunc() {\n    // Perform certain operations\n    return 0;\n}\n/* Cycle O(1) */\nfunction loop(n) {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* Recursion O(n) */\nfunction recur(n) {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>function constFunc(): number {\n    // Perform certain operations\n    return 0;\n}\n/* Cycle O(1) */\nfunction loop(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* Recursion O(n) */\nfunction recur(n: number): void {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n  // Perform certain operations\n  return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n  if (n == 1) return;\n  return recur(n - 1);\n}\n</code></pre> <pre><code>fn function() -&gt; i32 {\n    // Perform certain operations\n    return 0;\n}\n/* Cycle O(1) */\nfn loop(n: i32) {\n    for i in 0..n {\n        function();\n    }\n}\n/* Recursion O(n) */\nvoid recur(n: i32) {\n    if n == 1 {\n        return;\n    }\n    recur(n - 1);\n}\n</code></pre> <pre><code>int func() {\n    // Perform certain operations\n    return 0;\n}\n/* Cycle O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* Recursion O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>The time complexity of both <code>loop()</code> and <code>recur()</code> functions is \\(O(n)\\), but their space complexities differ.</p> <ul> <li>The <code>loop()</code> function calls <code>function()</code> \\(n\\) times in a loop, where each iteration's <code>function()</code> returns and releases its stack frame space, so the space complexity remains \\(O(1)\\).</li> <li>The recursive function <code>recur()</code> will have \\(n\\) instances of unreturned <code>recur()</code> existing simultaneously during its execution, thus occupying \\(O(n)\\) stack frame space.</li> </ul>"},{"location":"chapter_computational_complexity/space_complexity/#243-common-types","title":"2.4.3 \u00a0 Common Types","text":"<p>Let the size of the input data be \\(n\\), the following chart displays common types of space complexities (arranged from low to high).</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \\newline \\text{Constant Order} &lt; \\text{Logarithmic Order} &lt; \\text{Linear Order} &lt; \\text{Quadratic Order} &lt; \\text{Exponential Order} \\end{aligned} \\] <p></p> <p> Figure 2-16 \u00a0 Common Types of Space Complexity </p>"},{"location":"chapter_computational_complexity/space_complexity/#1-constant-order-o1","title":"1. \u00a0 Constant Order \\(O(1)\\)","text":"<p>Constant order is common in constants, variables, objects that are independent of the size of input data \\(n\\).</p> <p>Note that memory occupied by initializing variables or calling functions in a loop, which is released upon entering the next cycle, does not accumulate over space, thus the space complexity remains \\(O(1)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def function() -&gt; int:\n    \"\"\"\u51fd\u6570\"\"\"\n    # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n\ndef constant(n: int):\n    \"\"\"\u5e38\u6570\u9636\"\"\"\n    # \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    a = 0\n    nums = [0] * 10000\n    node = ListNode(0)\n    # \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in range(n):\n        c = 0\n    # \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in range(n):\n        function()\n</code></pre> space_complexity.cpp<pre><code>/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const int a = 0;\n    int b = 0;\n    vector&lt;int&gt; nums(10000);\n    ListNode node(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u51fd\u6570 */\nint function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    final int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new ListNode(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u51fd\u6570 */\nint Function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid Constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u51fd\u6570 */\nfunc function() int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc spaceConstant(n int) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a = 0\n    b := 0\n    nums := make([]int, 10000)\n    node := newNode(0)\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    var c int\n    for i := 0; i &lt; n; i++ {\n        c = 0\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n    b += 0\n    c += 0\n    nums[0] = 0\n    node.val = 0\n}\n</code></pre> space_complexity.swift<pre><code>/* \u51fd\u6570 */\n@discardableResult\nfunc function() -&gt; Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    let a = 0\n    var b = 0\n    let nums = Array(repeating: 0, count: 10000)\n    let node = ListNode(x: 0)\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in 0 ..&lt; n {\n        let c = 0\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n</code></pre> space_complexity.js<pre><code>/* \u51fd\u6570 */\nfunction constFunc() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u51fd\u6570 */\nfunction constFunc(): number {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n: number): void {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u51fd\u6570 */\nint function() {\n  // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n  return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n  // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n  final int a = 0;\n  int b = 0;\n  List&lt;int&gt; nums = List.filled(10000, 0);\n  ListNode node = ListNode(0);\n  // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n  for (var i = 0; i &lt; n; i++) {\n    int c = 0;\n  }\n  // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n  for (var i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u51fd\u6570 */\nfn function() -&gt; i32 {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\n#[allow(unused)]\nfn constant(n: i32) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const A: i32 = 0;\n    let b = 0;\n    let nums = vec![0; 10000];\n    let node = ListNode::new(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for i in 0..n {\n        let c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for i in 0..n {\n        function();\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const int a = 0;\n    int b = 0;\n    int nums[1000];\n    ListNode *node = newListNode(0);\n    free(node);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* \u51fd\u6570 */\nfun function(): Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfun constant(n: Int) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    val a = 0\n    var b = 0\n    val nums = Array(10000) { 0 }\n    val node = ListNode(0)\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (i in 0..&lt;n) {\n        val c = 0\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{function}\n\n[class]{}-[func]{constant}\n</code></pre> space_complexity.zig<pre><code>// \u51fd\u6570\nfn function() i32 {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n// \u5e38\u6570\u9636\nfn constant(n: i32) void {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a: i32 = 0;\n    var b: i32 = 0;\n    var nums = [_]i32{0}**10000;\n    var node = inc.ListNode(i32){.val = 0};\n    var i: i32 = 0;\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    while (i &lt; n) : (i += 1) {\n        var c: i32 = 0;\n        _ = c;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    i = 0;\n    while (i &lt; n) : (i += 1) {\n        _ = function();\n    }\n    _ = a;\n    _ = b;\n    _ = nums;\n    _ = node;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_computational_complexity/space_complexity/#2-linear-order-on","title":"2. \u00a0 Linear Order \\(O(n)\\)","text":"<p>Linear order is common in arrays, linked lists, stacks, queues, etc., where the number of elements is proportional to \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def linear(n: int):\n    \"\"\"\u7ebf\u6027\u9636\"\"\"\n    # \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    nums = [0] * n\n    # \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    hmap = dict[int, str]()\n    for i in range(n):\n        hmap[i] = str(i)\n</code></pre> space_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    vector&lt;int&gt; nums(n);\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    vector&lt;ListNode&gt; nodes;\n    for (int i = 0; i &lt; n; i++) {\n        nodes.push_back(ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    unordered_map&lt;int, string&gt; map;\n    for (int i = 0; i &lt; n; i++) {\n        map[i] = to_string(i);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    int[] nums = new int[n];\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    List&lt;ListNode&gt; nodes = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        nodes.add(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    Map&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        map.put(i, String.valueOf(i));\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7ebf\u6027\u9636 */\nvoid Linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    int[] nums = new int[n];\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    List&lt;ListNode&gt; nodes = [];\n    for (int i = 0; i &lt; n; i++) {\n        nodes.Add(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    Dictionary&lt;int, string&gt; map = [];\n    for (int i = 0; i &lt; n; i++) {\n        map.Add(i, i.ToString());\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u7ebf\u6027\u9636 */\nfunc spaceLinear(n int) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    _ = make([]int, n)\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    var nodes []*node\n    for i := 0; i &lt; n; i++ {\n        nodes = append(nodes, newNode(i))\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    m := make(map[int]string, n)\n    for i := 0; i &lt; n; i++ {\n        m[i] = strconv.Itoa(i)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    let nums = Array(repeating: 0, count: n)\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let nodes = (0 ..&lt; n).map { ListNode(x: $0) }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let map = Dictionary(uniqueKeysWithValues: (0 ..&lt; n).map { ($0, \"\\($0)\") })\n}\n</code></pre> space_complexity.js<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    const nums = new Array(n);\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const nodes = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): void {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    const nums = new Array(n);\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const nodes: ListNode[] = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n  // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n  List&lt;int&gt; nums = List.filled(n, 0);\n  // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n  List&lt;ListNode&gt; nodes = [];\n  for (var i = 0; i &lt; n; i++) {\n    nodes.add(ListNode(i));\n  }\n  // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n  Map&lt;int, String&gt; map = HashMap();\n  for (var i = 0; i &lt; n; i++) {\n    map.putIfAbsent(i, () =&gt; i.toString());\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7ebf\u6027\u9636 */\n#[allow(unused)]\nfn linear(n: i32) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    let mut nums = vec![0; n as usize];\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let mut nodes = Vec::new();\n    for i in 0..n {\n        nodes.push(ListNode::new(i))\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let mut map = HashMap::new();\n    for i in 0..n {\n        map.insert(i, i.to_string());\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u54c8\u5e0c\u8868 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n\n/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    int *nums = malloc(sizeof(int) * n);\n    free(nums);\n\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    ListNode **nodes = malloc(sizeof(ListNode *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        nodes[i] = newListNode(i);\n    }\n    // \u5185\u5b58\u91ca\u653e\n    for (int i = 0; i &lt; n; i++) {\n        free(nodes[i]);\n    }\n    free(nodes);\n\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    HashTable *h = NULL;\n    for (int i = 0; i &lt; n; i++) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = i;\n        tmp-&gt;val = i;\n        HASH_ADD_INT(h, key, tmp);\n    }\n\n    // \u5185\u5b58\u91ca\u653e\n    HashTable *curr, *tmp;\n    HASH_ITER(hh, h, curr, tmp) {\n        HASH_DEL(h, curr);\n        free(curr);\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* \u7ebf\u6027\u9636 */\nfun linear(n: Int) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    val nums = Array(n) { 0 }\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    val nodes = mutableListOf&lt;ListNode&gt;()\n    for (i in 0..&lt;n) {\n        nodes.add(ListNode(i))\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    val map = mutableMapOf&lt;Int, String&gt;()\n    for (i in 0..&lt;n) {\n        map[i] = i.toString()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{linear}\n</code></pre> space_complexity.zig<pre><code>// \u7ebf\u6027\u9636\nfn linear(comptime n: i32) !void {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    var nums = [_]i32{0}**n;\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    var nodes = std.ArrayList(i32).init(std.heap.page_allocator);\n    defer nodes.deinit();\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        try nodes.append(i);\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    var map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\n    defer map.deinit();\n    var j: i32 = 0;\n    while (j &lt; n) : (j += 1) {\n        const string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\n        defer std.heap.page_allocator.free(string);\n        try map.put(i, string);\n    }\n    _ = nums;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>As shown below, this function's recursive depth is \\(n\\), meaning there are \\(n\\) instances of unreturned <code>linear_recur()</code> function, using \\(O(n)\\) size of stack frame space:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def linear_recur(n: int):\n    \"\"\"\u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    print(\"\u9012\u5f52 n =\", n)\n    if n == 1:\n        return\n    linear_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n    cout &lt;&lt; \"\u9012\u5f52 n = \" &lt;&lt; n &lt;&lt; endl;\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n    System.out.println(\"\u9012\u5f52 n = \" + n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid LinearRecur(int n) {\n    Console.WriteLine(\"\u9012\u5f52 n = \" + n);\n    if (n == 1) return;\n    LinearRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceLinearRecur(n int) {\n    fmt.Println(\"\u9012\u5f52 n =\", n)\n    if n == 1 {\n        return\n    }\n    spaceLinearRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc linearRecur(n: Int) {\n    print(\"\u9012\u5f52 n = \\(n)\")\n    if n == 1 {\n        return\n    }\n    linearRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n) {\n    console.log(`\u9012\u5f52 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n: number): void {\n    console.log(`\u9012\u5f52 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n  print('\u9012\u5f52 n = $n');\n  if (n == 1) return;\n  linearRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn linear_recur(n: i32) {\n    println!(\"\u9012\u5f52 n = {}\", n);\n    if n == 1 {\n        return;\n    };\n    linear_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n    printf(\"\u9012\u5f52 n = %d\\r\\n\", n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.kt<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun linearRecur(n: Int) {\n    println(\"\u9012\u5f52 n = $n\")\n    if (n == 1)\n        return\n    linearRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{linear_recur}\n</code></pre> space_complexity.zig<pre><code>// \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn linearRecur(comptime n: i32) void {\n    std.debug.print(\"\u9012\u5f52 n = {}\\n\", .{n});\n    if (n == 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p></p> <p> Figure 2-17 \u00a0 Recursive Function Generating Linear Order Space Complexity </p>"},{"location":"chapter_computational_complexity/space_complexity/#3-quadratic-order-on2","title":"3. \u00a0 Quadratic Order \\(O(n^2)\\)","text":"<p>Quadratic order is common in matrices and graphs, where the number of elements is quadratic to \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def quadratic(n: int):\n    \"\"\"\u5e73\u65b9\u9636\"\"\"\n    # \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    num_matrix = [[0] * n for _ in range(n)]\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    vector&lt;vector&lt;int&gt;&gt; numMatrix;\n    for (int i = 0; i &lt; n; i++) {\n        vector&lt;int&gt; tmp;\n        for (int j = 0; j &lt; n; j++) {\n            tmp.push_back(0);\n        }\n        numMatrix.push_back(tmp);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    int[][] numMatrix = new int[n][n];\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    List&lt;List&lt;Integer&gt;&gt; numList = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            tmp.add(0);\n        }\n        numList.add(tmp);\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u9636 */\nvoid Quadratic(int n) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    int[,] numMatrix = new int[n, n];\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    List&lt;List&lt;int&gt;&gt; numList = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;int&gt; tmp = [];\n        for (int j = 0; j &lt; n; j++) {\n            tmp.Add(0);\n        }\n        numList.Add(tmp);\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u9636 */\nfunc spaceQuadratic(n int) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    numMatrix := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        numMatrix[i] = make([]int, n)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    let numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): void {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n  // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n  List&lt;List&lt;int&gt;&gt; numMatrix = List.generate(n, (_) =&gt; List.filled(n, 0));\n  // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n  List&lt;List&lt;int&gt;&gt; numList = [];\n  for (var i = 0; i &lt; n; i++) {\n    List&lt;int&gt; tmp = [];\n    for (int j = 0; j &lt; n; j++) {\n      tmp.add(0);\n    }\n    numList.add(tmp);\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u9636 */\n#[allow(unused)]\nfn quadratic(n: i32) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    let num_matrix = vec![vec![0; n as usize]; n as usize];\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    let mut num_list = Vec::new();\n    for i in 0..n {\n        let mut tmp = Vec::new();\n        for j in 0..n {\n            tmp.push(0);\n        }\n        num_list.push(tmp);\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    int **numMatrix = malloc(sizeof(int *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        int *tmp = malloc(sizeof(int) * n);\n        for (int j = 0; j &lt; n; j++) {\n            tmp[j] = 0;\n        }\n        numMatrix[i] = tmp;\n    }\n\n    // \u5185\u5b58\u91ca\u653e\n    for (int i = 0; i &lt; n; i++) {\n        free(numMatrix[i]);\n    }\n    free(numMatrix);\n}\n</code></pre> space_complexity.kt<pre><code>/* \u5e73\u65b9\u9636 */\nfun quadratic(n: Int) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    val numMatrix: Array&lt;Array&lt;Int&gt;?&gt; = arrayOfNulls(n)\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    val numList: MutableList&lt;MutableList&lt;Int&gt;&gt; = arrayListOf()\n    for (i in 0..&lt;n) {\n        val tmp = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            tmp.add(0)\n        }\n        numList.add(tmp)\n    }\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{quadratic}\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u9636\nfn quadratic(n: i32) !void {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    var nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\n    defer nodes.deinit();\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        var tmp = std.ArrayList(i32).init(std.heap.page_allocator);\n        defer tmp.deinit();\n        var j: i32 = 0;\n        while (j &lt; n) : (j += 1) {\n            try tmp.append(0);\n        }\n        try nodes.append(tmp);\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>As shown below, the recursive depth of this function is \\(n\\), and in each recursive call, an array is initialized with lengths \\(n\\), \\(n-1\\), \\(\\dots\\), \\(2\\), \\(1\\), averaging \\(n/2\\), thus overall occupying \\(O(n^2)\\) space:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def quadratic_recur(n: int) -&gt; int:\n    \"\"\"\u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n &lt;= 0:\n        return 0\n    # \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    nums = [0] * n\n    return quadratic_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    vector&lt;int&gt; nums(n);\n    cout &lt;&lt; \"\u9012\u5f52 n = \" &lt;&lt; n &lt;&lt; \" \u4e2d\u7684 nums \u957f\u5ea6 = \" &lt;&lt; nums.size() &lt;&lt; endl;\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    int[] nums = new int[n];\n    System.out.println(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.length);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint QuadraticRecur(int n) {\n    if (n &lt;= 0) return 0;\n    int[] nums = new int[n];\n    Console.WriteLine(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.Length);\n    return QuadraticRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceQuadraticRecur(n int) int {\n    if n &lt;= 0 {\n        return 0\n    }\n    nums := make([]int, n)\n    fmt.Printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d \\n\", n, len(nums))\n    return spaceQuadraticRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -&gt; Int {\n    if n &lt;= 0 {\n        return 0\n    }\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    let nums = Array(repeating: 0, count: n)\n    print(\"\u9012\u5f52 n = \\(n) \u4e2d\u7684 nums \u957f\u5ea6 = \\(nums.count)\")\n    return quadraticRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n) {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n: number): number {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n  if (n &lt;= 0) return 0;\n  List&lt;int&gt; nums = List.filled(n, 0);\n  print('\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}');\n  return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn quadratic_recur(n: i32) -&gt; i32 {\n    if n &lt;= 0 {\n        return 0;\n    };\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    let nums = vec![0; n as usize];\n    println!(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\", n, nums.len());\n    return quadratic_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    int *nums = malloc(sizeof(int) * n);\n    printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d\\r\\n\", n, n);\n    int res = quadraticRecur(n - 1);\n    free(nums);\n    return res;\n}\n</code></pre> space_complexity.kt<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\ntailrec fun quadraticRecur(n: Int): Int {\n    if (n &lt;= 0)\n        return 0\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    val nums = Array(n) { 0 }\n    println(\"\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.size}\")\n    return quadraticRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{quadratic_recur}\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn quadraticRecur(comptime n: i32) i32 {\n    if (n &lt;= 0) return 0;\n    var nums = [_]i32{0}**n;\n    std.debug.print(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\\n\", .{n, nums.len});\n    return quadraticRecur(n - 1);\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p></p> <p> Figure 2-18 \u00a0 Recursive Function Generating Quadratic Order Space Complexity </p>"},{"location":"chapter_computational_complexity/space_complexity/#4-exponential-order-o2n","title":"4. \u00a0 Exponential Order \\(O(2^n)\\)","text":"<p>Exponential order is common in binary trees. Observe the below image, a \"full binary tree\" with \\(n\\) levels has \\(2^n - 1\\) nodes, occupying \\(O(2^n)\\) space:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def build_tree(n: int) -&gt; TreeNode | None:\n    \"\"\"\u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\"\"\"\n    if n == 0:\n        return None\n    root = TreeNode(0)\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n    return root\n</code></pre> space_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return nullptr;\n    TreeNode *root = new TreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode buildTree(int n) {\n    if (n == 0)\n        return null;\n    TreeNode root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? BuildTree(int n) {\n    if (n == 0) return null;\n    TreeNode root = new(0) {\n        left = BuildTree(n - 1),\n        right = BuildTree(n - 1)\n    };\n    return root;\n}\n</code></pre> space_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n int) *TreeNode {\n    if n == 0 {\n        return nil\n    }\n    root := NewTreeNode(0)\n    root.Left = buildTree(n - 1)\n    root.Right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n: Int) -&gt; TreeNode? {\n    if n == 0 {\n        return nil\n    }\n    let root = TreeNode(x: 0)\n    root.left = buildTree(n: n - 1)\n    root.right = buildTree(n: n - 1)\n    return root\n}\n</code></pre> space_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n) {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n: number): TreeNode | null {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n) {\n  if (n == 0) return null;\n  TreeNode root = TreeNode(0);\n  root.left = buildTree(n - 1);\n  root.right = buildTree(n - 1);\n  return root;\n}\n</code></pre> space_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfn build_tree(n: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    if n == 0 {\n        return None;\n    };\n    let root = TreeNode::new(0);\n    root.borrow_mut().left = build_tree(n - 1);\n    root.borrow_mut().right = build_tree(n - 1);\n    return Some(root);\n}\n</code></pre> space_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return NULL;\n    TreeNode *root = newTreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.kt<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfun buildTree(n: Int): TreeNode? {\n    if (n == 0)\n        return null\n    val root = TreeNode(0)\n    root.left = buildTree(n - 1)\n    root.right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{build_tree}\n</code></pre> space_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\n    if (n == 0) return null;\n    const root = try mem_allocator.create(inc.TreeNode(i32));\n    root.init(0);\n    root.left = try buildTree(mem_allocator, n - 1);\n    root.right = try buildTree(mem_allocator, n - 1);\n    return root;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p></p> <p> Figure 2-19 \u00a0 Full Binary Tree Generating Exponential Order Space Complexity </p>"},{"location":"chapter_computational_complexity/space_complexity/#5-logarithmic-order-olog-n","title":"5. \u00a0 Logarithmic Order \\(O(\\log n)\\)","text":"<p>Logarithmic order is common in divide-and-conquer algorithms. For example, in merge sort, an array of length \\(n\\) is recursively divided in half each round, forming a recursion tree of height \\(\\log n\\), using \\(O(\\log n)\\) stack frame space.</p> <p>Another example is converting a number to a string. Given a positive integer \\(n\\), its number of digits is \\(\\log_{10} n + 1\\), corresponding to the length of the string, thus the space complexity is \\(O(\\log_{10} n + 1) = O(\\log n)\\).</p>"},{"location":"chapter_computational_complexity/space_complexity/#244-balancing-time-and-space","title":"2.4.4 \u00a0 Balancing Time and Space","text":"<p>Ideally, we aim for both time complexity and space complexity to be optimal. However, in practice, optimizing both simultaneously is often difficult.</p> <p>Lowering time complexity usually comes at the cost of increased space complexity, and vice versa. The approach of sacrificing memory space to improve algorithm speed is known as \"space-time tradeoff\"; the reverse is known as \"time-space tradeoff\".</p> <p>The choice depends on which aspect we value more. In most cases, time is more precious than space, so \"space-time tradeoff\" is often the more common strategy. Of course, controlling space complexity is also very important when dealing with large volumes of data.</p>"},{"location":"chapter_computational_complexity/summary/","title":"2.5 \u00a0 Summary","text":""},{"location":"chapter_computational_complexity/summary/#1-key-review","title":"1. \u00a0 Key Review","text":"<p>Algorithm Efficiency Assessment</p> <ul> <li>Time efficiency and space efficiency are the two main criteria for assessing the merits of an algorithm.</li> <li>We can assess algorithm efficiency through actual testing, but it's challenging to eliminate the influence of the test environment, and it consumes substantial computational resources.</li> <li>Complexity analysis can overcome the disadvantages of actual testing. Its results are applicable across all operating platforms and can reveal the efficiency of algorithms at different data scales.</li> </ul> <p>Time Complexity</p> <ul> <li>Time complexity measures the trend of an algorithm's running time with the increase in data volume, effectively assessing algorithm efficiency. However, it can fail in certain cases, such as with small input data volumes or when time complexities are the same, making it challenging to precisely compare the efficiency of algorithms.</li> <li>Worst-case time complexity is denoted using big O notation, representing the asymptotic upper bound, reflecting the growth level of the number of operations \\(T(n)\\) as \\(n\\) approaches infinity.</li> <li>Calculating time complexity involves two steps: first counting the number of operations, then determining the asymptotic upper bound.</li> <li>Common time complexities, arranged from low to high, include \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n \\log n)\\), \\(O(n^2)\\), \\(O(2^n)\\), and \\(O(n!)\\), among others.</li> <li>The time complexity of some algorithms is not fixed and depends on the distribution of input data. Time complexities are divided into worst, best, and average cases. The best case is rarely used because input data generally needs to meet strict conditions to achieve the best case.</li> <li>Average time complexity reflects the efficiency of an algorithm under random data inputs, closely resembling the algorithm's performance in actual applications. Calculating average time complexity requires accounting for the distribution of input data and the subsequent mathematical expectation.</li> </ul> <p>Space Complexity</p> <ul> <li>Space complexity, similar to time complexity, measures the trend of memory space occupied by an algorithm with the increase in data volume.</li> <li>The relevant memory space used during the algorithm's execution can be divided into input space, temporary space, and output space. Generally, input space is not included in space complexity calculations. Temporary space can be divided into temporary data, stack frame space, and instruction space, where stack frame space usually affects space complexity only in recursive functions.</li> <li>We usually focus only on the worst-case space complexity, which means calculating the space complexity of the algorithm under the worst input data and at the worst moment of operation.</li> <li>Common space complexities, arranged from low to high, include \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n^2)\\), and \\(O(2^n)\\), among others.</li> </ul>"},{"location":"chapter_computational_complexity/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: Is the space complexity of tail recursion \\(O(1)\\)?</p> <p>Theoretically, the space complexity of a tail-recursive function can be optimized to \\(O(1)\\). However, most programming languages (such as Java, Python, C++, Go, C#) do not support automatic optimization of tail recursion, so it's generally considered to have a space complexity of \\(O(n)\\).</p> <p>Q: What is the difference between the terms \"function\" and \"method\"?</p> <p>A \"function\" can be executed independently, with all parameters passed explicitly. A \"method\" is associated with an object and is implicitly passed to the object calling it, able to operate on the data contained within an instance of a class.</p> <p>Here are some examples from common programming languages:</p> <ul> <li>C is a procedural programming language without object-oriented concepts, so it only has functions. However, we can simulate object-oriented programming by creating structures (struct), and functions associated with these structures are equivalent to methods in other programming languages.</li> <li>Java and C# are object-oriented programming languages where code blocks (methods) are typically part of a class. Static methods behave like functions because they are bound to the class and cannot access specific instance variables.</li> <li>C++ and Python support both procedural programming (functions) and object-oriented programming (methods).</li> </ul> <p>Q: Does the \"Common Types of Space Complexity\" figure reflect the absolute size of occupied space?</p> <p>No, the figure shows space complexities, which reflect growth trends, not the absolute size of the occupied space.</p> <p>If you take \\(n = 8\\), you might find that the values of each curve don't correspond to their functions. This is because each curve includes a constant term, intended to compress the value range into a visually comfortable range.</p> <p>In practice, since we usually don't know the \"constant term\" complexity of each method, it's generally not possible to choose the best solution for \\(n = 8\\) based solely on complexity. However, for \\(n = 8^5\\), it's much easier to choose, as the growth trend becomes dominant.</p>"},{"location":"chapter_computational_complexity/time_complexity/","title":"2.3 \u00a0 Time Complexity","text":"<p>Time complexity is a concept used to measure how the run time of an algorithm increases with the size of the input data. Understanding time complexity is crucial for accurately assessing the efficiency of an algorithm.</p> <ol> <li>Determining the Running Platform: This includes hardware configuration, programming language, system environment, etc., all of which can affect the efficiency of code execution.</li> <li>Evaluating the Run Time for Various Computational Operations: For instance, an addition operation <code>+</code> might take 1 ns, a multiplication operation <code>*</code> might take 10 ns, a print operation <code>print()</code> might take 5 ns, etc.</li> <li>Counting All the Computational Operations in the Code: Summing the execution times of all these operations gives the total run time.</li> </ol> <p>For example, consider the following code with an input size of \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code># Under an operating platform\ndef algorithm(n: int):\n    a = 2      # 1 ns\n    a = a + 1  # 1 ns\n    a = a * 2  # 10 ns\n    # Cycle n times\n    for _ in range(n):  # 1 ns\n        print(0)        # 5 ns\n</code></pre> <pre><code>// Under a particular operating platform\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {  // 1 ns , every round i++ is executed\n        cout &lt;&lt; 0 &lt;&lt; endl;         // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {  // 1 ns , every round i++ is executed\n        System.out.println(0);     // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nvoid Algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {  // 1 ns , every round i++ is executed\n        Console.WriteLine(0);      // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nfunc algorithm(n int) {\n    a := 2     // 1 ns\n    a = a + 1  // 1 ns\n    a = a * 2  // 10 ns\n    // Loop n times\n    for i := 0; i &lt; n; i++ {  // 1 ns\n        fmt.Println(a)        // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nfunc algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // Loop n times\n    for _ in 0 ..&lt; n { // 1 ns\n        print(0) // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nfunction algorithm(n) {\n    var a = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // Loop n times\n    for(let i = 0; i &lt; n; i++) { // 1 ns , every round i++ is executed\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nfunction algorithm(n: number): void {\n    var a: number = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // Loop n times\n    for(let i = 0; i &lt; n; i++) { // 1 ns , every round i++ is executed\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nvoid algorithm(int n) {\n  int a = 2; // 1 ns\n  a = a + 1; // 1 ns\n  a = a * 2; // 10 ns\n  // Loop n times\n  for (int i = 0; i &lt; n; i++) { // 1 ns , every round i++ is executed\n    print(0); // 5 ns\n  }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nfn algorithm(n: i32) {\n    let mut a = 2;      // 1 ns\n    a = a + 1;          // 1 ns\n    a = a * 2;          // 10 ns\n    // Loop n times\n    for _ in 0..n {     // 1 ns for each round i++\n        println!(\"{}\", 0);  // 5 ns\n    }\n}\n</code></pre> <pre><code>// Under a particular operating platform\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {   // 1 ns , every round i++ is executed\n        printf(\"%d\", 0);            // 5 ns\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>// Under a particular operating platform\nfn algorithm(n: usize) void {\n    var a: i32 = 2; // 1 ns\n    a += 1; // 1 ns\n    a *= 2; // 10 ns\n    // Loop n times\n    for (0..n) |_| { // 1 ns\n        std.debug.print(\"{}\\n\", .{0}); // 5 ns\n    }\n}\n</code></pre> <p>Using the above method, the run time of the algorithm can be calculated as \\((6n + 12)\\) ns:</p> \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] <p>However, in practice, counting the run time of an algorithm is neither practical nor reasonable. First, we don't want to tie the estimated time to the running platform, as algorithms need to run on various platforms. Second, it's challenging to know the run time for each type of operation, making the estimation process difficult.</p>"},{"location":"chapter_computational_complexity/time_complexity/#231-assessing-time-growth-trend","title":"2.3.1 \u00a0 Assessing Time Growth Trend","text":"<p>Time complexity analysis does not count the algorithm's run time, but rather the growth trend of the run time as the data volume increases.</p> <p>Let's understand this concept of \"time growth trend\" with an example. Assume the input data size is \\(n\\), and consider three algorithms <code>A</code>, <code>B</code>, and <code>C</code>:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code># Time complexity of algorithm A: constant order\ndef algorithm_A(n: int):\n    print(0)\n# Time complexity of algorithm B: linear order\ndef algorithm_B(n: int):\n    for _ in range(n):\n        print(0)\n# Time complexity of algorithm C: constant order\ndef algorithm_C(n: int):\n    for _ in range(1000000):\n        print(0)\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nvoid algorithm_A(int n) {\n    cout &lt;&lt; 0 &lt;&lt; endl;\n}\n// Time complexity of algorithm B: linear order\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nvoid algorithm_A(int n) {\n    System.out.println(0);\n}\n// Time complexity of algorithm B: linear order\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        System.out.println(0);\n    }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        System.out.println(0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nvoid AlgorithmA(int n) {\n    Console.WriteLine(0);\n}\n// Time complexity of algorithm B: linear order\nvoid AlgorithmB(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Console.WriteLine(0);\n    }\n}\n// Time complexity of algorithm C: constant order\nvoid AlgorithmC(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        Console.WriteLine(0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfunc algorithm_A(n int) {\n    fmt.Println(0)\n}\n// Time complexity of algorithm B: linear order\nfunc algorithm_B(n int) {\n    for i := 0; i &lt; n; i++ {\n        fmt.Println(0)\n    }\n}\n// Time complexity of algorithm C: constant order\nfunc algorithm_C(n int) {\n    for i := 0; i &lt; 1000000; i++ {\n        fmt.Println(0)\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfunc algorithmA(n: Int) {\n    print(0)\n}\n\n// Time complexity of algorithm B: linear order\nfunc algorithmB(n: Int) {\n    for _ in 0 ..&lt; n {\n        print(0)\n    }\n}\n\n// Time complexity of algorithm C: constant order\nfunc algorithmC(n: Int) {\n    for _ in 0 ..&lt; 1_000_000 {\n        print(0)\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfunction algorithm_A(n) {\n    console.log(0);\n}\n// Time complexity of algorithm B: linear order\nfunction algorithm_B(n) {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// Time complexity of algorithm C: constant order\nfunction algorithm_C(n) {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfunction algorithm_A(n: number): void {\n    console.log(0);\n}\n// Time complexity of algorithm B: linear order\nfunction algorithm_B(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// Time complexity of algorithm C: constant order\nfunction algorithm_C(n: number): void {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nvoid algorithmA(int n) {\n  print(0);\n}\n// Time complexity of algorithm B: linear order\nvoid algorithmB(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    print(0);\n  }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithmC(int n) {\n  for (int i = 0; i &lt; 1000000; i++) {\n    print(0);\n  }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfn algorithm_A(n: i32) {\n    println!(\"{}\", 0);\n}\n// Time complexity of algorithm B: linear order\nfn algorithm_B(n: i32) {\n    for _ in 0..n {\n        println!(\"{}\", 0);\n    }\n}\n// Time complexity of algorithm C: constant order\nfn algorithm_C(n: i32) {\n    for _ in 0..1000000 {\n        println!(\"{}\", 0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nvoid algorithm_A(int n) {\n    printf(\"%d\", 0);\n}\n// Time complexity of algorithm B: linear order\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        printf(\"%d\", 0);\n    }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        printf(\"%d\", 0);\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfn algorithm_A(n: usize) void {\n    _ = n;\n    std.debug.print(\"{}\\n\", .{0});\n}\n// Time complexity of algorithm B: linear order\nfn algorithm_B(n: i32) void {\n    for (0..n) |_| {\n        std.debug.print(\"{}\\n\", .{0});\n    }\n}\n// Time complexity of algorithm C: constant order\nfn algorithm_C(n: i32) void {\n    _ = n;\n    for (0..1000000) |_| {\n        std.debug.print(\"{}\\n\", .{0});\n    }\n}\n</code></pre> <p>The following figure shows the time complexities of these three algorithms.</p> <ul> <li>Algorithm <code>A</code> has just one print operation, and its run time does not grow with \\(n\\). Its time complexity is considered \"constant order.\"</li> <li>Algorithm <code>B</code> involves a print operation looping \\(n\\) times, and its run time grows linearly with \\(n\\). Its time complexity is \"linear order.\"</li> <li>Algorithm <code>C</code> has a print operation looping 1,000,000 times. Although it takes a long time, it is independent of the input data size \\(n\\). Therefore, the time complexity of <code>C</code> is the same as <code>A</code>, which is \"constant order.\"</li> </ul> <p></p> <p> Figure 2-7 \u00a0 Time Growth Trend of Algorithms A, B, and C </p> <p>Compared to directly counting the run time of an algorithm, what are the characteristics of time complexity analysis?</p> <ul> <li>Time complexity effectively assesses algorithm efficiency. For instance, algorithm <code>B</code> has linearly growing run time, which is slower than algorithm <code>A</code> when \\(n &gt; 1\\) and slower than <code>C</code> when \\(n &gt; 1,000,000\\). In fact, as long as the input data size \\(n\\) is sufficiently large, a \"constant order\" complexity algorithm will always be better than a \"linear order\" one, demonstrating the essence of time growth trend.</li> <li>Time complexity analysis is more straightforward. Obviously, the running platform and the types of computational operations are irrelevant to the trend of run time growth. Therefore, in time complexity analysis, we can simply treat the execution time of all computational operations as the same \"unit time,\" simplifying the \"computational operation run time count\" to a \"computational operation count.\" This significantly reduces the complexity of estimation.</li> <li>Time complexity has its limitations. For example, although algorithms <code>A</code> and <code>C</code> have the same time complexity, their actual run times can be quite different. Similarly, even though algorithm <code>B</code> has a higher time complexity than <code>C</code>, it is clearly superior when the input data size \\(n\\) is small. In these cases, it's difficult to judge the efficiency of algorithms based solely on time complexity. Nonetheless, despite these issues, complexity analysis remains the most effective and commonly used method for evaluating algorithm efficiency.</li> </ul>"},{"location":"chapter_computational_complexity/time_complexity/#232-asymptotic-upper-bound","title":"2.3.2 \u00a0 Asymptotic Upper Bound","text":"<p>Consider a function with an input size of \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>def algorithm(n: int):\n    a = 1      # +1\n    a = a + 1  # +1\n    a = a * 2  # +1\n    # Cycle n times\n    for i in range(n):  # +1\n        print(0)        # +1\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) { // +1 (execute i ++ every round)\n        cout &lt;&lt; 0 &lt;&lt; endl;    // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) { // +1 (execute i ++ every round)\n        System.out.println(0);    // +1\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {   // +1 (execute i ++ every round)\n        Console.WriteLine(0);   // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1      // +1\n    a = a + 1   // +1\n    a = a * 2   // +1\n    // Loop n times\n    for i := 0; i &lt; n; i++ {   // +1\n        fmt.Println(a)         // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // Loop n times\n    for _ in 0 ..&lt; n { // +1\n        print(0) // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    var a = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // Loop n times\n    for(let i = 0; i &lt; n; i++){ // +1 (execute i ++ every round)\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void{\n    var a: number = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // Loop n times\n    for(let i = 0; i &lt; n; i++){ // +1 (execute i ++ every round)\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +1\n  a = a + 1; // +1\n  a = a * 2; // +1\n  // Loop n times\n  for (int i = 0; i &lt; n; i++) { // +1 (execute i ++ every round)\n    print(0); // +1\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;   // +1\n    a = a + 1;      // +1\n    a = a * 2;      // +1\n\n    // Loop n times\n    for _ in 0..n { // +1 (execute i ++ every round)\n        println!(\"{}\", 0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {   // +1 (execute i ++ every round)\n        printf(\"%d\", 0);            // +1\n    }\n} \n</code></pre> <pre><code>\n</code></pre> <pre><code>fn algorithm(n: usize) void {\n    var a: i32 = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // Loop n times\n    for (0..n) |_| { // +1 (execute i ++ every round)\n        std.debug.print(\"{}\\n\", .{0}); // +1\n    }\n}\n</code></pre> <p>Given a function that represents the number of operations of an algorithm as a function of the input size \\(n\\), denoted as \\(T(n)\\), consider the following example:</p> \\[ T(n) = 3 + 2n \\] <p>Since \\(T(n)\\) is a linear function, its growth trend is linear, and therefore, its time complexity is of linear order, denoted as \\(O(n)\\). This mathematical notation, known as \"big-O notation,\" represents the \"asymptotic upper bound\" of the function \\(T(n)\\).</p> <p>In essence, time complexity analysis is about finding the asymptotic upper bound of the \"number of operations \\(T(n)\\)\". It has a precise mathematical definition.</p> <p>Asymptotic Upper Bound</p> <p>If there exist positive real numbers \\(c\\) and \\(n_0\\) such that for all \\(n &gt; n_0\\), \\(T(n) \\leq c \\cdot f(n)\\), then \\(f(n)\\) is considered an asymptotic upper bound of \\(T(n)\\), denoted as \\(T(n) = O(f(n))\\).</p> <p>As illustrated below, calculating the asymptotic upper bound involves finding a function \\(f(n)\\) such that, as \\(n\\) approaches infinity, \\(T(n)\\) and \\(f(n)\\) have the same growth order, differing only by a constant factor \\(c\\).</p> <p></p> <p> Figure 2-8 \u00a0 Asymptotic Upper Bound of a Function </p>"},{"location":"chapter_computational_complexity/time_complexity/#233-calculation-method","title":"2.3.3 \u00a0 Calculation Method","text":"<p>While the concept of asymptotic upper bound might seem mathematically dense, you don't need to fully grasp it right away. Let's first understand the method of calculation, which can be practiced and comprehended over time.</p> <p>Once \\(f(n)\\) is determined, we obtain the time complexity \\(O(f(n))\\). But how do we determine the asymptotic upper bound \\(f(n)\\)? This process generally involves two steps: counting the number of operations and determining the asymptotic upper bound.</p>"},{"location":"chapter_computational_complexity/time_complexity/#1-step-1-counting-the-number-of-operations","title":"1. \u00a0 Step 1: Counting the Number of Operations","text":"<p>This step involves going through the code line by line. However, due to the presence of the constant \\(c\\) in \\(c \\cdot f(n)\\), all coefficients and constant terms in \\(T(n)\\) can be ignored. This principle allows for simplification techniques in counting operations.</p> <ol> <li>Ignore constant terms in \\(T(n)\\), as they do not affect the time complexity being independent of \\(n\\).</li> <li>Omit all coefficients. For example, looping \\(2n\\), \\(5n + 1\\) times, etc., can be simplified to \\(n\\) times since the coefficient before \\(n\\) does not impact the time complexity.</li> <li>Use multiplication for nested loops. The total number of operations equals the product of the number of operations in each loop, applying the simplification techniques from points 1 and 2 for each loop level.</li> </ol> <p>Given a function, we can use these techniques to count operations:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>def algorithm(n: int):\n    a = 1      # +0 (trick 1)\n    a = a + n  # +0 (trick 1)\n    # +n (technique 2)\n    for i in range(5 * n + 1):\n        print(0)\n    # +n*n (technique 3)\n    for i in range(2 * n):\n        for j in range(n + 1):\n            print(0)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0 (trick 1)\n    a = a + n;  // +0 (trick 1)\n    // +n (technique 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n    // +n*n (technique 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            cout &lt;&lt; 0 &lt;&lt; endl;\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0 (trick 1)\n    a = a + n;  // +0 (trick 1)\n    // +n (technique 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        System.out.println(0);\n    }\n    // +n*n (technique 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            System.out.println(0);\n        }\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +0 (trick 1)\n    a = a + n;  // +0 (trick 1)\n    // +n (technique 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        Console.WriteLine(0);\n    }\n    // +n*n (technique 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            Console.WriteLine(0);\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1     // +0 (trick 1)\n    a = a + n  // +0 (trick 1)\n    // +n (technique 2)\n    for i := 0; i &lt; 5 * n + 1; i++ {\n        fmt.Println(0)\n    }\n    // +n*n (technique 3)\n    for i := 0; i &lt; 2 * n; i++ {\n        for j := 0; j &lt; n + 1; j++ {\n            fmt.Println(0)\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +0 (trick 1)\n    a = a + n // +0 (trick 1)\n    // +n (technique 2)\n    for _ in 0 ..&lt; (5 * n + 1) {\n        print(0)\n    }\n    // +n*n (technique 3)\n    for _ in 0 ..&lt; (2 * n) {\n        for _ in 0 ..&lt; (n + 1) {\n            print(0)\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    let a = 1;  // +0 (trick 1)\n    a = a + n;  // +0 (trick 1)\n    // +n (technique 2)\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n (technique 3)\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    let a = 1;  // +0 (trick 1)\n    a = a + n;  // +0 (trick 1)\n    // +n (technique 2)\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n (technique 3)\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +0 (trick 1)\n  a = a + n; // +0 (trick 1)\n  // +n (technique 2)\n  for (int i = 0; i &lt; 5 * n + 1; i++) {\n    print(0);\n  }\n  // +n*n (technique 3)\n  for (int i = 0; i &lt; 2 * n; i++) {\n    for (int j = 0; j &lt; n + 1; j++) {\n      print(0);\n    }\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;     // +0 (trick 1)\n    a = a + n;        // +0 (trick 1)\n\n    // +n (technique 2)\n    for i in 0..(5 * n + 1) {\n        println!(\"{}\", 0);\n    }\n\n    // +n*n (technique 3)\n    for i in 0..(2 * n) {\n        for j in 0..(n + 1) {\n            println!(\"{}\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0 (trick 1)\n    a = a + n;  // +0 (trick 1)\n    // +n (technique 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        printf(\"%d\", 0);\n    }\n    // +n*n (technique 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            printf(\"%d\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>fn algorithm(n: usize) void {\n    var a: i32 = 1;     // +0 (trick 1)\n    a = a + @as(i32, @intCast(n));        // +0 (trick 1)\n\n    // +n (technique 2)\n    for(0..(5 * n + 1)) |_| {\n        std.debug.print(\"{}\\n\", .{0});\n    }\n\n    // +n*n (technique 3)\n    for(0..(2 * n)) |_| {\n        for(0..(n + 1)) |_| {\n            std.debug.print(\"{}\\n\", .{0});\n        }\n    }\n}\n</code></pre> <p>The formula below shows the counting results before and after simplification, both leading to a time complexity of \\(O(n^2)\\):</p> \\[ \\begin{aligned} T(n) &amp; = 2n(n + 1) + (5n + 1) + 2 &amp; \\text{Complete Count (-.-|||)} \\newline &amp; = 2n^2 + 7n + 3 \\newline T(n) &amp; = n^2 + n &amp; \\text{Simplified Count (o.O)} \\end{aligned} \\]"},{"location":"chapter_computational_complexity/time_complexity/#2-step-2-determining-the-asymptotic-upper-bound","title":"2. \u00a0 Step 2: Determining the Asymptotic Upper Bound","text":"<p>The time complexity is determined by the highest order term in \\(T(n)\\). This is because, as \\(n\\) approaches infinity, the highest order term dominates, rendering the influence of other terms negligible.</p> <p>The following table illustrates examples of different operation counts and their corresponding time complexities. Some exaggerated values are used to emphasize that coefficients cannot alter the order of growth. When \\(n\\) becomes very large, these constants become insignificant.</p> <p> Table: Time Complexity for Different Operation Counts </p> Operation Count \\(T(n)\\) Time Complexity \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#234-common-types-of-time-complexity","title":"2.3.4 \u00a0 Common Types of Time Complexity","text":"<p>Let's consider the input data size as \\(n\\). The common types of time complexities are illustrated below, arranged from lowest to highest:</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n \\log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!) \\newline \\text{Constant Order} &lt; \\text{Logarithmic Order} &lt; \\text{Linear Order} &lt; \\text{Linear-Logarithmic Order} &lt; \\text{Quadratic Order} &lt; \\text{Exponential Order} &lt; \\text{Factorial Order} \\end{aligned} \\] <p></p> <p> Figure 2-9 \u00a0 Common Types of Time Complexity </p>"},{"location":"chapter_computational_complexity/time_complexity/#1-constant-order-o1","title":"1. \u00a0 Constant Order \\(O(1)\\)","text":"<p>Constant order means the number of operations is independent of the input data size \\(n\\). In the following function, although the number of operations <code>size</code> might be large, the time complexity remains \\(O(1)\\) as it's unrelated to \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def constant(n: int) -&gt; int:\n    \"\"\"\u5e38\u6570\u9636\"\"\"\n    count = 0\n    size = 100000\n    for _ in range(size):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e38\u6570\u9636 */\nint Constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e38\u6570\u9636 */\nfunc constant(n int) int {\n    count := 0\n    size := 100000\n    for i := 0; i &lt; size; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) -&gt; Int {\n    var count = 0\n    let size = 100_000\n    for _ in 0 ..&lt; size {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e38\u6570\u9636 */\nfunction constant(n: number): number {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n  int count = 0;\n  int size = 100000;\n  for (var i = 0; i &lt; size; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e38\u6570\u9636 */\nfn constant(n: i32) -&gt; i32 {\n    _ = n;\n    let mut count = 0;\n    let size = 100_000;\n    for _ in 0..size {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    int i = 0;\n    for (int i = 0; i &lt; size; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e38\u6570\u9636 */\nfun constant(n: Int): Int {\n    var count = 0\n    val size = 10_0000\n    for (i in 0..&lt;size)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{constant}\n</code></pre> time_complexity.zig<pre><code>// \u5e38\u6570\u9636\nfn constant(n: i32) i32 {\n    _ = n;\n    var count: i32 = 0;\n    const size: i32 = 100_000;\n    var i: i32 = 0;\n    while(i&lt;size) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_computational_complexity/time_complexity/#2-linear-order-on","title":"2. \u00a0 Linear Order \\(O(n)\\)","text":"<p>Linear order indicates the number of operations grows linearly with the input data size \\(n\\). Linear order commonly appears in single-loop structures:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def linear(n: int) -&gt; int:\n    \"\"\"\u7ebf\u6027\u9636\"\"\"\n    count = 0\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u9636 */\nint Linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n int) int {\n    count := 0\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) -&gt; Int {\n    var count = 0\n    for _ in 0 ..&lt; n {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): number {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n  int count = 0;\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u9636 */\nfn linear(n: i32) -&gt; i32 {\n    let mut count = 0;\n    for _ in 0..n {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7ebf\u6027\u9636 */\nfun linear(n: Int): Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (i in 0..&lt;n)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{linear}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u9636\nfn linear(n: i32) i32 {\n    var count: i32 = 0;\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Operations like array traversal and linked list traversal have a time complexity of \\(O(n)\\), where \\(n\\) is the length of the array or list:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def array_traversal(nums: list[int]) -&gt; int:\n    \"\"\"\u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\"\"\"\n    count = 0\n    # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for num in nums:\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(vector&lt;int&gt; &amp;nums) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint ArrayTraversal(int[] nums) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    foreach (int num in nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums []int) int {\n    count := 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for range nums {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums: [Int]) -&gt; Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for _ in nums {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums) {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums: number[]): number {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(List&lt;int&gt; nums) {\n  int count = 0;\n  // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n  for (var _num in nums) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfn array_traversal(nums: &amp;[i32]) -&gt; i32 {\n    let mut count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for _ in nums {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int *nums, int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfun arrayTraversal(nums: IntArray): Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (num in nums) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{array_traversal}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\nfn arrayTraversal(nums: []i32) i32 {\n    var count: i32 = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (nums) |_| {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>It's important to note that the input data size \\(n\\) should be determined based on the type of input data. For example, in the first example, \\(n\\) represents the input data size, while in the second example, the length of the array \\(n\\) is the data size.</p>"},{"location":"chapter_computational_complexity/time_complexity/#3-quadratic-order-on2","title":"3. \u00a0 Quadratic Order \\(O(n^2)\\)","text":"<p>Quadratic order means the number of operations grows quadratically with the input data size \\(n\\). Quadratic order typically appears in nested loops, where both the outer and inner loops have a time complexity of \\(O(n)\\), resulting in an overall complexity of \\(O(n^2)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def quadratic(n: int) -&gt; int:\n    \"\"\"\u5e73\u65b9\u9636\"\"\"\n    count = 0\n    # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for i in range(n):\n        for j in range(n):\n            count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u9636 */\nint Quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n int) int {\n    count := 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; n; j++ {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) -&gt; Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; n {\n            count += 1\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): number {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n  int count = 0;\n  // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n  for (int i = 0; i &lt; n; i++) {\n    for (int j = 0; j &lt; n; j++) {\n      count++;\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u9636 */\nfn quadratic(n: i32) -&gt; i32 {\n    let mut count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for _ in 0..n {\n        for _ in 0..n {\n            count += 1;\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e73\u65b9\u9636 */\nfun quadratic(n: Int): Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;n) {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{quadratic}\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u9636\nfn quadratic(n: i32) i32 {\n    var count: i32 = 0;\n    var i: i32 = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    while (i &lt; n) : (i += 1) {\n        var j: i32 = 0;\n        while (j &lt; n) : (j += 1) {\n            count += 1;\n        }\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The following image compares constant order, linear order, and quadratic order time complexities.</p> <p></p> <p> Figure 2-10 \u00a0 Constant, Linear, and Quadratic Order Time Complexities </p> <p>For instance, in bubble sort, the outer loop runs \\(n - 1\\) times, and the inner loop runs \\(n-1\\), \\(n-2\\), ..., \\(2\\), \\(1\\) times, averaging \\(n / 2\\) times, resulting in a time complexity of \\(O((n - 1) n / 2) = O(n^2)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def bubble_sort(nums: list[int]) -&gt; int:\n    \"\"\"\u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\"\"\"\n    count = 0  # \u8ba1\u6570\u5668\n    # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in range(len(nums) - 1, 0, -1):\n        # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                tmp: int = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3  # \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(vector&lt;int&gt; &amp;nums) {\n    int count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\n    int count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint BubbleSort(int[] nums) {\n    int count = 0;  // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                count += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\n    count := 0 // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                tmp := nums[j]\n                nums[j] = nums[j+1]\n                nums[j+1] = tmp\n                count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -&gt; Int {\n    var count = 0 // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\n    let count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\n    let count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(List&lt;int&gt; nums) {\n  int count = 0; // \u8ba1\u6570\u5668\n  // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n  for (var i = nums.length - 1; i &gt; 0; i--) {\n    // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n    for (var j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n      }\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfn bubble_sort(nums: &amp;mut [i32]) -&gt; i32 {\n    let mut count = 0; // \u8ba1\u6570\u5668\n\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in (1..nums.len()).rev() {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int *nums, int n) {\n    int count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = n - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfun bubbleSort(nums: IntArray): Int {\n    var count = 0\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j] = nums[j + 1].also { nums[j + 1] = nums[j] }\n                count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{bubble_sort}\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\n    var count: i32 = 0;  // \u8ba1\u6570\u5668 \n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    var i: i32 = @as(i32, @intCast(nums.len)) - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var j: usize = 0;\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_computational_complexity/time_complexity/#4-exponential-order-o2n","title":"4. \u00a0 Exponential Order \\(O(2^n)\\)","text":"<p>Biological \"cell division\" is a classic example of exponential order growth: starting with one cell, it becomes two after one division, four after two divisions, and so on, resulting in \\(2^n\\) cells after \\(n\\) divisions.</p> <p>The following image and code simulate the cell division process, with a time complexity of \\(O(2^n)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def exponential(n: int) -&gt; int:\n    \"\"\"\u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n    count = 0\n    base = 1\n    # \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in range(n):\n        for _ in range(base):\n            count += 1\n        base *= 2\n    # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Exponential(int n) {\n    int count = 0, bas = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc exponential(n int) int {\n    count, base := 0, 1\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; base; j++ {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc exponential(n: Int) -&gt; Int {\n    var count = 0\n    var base = 1\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; base {\n            count += 1\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n) {\n    let count = 0,\n        base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n: number): number {\n    let count = 0,\n        base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n  int count = 0, base = 1;\n  // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n  for (var i = 0; i &lt; n; i++) {\n    for (var j = 0; j &lt; base; j++) {\n      count++;\n    }\n    base *= 2;\n  }\n  // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn exponential(n: i32) -&gt; i32 {\n    let mut count = 0;\n    let mut base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0..n {\n        for _ in 0..base {\n            count += 1\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n    int count = 0;\n    int bas = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfun exponential(n: Int): Int {\n    var count = 0\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    var base = 1\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;base) {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{exponential}\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn exponential(n: i32) i32 {\n    var count: i32 = 0;\n    var bas: i32 = 1;\n    var i: i32 = 0;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    while (i &lt; n) : (i += 1) {\n        var j: i32 = 0;\n        while (j &lt; bas) : (j += 1) {\n            count += 1;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p></p> <p> Figure 2-11 \u00a0 Exponential Order Time Complexity </p> <p>In practice, exponential order often appears in recursive functions. For example, in the code below, it recursively splits into two halves, stopping after \\(n\\) divisions:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def exp_recur(n: int) -&gt; int:\n    \"\"\"\u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n == 1:\n        return 1\n    return exp_recur(n - 1) + exp_recur(n - 1) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint ExpRecur(int n) {\n    if (n == 1) return 1;\n    return ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc expRecur(n int) int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n-1) + expRecur(n-1) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc expRecur(n: Int) -&gt; Int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n) {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n: number): number {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n  if (n == 1) return 1;\n  return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn exp_recur(n: i32) -&gt; i32 {\n    if n == 1 {\n        return 1;\n    }\n    exp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun expRecur(n: Int): Int {\n    if (n == 1) {\n        return 1\n    }\n    return expRecur(n - 1) + expRecur(n - 1) + 1\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{exp_recur}\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn expRecur(n: i32) i32 {\n    if (n == 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Exponential order growth is extremely rapid and is commonly seen in exhaustive search methods (brute force, backtracking, etc.). For large-scale problems, exponential order is unacceptable, often requiring dynamic programming or greedy algorithms as solutions.</p>"},{"location":"chapter_computational_complexity/time_complexity/#5-logarithmic-order-olog-n","title":"5. \u00a0 Logarithmic Order \\(O(\\log n)\\)","text":"<p>In contrast to exponential order, logarithmic order reflects situations where \"the size is halved each round.\" Given an input data size \\(n\\), since the size is halved each round, the number of iterations is \\(\\log_2 n\\), the inverse function of \\(2^n\\).</p> <p>The following image and code simulate the \"halving each round\" process, with a time complexity of \\(O(\\log_2 n)\\), commonly abbreviated as \\(O(\\log n)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def logarithmic(n: int) -&gt; int:\n    \"\"\"\u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n    count = 0\n    while n &gt; 1:\n        n = n / 2\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n /= 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc logarithmic(n int) int {\n    count := 0\n    for n &gt; 1 {\n        n = n / 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc logarithmic(n: Int) -&gt; Int {\n    var count = 0\n    var n = n\n    while n &gt; 1 {\n        n = n / 2\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n) {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n: number): number {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n  int count = 0;\n  while (n &gt; 1) {\n    n = n ~/ 2;\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn logarithmic(mut n: i32) -&gt; i32 {\n    let mut count = 0;\n    while n &gt; 1 {\n        n = n / 2;\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfun logarithmic(n: Int): Int {\n    var n1 = n\n    var count = 0\n    while (n1 &gt; 1) {\n        n1 /= 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{logarithmic}\n</code></pre> time_complexity.zig<pre><code>// \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn logarithmic(n: i32) i32 {\n    var count: i32 = 0;\n    var n_var = n;\n    while (n_var &gt; 1)\n    {\n        n_var = n_var / 2;\n        count +=1;\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p></p> <p> Figure 2-12 \u00a0 Logarithmic Order Time Complexity </p> <p>Like exponential order, logarithmic order also frequently appears in recursive functions. The code below forms a recursive tree of height \\(\\log_2 n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def log_recur(n: int) -&gt; int:\n    \"\"\"\u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n &lt;= 1:\n        return 0\n    return log_recur(n / 2) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint LogRecur(int n) {\n    if (n &lt;= 1) return 0;\n    return LogRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc logRecur(n int) int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n/2) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc logRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n: n / 2) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n) {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n: number): number {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n  if (n &lt;= 1) return 0;\n  return logRecur(n ~/ 2) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 0;\n    }\n    log_recur(n / 2) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun logRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 0\n    return logRecur(n / 2) + 1\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{log_recur}\n</code></pre> time_complexity.zig<pre><code>// \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn logRecur(n: i32) i32 {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Logarithmic order is typical in algorithms based on the divide-and-conquer strategy, embodying the \"split into many\" and \"simplify complex problems\" approach. It's slow-growing and is the most ideal time complexity after constant order.</p> <p>What is the base of \\(O(\\log n)\\)?</p> <p>Technically, \"splitting into \\(m\\)\" corresponds to a time complexity of \\(O(\\log_m n)\\). Using the logarithm base change formula, we can equate different logarithmic complexities:</p> \\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\] <p>This means the base \\(m\\) can be changed without affecting the complexity. Therefore, we often omit the base \\(m\\) and simply denote logarithmic order as \\(O(\\log n)\\).</p>"},{"location":"chapter_computational_complexity/time_complexity/#6-linear-logarithmic-order-on-log-n","title":"6. \u00a0 Linear-Logarithmic Order \\(O(n \\log n)\\)","text":"<p>Linear-logarithmic order often appears in nested loops, with the complexities of the two loops being \\(O(\\log n)\\) and \\(O(n)\\) respectively. The related code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def linear_log_recur(n: int) -&gt; int:\n    \"\"\"\u7ebf\u6027\u5bf9\u6570\u9636\"\"\"\n    if n &lt;= 1:\n        return 1\n    count: int = linear_log_recur(n // 2) + linear_log_recur(n // 2)\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint LinearLogRecur(int n) {\n    if (n &lt;= 1) return 1;\n    int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n int) int {\n    if n &lt;= 1 {\n        return 1\n    }\n    count := linearLogRecur(n/2) + linearLogRecur(n/2)\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 1\n    }\n    var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\n    for _ in stride(from: 0, to: n, by: 1) {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n) {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n: number): number {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n  if (n &lt;= 1) return 1;\n  int count = linearLogRecur(n ~/ 2) + linearLogRecur(n ~/ 2);\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfn linear_log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 1;\n    }\n    let mut count = linear_log_recur(n / 2) + linear_log_recur(n / 2);\n    for _ in 0..n as i32 {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfun linearLogRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 1\n    var count = linearLogRecur(n / 2) + linearLogRecur(n / 2)\n    for (i in 0..&lt;n.toInt()) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{linear_log_recur}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u5bf9\u6570\u9636\nfn linearLogRecur(n: i32) i32 {\n    if (n &lt;= 1) return 1;\n    var count: i32 = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The image below demonstrates how linear-logarithmic order is generated. Each level of a binary tree has \\(n\\) operations, and the tree has \\(\\log_2 n + 1\\) levels, resulting in a time complexity of \\(O(n \\log n)\\).</p> <p></p> <p> Figure 2-13 \u00a0 Linear-Logarithmic Order Time Complexity </p> <p>Mainstream sorting algorithms typically have a time complexity of \\(O(n \\log n)\\), such as quicksort, mergesort, and heapsort.</p>"},{"location":"chapter_computational_complexity/time_complexity/#7-factorial-order-on","title":"7. \u00a0 Factorial Order \\(O(n!)\\)","text":"<p>Factorial order corresponds to the mathematical problem of \"full permutation.\" Given \\(n\\) distinct elements, the total number of possible permutations is:</p> \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\] <p>Factorials are typically implemented using recursion. As shown in the image and code below, the first level splits into \\(n\\) branches, the second level into \\(n - 1\\) branches, and so on, stopping after the \\(n\\)th level:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def factorial_recur(n: int) -&gt; int:\n    \"\"\"\u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n == 0:\n        return 1\n    count = 0\n    # \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for _ in range(n):\n        count += factorial_recur(n - 1)\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint FactorialRecur(int n) {\n    if (n == 0) return 1;\n    int count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (int i = 0; i &lt; n; i++) {\n        count += FactorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n int) int {\n    if n == 0 {\n        return 1\n    }\n    count := 0\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for i := 0; i &lt; n; i++ {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n: Int) -&gt; Int {\n    if n == 0 {\n        return 1\n    }\n    var count = 0\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for _ in 0 ..&lt; n {\n        count += factorialRecur(n: n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n) {\n    if (n === 0) return 1;\n    let count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n: number): number {\n    if (n === 0) return 1;\n    let count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n  if (n == 0) return 1;\n  int count = 0;\n  // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n  for (var i = 0; i &lt; n; i++) {\n    count += factorialRecur(n - 1);\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn factorial_recur(n: i32) -&gt; i32 {\n    if n == 0 {\n        return 1;\n    }\n    let mut count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for _ in 0..n {\n        count += factorial_recur(n - 1);\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun factorialRecur(n: Int): Int {\n    if (n == 0)\n        return 1\n    var count = 0\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (i in 0..&lt;n) {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{factorial_recur}\n</code></pre> time_complexity.zig<pre><code>// \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn factorialRecur(n: i32) i32 {\n    if (n == 0) return 1;\n    var count: i32 = 0;\n    var i: i32 = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    while (i &lt; n) : (i += 1) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p></p> <p> Figure 2-14 \u00a0 Factorial Order Time Complexity </p> <p>Note that factorial order grows even faster than exponential order; it's unacceptable for larger \\(n\\) values.</p>"},{"location":"chapter_computational_complexity/time_complexity/#235-worst-best-and-average-time-complexities","title":"2.3.5 \u00a0 Worst, Best, and Average Time Complexities","text":"<p>The time efficiency of an algorithm is often not fixed but depends on the distribution of the input data. Assume we have an array <code>nums</code> of length \\(n\\), consisting of numbers from \\(1\\) to \\(n\\), each appearing only once, but in a randomly shuffled order. The task is to return the index of the element \\(1\\). We can draw the following conclusions:</p> <ul> <li>When <code>nums = [?, ?, ..., 1]</code>, that is, when the last element is \\(1\\), it requires a complete traversal of the array, achieving the worst-case time complexity of \\(O(n)\\).</li> <li>When <code>nums = [1, ?, ?, ...]</code>, that is, when the first element is \\(1\\), no matter the length of the array, no further traversal is needed, achieving the best-case time complexity of \\(\\Omega(1)\\).</li> </ul> <p>The \"worst-case time complexity\" corresponds to the asymptotic upper bound, denoted by the big \\(O\\) notation. Correspondingly, the \"best-case time complexity\" corresponds to the asymptotic lower bound, denoted by \\(\\Omega\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig worst_best_time_complexity.py<pre><code>def random_numbers(n: int) -&gt; list[int]:\n    \"\"\"\u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a: 1, 2, ..., n \uff0c\u987a\u5e8f\u88ab\u6253\u4e71\"\"\"\n    # \u751f\u6210\u6570\u7ec4 nums =: 1, 2, 3, ..., n\n    nums = [i for i in range(1, n + 1)]\n    # \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    random.shuffle(nums)\n    return nums\n\ndef find_one(nums: list[int]) -&gt; int:\n    \"\"\"\u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\"\"\"\n    for i in range(len(nums)):\n        # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1:\n            return i\n    return -1\n</code></pre> worst_best_time_complexity.cpp<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nvector&lt;int&gt; randomNumbers(int n) {\n    vector&lt;int&gt; nums(n);\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u4f7f\u7528\u7cfb\u7edf\u65f6\u95f4\u751f\u6210\u968f\u673a\u79cd\u5b50\n    unsigned seed = chrono::system_clock::now().time_since_epoch().count();\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    shuffle(nums.begin(), nums.end(), default_random_engine(seed));\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector&lt;int&gt; &amp;nums) {\n    for (int i = 0; i &lt; nums.size(); i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.java<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\n    Integer[] nums = new Integer[n];\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    Collections.shuffle(Arrays.asList(nums));\n    // Integer[] -&gt; int[]\n    int[] res = new int[n];\n    for (int i = 0; i &lt; n; i++) {\n        res[i] = nums[i];\n    }\n    return res;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.cs<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] RandomNumbers(int n) {\n    int[] nums = new int[n];\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (int i = 0; i &lt; nums.Length; i++) {\n        int index = new Random().Next(i, nums.Length);\n        (nums[i], nums[index]) = (nums[index], nums[i]);\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint FindOne(int[] nums) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.go<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n int) []int {\n    nums := make([]int, n)\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for i := 0; i &lt; n; i++ {\n        nums[i] = i + 1\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    rand.Shuffle(len(nums), func(i, j int) {\n        nums[i], nums[j] = nums[j], nums[i]\n    })\n    return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\n    for i := 0; i &lt; len(nums); i++ {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.swift<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n: Int) -&gt; [Int] {\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    var nums = Array(1 ... n)\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    nums.shuffle()\n    return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -&gt; Int {\n    for i in nums.indices {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.js<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n) {\n    const nums = Array(n);\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.ts<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n: number): number[] {\n    const nums = Array(n);\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.dart<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nList&lt;int&gt; randomNumbers(int n) {\n  final nums = List.filled(n, 0);\n  // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n  for (var i = 0; i &lt; n; i++) {\n    nums[i] = i + 1;\n  }\n  // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n  nums.shuffle();\n\n  return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(List&lt;int&gt; nums) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n    // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n    if (nums[i] == 1) return i;\n  }\n\n  return -1;\n}\n</code></pre> worst_best_time_complexity.rs<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfn random_numbers(n: i32) -&gt; Vec&lt;i32&gt; {\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    let mut nums = (1..=n).collect::&lt;Vec&lt;i32&gt;&gt;();\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    nums.shuffle(&amp;mut thread_rng());\n    nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfn find_one(nums: &amp;[i32]) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1 {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> worst_best_time_complexity.c<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint *randomNumbers(int n) {\n    // \u5206\u914d\u5806\u533a\u5185\u5b58\uff08\u521b\u5efa\u4e00\u7ef4\u53ef\u53d8\u957f\u6570\u7ec4\uff1a\u6570\u7ec4\u4e2d\u5143\u7d20\u6570\u91cf\u4e3a n \uff0c\u5143\u7d20\u7c7b\u578b\u4e3a int \uff09\n    int *nums = (int *)malloc(n * sizeof(int));\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (int i = n - 1; i &gt; 0; i--) {\n        int j = rand() % (i + 1);\n        int temp = nums[i];\n        nums[i] = nums[j];\n        nums[j] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int *nums, int n) {\n    for (int i = 0; i &lt; n; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.kt<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfun randomNumbers(n: Int): Array&lt;Int?&gt; {\n    val nums = IntArray(n)\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (i in 0..&lt;n) {\n        nums[i] = i + 1\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    val mutableList = nums.toMutableList()\n    mutableList.shuffle()\n    // Integer[] -&gt; int[]\n    val res = arrayOfNulls&lt;Int&gt;(n)\n    for (i in 0..&lt;n) {\n        res[i] = mutableList[i]\n    }\n    return res\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfun findOne(nums: Array&lt;Int?&gt;): Int {\n    for (i in nums.indices) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.rb<pre><code>[class]{}-[func]{random_numbers}\n\n[class]{}-[func]{find_one}\n</code></pre> worst_best_time_complexity.zig<pre><code>// \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71\nfn randomNumbers(comptime n: usize) [n]i32 {\n    var nums: [n]i32 = undefined;\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (&amp;nums, 0..) |*num, i| {\n        num.* = @as(i32, @intCast(i)) + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    const rand = std.crypto.random;\n    rand.shuffle(i32, &amp;nums);\n    return nums;\n}\n\n// \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\nfn findOne(nums: []i32) i32 {\n    for (nums, 0..) |num, i| {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (num == 1) return @intCast(i);\n    }\n    return -1;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>It's important to note that the best-case time complexity is rarely used in practice, as it is usually only achievable under very low probabilities and might be misleading. The worst-case time complexity is more practical as it provides a safety value for efficiency, allowing us to confidently use the algorithm.</p> <p>From the above example, it's clear that both the worst-case and best-case time complexities only occur under \"special data distributions,\" which may have a small probability of occurrence and may not accurately reflect the algorithm's run efficiency. In contrast, the average time complexity can reflect the algorithm's efficiency under random input data, denoted by the \\(\\Theta\\) notation.</p> <p>For some algorithms, we can simply estimate the average case under a random data distribution. For example, in the aforementioned example, since the input array is shuffled, the probability of element \\(1\\) appearing at any index is equal. Therefore, the average number of loops for the algorithm is half the length of the array \\(n / 2\\), giving an average time complexity of \\(\\Theta(n / 2) = \\Theta(n)\\).</p> <p>However, calculating the average time complexity for more complex algorithms can be quite difficult, as it's challenging to analyze the overall mathematical expectation under the data distribution. In such cases, we usually use the worst-case time complexity as the standard for judging the efficiency of the algorithm.</p> <p>Why is the \\(\\Theta\\) symbol rarely seen?</p> <p>Possibly because the \\(O\\) notation is more commonly spoken, it is often used to represent the average time complexity. However, strictly speaking, this practice is not accurate. In this book and other materials, if you encounter statements like \"average time complexity \\(O(n)\\)\", please understand it directly as \\(\\Theta(n)\\).</p>"},{"location":"chapter_data_structure/","title":"Chapter 3. \u00a0 Data Structures","text":"<p>Abstract</p> <p>Data structures serve as a robust and diverse framework.</p> <p>They offer a blueprint for the orderly organization of data, upon which algorithms come to life.</p>"},{"location":"chapter_data_structure/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>3.1 \u00a0 Classification of Data Structures</li> <li>3.2 \u00a0 Fundamental Data Types</li> <li>3.3 \u00a0 Number Encoding *</li> <li>3.4 \u00a0 Character Encoding *</li> <li>3.5 \u00a0 Summary</li> </ul>"},{"location":"chapter_data_structure/basic_data_types/","title":"3.2 \u00a0 Basic Data Types","text":"<p>When discussing data in computers, various forms like text, images, videos, voice and 3D models comes to mind. Despite their different organizational forms, they are all composed of various basic data types.</p> <p>Basic data types are those that the CPU can directly operate on and are directly used in algorithms, mainly including the following.</p> <ul> <li>Integer types: <code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>.</li> <li>Floating-point types: <code>float</code>, <code>double</code>, used to represent decimals.</li> <li>Character type: <code>char</code>, used to represent letters, punctuation, and even emojis in various languages.</li> <li>Boolean type: <code>bool</code>, used to represent \"yes\" or \"no\" decisions.</li> </ul> <p>Basic data types are stored in computers in binary form. One binary digit is 1 bit. In most modern operating systems, 1 byte consists of 8 bits.</p> <p>The range of values for basic data types depends on the size of the space they occupy. Below, we take Java as an example.</p> <ul> <li>The integer type <code>byte</code> occupies 1 byte = 8 bits and can represent \\(2^8\\) numbers.</li> <li>The integer type <code>int</code> occupies 4 bytes = 32 bits and can represent \\(2^{32}\\) numbers.</li> </ul> <p>The following table lists the space occupied, value range, and default values of various basic data types in Java. While memorizing this table isn't necessary, having a general understanding of it and referencing it when required is recommended.</p> <p> Table 3-1 \u00a0 Space Occupied and Value Range of Basic Data Types </p> Type Symbol Space Occupied Minimum Value Maximum Value Default Value Integer <code>byte</code> 1 byte \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) 0 <code>short</code> 2 bytes \\(-2^{15}\\) \\(2^{15} - 1\\) 0 <code>int</code> 4 bytes \\(-2^{31}\\) \\(2^{31} - 1\\) 0 <code>long</code> 8 bytes \\(-2^{63}\\) \\(2^{63} - 1\\) 0 Float <code>float</code> 4 bytes \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0\\text{f}\\) <code>double</code> 8 bytes \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) 0.0 Char <code>char</code> 2 bytes 0 \\(2^{16} - 1\\) 0 Boolean <code>bool</code> 1 byte \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\) <p>Please note that the above table is specific to Java's basic data types. Every programming language has its own data type definitions, which might differ in space occupied, value ranges, and default values.</p> <ul> <li>In Python, the integer type <code>int</code> can be of any size, limited only by available memory; the floating-point <code>float</code> is double precision 64-bit; there is no <code>char</code> type, as a single character is actually a string <code>str</code> of length 1.</li> <li>C and C++ do not specify the size of basic data types, it varies with implementation and platform. The above table follows the LP64 data model, used for Unix 64-bit operating systems including Linux and macOS.</li> <li>The size of <code>char</code> in C and C++ is 1 byte, while in most programming languages, it depends on the specific character encoding method, as detailed in the \"Character Encoding\" chapter.</li> <li>Even though representing a boolean only requires 1 bit (0 or 1), it is usually stored in memory as 1 byte. This is because modern computer CPUs typically use 1 byte as the smallest addressable memory unit.</li> </ul> <p>So, what is the connection between basic data types and data structures? We know that data structures are ways to organize and store data in computers. The focus here is on \"structure\" rather than \"data\".</p> <p>If we want to represent \"a row of numbers\", we naturally think of using an array. This is because the linear structure of an array can represent the adjacency and the ordering of the numbers, but whether the stored content is an integer <code>int</code>, a decimal <code>float</code>, or a character <code>char</code>, is irrelevant to the \"data structure\".</p> <p>In other words, basic data types provide the \"content type\" of data, while data structures provide the \"way of organizing\" data. For example, in the following code, we use the same data structure (array) to store and represent different basic data types, including <code>int</code>, <code>float</code>, <code>char</code>, <code>bool</code>, etc.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code># Using various basic data types to initialize arrays\nnumbers: list[int] = [0] * 5\ndecimals: list[float] = [0.0] * 5\n# Python's characters are actually strings of length 1\ncharacters: list[str] = ['0'] * 5\nbools: list[bool] = [False] * 5\n# Python's lists can freely store various basic data types and object references\ndata = [0, 0.0, 'a', False, ListNode(0)]\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool bools[5];\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] bools = new boolean[5];\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] bools = new bool[5];\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar bools = [5]bool{}\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nlet numbers = Array(repeating: 0, count: 5)\nlet decimals = Array(repeating: 0.0, count: 5)\nlet characters: [Character] = Array(repeating: \"a\", count: 5)\nlet bools = Array(repeating: false, count: 5)\n</code></pre> <pre><code>// JavaScript's arrays can freely store various basic data types and objects\nconst array = [0, 0.0, 'a', false];\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst bools: boolean[] = [];\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nList&lt;int&gt; numbers = List.filled(5, 0);\nList&lt;double&gt; decimals = List.filled(5, 0.0);\nList&lt;String&gt; characters = List.filled(5, 'a');\nList&lt;bool&gt; bools = List.filled(5, false);\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nlet numbers: Vec&lt;i32&gt; = vec![0; 5];\nlet decimals: Vec&lt;f32&gt; = vec![0.0, 5];\nlet characters: Vec&lt;char&gt; = vec!['0'; 5];\nlet bools: Vec&lt;bool&gt; = vec![false; 5];\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool bools[10];\n</code></pre> <pre><code>\n</code></pre> <pre><code>// Using various basic data types to initialize arrays\nvar numbers: [5]i32 = undefined;\nvar decimals: [5]f32 = undefined;\nvar characters: [5]u8 = undefined;\nvar bools: [5]bool = undefined;\n</code></pre>"},{"location":"chapter_data_structure/character_encoding/","title":"3.4 \u00a0 Character Encoding *","text":"<p>In the computer system, all data is stored in binary form, and characters (represented by char) are no exception. To represent characters, we need to develop a \"character set\" that defines a one-to-one mapping between each character and binary numbers. With the character set, computers can convert binary numbers to characters by looking up the table.</p>"},{"location":"chapter_data_structure/character_encoding/#341-ascii-character-set","title":"3.4.1 \u00a0 ASCII Character Set","text":"<p>The \"ASCII code\" is one of the earliest character sets, officially known as the American Standard Code for Information Interchange. It uses 7 binary digits (the lower 7 bits of a byte) to represent a character, allowing for a maximum of 128 different characters. As shown in the Figure 3-6 , ASCII includes uppercase and lowercase English letters, numbers 0 ~ 9, various punctuation marks, and certain control characters (such as newline and tab).</p> <p></p> <p> Figure 3-6 \u00a0 ASCII Code </p> <p>However, ASCII can only represent English characters. With the globalization of computers, a character set called \"EASCII\" was developed to represent more languages. It expands from the 7-bit structure of ASCII to 8 bits, enabling the representation of 256 characters.</p> <p>Globally, various region-specific EASCII character sets have been introduced. The first 128 characters of these sets are consistent with the ASCII, while the remaining 128 characters are defined differently to accommodate the requirements of different languages.</p>"},{"location":"chapter_data_structure/character_encoding/#342-gbk-character-set","title":"3.4.2 \u00a0 GBK Character Set","text":"<p>Later, it was found that EASCII still could not meet the character requirements of many languages. For instance, there are nearly a hundred thousand Chinese characters, with several thousand used regularly. In 1980, the Standardization Administration of China released the \"GB2312\" character set, which included 6763 Chinese characters, essentially fulfilling the computer processing needs for the Chinese language.</p> <p>However, GB2312 could not handle some rare and traditional characters. The \"GBK\" character set expands GB2312 and includes 21886 Chinese characters. In the GBK encoding scheme, ASCII characters are represented with one byte, while Chinese characters use two bytes.</p>"},{"location":"chapter_data_structure/character_encoding/#343-unicode-character-set","title":"3.4.3 \u00a0 Unicode Character Set","text":"<p>With the rapid evolution of computer technology and a plethora of character sets and encoding standards, numerous problems arose. On the one hand, these character sets generally only defined characters for specific languages and could not function properly in multilingual environments. On the other hand, the existence of multiple character set standards for the same language caused garbled text when information was exchanged between computers using different encoding standards.</p> <p>Researchers of that era thought: What if a comprehensive character set encompassing all global languages and symbols was developed? Wouldn't this resolve the issues associated with cross-linguistic environments and garbled text? Inspired by this idea, the extensive character set, Unicode, was born.</p> <p>\"Unicode\" is referred to as \"\u7edf\u4e00\u7801\" (Unified Code) in Chinese, theoretically capable of accommodating over a million characters. It aims to incorporate characters from all over the world into a single set, providing a universal character set for processing and displaying various languages and reducing the issues of garbled text due to different encoding standards.</p> <p>Since its release in 1991, Unicode has continually expanded to include new languages and characters. As of September 2022, Unicode contains 149,186 characters, including characters, symbols, and even emojis from various languages. In the vast Unicode character set, commonly used characters occupy 2 bytes, while some rare characters may occupy 3 or even 4 bytes.</p> <p>Unicode is a universal character set that assigns a number (called a \"code point\") to each character, but it does not specify how these character code points should be stored in a computer system. One might ask: How does a system interpret Unicode code points of varying lengths within a text? For example, given a 2-byte code, how does the system determine if it represents a single 2-byte character or two 1-byte characters?</p> <p>A straightforward solution to this problem is to store all characters as equal-length encodings. As shown in the Figure 3-7 , each character in \"Hello\" occupies 1 byte, while each character in \"\u7b97\u6cd5\" (algorithm) occupies 2 bytes. We could encode all characters in \"Hello \u7b97\u6cd5\" as 2 bytes by padding the higher bits with zeros. This method would enable the system to interpret a character every 2 bytes, recovering the content of the phrase.</p> <p></p> <p> Figure 3-7 \u00a0 Unicode Encoding Example </p> <p>However, as ASCII has shown us, encoding English only requires 1 byte. Using the above approach would double the space occupied by English text compared to ASCII encoding, which is a waste of memory space. Therefore, a more efficient Unicode encoding method is needed.</p>"},{"location":"chapter_data_structure/character_encoding/#344-utf-8-encoding","title":"3.4.4 \u00a0 UTF-8 Encoding","text":"<p>Currently, UTF-8 has become the most widely used Unicode encoding method internationally. It is a variable-length encoding, using 1 to 4 bytes to represent a character, depending on the complexity of the character. ASCII characters need only 1 byte, Latin and Greek letters require 2 bytes, commonly used Chinese characters need 3 bytes, and some other rare characters need 4 bytes.</p> <p>The encoding rules for UTF-8 are not complex and can be divided into two cases:</p> <ul> <li>For 1-byte characters, set the highest bit to \\(0\\), and the remaining 7 bits to the Unicode code point. Notably, ASCII characters occupy the first 128 code points in the Unicode set. This means that UTF-8 encoding is backward compatible with ASCII. This implies that UTF-8 can be used to parse ancient ASCII text.</li> <li>For characters of length \\(n\\) bytes (where \\(n &gt; 1\\)), set the highest \\(n\\) bits of the first byte to \\(1\\), and the \\((n + 1)^{\\text{th}}\\) bit to \\(0\\); starting from the second byte, set the highest 2 bits of each byte to \\(10\\); the rest of the bits are used to fill the Unicode code point.</li> </ul> <p>The Figure 3-8  shows the UTF-8 encoding for \"Hello\u7b97\u6cd5\". It can be observed that since the highest \\(n\\) bits are set to \\(1\\), the system can determine the length of the character as \\(n\\) by counting the number of highest bits set to \\(1\\).</p> <p>But why set the highest 2 bits of the remaining bytes to \\(10\\)? Actually, this \\(10\\) serves as a kind of checksum. If the system starts parsing text from an incorrect byte, the \\(10\\) at the beginning of the byte can help the system quickly detect anomalies.</p> <p>The reason for using \\(10\\) as a checksum is that, under UTF-8 encoding rules, it's impossible for the highest two bits of a character to be \\(10\\). This can be proven by contradiction: If the highest two bits of a character are \\(10\\), it indicates that the character's length is \\(1\\), corresponding to ASCII. However, the highest bit of an ASCII character should be \\(0\\), which contradicts the assumption.</p> <p></p> <p> Figure 3-8 \u00a0 UTF-8 Encoding Example </p> <p>Apart from UTF-8, other common encoding methods include:</p> <ul> <li>UTF-16 Encoding: Uses 2 or 4 bytes to represent a character. All ASCII characters and commonly used non-English characters are represented with 2 bytes; a few characters require 4 bytes. For 2-byte characters, the UTF-16 encoding equals the Unicode code point.</li> <li>UTF-32 Encoding: Every character uses 4 bytes. This means UTF-32 occupies more space than UTF-8 and UTF-16, especially for texts with a high proportion of ASCII characters.</li> </ul> <p>From the perspective of storage space, using UTF-8 to represent English characters is very efficient because it only requires 1 byte; using UTF-16 to encode some non-English characters (such as Chinese) can be more efficient because it only requires 2 bytes, while UTF-8 might need 3 bytes.</p> <p>From a compatibility perspective, UTF-8 is the most versatile, with many tools and libraries supporting UTF-8 as a priority.</p>"},{"location":"chapter_data_structure/character_encoding/#345-character-encoding-in-programming-languages","title":"3.4.5 \u00a0 Character Encoding in Programming Languages","text":"<p>Historically, many programming languages utilized fixed-length encodings such as UTF-16 or UTF-32 for processing strings during program execution. This allows strings to be handled as arrays, offering several advantages:</p> <ul> <li>Random Access: Strings encoded in UTF-16 can be accessed randomly with ease. For UTF-8, which is a variable-length encoding, locating the \\(i^{th}\\) character requires traversing the string from the start to the \\(i^{th}\\) position, taking \\(O(n)\\) time.</li> <li>Character Counting: Similar to random access, counting the number of characters in a UTF-16 encoded string is an \\(O(1)\\) operation. However, counting characters in a UTF-8 encoded string requires traversing the entire string.</li> <li>String Operations: Many string operations like splitting, concatenating, inserting, and deleting are easier on UTF-16 encoded strings. These operations generally require additional computation on UTF-8 encoded strings to ensure the validity of the UTF-8 encoding.</li> </ul> <p>The design of character encoding schemes in programming languages is an interesting topic involving various factors:</p> <ul> <li>Java\u2019s <code>String</code> type uses UTF-16 encoding, with each character occupying 2 bytes. This was based on the initial belief that 16 bits were sufficient to represent all possible characters and proven incorrect later. As the Unicode standard expanded beyond 16 bits, characters in Java may now be represented by a pair of 16-bit values, known as \u201csurrogate pairs.\u201d</li> <li>JavaScript and TypeScript use UTF-16 encoding for similar reasons as Java. When JavaScript was first introduced by Netscape in 1995, Unicode was still in its early stages, and 16-bit encoding was sufficient to represent all Unicode characters.</li> <li>C# uses UTF-16 encoding, largely because the .NET platform, designed by Microsoft, and many Microsoft technologies, including the Windows operating system, extensively use UTF-16 encoding.</li> </ul> <p>Due to the underestimation of character counts, these languages had to use \"surrogate pairs\" to represent Unicode characters exceeding 16 bits. This approach has its drawbacks: strings containing surrogate pairs may have characters occupying 2 or 4 bytes, losing the advantage of fixed-length encoding. Additionally, handling surrogate pairs adds complexity and debugging difficulty to programming.</p> <p>Addressing these challenges, some languages have adopted alternative encoding strategies:</p> <ul> <li>Python\u2019s <code>str</code> type uses Unicode encoding with a flexible representation where the storage length of characters depends on the largest Unicode code point in the string. If all characters are ASCII, each character occupies 1 byte, 2 bytes for characters within the Basic Multilingual Plane (BMP), and 4 bytes for characters beyond the BMP.</li> <li>Go\u2019s <code>string</code> type internally uses UTF-8 encoding. Go also provides the <code>rune</code> type for representing individual Unicode code points.</li> <li>Rust\u2019s <code>str</code> and <code>String</code> types use UTF-8 encoding internally. Rust also offers the <code>char</code> type for individual Unicode code points.</li> </ul> <p>It\u2019s important to note that the above discussion pertains to how strings are stored in programming languages, which is different from how strings are stored in files or transmitted over networks. For file storage or network transmission, strings are usually encoded in UTF-8 format for optimal compatibility and space efficiency.</p>"},{"location":"chapter_data_structure/classification_of_data_structure/","title":"3.1 \u00a0 Classification of Data Structures","text":"<p>Common data structures include arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graphs. They can be classified into \"logical structure\" and \"physical structure\".</p>"},{"location":"chapter_data_structure/classification_of_data_structure/#311-logical-structure-linear-and-non-linear","title":"3.1.1 \u00a0 Logical Structure: Linear and Non-Linear","text":"<p>The logical structures reveal the logical relationships between data elements. In arrays and linked lists, data are arranged in a specific sequence, demonstrating the linear relationship between data; while in trees, data are arranged hierarchically from the top down, showing the derived relationship between \"ancestors\" and \"descendants\"; and graphs are composed of nodes and edges, reflecting the intricate network relationship.</p> <p>As shown in the Figure 3-1 , logical structures can be divided into two major categories: \"linear\" and \"non-linear\". Linear structures are more intuitive, indicating data is arranged linearly in logical relationships; non-linear structures, conversely, are arranged non-linearly.</p> <ul> <li>Linear Data Structures: Arrays, Linked Lists, Stacks, Queues, Hash Tables.</li> <li>Non-Linear Data Structures: Trees, Heaps, Graphs, Hash Tables.</li> </ul> <p></p> <p> Figure 3-1 \u00a0 Linear and Non-Linear Data Structures </p> <p>Non-linear data structures can be further divided into tree structures and network structures.</p> <ul> <li>Linear Structures: Arrays, linked lists, queues, stacks, and hash tables, where elements have a one-to-one sequential relationship.</li> <li>Tree Structures: Trees, Heaps, Hash Tables, where elements have a one-to-many relationship.</li> <li>Network Structures: Graphs, where elements have a many-to-many relationships.</li> </ul>"},{"location":"chapter_data_structure/classification_of_data_structure/#312-physical-structure-contiguous-and-dispersed","title":"3.1.2 \u00a0 Physical Structure: Contiguous and Dispersed","text":"<p>During the execution of an algorithm, the data being processed is stored in memory. The Figure 3-2  shows a computer memory stick where each black square is a physical memory space. We can think of memory as a vast Excel spreadsheet, with each cell capable of storing a certain amount of data.</p> <p>The system accesses the data at the target location by means of a memory address. As shown in the Figure 3-2 , the computer assigns a unique identifier to each cell in the table according to specific rules, ensuring that each memory space has a unique memory address. With these addresses, the program can access the data stored in memory.</p> <p></p> <p> Figure 3-2 \u00a0 Memory Stick, Memory Spaces, Memory Addresses </p> <p>Tip</p> <p>It's worth noting that comparing memory to an Excel spreadsheet is a simplified analogy. The actual working mechanism of memory is more complex, involving concepts like address space, memory management, cache mechanisms, virtual memory, and physical memory.</p> <p>Memory is a shared resource for all programs. When a block of memory is occupied by one program, it cannot be simultaneously used by other programs. Therefore, considering memory resources is crucial in designing data structures and algorithms. For instance, the algorithm's peak memory usage should not exceed the remaining free memory of the system; if there is a lack of contiguous memory blocks, then the data structure chosen must be able to be stored in non-contiguous memory blocks.</p> <p>As illustrated in the Figure 3-3 , the physical structure reflects the way data is stored in computer memory and it can be divided into contiguous space storage (arrays) and non-contiguous space storage (linked lists). The two types of physical structures exhibit complementary characteristics in terms of time efficiency and space efficiency.</p> <p></p> <p> Figure 3-3 \u00a0 Contiguous Space Storage and Dispersed Space Storage </p> <p>It is worth noting that all data structures are implemented based on arrays, linked lists, or a combination of both. For example, stacks and queues can be implemented using either arrays or linked lists; while implementations of hash tables may involve both arrays and linked lists. - Array-based implementations: Stacks, Queues, Hash Tables, Trees, Heaps, Graphs, Matrices, Tensors (arrays with dimensions \\(\\geq 3\\)). - Linked-list-based implementations: Stacks, Queues, Hash Tables, Trees, Heaps, Graphs, etc.</p> <p>Data structures implemented based on arrays are also called \u201cStatic Data Structures,\u201d meaning their length cannot be changed after initialization. Conversely, those based on linked lists are called \u201cDynamic Data Structures,\u201d which can still adjust their size during program execution.</p> <p>Tip</p> <p>If you find it challenging to comprehend the physical structure, it is recommended that you read the next chapter, \"Arrays and Linked Lists,\" and revisit this section later.</p>"},{"location":"chapter_data_structure/number_encoding/","title":"3.3 \u00a0 Number Encoding *","text":"<p>Note</p> <p>In this book, chapters marked with an asterisk '*' are optional readings. If you are short on time or find them challenging, you may skip these initially and return to them after completing the essential chapters.</p>"},{"location":"chapter_data_structure/number_encoding/#331-integer-encoding","title":"3.3.1 \u00a0 Integer Encoding","text":"<p>In the table from the previous section, we observed that all integer types can represent one more negative number than positive numbers, such as the <code>byte</code> range of \\([-128, 127]\\). This phenomenon seems counterintuitive, and its underlying reason involves knowledge of sign-magnitude, one's complement, and two's complement encoding.</p> <p>Firstly, it's important to note that numbers are stored in computers using the two's complement form. Before analyzing why this is the case, let's define these three encoding methods:</p> <ul> <li>Sign-magnitude: The highest bit of a binary representation of a number is considered the sign bit, where \\(0\\) represents a positive number and \\(1\\) represents a negative number. The remaining bits represent the value of the number.</li> <li>One's complement: The one's complement of a positive number is the same as its sign-magnitude. For negative numbers, it's obtained by inverting all bits except the sign bit.</li> <li>Two's complement: The two's complement of a positive number is the same as its sign-magnitude. For negative numbers, it's obtained by adding \\(1\\) to their one's complement.</li> </ul> <p>The following diagram illustrates the conversions among sign-magnitude, one's complement, and two's complement:</p> <p></p> <p> Figure 3-4 \u00a0 Conversions between Sign-Magnitude, One's Complement, and Two's Complement </p> <p>Although sign-magnitude is the most intuitive, it has limitations. For one, negative numbers in sign-magnitude cannot be directly used in calculations. For example, in sign-magnitude, calculating \\(1 + (-2)\\) results in \\(-3\\), which is incorrect.</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 + 1000 \\; 0010 \\newline &amp; = 1000 \\; 0011 \\newline &amp; \\rightarrow -3 \\end{aligned} \\] <p>To address this, computers introduced the one's complement. If we convert to one's complement and calculate \\(1 + (-2)\\), then convert the result back to sign-magnitude, we get the correct result of \\(-1\\).</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 \\; \\text{(Sign-magnitude)} + 1000 \\; 0010 \\; \\text{(Sign-magnitude)} \\newline &amp; = 0000 \\; 0001 \\; \\text{(One's complement)} + 1111 \\; 1101 \\; \\text{(One's complement)} \\newline &amp; = 1111 \\; 1110 \\; \\text{(One's complement)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(Sign-magnitude)} \\newline &amp; \\rightarrow -1 \\end{aligned} \\] <p>Additionally, there are two representations of zero in sign-magnitude: \\(+0\\) and \\(-0\\). This means two different binary encodings for zero, which could lead to ambiguity. For example, in conditional checks, not differentiating between positive and negative zero might result in incorrect outcomes. Addressing this ambiguity would require additional checks, potentially reducing computational efficiency.</p> \\[ \\begin{aligned} +0 &amp; \\rightarrow 0000 \\; 0000 \\newline -0 &amp; \\rightarrow 1000 \\; 0000 \\end{aligned} \\] <p>Like sign-magnitude, one's complement also suffers from the positive and negative zero ambiguity. Therefore, computers further introduced the two's complement. Let's observe the conversion process for negative zero in sign-magnitude, one's complement, and two's complement:</p> \\[ \\begin{aligned} -0 \\rightarrow \\; &amp; 1000 \\; 0000 \\; \\text{(Sign-magnitude)} \\newline = \\; &amp; 1111 \\; 1111 \\; \\text{(One's complement)} \\newline = 1 \\; &amp; 0000 \\; 0000 \\; \\text{(Two's complement)} \\newline \\end{aligned} \\] <p>Adding \\(1\\) to the one's complement of negative zero produces a carry, but with <code>byte</code> length being only 8 bits, the carried-over \\(1\\) to the 9<sup>th</sup> bit is discarded. Therefore, the two's complement of negative zero is \\(0000 \\; 0000\\), the same as positive zero, thus resolving the ambiguity.</p> <p>One last puzzle is the \\([-128, 127]\\) range for <code>byte</code>, with an additional negative number, \\(-128\\). We observe that for the interval \\([-127, +127]\\), all integers have corresponding sign-magnitude, one's complement, and two's complement, allowing for mutual conversion between them.</p> <p>However, the two's complement \\(1000 \\; 0000\\) is an exception without a corresponding sign-magnitude. According to the conversion method, its sign-magnitude would be \\(0000 \\; 0000\\), indicating zero. This presents a contradiction because its two's complement should represent itself. Computers designate this special two's complement \\(1000 \\; 0000\\) as representing \\(-128\\). In fact, the calculation of \\((-1) + (-127)\\) in two's complement results in \\(-128\\).</p> \\[ \\begin{aligned} &amp; (-127) + (-1) \\newline &amp; \\rightarrow 1111 \\; 1111 \\; \\text{(Sign-magnitude)} + 1000 \\; 0001 \\; \\text{(Sign-magnitude)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(One's complement)} + 1111 \\; 1110 \\; \\text{(One's complement)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(Two's complement)} + 1111 \\; 1111 \\; \\text{(Two's complement)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(Two's complement)} \\newline &amp; \\rightarrow -128 \\end{aligned} \\] <p>As you might have noticed, all these calculations are additions, hinting at an important fact: computers' internal hardware circuits are primarily designed around addition operations. This is because addition is simpler to implement in hardware compared to other operations like multiplication, division, and subtraction, allowing for easier parallelization and faster computation.</p> <p>It's important to note that this doesn't mean computers can only perform addition. By combining addition with basic logical operations, computers can execute a variety of other mathematical operations. For example, the subtraction \\(a - b\\) can be translated into \\(a + (-b)\\); multiplication and division can be translated into multiple additions or subtractions.</p> <p>We can now summarize the reason for using two's complement in computers: with two's complement representation, computers can use the same circuits and operations to handle both positive and negative number addition, eliminating the need for special hardware circuits for subtraction and avoiding the ambiguity of positive and negative zero. This greatly simplifies hardware design and enhances computational efficiency.</p> <p>The design of two's complement is quite ingenious, and due to space constraints, we'll stop here. Interested readers are encouraged to explore further.</p>"},{"location":"chapter_data_structure/number_encoding/#332-floating-point-number-encoding","title":"3.3.2 \u00a0 Floating-Point Number Encoding","text":"<p>You might have noticed something intriguing: despite having the same length of 4 bytes, why does a <code>float</code> have a much larger range of values compared to an <code>int</code>? This seems counterintuitive, as one would expect the range to shrink for <code>float</code> since it needs to represent fractions.</p> <p>In fact, this is due to the different representation method used by floating-point numbers (<code>float</code>). Let's consider a 32-bit binary number as:</p> \\[ b_{31} b_{30} b_{29} \\ldots b_2 b_1 b_0 \\] <p>According to the IEEE 754 standard, a 32-bit <code>float</code> consists of the following three parts:</p> <ul> <li>Sign bit \\(\\mathrm{S}\\): Occupies 1 bit, corresponding to \\(b_{31}\\).</li> <li>Exponent bit \\(\\mathrm{E}\\): Occupies 8 bits, corresponding to \\(b_{30} b_{29} \\ldots b_{23}\\).</li> <li>Fraction bit \\(\\mathrm{N}\\): Occupies 23 bits, corresponding to \\(b_{22} b_{21} \\ldots b_0\\).</li> </ul> <p>The value of a binary <code>float</code> number is calculated as:</p> \\[ \\text{val} = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2 - 127} \\times \\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\] <p>Converted to a decimal formula, this becomes:</p> \\[ \\text{val} = (-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} - 127} \\times (1 + \\mathrm{N}) \\] <p>The range of each component is:</p> \\[ \\begin{aligned} \\mathrm{S} \\in &amp; \\{ 0, 1\\}, \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = &amp; (1 + \\sum_{i=1}^{23} b_{23-i} \\times 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\] <p></p> <p> Figure 3-5 \u00a0 Example Calculation of a float in IEEE 754 Standard </p> <p>Observing the diagram, given an example data \\(\\mathrm{S} = 0\\), \\(\\mathrm{E} = 124\\), \\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\), we have:</p> \\[ \\text{val} = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\] <p>Now we can answer the initial question: The representation of <code>float</code> includes an exponent bit, leading to a much larger range than <code>int</code>. Based on the above calculation, the maximum positive number representable by <code>float</code> is approximately \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\), and the minimum negative number is obtained by switching the sign bit.</p> <p>However, the trade-off for <code>float</code>'s expanded range is a sacrifice in precision. The integer type <code>int</code> uses all 32 bits to represent the number, with values evenly distributed; but due to the exponent bit, the larger the value of a <code>float</code>, the greater the difference between adjacent numbers.</p> <p>As shown in the Table 3-2 , exponent bits \\(E = 0\\) and \\(E = 255\\) have special meanings, used to represent zero, infinity, \\(\\mathrm{NaN}\\), etc.</p> <p> Table 3-2 \u00a0 Meaning of Exponent Bits </p> Exponent Bit E Fraction Bit \\(\\mathrm{N} = 0\\) Fraction Bit \\(\\mathrm{N} \\ne 0\\) Calculation Formula \\(0\\) \\(\\pm 0\\) Subnormal Numbers \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) Normal Numbers Normal Numbers \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\) <p>It's worth noting that subnormal numbers significantly improve the precision of floating-point numbers. The smallest positive normal number is \\(2^{-126}\\), and the smallest positive subnormal number is \\(2^{-126} \\times 2^{-23}\\).</p> <p>Double-precision <code>double</code> also uses a similar representation method to <code>float</code>, which is not elaborated here for brevity.</p>"},{"location":"chapter_data_structure/summary/","title":"3.5 \u00a0 Summary","text":""},{"location":"chapter_data_structure/summary/#1-key-review","title":"1. \u00a0 Key Review","text":"<ul> <li>Data structures can be categorized from two perspectives: logical structure and physical structure. Logical structure describes the logical relationships between data elements, while physical structure describes how data is stored in computer memory.</li> <li>Common logical structures include linear, tree-like, and network structures. We generally classify data structures into linear (arrays, linked lists, stacks, queues) and non-linear (trees, graphs, heaps) based on their logical structure. The implementation of hash tables may involve both linear and non-linear data structures.</li> <li>When a program runs, data is stored in computer memory. Each memory space has a corresponding memory address, and the program accesses data through these addresses.</li> <li>Physical structures are primarily divided into contiguous space storage (arrays) and dispersed space storage (linked lists). All data structures are implemented using arrays, linked lists, or a combination of both.</li> <li>Basic data types in computers include integers (<code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>), floating-point numbers (<code>float</code>, <code>double</code>), characters (<code>char</code>), and booleans (<code>boolean</code>). Their range depends on the size of the space occupied and the representation method.</li> <li>Original code, complement code, and two's complement code are three methods of encoding numbers in computers, and they can be converted into each other. The highest bit of the original code of an integer is the sign bit, and the remaining bits represent the value of the number.</li> <li>Integers are stored in computers in the form of two's complement. In this representation, the computer can treat the addition of positive and negative numbers uniformly, without the need for special hardware circuits for subtraction, and there is no ambiguity of positive and negative zero.</li> <li>The encoding of floating-point numbers consists of 1 sign bit, 8 exponent bits, and 23 fraction bits. Due to the presence of the exponent bit, the range of floating-point numbers is much greater than that of integers, but at the cost of sacrificing precision.</li> <li>ASCII is the earliest English character set, 1 byte in length, and includes 127 characters. The GBK character set is a commonly used Chinese character set, including more than 20,000 Chinese characters. Unicode strives to provide a complete character set standard, including characters from various languages worldwide, thus solving the problem of garbled characters caused by inconsistent character encoding methods.</li> <li>UTF-8 is the most popular Unicode encoding method, with excellent universality. It is a variable-length encoding method with good scalability and effectively improves the efficiency of space usage. UTF-16 and UTF-32 are fixed-length encoding methods. When encoding Chinese characters, UTF-16 occupies less space than UTF-8. Programming languages like Java and C# use UTF-16 encoding by default.</li> </ul>"},{"location":"chapter_data_structure/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: Why does a hash table contain both linear and non-linear data structures?</p> <p>The underlying structure of a hash table is an array. To resolve hash collisions, we may use \"chaining\": each bucket in the array points to a linked list, which, when exceeding a certain threshold, might be transformed into a tree (usually a red-black tree). From a storage perspective, the foundation of a hash table is an array, where each bucket slot might contain a value, a linked list, or a tree. Therefore, hash tables may contain both linear data structures (arrays, linked lists) and non-linear data structures (trees).</p> <p>Q: Is the length of the <code>char</code> type 1 byte?</p> <p>The length of the <code>char</code> type is determined by the encoding method used by the programming language. For example, Java, JavaScript, TypeScript, and C# all use UTF-16 encoding (to save Unicode code points), so the length of the char type is 2 bytes.</p> <p>Q: Is there ambiguity in calling data structures based on arrays \"static data structures\"? Because operations like push and pop on stacks are \"dynamic\".</p> <p>While stacks indeed allow for dynamic data operations, the data structure itself remains \"static\" (with unchangeable length). Even though data structures based on arrays can dynamically add or remove elements, their capacity is fixed. If the data volume exceeds the pre-allocated size, a new, larger array needs to be created, and the contents of the old array copied into it.</p> <p>Q: When building stacks (queues) without specifying their size, why are they considered \"static data structures\"?</p> <p>In high-level programming languages, we don't need to manually specify the initial capacity of stacks (queues); this task is automatically handled internally by the class. For example, the initial capacity of Java's ArrayList is usually 10. Furthermore, the expansion operation is also implemented automatically. See the subsequent \"List\" chapter for details.</p>"},{"location":"chapter_graph/","title":"Chapter 9. \u00a0 Graph","text":"<p>Abstract</p> <p>In the journey of life, we are like individual nodes, connected by countless invisible edges.</p> <p>Every encountering and parting leaves a unique mark on this vast network graph.</p>"},{"location":"chapter_graph/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>9.1 \u00a0 Graph</li> <li>9.2 \u00a0 Basic Graph Operations</li> <li>9.3 \u00a0 Graph Traversal</li> <li>9.4 \u00a0 Summary</li> </ul>"},{"location":"chapter_graph/graph/","title":"9.1 \u00a0 Graph","text":"<p>A \"graph\" is a type of nonlinear data structure, consisting of \"vertices\" and \"edges\". A graph \\(G\\) can be abstractly represented as a collection of a set of vertices \\(V\\) and a set of edges \\(E\\). The following example shows a graph containing 5 vertices and 7 edges.</p> \\[ \\begin{aligned} V &amp; = \\{ 1, 2, 3, 4, 5 \\} \\newline E &amp; = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G &amp; = \\{ V, E \\} \\newline \\end{aligned} \\] <p>If vertices are viewed as nodes and edges as references (pointers) connecting the nodes, graphs can be seen as a data structure that extends from linked lists. As shown below, compared to linear relationships (linked lists) and divide-and-conquer relationships (trees), network relationships (graphs) are more complex due to their higher degree of freedom.</p> <p></p> <p> Figure 9-1 \u00a0 Relationship between linked lists, trees, and graphs </p>"},{"location":"chapter_graph/graph/#911-common-types-of-graphs","title":"9.1.1 \u00a0 Common types of graphs","text":"<p>Based on whether edges have direction, graphs can be divided into \"undirected graphs\" and \"directed graphs\", as shown below.</p> <ul> <li>In undirected graphs, edges represent a \"bidirectional\" connection between two vertices, for example, the \"friendship\" in WeChat or QQ.</li> <li>In directed graphs, edges have directionality, that is, the edges \\(A \\rightarrow B\\) and \\(A \\leftarrow B\\) are independent of each other, for example, the \"follow\" and \"be followed\" relationship on Weibo or TikTok.</li> </ul> <p></p> <p> Figure 9-2 \u00a0 Directed and undirected graphs </p> <p>Based on whether all vertices are connected, graphs can be divided into \"connected graphs\" and \"disconnected graphs\", as shown below.</p> <ul> <li>For connected graphs, it is possible to reach any other vertex starting from a certain vertex.</li> <li>For disconnected graphs, there is at least one vertex that cannot be reached from a certain starting vertex.</li> </ul> <p></p> <p> Figure 9-3 \u00a0 Connected and disconnected graphs </p> <p>We can also add a \"weight\" variable to edges, resulting in \"weighted graphs\" as shown below. For example, in mobile games like \"Honor of Kings\", the system calculates the \"closeness\" between players based on shared gaming time, and this closeness network can be represented with a weighted graph.</p> <p></p> <p> Figure 9-4 \u00a0 Weighted and unweighted graphs </p> <p>Graph data structures include the following commonly used terms.</p> <ul> <li>\"Adjacency\": When there is an edge connecting two vertices, these two vertices are said to be \"adjacent\". In the above figure, the adjacent vertices of vertex 1 are vertices 2, 3, and 5.</li> <li>\"Path\": The sequence of edges passed from vertex A to vertex B is called a \"path\" from A to B. In the above figure, the edge sequence 1-5-2-4 is a path from vertex 1 to vertex 4.</li> <li>\"Degree\": The number of edges a vertex has. For directed graphs, \"in-degree\" refers to how many edges point to the vertex, and \"out-degree\" refers to how many edges point out from the vertex.</li> </ul>"},{"location":"chapter_graph/graph/#912-representation-of-graphs","title":"9.1.2 \u00a0 Representation of graphs","text":"<p>Common representations of graphs include \"adjacency matrices\" and \"adjacency lists\". The following examples use undirected graphs.</p>"},{"location":"chapter_graph/graph/#1-adjacency-matrix","title":"1. \u00a0 Adjacency matrix","text":"<p>Let the number of vertices in the graph be \\(n\\), the \"adjacency matrix\" uses an \\(n \\times n\\) matrix to represent the graph, where each row (column) represents a vertex, and the matrix elements represent edges, with \\(1\\) or \\(0\\) indicating whether there is an edge between two vertices.</p> <p>As shown below, let the adjacency matrix be \\(M\\), and the list of vertices be \\(V\\), then the matrix element \\(M[i, j] = 1\\) indicates there is an edge between vertex \\(V[i]\\) and vertex \\(V[j]\\), conversely \\(M[i, j] = 0\\) indicates there is no edge between the two vertices.</p> <p></p> <p> Figure 9-5 \u00a0 Representation of a graph with an adjacency matrix </p> <p>Adjacency matrices have the following characteristics.</p> <ul> <li>A vertex cannot be connected to itself, so the elements on the main diagonal of the adjacency matrix are meaningless.</li> <li>For undirected graphs, edges in both directions are equivalent, thus the adjacency matrix is symmetric about the main diagonal.</li> <li>By replacing the elements of the adjacency matrix from \\(1\\) and \\(0\\) to weights, it can represent weighted graphs.</li> </ul> <p>When representing graphs with adjacency matrices, it is possible to directly access matrix elements to obtain edges, thus operations of addition, deletion, lookup, and modification are very efficient, all with a time complexity of \\(O(1)\\). However, the space complexity of the matrix is \\(O(n^2)\\), which consumes more memory.</p>"},{"location":"chapter_graph/graph/#2-adjacency-list","title":"2. \u00a0 Adjacency list","text":"<p>The \"adjacency list\" uses \\(n\\) linked lists to represent the graph, with each linked list node representing a vertex. The \\(i\\)-th linked list corresponds to vertex \\(i\\) and contains all adjacent vertices (vertices connected to that vertex). The Figure 9-6  shows an example of a graph stored using an adjacency list.</p> <p></p> <p> Figure 9-6 \u00a0 Representation of a graph with an adjacency list </p> <p>The adjacency list only stores actual edges, and the total number of edges is often much less than \\(n^2\\), making it more space-efficient. However, finding edges in the adjacency list requires traversing the linked list, so its time efficiency is not as good as that of the adjacency matrix.</p> <p>Observing the above figure, the structure of the adjacency list is very similar to the \"chaining\" in hash tables, hence we can use similar methods to optimize efficiency. For example, when the linked list is long, it can be transformed into an AVL tree or red-black tree, thus optimizing the time efficiency from \\(O(n)\\) to \\(O(\\log n)\\); the linked list can also be transformed into a hash table, thus reducing the time complexity to \\(O(1)\\).</p>"},{"location":"chapter_graph/graph/#913-common-applications-of-graphs","title":"9.1.3 \u00a0 Common applications of graphs","text":"<p>As shown in the Table 9-1 , many real-world systems can be modeled with graphs, and corresponding problems can be reduced to graph computing problems.</p> <p> Table 9-1 \u00a0 Common graphs in real life </p> Vertices Edges Graph Computing Problem Social Networks Users Friendships Potential Friend Recommendations Subway Lines Stations Connectivity Between Stations Shortest Route Recommendations Solar System Celestial Bodies Gravitational Forces Between Celestial Bodies Planetary Orbit Calculations"},{"location":"chapter_graph/graph_operations/","title":"9.2 \u00a0 Basic operations on graphs","text":"<p>The basic operations on graphs can be divided into operations on \"edges\" and operations on \"vertices\". Under the two representation methods of \"adjacency matrix\" and \"adjacency list\", the implementation methods are different.</p>"},{"location":"chapter_graph/graph_operations/#921-implementation-based-on-adjacency-matrix","title":"9.2.1 \u00a0 Implementation based on adjacency matrix","text":"<p>Given an undirected graph with \\(n\\) vertices, the various operations are implemented as shown in the Figure 9-7 .</p> <ul> <li>Adding or removing an edge: Directly modify the specified edge in the adjacency matrix, using \\(O(1)\\) time. Since it is an undirected graph, it is necessary to update the edges in both directions simultaneously.</li> <li>Adding a vertex: Add a row and a column at the end of the adjacency matrix and fill them all with \\(0\\)s, using \\(O(n)\\) time.</li> <li>Removing a vertex: Delete a row and a column in the adjacency matrix. The worst case is when the first row and column are removed, requiring \\((n-1)^2\\) elements to be \"moved up and to the left\", thus using \\(O(n^2)\\) time.</li> <li>Initialization: Pass in \\(n\\) vertices, initialize a vertex list <code>vertices</code> of length \\(n\\), using \\(O(n)\\) time; initialize an \\(n \\times n\\) size adjacency matrix <code>adjMat</code>, using \\(O(n^2)\\) time.</li> </ul> Initialize adjacency matrixAdd an edgeRemove an edgeAdd a vertexRemove a vertex <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-7 \u00a0 Initialization, adding and removing edges, adding and removing vertices in adjacency matrix </p> <p>Below is the implementation code for graphs represented using an adjacency matrix:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_adjacency_matrix.py<pre><code>class GraphAdjMat:\n    \"\"\"\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n\n    def __init__(self, vertices: list[int], edges: list[list[int]]):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        # \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n        self.vertices: list[int] = []\n        # \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n        self.adj_mat: list[list[int]] = []\n        # \u6dfb\u52a0\u9876\u70b9\n        for val in vertices:\n            self.add_vertex(val)\n        # \u6dfb\u52a0\u8fb9\n        # \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for e in edges:\n            self.add_edge(e[0], e[1])\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\n        return len(self.vertices)\n\n    def add_vertex(self, val: int):\n        \"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\n        n = self.size()\n        # \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        self.vertices.append(val)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        new_row = [0] * n\n        self.adj_mat.append(new_row)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for row in self.adj_mat:\n            row.append(0)\n\n    def remove_vertex(self, index: int):\n        \"\"\"\u5220\u9664\u9876\u70b9\"\"\"\n        if index &gt;= self.size():\n            raise IndexError()\n        # \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        self.vertices.pop(index)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        self.adj_mat.pop(index)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for row in self.adj_mat:\n            row.pop(index)\n\n    def add_edge(self, i: int, j: int):\n        \"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n        # \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        # \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        # \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        self.adj_mat[i][j] = 1\n        self.adj_mat[j][i] = 1\n\n    def remove_edge(self, i: int, j: int):\n        \"\"\"\u5220\u9664\u8fb9\"\"\"\n        # \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        # \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        self.adj_mat[i][j] = 0\n        self.adj_mat[j][i] = 0\n\n    def print(self):\n        \"\"\"\u6253\u5370\u90bb\u63a5\u77e9\u9635\"\"\"\n        print(\"\u9876\u70b9\u5217\u8868 =\", self.vertices)\n        print(\"\u90bb\u63a5\u77e9\u9635 =\")\n        print_matrix(self.adj_mat)\n</code></pre> graph_adjacency_matrix.cpp<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    vector&lt;int&gt; vertices;       // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    vector&lt;vector&lt;int&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    GraphAdjMat(const vector&lt;int&gt; &amp;vertices, const vector&lt;vector&lt;int&gt;&gt; &amp;edges) {\n        // \u6dfb\u52a0\u9876\u70b9\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (const vector&lt;int&gt; &amp;edge : edges) {\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int size() const {\n        return vertices.size();\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    void addVertex(int val) {\n        int n = size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.push_back(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        adjMat.emplace_back(vector&lt;int&gt;(n, 0));\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.push_back(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    void removeVertex(int index) {\n        if (index &gt;= size()) {\n            throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.erase(vertices.begin() + index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.erase(adjMat.begin() + index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.erase(row.begin() + index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    void addEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    void removeEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n        }\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    void print() {\n        cout &lt;&lt; \"\u9876\u70b9\u5217\u8868 = \";\n        printVector(vertices);\n        cout &lt;&lt; \"\u90bb\u63a5\u77e9\u9635 =\" &lt;&lt; endl;\n        printVectorMatrix(adjMat);\n    }\n};\n</code></pre> graph_adjacency_matrix.java<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    List&lt;Integer&gt; vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    List&lt;List&lt;Integer&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = new ArrayList&lt;&gt;();\n        this.adjMat = new ArrayList&lt;&gt;();\n        // \u6dfb\u52a0\u9876\u70b9\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (int[] e : edges) {\n            addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    public int size() {\n        return vertices.size();\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void addVertex(int val) {\n        int n = size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.add(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        List&lt;Integer&gt; newRow = new ArrayList&lt;&gt;(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.add(0);\n        }\n        adjMat.add(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.add(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void removeVertex(int index) {\n        if (index &gt;= size())\n            throw new IndexOutOfBoundsException();\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.remove(index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void addEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat.get(i).set(j, 1);\n        adjMat.get(j).set(i, 1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void removeEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        adjMat.get(i).set(j, 0);\n        adjMat.get(j).set(i, 0);\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    public void print() {\n        System.out.print(\"\u9876\u70b9\u5217\u8868 = \");\n        System.out.println(vertices);\n        System.out.println(\"\u90bb\u63a5\u77e9\u9635 =\");\n        PrintUtil.printMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.cs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    List&lt;int&gt; vertices;     // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    List&lt;List&lt;int&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u6dfb\u52a0\u9876\u70b9\n        foreach (int val in vertices) {\n            AddVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        foreach (int[] e in edges) {\n            AddEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int Size() {\n        return vertices.Count;\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void AddVertex(int val) {\n        int n = Size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.Add(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        List&lt;int&gt; newRow = new(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.Add(0);\n        }\n        adjMat.Add(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.Add(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void RemoveVertex(int index) {\n        if (index &gt;= Size())\n            throw new IndexOutOfRangeException();\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.RemoveAt(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.RemoveAt(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.RemoveAt(index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void AddEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void RemoveEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    public void Print() {\n        Console.Write(\"\u9876\u70b9\u5217\u8868 = \");\n        PrintUtil.PrintList(vertices);\n        Console.WriteLine(\"\u90bb\u63a5\u77e9\u9635 =\");\n        PrintUtil.PrintMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.go<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjMat struct {\n    // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    vertices []int\n    // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    adjMat [][]int\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n    // \u6dfb\u52a0\u9876\u70b9\n    n := len(vertices)\n    adjMat := make([][]int, n)\n    for i := range adjMat {\n        adjMat[i] = make([]int, n)\n    }\n    // \u521d\u59cb\u5316\u56fe\n    g := &amp;graphAdjMat{\n        vertices: vertices,\n        adjMat:   adjMat,\n    }\n    // \u6dfb\u52a0\u8fb9\n    // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    for i := range edges {\n        g.addEdge(edges[i][0], edges[i][1])\n    }\n    return g\n}\n\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjMat) size() int {\n    return len(g.vertices)\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjMat) addVertex(val int) {\n    n := g.size()\n    // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n    g.vertices = append(g.vertices, val)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n    newRow := make([]int, n)\n    g.adjMat = append(g.adjMat, newRow)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i], 0)\n    }\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjMat) removeVertex(index int) {\n    if index &gt;= g.size() {\n        return\n    }\n    // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n    g.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n    g.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n    }\n}\n\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) addEdge(i, j int) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n    g.adjMat[i][j] = 1\n    g.adjMat[j][i] = 1\n}\n\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    g.adjMat[i][j] = 0\n    g.adjMat[j][i] = 0\n}\n\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc (g *graphAdjMat) print() {\n    fmt.Printf(\"\\t\u9876\u70b9\u5217\u8868 = %v\\n\", g.vertices)\n    fmt.Printf(\"\\t\u90bb\u63a5\u77e9\u9635 = \\n\")\n    for i := range g.adjMat {\n        fmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n    }\n}\n</code></pre> graph_adjacency_matrix.swift<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    private var vertices: [Int] // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    private var adjMat: [[Int]] // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init(vertices: [Int], edges: [[Int]]) {\n        self.vertices = []\n        adjMat = []\n        // \u6dfb\u52a0\u9876\u70b9\n        for val in vertices {\n            addVertex(val: val)\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for e in edges {\n            addEdge(i: e[0], j: e[1])\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    func size() -&gt; Int {\n        vertices.count\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    func addVertex(val: Int) {\n        let n = size()\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.append(val)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        let newRow = Array(repeating: 0, count: n)\n        adjMat.append(newRow)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for i in adjMat.indices {\n            adjMat[i].append(0)\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    func removeVertex(index: Int) {\n        if index &gt;= size() {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.remove(at: index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.remove(at: index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for i in adjMat.indices {\n            adjMat[i].remove(at: index)\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    func addEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    func removeEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    func print() {\n        Swift.print(\"\u9876\u70b9\u5217\u8868 = \", terminator: \"\")\n        Swift.print(vertices)\n        Swift.print(\"\u90bb\u63a5\u77e9\u9635 =\")\n        PrintUtil.printMatrix(matrix: adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.js<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    constructor(vertices, edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u6dfb\u52a0\u9876\u70b9\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size() {\n        return this.vertices.length;\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(val) {\n        const n = this.size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        this.vertices.push(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        const newRow = [];\n        for (let j = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(index) {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        this.vertices.splice(index, 1);\n\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        this.adjMat.splice(index, 1);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    addEdge(i, j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    removeEdge(i, j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    print() {\n        console.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\n        console.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.ts<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    vertices: number[]; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    adjMat: number[][]; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    constructor(vertices: number[], edges: number[][]) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u6dfb\u52a0\u9876\u70b9\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size(): number {\n        return this.vertices.length;\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(val: number): void {\n        const n: number = this.size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        this.vertices.push(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        const newRow: number[] = [];\n        for (let j: number = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(index: number): void {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        this.vertices.splice(index, 1);\n\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        this.adjMat.splice(index, 1);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    addEdge(i: number, j: number): void {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    removeEdge(i: number, j: number): void {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    print(): void {\n        console.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\n        console.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.dart<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n  List&lt;int&gt; vertices = []; // \u9876\u70b9\u5143\u7d20\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n  List&lt;List&lt;int&gt;&gt; adjMat = []; //\u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  GraphAdjMat(List&lt;int&gt; vertices, List&lt;List&lt;int&gt;&gt; edges) {\n    this.vertices = [];\n    this.adjMat = [];\n    // \u6dfb\u52a0\u9876\u70b9\n    for (int val in vertices) {\n      addVertex(val);\n    }\n    // \u6dfb\u52a0\u8fb9\n    // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    for (List&lt;int&gt; e in edges) {\n      addEdge(e[0], e[1]);\n    }\n  }\n\n  /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n  int size() {\n    return vertices.length;\n  }\n\n  /* \u6dfb\u52a0\u9876\u70b9 */\n  void addVertex(int val) {\n    int n = size();\n    // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n    vertices.add(val);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n    List&lt;int&gt; newRow = List.filled(n, 0, growable: true);\n    adjMat.add(newRow);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n    for (List&lt;int&gt; row in adjMat) {\n      row.add(0);\n    }\n  }\n\n  /* \u5220\u9664\u9876\u70b9 */\n  void removeVertex(int index) {\n    if (index &gt;= size()) {\n      throw IndexError;\n    }\n    // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n    vertices.removeAt(index);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n    adjMat.removeAt(index);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n    for (List&lt;int&gt; row in adjMat) {\n      row.removeAt(index);\n    }\n  }\n\n  /* \u6dfb\u52a0\u8fb9 */\n  // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n  void addEdge(int i, int j) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n    adjMat[i][j] = 1;\n    adjMat[j][i] = 1;\n  }\n\n  /* \u5220\u9664\u8fb9 */\n  // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n  void removeEdge(int i, int j) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    adjMat[i][j] = 0;\n    adjMat[j][i] = 0;\n  }\n\n  /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n  void printAdjMat() {\n    print(\"\u9876\u70b9\u5217\u8868 = $vertices\");\n    print(\"\u90bb\u63a5\u77e9\u9635 = \");\n    printMatrix(adjMat);\n  }\n}\n</code></pre> graph_adjacency_matrix.rs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjMat {\n    // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    pub vertices: Vec&lt;i32&gt;,\n    // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    pub adj_mat: Vec&lt;Vec&lt;i32&gt;&gt;,\n}\n\nimpl GraphAdjMat {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(vertices: Vec&lt;i32&gt;, edges: Vec&lt;[usize; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjMat {\n            vertices: vec![],\n            adj_mat: vec![],\n        };\n        // \u6dfb\u52a0\u9876\u70b9\n        for val in vertices {\n            graph.add_vertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for edge in edges {\n            graph.add_edge(edge[0], edge[1])\n        }\n\n        graph\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    pub fn size(&amp;self) -&gt; usize {\n        self.vertices.len()\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    pub fn add_vertex(&amp;mut self, val: i32) {\n        let n = self.size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        self.vertices.push(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        self.adj_mat.push(vec![0; n]);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for row in &amp;mut self.adj_mat {\n            row.push(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    pub fn remove_vertex(&amp;mut self, index: usize) {\n        if index &gt;= self.size() {\n            panic!(\"index error\")\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        self.vertices.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        self.adj_mat.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for row in &amp;mut self.adj_mat {\n            row.remove(index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    pub fn add_edge(&amp;mut self, i: usize, j: usize) {\n        // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        self.adj_mat[i][j] = 1;\n        self.adj_mat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    pub fn remove_edge(&amp;mut self, i: usize, j: usize) {\n        // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        self.adj_mat[i][j] = 0;\n        self.adj_mat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    pub fn print(&amp;self) {\n        println!(\"\u9876\u70b9\u5217\u8868 = {:?}\", self.vertices);\n        println!(\"\u90bb\u63a5\u77e9\u9635 =\");\n        println!(\"[\");\n        for row in &amp;self.adj_mat {\n            println!(\"  {:?},\", row);\n        }\n        println!(\"]\")\n    }\n}\n</code></pre> graph_adjacency_matrix.c<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7ed3\u6784\u4f53 */\ntypedef struct {\n    int vertices[MAX_SIZE];\n    int adjMat[MAX_SIZE][MAX_SIZE];\n    int size;\n} GraphAdjMat;\n\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjMat *newGraphAdjMat() {\n    GraphAdjMat *graph = (GraphAdjMat *)malloc(sizeof(GraphAdjMat));\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        for (int j = 0; j &lt; MAX_SIZE; j++) {\n            graph-&gt;adjMat[i][j] = 0;\n        }\n    }\n    return graph;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjMat(GraphAdjMat *graph) {\n    free(graph);\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjMat *graph, int val) {\n    if (graph-&gt;size == MAX_SIZE) {\n        fprintf(stderr, \"\u56fe\u7684\u9876\u70b9\u6570\u91cf\u5df2\u8fbe\u6700\u5927\u503c\\n\");\n        return;\n    }\n    // \u6dfb\u52a0\u7b2c n \u4e2a\u9876\u70b9\uff0c\u5e76\u5c06\u7b2c n \u884c\u548c\u5217\u7f6e\u96f6\n    int n = graph-&gt;size;\n    graph-&gt;vertices[n] = val;\n    for (int i = 0; i &lt;= n; i++) {\n        graph-&gt;adjMat[n][i] = graph-&gt;adjMat[i][n] = 0;\n    }\n    graph-&gt;size++;\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjMat *graph, int index) {\n    if (index &lt; 0 || index &gt;= graph-&gt;size) {\n        fprintf(stderr, \"\u9876\u70b9\u7d22\u5f15\u8d8a\u754c\\n\");\n        return;\n    }\n    // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        graph-&gt;vertices[i] = graph-&gt;vertices[i + 1];\n    }\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        for (int j = 0; j &lt; graph-&gt;size; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i + 1][j];\n        }\n    }\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        for (int j = index; j &lt; graph-&gt;size - 1; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i][j + 1];\n        }\n    }\n    graph-&gt;size--;\n}\n\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 1;\n    graph-&gt;adjMat[j][i] = 1;\n}\n\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 0;\n    graph-&gt;adjMat[j][i] = 0;\n}\n\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid printGraphAdjMat(GraphAdjMat *graph) {\n    printf(\"\u9876\u70b9\u5217\u8868 = \");\n    printArray(graph-&gt;vertices, graph-&gt;size);\n    printf(\"\u90bb\u63a5\u77e9\u9635 =\\n\");\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        printArray(graph-&gt;adjMat[i], graph-&gt;size);\n    }\n}\n</code></pre> graph_adjacency_matrix.kt<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat(vertices: IntArray, edges: Array&lt;IntArray&gt;) {\n    val vertices: MutableList&lt;Int&gt; = ArrayList() // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    val adjMat: MutableList&lt;MutableList&lt;Int&gt;&gt; = ArrayList() // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    init {\n        // \u6dfb\u52a0\u9876\u70b9\n        for (vertex in vertices) {\n            addVertex(vertex)\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (edge in edges) {\n            addEdge(edge[0], edge[1])\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    fun size(): Int {\n        return vertices.size\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    fun addVertex(value: Int) {\n        val n = size()\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.add(value)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        val newRow: MutableList&lt;Int&gt; = mutableListOf()\n        for (j in 0..&lt;n) {\n            newRow.add(0)\n        }\n        adjMat.add(newRow)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (row in adjMat) {\n            row.add(0)\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    fun removeVertex(index: Int) {\n        if (index &gt;= size()) throw IndexOutOfBoundsException()\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.removeAt(index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.removeAt(index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (row in adjMat) {\n            row.removeAt(index)\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    fun addEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) throw java.lang.IndexOutOfBoundsException()\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    fun removeEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) throw java.lang.IndexOutOfBoundsException()\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    fun print() {\n        print(\"\u9876\u70b9\u5217\u8868 = \")\n        println(vertices);\n        println(\"\u90bb\u63a5\u77e9\u9635 =\");\n        printMatrix(adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.rb<pre><code>[class]{GraphAdjMat}-[func]{}\n</code></pre> graph_adjacency_matrix.zig<pre><code>[class]{GraphAdjMat}-[func]{}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_graph/graph_operations/#922-implementation-based-on-adjacency-list","title":"9.2.2 \u00a0 Implementation based on adjacency list","text":"<p>Given an undirected graph with a total of \\(n\\) vertices and \\(m\\) edges, the various operations can be implemented as shown in the Figure 9-8 .</p> <ul> <li>Adding an edge: Simply add the edge at the end of the corresponding vertex's linked list, using \\(O(1)\\) time. Because it is an undirected graph, it is necessary to add edges in both directions simultaneously.</li> <li>Removing an edge: Find and remove the specified edge in the corresponding vertex's linked list, using \\(O(m)\\) time. In an undirected graph, it is necessary to remove edges in both directions simultaneously.</li> <li>Adding a vertex: Add a linked list in the adjacency list and make the new vertex the head node of the list, using \\(O(1)\\) time.</li> <li>Removing a vertex: It is necessary to traverse the entire adjacency list, removing all edges that include the specified vertex, using \\(O(n + m)\\) time.</li> <li>Initialization: Create \\(n\\) vertices and \\(2m\\) edges in the adjacency list, using \\(O(n + m)\\) time.</li> </ul> Initialize adjacency listAdd an edgeRemove an edgeAdd a vertexRemove a vertex <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-8 \u00a0 Initialization, adding and removing edges, adding and removing vertices in adjacency list </p> <p>Below is the adjacency list code implementation. Compared to the above diagram, the actual code has the following differences.</p> <ul> <li>For convenience in adding and removing vertices, and to simplify the code, we use lists (dynamic arrays) instead of linked lists.</li> <li>Use a hash table to store the adjacency list, <code>key</code> being the vertex instance, <code>value</code> being the list (linked list) of adjacent vertices of that vertex.</li> </ul> <p>Additionally, we use the <code>Vertex</code> class to represent vertices in the adjacency list. The reason for this is: if, like with the adjacency matrix, list indexes were used to distinguish different vertices, then suppose you want to delete the vertex at index \\(i\\), you would need to traverse the entire adjacency list and decrement all indexes greater than \\(i\\) by \\(1\\), which is very inefficient. However, if each vertex is a unique <code>Vertex</code> instance, then deleting a vertex does not require any changes to other vertices.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_adjacency_list.py<pre><code>class GraphAdjList:\n    \"\"\"\u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n\n    def __init__(self, edges: list[list[Vertex]]):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        # \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        self.adj_list = dict[Vertex, list[Vertex]]()\n        # \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for edge in edges:\n            self.add_vertex(edge[0])\n            self.add_vertex(edge[1])\n            self.add_edge(edge[0], edge[1])\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\n        return len(self.adj_list)\n\n    def add_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # \u6dfb\u52a0\u8fb9 vet1 - vet2\n        self.adj_list[vet1].append(vet2)\n        self.adj_list[vet2].append(vet1)\n\n    def remove_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"\u5220\u9664\u8fb9\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # \u5220\u9664\u8fb9 vet1 - vet2\n        self.adj_list[vet1].remove(vet2)\n        self.adj_list[vet2].remove(vet1)\n\n    def add_vertex(self, vet: Vertex):\n        \"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\n        if vet in self.adj_list:\n            return\n        # \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        self.adj_list[vet] = []\n\n    def remove_vertex(self, vet: Vertex):\n        \"\"\"\u5220\u9664\u9876\u70b9\"\"\"\n        if vet not in self.adj_list:\n            raise ValueError()\n        # \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        self.adj_list.pop(vet)\n        # \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for vertex in self.adj_list:\n            if vet in self.adj_list[vertex]:\n                self.adj_list[vertex].remove(vet)\n\n    def print(self):\n        \"\"\"\u6253\u5370\u90bb\u63a5\u8868\"\"\"\n        print(\"\u90bb\u63a5\u8868 =\")\n        for vertex in self.adj_list:\n            tmp = [v.val for v in self.adj_list[vertex]]\n            print(f\"{vertex.val}: {tmp},\")\n</code></pre> graph_adjacency_list.cpp<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n  public:\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    unordered_map&lt;Vertex *, vector&lt;Vertex *&gt;&gt; adjList;\n\n    /* \u5728 vector \u4e2d\u5220\u9664\u6307\u5b9a\u8282\u70b9 */\n    void remove(vector&lt;Vertex *&gt; &amp;vec, Vertex *vet) {\n        for (int i = 0; i &lt; vec.size(); i++) {\n            if (vec[i] == vet) {\n                vec.erase(vec.begin() + i);\n                break;\n            }\n        }\n    }\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    GraphAdjList(const vector&lt;vector&lt;Vertex *&gt;&gt; &amp;edges) {\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (const vector&lt;Vertex *&gt; &amp;edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int size() {\n        return adjList.size();\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    void addEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1].push_back(vet2);\n        adjList[vet2].push_back(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    void removeEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        remove(adjList[vet1], vet2);\n        remove(adjList[vet2], vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    void addVertex(Vertex *vet) {\n        if (adjList.count(vet))\n            return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList[vet] = vector&lt;Vertex *&gt;();\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    void removeVertex(Vertex *vet) {\n        if (!adjList.count(vet))\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.erase(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (auto &amp;adj : adjList) {\n            remove(adj.second, vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    void print() {\n        cout &lt;&lt; \"\u90bb\u63a5\u8868 =\" &lt;&lt; endl;\n        for (auto &amp;adj : adjList) {\n            const auto &amp;key = adj.first;\n            const auto &amp;vec = adj.second;\n            cout &lt;&lt; key-&gt;val &lt;&lt; \": \";\n            printVector(vetsToVals(vec));\n        }\n    }\n};\n</code></pre> graph_adjacency_list.java<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public GraphAdjList(Vertex[][] edges) {\n        this.adjList = new HashMap&lt;&gt;();\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (Vertex[] edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    public int size() {\n        return adjList.size();\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    public void addEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList.get(vet1).add(vet2);\n        adjList.get(vet2).add(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    public void removeEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList.get(vet1).remove(vet2);\n        adjList.get(vet2).remove(vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void addVertex(Vertex vet) {\n        if (adjList.containsKey(vet))\n            return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList.put(vet, new ArrayList&lt;&gt;());\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void removeVertex(Vertex vet) {\n        if (!adjList.containsKey(vet))\n            throw new IllegalArgumentException();\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.remove(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (List&lt;Vertex&gt; list : adjList.values()) {\n            list.remove(vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    public void print() {\n        System.out.println(\"\u90bb\u63a5\u8868 =\");\n        for (Map.Entry&lt;Vertex, List&lt;Vertex&gt;&gt; pair : adjList.entrySet()) {\n            List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n            for (Vertex vertex : pair.getValue())\n                tmp.add(vertex.val);\n            System.out.println(pair.getKey().val + \": \" + tmp + \",\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.cs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    public Dictionary&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* \u6784\u9020\u51fd\u6570 */\n    public GraphAdjList(Vertex[][] edges) {\n        adjList = [];\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        foreach (Vertex[] edge in edges) {\n            AddVertex(edge[0]);\n            AddVertex(edge[1]);\n            AddEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int Size() {\n        return adjList.Count;\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    public void AddEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1].Add(vet2);\n        adjList[vet2].Add(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    public void RemoveEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList[vet1].Remove(vet2);\n        adjList[vet2].Remove(vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void AddVertex(Vertex vet) {\n        if (adjList.ContainsKey(vet))\n            return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList.Add(vet, []);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void RemoveVertex(Vertex vet) {\n        if (!adjList.ContainsKey(vet))\n            throw new InvalidOperationException();\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.Remove(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        foreach (List&lt;Vertex&gt; list in adjList.Values) {\n            list.Remove(vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    public void Print() {\n        Console.WriteLine(\"\u90bb\u63a5\u8868 =\");\n        foreach (KeyValuePair&lt;Vertex, List&lt;Vertex&gt;&gt; pair in adjList) {\n            List&lt;int&gt; tmp = [];\n            foreach (Vertex vertex in pair.Value)\n                tmp.Add(vertex.val);\n            Console.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.go<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjList struct {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    adjList map[Vertex][]Vertex\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\n    g := &amp;graphAdjList{\n        adjList: make(map[Vertex][]Vertex),\n    }\n    // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n    for _, edge := range edges {\n        g.addVertex(edge[0])\n        g.addVertex(edge[1])\n        g.addEdge(edge[0], edge[1])\n    }\n    return g\n}\n\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjList) size() int {\n    return len(g.adjList)\n}\n\n/* \u6dfb\u52a0\u8fb9 */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // \u6dfb\u52a0\u8fb9 vet1 - vet2, \u6dfb\u52a0\u533f\u540d struct{},\n    g.adjList[vet1] = append(g.adjList[vet1], vet2)\n    g.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n\n/* \u5220\u9664\u8fb9 */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // \u5220\u9664\u8fb9 vet1 - vet2\n    g.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2)\n    g.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1)\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if ok {\n        return\n    }\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n    g.adjList[vet] = make([]Vertex, 0)\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if !ok {\n        panic(\"error\")\n    }\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n    delete(g.adjList, vet)\n    // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n    for v, list := range g.adjList {\n        g.adjList[v] = DeleteSliceElms(list, vet)\n    }\n}\n\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nfunc (g *graphAdjList) print() {\n    var builder strings.Builder\n    fmt.Printf(\"\u90bb\u63a5\u8868 = \\n\")\n    for k, v := range g.adjList {\n        builder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\n        for _, vet := range v {\n            builder.WriteString(strconv.Itoa(vet.Val) + \" \")\n        }\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> graph_adjacency_list.swift<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    public private(set) var adjList: [Vertex: [Vertex]]\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public init(edges: [[Vertex]]) {\n        adjList = [:]\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for edge in edges {\n            addVertex(vet: edge[0])\n            addVertex(vet: edge[1])\n            addEdge(vet1: edge[0], vet2: edge[1])\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    public func size() -&gt; Int {\n        adjList.count\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    public func addEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"\u53c2\u6570\u9519\u8bef\")\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1]?.append(vet2)\n        adjList[vet2]?.append(vet1)\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    public func removeEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"\u53c2\u6570\u9519\u8bef\")\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList[vet1]?.removeAll { $0 == vet2 }\n        adjList[vet2]?.removeAll { $0 == vet1 }\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public func addVertex(vet: Vertex) {\n        if adjList[vet] != nil {\n            return\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList[vet] = []\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public func removeVertex(vet: Vertex) {\n        if adjList[vet] == nil {\n            fatalError(\"\u53c2\u6570\u9519\u8bef\")\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.removeValue(forKey: vet)\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for key in adjList.keys {\n            adjList[key]?.removeAll { $0 == vet }\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    public func print() {\n        Swift.print(\"\u90bb\u63a5\u8868 =\")\n        for (vertex, list) in adjList {\n            let list = list.map { $0.val }\n            Swift.print(\"\\(vertex.val): \\(list),\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.js<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    adjList;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(edges) {\n        this.adjList = new Map();\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size() {\n        return this.adjList.size;\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    addEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    removeEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(vet) {\n        if (this.adjList.has(vet)) return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        this.adjList.set(vet, []);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(vet) {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        this.adjList.delete(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (const set of this.adjList.values()) {\n            const index = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    print() {\n        console.log('\u90bb\u63a5\u8868 =');\n        for (const [key, value] of this.adjList) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.ts<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    adjList: Map&lt;Vertex, Vertex[]&gt;;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(edges: Vertex[][]) {\n        this.adjList = new Map();\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size(): number {\n        return this.adjList.size;\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    addEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    removeEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(vet: Vertex): void {\n        if (this.adjList.has(vet)) return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        this.adjList.set(vet, []);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(vet: Vertex): void {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        this.adjList.delete(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (const set of this.adjList.values()) {\n            const index: number = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    print(): void {\n        console.log('\u90bb\u63a5\u8868 =');\n        for (const [key, value] of this.adjList.entries()) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.dart<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n  // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n  Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList = {};\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  GraphAdjList(List&lt;List&lt;Vertex&gt;&gt; edges) {\n    for (List&lt;Vertex&gt; edge in edges) {\n      addVertex(edge[0]);\n      addVertex(edge[1]);\n      addEdge(edge[0], edge[1]);\n    }\n  }\n\n  /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n  int size() {\n    return adjList.length;\n  }\n\n  /* \u6dfb\u52a0\u8fb9 */\n  void addEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // \u6dfb\u52a0\u8fb9 vet1 - vet2\n    adjList[vet1]!.add(vet2);\n    adjList[vet2]!.add(vet1);\n  }\n\n  /* \u5220\u9664\u8fb9 */\n  void removeEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // \u5220\u9664\u8fb9 vet1 - vet2\n    adjList[vet1]!.remove(vet2);\n    adjList[vet2]!.remove(vet1);\n  }\n\n  /* \u6dfb\u52a0\u9876\u70b9 */\n  void addVertex(Vertex vet) {\n    if (adjList.containsKey(vet)) return;\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n    adjList[vet] = [];\n  }\n\n  /* \u5220\u9664\u9876\u70b9 */\n  void removeVertex(Vertex vet) {\n    if (!adjList.containsKey(vet)) {\n      throw ArgumentError;\n    }\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n    adjList.remove(vet);\n    // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n    adjList.forEach((key, value) {\n      value.remove(vet);\n    });\n  }\n\n  /* \u6253\u5370\u90bb\u63a5\u8868 */\n  void printAdjList() {\n    print(\"\u90bb\u63a5\u8868 =\");\n    adjList.forEach((key, value) {\n      List&lt;int&gt; tmp = [];\n      for (Vertex vertex in value) {\n        tmp.add(vertex.val);\n      }\n      print(\"${key.val}: $tmp,\");\n    });\n  }\n}\n</code></pre> graph_adjacency_list.rs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    pub adj_list: HashMap&lt;Vertex, Vec&lt;Vertex&gt;&gt;,\n}\n\nimpl GraphAdjList {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(edges: Vec&lt;[Vertex; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjList {\n            adj_list: HashMap::new(),\n        };\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for edge in edges {\n            graph.add_vertex(edge[0]);\n            graph.add_vertex(edge[1]);\n            graph.add_edge(edge[0], edge[1]);\n        }\n\n        graph\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    #[allow(unused)]\n    pub fn size(&amp;self) -&gt; usize {\n        self.adj_list.len()\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    pub fn add_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n        {\n            panic!(\"value error\");\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        self.adj_list.get_mut(&amp;vet1).unwrap().push(vet2);\n        self.adj_list.get_mut(&amp;vet2).unwrap().push(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    #[allow(unused)]\n    pub fn remove_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n        {\n            panic!(\"value error\");\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        self.adj_list\n            .get_mut(&amp;vet1)\n            .unwrap()\n            .retain(|&amp;vet| vet != vet2);\n        self.adj_list\n            .get_mut(&amp;vet2)\n            .unwrap()\n            .retain(|&amp;vet| vet != vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    pub fn add_vertex(&amp;mut self, vet: Vertex) {\n        if self.adj_list.contains_key(&amp;vet) {\n            return;\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        self.adj_list.insert(vet, vec![]);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    #[allow(unused)]\n    pub fn remove_vertex(&amp;mut self, vet: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet) {\n            panic!(\"value error\");\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        self.adj_list.remove(&amp;vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for list in self.adj_list.values_mut() {\n            list.retain(|&amp;v| v != vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    pub fn print(&amp;self) {\n        println!(\"\u90bb\u63a5\u8868 =\");\n        for (vertex, list) in &amp;self.adj_list {\n            let list = list.iter().map(|vertex| vertex.val).collect::&lt;Vec&lt;i32&gt;&gt;();\n            println!(\"{}: {:?},\", vertex.val, list);\n        }\n    }\n}\n</code></pre> graph_adjacency_list.c<pre><code>/* \u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct AdjListNode {\n    Vertex *vertex;           // \u9876\u70b9\n    struct AdjListNode *next; // \u540e\u7ee7\u8282\u70b9\n} AdjListNode;\n\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* \u6dfb\u52a0\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid addEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *node = (AdjListNode *)malloc(sizeof(AdjListNode));\n    node-&gt;vertex = vet;\n    // \u5934\u63d2\u6cd5\n    node-&gt;next = head-&gt;next;\n    head-&gt;next = node;\n}\n\n/* \u5220\u9664\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid removeEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *pre = head;\n    AdjListNode *cur = head-&gt;next;\n    // \u5728\u94fe\u8868\u4e2d\u641c\u7d22 vet \u5bf9\u5e94\u8282\u70b9\n    while (cur != NULL &amp;&amp; cur-&gt;vertex != vet) {\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n    if (cur == NULL)\n        return;\n    // \u5c06 vet \u5bf9\u5e94\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\n    pre-&gt;next = cur-&gt;next;\n    // \u91ca\u653e\u5185\u5b58\n    free(cur);\n}\n\n/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntypedef struct {\n    AdjListNode *heads[MAX_SIZE]; // \u8282\u70b9\u6570\u7ec4\n    int size;                     // \u8282\u70b9\u6570\u91cf\n} GraphAdjList;\n\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjList *newGraphAdjList() {\n    GraphAdjList *graph = (GraphAdjList *)malloc(sizeof(GraphAdjList));\n    if (!graph) {\n        return NULL;\n    }\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        graph-&gt;heads[i] = NULL;\n    }\n    return graph;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjList(GraphAdjList *graph) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        AdjListNode *cur = graph-&gt;heads[i];\n        while (cur != NULL) {\n            AdjListNode *next = cur-&gt;next;\n            if (cur != graph-&gt;heads[i]) {\n                free(cur);\n            }\n            cur = next;\n        }\n        free(graph-&gt;heads[i]-&gt;vertex);\n        free(graph-&gt;heads[i]);\n    }\n    free(graph);\n}\n\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL &amp;&amp; head1 != head2);\n    // \u6dfb\u52a0\u8fb9 vet1 - vet2\n    addEdgeHelper(head1, vet2);\n    addEdgeHelper(head2, vet1);\n}\n\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL);\n    // \u5220\u9664\u8fb9 vet1 - vet2\n    removeEdgeHelper(head1, head2-&gt;vertex);\n    removeEdgeHelper(head2, head1-&gt;vertex);\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjList *graph, Vertex *vet) {\n    assert(graph != NULL &amp;&amp; graph-&gt;size &lt; MAX_SIZE);\n    AdjListNode *head = (AdjListNode *)malloc(sizeof(AdjListNode));\n    head-&gt;vertex = vet;\n    head-&gt;next = NULL;\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n    graph-&gt;heads[graph-&gt;size++] = head;\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjList *graph, Vertex *vet) {\n    AdjListNode *node = findNode(graph, vet);\n    assert(node != NULL);\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n    AdjListNode *cur = node, *pre = NULL;\n    while (cur) {\n        pre = cur;\n        cur = cur-&gt;next;\n        free(pre);\n    }\n    // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        cur = graph-&gt;heads[i];\n        pre = NULL;\n        while (cur) {\n            pre = cur;\n            cur = cur-&gt;next;\n            if (cur &amp;&amp; cur-&gt;vertex == vet) {\n                pre-&gt;next = cur-&gt;next;\n                free(cur);\n                break;\n            }\n        }\n    }\n    // \u5c06\u8be5\u9876\u70b9\u4e4b\u540e\u7684\u9876\u70b9\u5411\u524d\u79fb\u52a8\uff0c\u4ee5\u586b\u8865\u7a7a\u7f3a\n    int i;\n    for (i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i] == node)\n            break;\n    }\n    for (int j = i; j &lt; graph-&gt;size - 1; j++) {\n        graph-&gt;heads[j] = graph-&gt;heads[j + 1];\n    }\n    graph-&gt;size--;\n    free(vet);\n}\n</code></pre> graph_adjacency_list.kt<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList(edges: Array&lt;Array&lt;Vertex?&gt;&gt;) {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    val adjList: MutableMap&lt;Vertex, MutableList&lt;Vertex&gt;&gt; = HashMap()\n\n    /* \u6784\u9020\u51fd\u6570 */\n    init {\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (edge in edges) {\n            addVertex(edge[0]!!);\n            addVertex(edge[1]!!);\n            addEdge(edge[0]!!, edge[1]!!);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    fun size(): Int {\n        return adjList.size\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    fun addEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1]?.add(vet2)\n        adjList[vet2]?.add(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    fun removeEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList[vet1]?.remove(vet2);\n        adjList[vet2]?.remove(vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    fun addVertex(vet: Vertex) {\n        if (adjList.containsKey(vet))\n            return\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList[vet] = mutableListOf()\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    fun removeVertex(vet: Vertex) {\n        if (!adjList.containsKey(vet))\n            throw IllegalArgumentException()\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.remove(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (list in adjList.values) {\n            list.remove(vet)\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    fun print() {\n        println(\"\u90bb\u63a5\u8868 =\")\n        for (pair in adjList.entries) {\n            val tmp = ArrayList&lt;Int&gt;()\n            for (vertex in pair.value) {\n                tmp.add(vertex.value)\n            }\n            println(\"${pair.key.value}: $tmp,\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.rb<pre><code>[class]{GraphAdjList}-[func]{}\n</code></pre> graph_adjacency_list.zig<pre><code>[class]{GraphAdjList}-[func]{}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_graph/graph_operations/#923-efficiency-comparison","title":"9.2.3 \u00a0 Efficiency comparison","text":"<p>Assuming there are \\(n\\) vertices and \\(m\\) edges in the graph, the Table 9-2  compares the time efficiency and space efficiency of the adjacency matrix and adjacency list.</p> <p> Table 9-2 \u00a0 Comparison of adjacency matrix and adjacency list </p> Adjacency matrix Adjacency list (Linked list) Adjacency list (Hash table) Determine adjacency \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) Add an edge \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) Remove an edge \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) Add a vertex \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) Remove a vertex \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) Memory space usage \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\) <p>Observing the Table 9-2 , it seems that the adjacency list (hash table) has the best time efficiency and space efficiency. However, in practice, operating on edges in the adjacency matrix is more efficient, requiring only a single array access or assignment operation. Overall, the adjacency matrix exemplifies the principle of \"space for time\", while the adjacency list exemplifies \"time for space\".</p>"},{"location":"chapter_graph/graph_traversal/","title":"9.3 \u00a0 Graph traversal","text":"<p>Trees represent a \"one-to-many\" relationship, while graphs have a higher degree of freedom and can represent any \"many-to-many\" relationship. Therefore, we can consider trees as a special case of graphs. Clearly, tree traversal operations are also a special case of graph traversal operations.</p> <p>Both graphs and trees require the application of search algorithms to implement traversal operations. Graph traversal can be divided into two types: \"Breadth-First Search (BFS)\" and \"Depth-First Search (DFS)\".</p>"},{"location":"chapter_graph/graph_traversal/#931-breadth-first-search","title":"9.3.1 \u00a0 Breadth-first search","text":"<p>Breadth-first search is a near-to-far traversal method, starting from a certain node, always prioritizing the visit to the nearest vertices and expanding outwards layer by layer. As shown in the Figure 9-9 , starting from the top left vertex, first traverse all adjacent vertices of that vertex, then traverse all adjacent vertices of the next vertex, and so on, until all vertices have been visited.</p> <p></p> <p> Figure 9-9 \u00a0 Breadth-first traversal of a graph </p>"},{"location":"chapter_graph/graph_traversal/#1-algorithm-implementation","title":"1. \u00a0 Algorithm implementation","text":"<p>BFS is usually implemented with the help of a queue, as shown in the code below. The queue has a \"first in, first out\" property, which aligns with the BFS idea of traversing \"from near to far\".</p> <ol> <li>Add the starting vertex <code>startVet</code> to the queue and start the loop.</li> <li>In each iteration of the loop, pop the vertex at the front of the queue and record it as visited, then add all adjacent vertices of that vertex to the back of the queue.</li> <li>Repeat step <code>2.</code> until all vertices have been visited.</li> </ol> <p>To prevent revisiting vertices, we use a hash table <code>visited</code> to record which nodes have been visited.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_bfs.py<pre><code>def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\n    # \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    # \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res = []\n    # \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited = set[Vertex]([start_vet])\n    # \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    que = deque[Vertex]([start_vet])\n    # \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while len(que) &gt; 0:\n        vet = que.popleft()  # \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        # \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for adj_vet in graph.adj_list[vet]:\n            if adj_vet in visited:\n                continue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            que.append(adj_vet)  # \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adj_vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    # \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n</code></pre> graph_bfs.cpp<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector&lt;Vertex *&gt; graphBFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    vector&lt;Vertex *&gt; res;\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    unordered_set&lt;Vertex *&gt; visited = {startVet};\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    queue&lt;Vertex *&gt; que;\n    que.push(startVet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!que.empty()) {\n        Vertex *vet = que.front();\n        que.pop();          // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push_back(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (auto adjVet : graph.adjList[vet]) {\n            if (visited.count(adjVet))\n                continue;            // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            que.push(adjVet);        // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.emplace(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.java<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    visited.add(startVet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    Queue&lt;Vertex&gt; que = new LinkedList&lt;&gt;();\n    que.offer(startVet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!que.isEmpty()) {\n        Vertex vet = que.poll(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.add(vet);            // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (Vertex adjVet : graph.adjList.get(vet)) {\n            if (visited.contains(adjVet))\n                continue;        // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            que.offer(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.cs<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; GraphBFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    HashSet&lt;Vertex&gt; visited = [startVet];\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    Queue&lt;Vertex&gt; que = new();\n    que.Enqueue(startVet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (que.Count &gt; 0) {\n        Vertex vet = que.Dequeue(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.Add(vet);               // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        foreach (Vertex adjVet in graph.adjList[vet]) {\n            if (visited.Contains(adjVet)) {\n                continue;          // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.Enqueue(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.Add(adjVet);   // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.go<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res := make([]Vertex, 0)\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited := make(map[Vertex]struct{})\n    visited[startVet] = struct{}{}\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS, \u4f7f\u7528\u5207\u7247\u6a21\u62df\u961f\u5217\n    queue := make([]Vertex, 0)\n    queue = append(queue, startVet)\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    for len(queue) &gt; 0 {\n        // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        vet := queue[0]\n        queue = queue[1:]\n        // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        res = append(res, vet)\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for _, adjVet := range g.adjList[vet] {\n            _, isExist := visited[adjVet]\n            // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            if !isExist {\n                queue = append(queue, adjVet)\n                visited[adjVet] = struct{}{}\n            }\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.swift<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    var res: [Vertex] = []\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    var visited: Set&lt;Vertex&gt; = [startVet]\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    var que: [Vertex] = [startVet]\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while !que.isEmpty {\n        let vet = que.removeFirst() // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for adjVet in graph.adjList[vet] ?? [] {\n            if visited.contains(adjVet) {\n                continue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.append(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.insert(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.js<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph, startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited = new Set();\n    visited.add(startVet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    const que = [startVet];\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (que.length) {\n        const vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.ts<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res: Vertex[] = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited: Set&lt;Vertex&gt; = new Set();\n    visited.add(startVet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    const que = [startVet];\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (que.length) {\n        const vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\n            visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.dart<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n  // \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n  // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n  List&lt;Vertex&gt; res = [];\n  // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n  Set&lt;Vertex&gt; visited = {};\n  visited.add(startVet);\n  // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n  Queue&lt;Vertex&gt; que = Queue();\n  que.add(startVet);\n  // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n  while (que.isNotEmpty) {\n    Vertex vet = que.removeFirst(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n    res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (Vertex adjVet in graph.adjList[vet]!) {\n      if (visited.contains(adjVet)) {\n        continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n      }\n      que.add(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n      visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    }\n  }\n  // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n  return res;\n}\n</code></pre> graph_bfs.rs<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    let mut res = vec![];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    let mut visited = HashSet::new();\n    visited.insert(start_vet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    let mut que = VecDeque::new();\n    que.push_back(start_vet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while !que.is_empty() {\n        let vet = que.pop_front().unwrap(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n            for &amp;adj_vet in adj_vets {\n                if visited.contains(&amp;adj_vet) {\n                    continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n                }\n                que.push_back(adj_vet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n                visited.insert(adj_vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n            }\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res\n}\n</code></pre> graph_bfs.c<pre><code>/* \u8282\u70b9\u961f\u5217\u7ed3\u6784\u4f53 */\ntypedef struct {\n    Vertex *vertices[MAX_SIZE];\n    int front, rear, size;\n} Queue;\n\n/* \u6784\u9020\u51fd\u6570 */\nQueue *newQueue() {\n    Queue *q = (Queue *)malloc(sizeof(Queue));\n    q-&gt;front = q-&gt;rear = q-&gt;size = 0;\n    return q;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nint isEmpty(Queue *q) {\n    return q-&gt;size == 0;\n}\n\n/* \u5165\u961f\u64cd\u4f5c */\nvoid enqueue(Queue *q, Vertex *vet) {\n    q-&gt;vertices[q-&gt;rear] = vet;\n    q-&gt;rear = (q-&gt;rear + 1) % MAX_SIZE;\n    q-&gt;size++;\n}\n\n/* \u51fa\u961f\u64cd\u4f5c */\nVertex *dequeue(Queue *q) {\n    Vertex *vet = q-&gt;vertices[q-&gt;front];\n    q-&gt;front = (q-&gt;front + 1) % MAX_SIZE;\n    q-&gt;size--;\n    return vet;\n}\n\n/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **visited, int size, Vertex *vet) {\n    // \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\n    for (int i = 0; i &lt; size; i++) {\n        if (visited[i] == vet)\n            return 1;\n    }\n    return 0;\n}\n\n/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphBFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize, Vertex **visited, int *visitedSize) {\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    Queue *queue = newQueue();\n    enqueue(queue, startVet);\n    visited[(*visitedSize)++] = startVet;\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!isEmpty(queue)) {\n        Vertex *vet = dequeue(queue); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res[(*resSize)++] = vet;      // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        AdjListNode *node = findNode(graph, vet);\n        while (node != NULL) {\n            // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            if (!isVisited(visited, *visitedSize, node-&gt;vertex)) {\n                enqueue(queue, node-&gt;vertex);             // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n                visited[(*visitedSize)++] = node-&gt;vertex; // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n            }\n            node = node-&gt;next;\n        }\n    }\n    // \u91ca\u653e\u5185\u5b58\n    free(queue);\n}\n</code></pre> graph_bfs.kt<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfun graphBFS(graph: GraphAdjList, startVet: Vertex): List&lt;Vertex&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    val res: MutableList&lt;Vertex&gt; = ArrayList()\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    val visited: MutableSet&lt;Vertex&gt; = HashSet()\n    visited.add(startVet)\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    val que: Queue&lt;Vertex&gt; = LinkedList()\n    que.offer(startVet)\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!que.isEmpty()) {\n        val vet = que.poll() // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.add(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (adjVet in graph.adjList[vet]!!) {\n            if (visited.contains(adjVet)) continue  // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n\n            que.offer(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.rb<pre><code>[class]{}-[func]{graph_bfs}\n</code></pre> graph_bfs.zig<pre><code>[class]{}-[func]{graphBFS}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The code is relatively abstract, it is suggested to compare with the following figure to deepen the understanding.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-10 \u00a0 Steps of breadth-first search of a graph </p> <p>Is the sequence of breadth-first traversal unique?</p> <p>Not unique. Breadth-first traversal only requires traversing in a \"from near to far\" order, and the traversal order of multiple vertices at the same distance can be arbitrarily shuffled. For example, in the above figure, the visitation order of vertices \\(1\\) and \\(3\\) can be switched, as can the order of vertices \\(2\\), \\(4\\), and \\(6\\).</p>"},{"location":"chapter_graph/graph_traversal/#2-complexity-analysis","title":"2. \u00a0 Complexity analysis","text":"<p>Time complexity: All vertices will be enqueued and dequeued once, using \\(O(|V|)\\) time; in the process of traversing adjacent vertices, since it is an undirected graph, all edges will be visited \\(2\\) times, using \\(O(2|E|)\\) time; overall using \\(O(|V| + |E|)\\) time.</p> <p>Space complexity: The maximum number of vertices in list <code>res</code>, hash table <code>visited</code>, and queue <code>que</code> is \\(|V|\\), using \\(O(|V|)\\) space.</p>"},{"location":"chapter_graph/graph_traversal/#932-depth-first-search","title":"9.3.2 \u00a0 Depth-first search","text":"<p>Depth-first search is a traversal method that prioritizes going as far as possible and then backtracks when no further paths are available. As shown in the Figure 9-11 , starting from the top left vertex, visit some adjacent vertex of the current vertex until no further path is available, then return and continue until all vertices are traversed.</p> <p></p> <p> Figure 9-11 \u00a0 Depth-first traversal of a graph </p>"},{"location":"chapter_graph/graph_traversal/#1-algorithm-implementation_1","title":"1. \u00a0 Algorithm implementation","text":"<p>This \"go as far as possible and then return\" algorithm paradigm is usually implemented based on recursion. Similar to breadth-first search, in depth-first search, we also need the help of a hash table <code>visited</code> to record the visited vertices to avoid revisiting.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_dfs.py<pre><code>def dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n    \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570\"\"\"\n    res.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    # \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for adjVet in graph.adj_list[vet]:\n        if adjVet in visited:\n            continue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        # \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet)\n\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\n    # \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    # \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res = []\n    # \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited = set[Vertex]()\n    dfs(graph, visited, res, start_vet)\n    return res\n</code></pre> graph_dfs.cpp<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList &amp;graph, unordered_set&lt;Vertex *&gt; &amp;visited, vector&lt;Vertex *&gt; &amp;res, Vertex *vet) {\n    res.push_back(vet);   // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.emplace(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (Vertex *adjVet : graph.adjList[vet]) {\n        if (visited.count(adjVet))\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector&lt;Vertex *&gt; graphDFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    vector&lt;Vertex *&gt; res;\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    unordered_set&lt;Vertex *&gt; visited;\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.java<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList graph, Set&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (Vertex adjVet : graph.adjList.get(vet)) {\n        if (visited.contains(adjVet))\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.cs<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid DFS(GraphAdjList graph, HashSet&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.Add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.Add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    foreach (Vertex adjVet in graph.adjList[vet]) {\n        if (visited.Contains(adjVet)) {\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9                             \n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        DFS(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; GraphDFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    HashSet&lt;Vertex&gt; visited = [];\n    DFS(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.go<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n    // append \u64cd\u4f5c\u4f1a\u8fd4\u56de\u65b0\u7684\u7684\u5f15\u7528\uff0c\u5fc5\u987b\u8ba9\u539f\u5f15\u7528\u91cd\u65b0\u8d4b\u503c\u4e3a\u65b0slice\u7684\u5f15\u7528\n    *res = append(*res, vet)\n    visited[vet] = struct{}{}\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for _, adjVet := range g.adjList[vet] {\n        _, isExist := visited[adjVet]\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        if !isExist {\n            dfs(g, visited, res, adjVet)\n        }\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res := make([]Vertex, 0)\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited := make(map[Vertex]struct{})\n    dfs(g, visited, &amp;res, startVet)\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_dfs.swift<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfunc dfs(graph: GraphAdjList, visited: inout Set&lt;Vertex&gt;, res: inout [Vertex], vet: Vertex) {\n    res.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.insert(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for adjVet in graph.adjList[vet] ?? [] {\n        if visited.contains(adjVet) {\n            continue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: adjVet)\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    var res: [Vertex] = []\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    var visited: Set&lt;Vertex&gt; = []\n    dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: startVet)\n    return res\n}\n</code></pre> graph_dfs.js<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction dfs(graph, visited, res, vet) {\n    res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph, startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.ts<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfunction dfs(\n    graph: GraphAdjList,\n    visited: Set&lt;Vertex&gt;,\n    res: Vertex[],\n    vet: Vertex\n): void {\n    res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res: Vertex[] = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited: Set&lt;Vertex&gt; = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.dart<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(\n  GraphAdjList graph,\n  Set&lt;Vertex&gt; visited,\n  List&lt;Vertex&gt; res,\n  Vertex vet,\n) {\n  res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n  visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n  // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n  for (Vertex adjVet in graph.adjList[vet]!) {\n    if (visited.contains(adjVet)) {\n      continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n    }\n    // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n    dfs(graph, visited, res, adjVet);\n  }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n  // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n  List&lt;Vertex&gt; res = [];\n  // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n  Set&lt;Vertex&gt; visited = {};\n  dfs(graph, visited, res, startVet);\n  return res;\n}\n</code></pre> graph_dfs.rs<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfn dfs(graph: &amp;GraphAdjList, visited: &amp;mut HashSet&lt;Vertex&gt;, res: &amp;mut Vec&lt;Vertex&gt;, vet: Vertex) {\n    res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.insert(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n                         // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n        for &amp;adj_vet in adj_vets {\n            if visited.contains(&amp;adj_vet) {\n                continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n            dfs(graph, visited, res, adj_vet);\n        }\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    let mut res = vec![];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    let mut visited = HashSet::new();\n    dfs(&amp;graph, &amp;mut visited, &amp;mut res, start_vet);\n\n    res\n}\n</code></pre> graph_dfs.c<pre><code>/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **res, int size, Vertex *vet) {\n    // \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\n    for (int i = 0; i &lt; size; i++) {\n        if (res[i] == vet) {\n            return 1;\n        }\n    }\n    return 0;\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList *graph, Vertex **res, int *resSize, Vertex *vet) {\n    // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    res[(*resSize)++] = vet;\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    AdjListNode *node = findNode(graph, vet);\n    while (node != NULL) {\n        // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        if (!isVisited(res, *resSize, node-&gt;vertex)) {\n            // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n            dfs(graph, res, resSize, node-&gt;vertex);\n        }\n        node = node-&gt;next;\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphDFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize) {\n    dfs(graph, res, resSize, startVet);\n}\n</code></pre> graph_dfs.kt<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfun dfs(\n    graph: GraphAdjList,\n    visited: MutableSet&lt;Vertex?&gt;,\n    res: MutableList&lt;Vertex?&gt;,\n    vet: Vertex?\n) {\n    res.add(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (adjVet in graph.adjList[vet]!!) {\n        if (visited.contains(adjVet)) continue  // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet)\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfun graphDFS(\n    graph: GraphAdjList,\n    startVet: Vertex?\n): List&lt;Vertex?&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    val res: MutableList&lt;Vertex?&gt; = ArrayList()\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    val visited: MutableSet&lt;Vertex?&gt; = HashSet()\n    dfs(graph, visited, res, startVet)\n    return res\n}\n</code></pre> graph_dfs.rb<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{graph_dfs}\n</code></pre> graph_dfs.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{graphDFS}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The algorithm process of depth-first search is shown in the following figure.</p> <ul> <li>Dashed lines represent downward recursion, indicating that a new recursive method has been initiated to visit a new vertex.</li> <li>Curved dashed lines represent upward backtracking, indicating that this recursive method has returned to the position where this method was initiated.</li> </ul> <p>To deepen the understanding, it is suggested to combine the following figure with the code to simulate (or draw) the entire DFS process in your mind, including when each recursive method is initiated and when it returns.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-12 \u00a0 Steps of depth-first search of a graph </p> <p>Is the sequence of depth-first traversal unique?</p> <p>Similar to breadth-first traversal, the order of the depth-first traversal sequence is also not unique. Given a certain vertex, exploring in any direction first is possible, that is, the order of adjacent vertices can be arbitrarily shuffled, all being part of depth-first traversal.</p> <p>Taking tree traversal as an example, \"root \\(\\rightarrow\\) left \\(\\rightarrow\\) right\", \"left \\(\\rightarrow\\) root \\(\\rightarrow\\) right\", \"left \\(\\rightarrow\\) right \\(\\rightarrow\\) root\" correspond to preorder, inorder, and postorder traversals, respectively. They showcase three types of traversal priorities, yet all three are considered depth-first traversal.</p>"},{"location":"chapter_graph/graph_traversal/#2-complexity-analysis_1","title":"2. \u00a0 Complexity analysis","text":"<p>Time complexity: All vertices will be visited once, using \\(O(|V|)\\) time; all edges will be visited twice, using \\(O(2|E|)\\) time; overall using \\(O(|V| + |E|)\\) time.</p> <p>Space complexity: The maximum number of vertices in list <code>res</code>, hash table <code>visited</code> is \\(|V|\\), and the maximum recursion depth is \\(|V|\\), therefore using \\(O(|V|)\\) space.</p>"},{"location":"chapter_graph/summary/","title":"9.4 \u00a0 Summary","text":""},{"location":"chapter_graph/summary/#1-key-review","title":"1. \u00a0 Key review","text":"<ul> <li>A graph consists of vertices and edges and can be represented as a set comprising a group of vertices and a group of edges.</li> <li>Compared to linear relationships (linked lists) and divide-and-conquer relationships (trees), network relationships (graphs) have a higher degree of freedom and are therefore more complex.</li> <li>The edges of a directed graph have directionality, any vertex in a connected graph is reachable, and each edge in a weighted graph contains a weight variable.</li> <li>Adjacency matrices use matrices to represent graphs, with each row (column) representing a vertex and matrix elements representing edges, using \\(1\\) or \\(0\\) to indicate the presence or absence of an edge between two vertices. Adjacency matrices are highly efficient for add, delete, find, and modify operations, but they consume more space.</li> <li>Adjacency lists use multiple linked lists to represent graphs, with the \\(i^{th}\\) list corresponding to vertex \\(i\\), containing all its adjacent vertices. Adjacency lists save more space compared to adjacency matrices, but since it is necessary to traverse the list to find edges, their time efficiency is lower.</li> <li>When the linked lists in the adjacency list are too long, they can be converted into red-black trees or hash tables to improve query efficiency.</li> <li>From the perspective of algorithmic thinking, adjacency matrices embody the principle of \"space for time,\" while adjacency lists embody \"time for space.\"</li> <li>Graphs can be used to model various real systems, such as social networks, subway routes, etc.</li> <li>A tree is a special case of a graph, and tree traversal is also a special case of graph traversal.</li> <li>Breadth-first traversal of a graph is a search method that expands layer by layer from near to far, usually implemented with a queue.</li> <li>Depth-first traversal of a graph is a search method that prefers to go as deep as possible and backtracks when no further paths are available, often based on recursion.</li> </ul>"},{"location":"chapter_graph/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: Is a path defined as a sequence of vertices or a sequence of edges?</p> <p>Definitions vary between different language versions on Wikipedia: the English version defines a path as \"a sequence of edges,\" while the Chinese version defines it as \"a sequence of vertices.\" Here is the original text from the English version: In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.</p> <p>In this document, a path is considered a sequence of edges, rather than a sequence of vertices. This is because there might be multiple edges connecting two vertices, in which case each edge corresponds to a path.</p> <p>Q: In a disconnected graph, are there points that cannot be traversed to?</p> <p>In a disconnected graph, starting from a certain vertex, there is at least one vertex that cannot be reached. Traversing a disconnected graph requires setting multiple starting points to traverse all connected components of the graph.</p> <p>Q: In an adjacency list, does the order of \"all vertices connected to that vertex\" matter?</p> <p>It can be in any order. However, in practical applications, it might be necessary to sort according to certain rules, such as the order in which vertices are added, or the order of vertex values, etc., to facilitate the quick search for vertices with certain extremal values.</p>"},{"location":"chapter_hashing/","title":"Chapter 6. \u00a0 Hash Table","text":"<p>Abstract</p> <p>In the world of computing, a hash table is akin to an intelligent librarian.</p> <p>It understands how to compute index numbers, enabling swift retrieval of the desired book.</p>"},{"location":"chapter_hashing/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>6.1 \u00a0 Hash Table</li> <li>6.2 \u00a0 Hash Collision</li> <li>6.3 \u00a0 Hash Algorithm</li> <li>6.4 \u00a0 Summary</li> </ul>"},{"location":"chapter_hashing/hash_algorithm/","title":"6.3 \u00a0 Hash Algorithms","text":"<p>The previous two sections introduced the working principle of hash tables and the methods to handle hash collisions. However, both open addressing and chaining can only ensure that the hash table functions normally when collisions occur, but cannot reduce the frequency of hash collisions.</p> <p>If hash collisions occur too frequently, the performance of the hash table will deteriorate drastically. As shown in the Figure 6-8 , for a chaining hash table, in the ideal case, the key-value pairs are evenly distributed across the buckets, achieving optimal query efficiency; in the worst case, all key-value pairs are stored in the same bucket, degrading the time complexity to \\(O(n)\\).</p> <p></p> <p> Figure 6-8 \u00a0 Ideal and Worst Cases of Hash Collisions </p> <p>The distribution of key-value pairs is determined by the hash function. Recalling the steps of calculating a hash function, first compute the hash value, then modulo it by the array length:</p> <pre><code>index = hash(key) % capacity\n</code></pre> <p>Observing the above formula, when the hash table capacity <code>capacity</code> is fixed, the hash algorithm <code>hash()</code> determines the output value, thereby determining the distribution of key-value pairs in the hash table.</p> <p>This means that, to reduce the probability of hash collisions, we should focus on the design of the hash algorithm <code>hash()</code>.</p>"},{"location":"chapter_hashing/hash_algorithm/#631-goals-of-hash-algorithms","title":"6.3.1 \u00a0 Goals of Hash Algorithms","text":"<p>To achieve a \"fast and stable\" hash table data structure, hash algorithms should have the following characteristics:</p> <ul> <li>Determinism: For the same input, the hash algorithm should always produce the same output. Only then can the hash table be reliable.</li> <li>High Efficiency: The process of computing the hash value should be fast enough. The smaller the computational overhead, the more practical the hash table.</li> <li>Uniform Distribution: The hash algorithm should ensure that key-value pairs are evenly distributed in the hash table. The more uniform the distribution, the lower the probability of hash collisions.</li> </ul> <p>In fact, hash algorithms are not only used to implement hash tables but are also widely applied in other fields.</p> <ul> <li>Password Storage: To protect the security of user passwords, systems usually do not store the plaintext passwords but rather the hash values of the passwords. When a user enters a password, the system calculates the hash value of the input and compares it with the stored hash value. If they match, the password is considered correct.</li> <li>Data Integrity Check: The data sender can calculate the hash value of the data and send it along; the receiver can recalculate the hash value of the received data and compare it with the received hash value. If they match, the data is considered intact.</li> </ul> <p>For cryptographic applications, to prevent reverse engineering such as deducing the original password from the hash value, hash algorithms need higher-level security features.</p> <ul> <li>Unidirectionality: It should be impossible to deduce any information about the input data from the hash value.</li> <li>Collision Resistance: It should be extremely difficult to find two different inputs that produce the same hash value.</li> <li>Avalanche Effect: Minor changes in the input should lead to significant and unpredictable changes in the output.</li> </ul> <p>Note that \"Uniform Distribution\" and \"Collision Resistance\" are two separate concepts. Satisfying uniform distribution does not necessarily mean collision resistance. For example, under random input <code>key</code>, the hash function <code>key % 100</code> can produce a uniformly distributed output. However, this hash algorithm is too simple, and all <code>key</code> with the same last two digits will have the same output, making it easy to deduce a usable <code>key</code> from the hash value, thereby cracking the password.</p>"},{"location":"chapter_hashing/hash_algorithm/#632-design-of-hash-algorithms","title":"6.3.2 \u00a0 Design of Hash Algorithms","text":"<p>The design of hash algorithms is a complex issue that requires consideration of many factors. However, for some less demanding scenarios, we can also design some simple hash algorithms.</p> <ul> <li>Additive Hash: Add up the ASCII codes of each character in the input and use the total sum as the hash value.</li> <li>Multiplicative Hash: Utilize the non-correlation of multiplication, multiplying each round by a constant, accumulating the ASCII codes of each character into the hash value.</li> <li>XOR Hash: Accumulate the hash value by XORing each element of the input data.</li> <li>Rotating Hash: Accumulate the ASCII code of each character into a hash value, performing a rotation operation on the hash value before each accumulation.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig simple_hash.py<pre><code>def add_hash(key: str) -&gt; int:\n    \"\"\"\u52a0\u6cd5\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash += ord(c)\n    return hash % modulus\n\ndef mul_hash(key: str) -&gt; int:\n    \"\"\"\u4e58\u6cd5\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = 31 * hash + ord(c)\n    return hash % modulus\n\ndef xor_hash(key: str) -&gt; int:\n    \"\"\"\u5f02\u6216\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash ^= ord(c)\n    return hash % modulus\n\ndef rot_hash(key: str) -&gt; int:\n    \"\"\"\u65cb\u8f6c\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ ord(c)\n    return hash % modulus\n</code></pre> simple_hash.cpp<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (31 * hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash ^= (int)c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.java<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (31 * hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\n    int hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash ^= (int) c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n</code></pre> simple_hash.cs<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint AddHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint MulHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (31 * hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint XorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash ^= c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint RotHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.go<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (31*hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key string) int {\n    hash := 0\n    modulus := 1000000007\n    for _, b := range []byte(key) {\n        fmt.Println(int(b))\n        hash ^= int(b)\n        hash = (31*hash + int(b)) % modulus\n    }\n    return hash &amp; modulus\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ int64(b)) % modulus\n    }\n    return int(hash)\n}\n</code></pre> simple_hash.swift<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (31 * hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash ^= Int(scalar.value)\n        }\n    }\n    return hash &amp; MODULUS\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n</code></pre> simple_hash.js<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.ts<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.dart<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (31 * hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash ^= key.codeUnitAt(i);\n  }\n  return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n</code></pre> simple_hash.rs<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfn add_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfn mul_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (31 * hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfn xor_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash ^= c as i64;\n    }\n\n    (hash &amp; MODULUS) as i32\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfn rot_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n</code></pre> simple_hash.c<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (31 * hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(char *key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash ^= (unsigned char)key[i];\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (unsigned char)key[i]) % MODULUS;\n    }\n\n    return (int)hash;\n}\n</code></pre> simple_hash.kt<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfun addHash(key: String): Int {\n    var hash = 0L\n    for (c in key.toCharArray()) {\n        hash = (hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfun mulHash(key: String): Int {\n    var hash = 0L\n    for (c in key.toCharArray()) {\n        hash = (31 * hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfun xorHash(key: String): Int {\n    var hash = 0\n    for (c in key.toCharArray()) {\n        hash = hash xor c.code\n    }\n    return hash and MODULUS\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfun rotHash(key: String): Int {\n    var hash = 0L\n    for (c in key.toCharArray()) {\n        hash = ((hash shl 4) xor (hash shr 28) xor c.code.toLong()) % MODULUS\n    }\n    return hash.toInt()\n}\n</code></pre> simple_hash.rb<pre><code>[class]{}-[func]{add_hash}\n\n[class]{}-[func]{mul_hash}\n\n[class]{}-[func]{xor_hash}\n\n[class]{}-[func]{rot_hash}\n</code></pre> simple_hash.zig<pre><code>[class]{}-[func]{addHash}\n\n[class]{}-[func]{mulHash}\n\n[class]{}-[func]{xorHash}\n\n[class]{}-[func]{rotHash}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>It is observed that the last step of each hash algorithm is to take the modulus of the large prime number \\(1000000007\\) to ensure that the hash value is within an appropriate range. It is worth pondering why emphasis is placed on modulo a prime number, or what are the disadvantages of modulo a composite number? This is an interesting question.</p> <p>To conclude: Using a large prime number as the modulus can maximize the uniform distribution of hash values. Since a prime number does not share common factors with other numbers, it can reduce the periodic patterns caused by the modulo operation, thus avoiding hash collisions.</p> <p>For example, suppose we choose the composite number \\(9\\) as the modulus, which can be divided by \\(3\\), then all <code>key</code> divisible by \\(3\\) will be mapped to hash values \\(0\\), \\(3\\), \\(6\\).</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 9 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\dots \\} \\end{aligned} \\] <p>If the input <code>key</code> happens to have this kind of arithmetic sequence distribution, then the hash values will cluster, thereby exacerbating hash collisions. Now, suppose we replace <code>modulus</code> with the prime number \\(13\\), since there are no common factors between <code>key</code> and <code>modulus</code>, the uniformity of the output hash values will be significantly improved.</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 13 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\dots \\} \\end{aligned} \\] <p>It is worth noting that if the <code>key</code> is guaranteed to be randomly and uniformly distributed, then choosing a prime number or a composite number as the modulus can both produce uniformly distributed hash values. However, when the distribution of <code>key</code> has some periodicity, modulo a composite number is more likely to result in clustering.</p> <p>In summary, we usually choose a prime number as the modulus, and this prime number should be large enough to eliminate periodic patterns as much as possible, enhancing the robustness of the hash algorithm.</p>"},{"location":"chapter_hashing/hash_algorithm/#633-common-hash-algorithms","title":"6.3.3 \u00a0 Common Hash Algorithms","text":"<p>It is not hard to see that the simple hash algorithms mentioned above are quite \"fragile\" and far from reaching the design goals of hash algorithms. For example, since addition and XOR obey the commutative law, additive hash and XOR hash cannot distinguish strings with the same content but in different order, which may exacerbate hash collisions and cause security issues.</p> <p>In practice, we usually use some standard hash algorithms, such as MD5, SHA-1, SHA-2, and SHA-3. They can map input data of any length to a fixed-length hash value.</p> <p>Over the past century, hash algorithms have been in a continuous process of upgrading and optimization. Some researchers strive to improve the performance of hash algorithms, while others, including hackers, are dedicated to finding security issues in hash algorithms. The Table 6-2  shows hash algorithms commonly used in practical applications.</p> <ul> <li>MD5 and SHA-1 have been successfully attacked multiple times and are thus abandoned in various security applications.</li> <li>SHA-2 series, especially SHA-256, is one of the most secure hash algorithms to date, with no successful attacks reported, hence commonly used in various security applications and protocols.</li> <li>SHA-3 has lower implementation costs and higher computational efficiency compared to SHA-2, but its current usage coverage is not as extensive as the SHA-2 series.</li> </ul> <p> Table 6-2 \u00a0 Common Hash Algorithms </p> MD5 SHA-1 SHA-2 SHA-3 Release Year 1992 1995 2002 2008 Output Length 128 bit 160 bit 256/512 bit 224/256/384/512 bit Hash Collisions Frequent Frequent Rare Rare Security Level Low, has been successfully attacked Low, has been successfully attacked High High Applications Abandoned, still used for data integrity checks Abandoned Cryptocurrency transaction verification, digital signatures, etc. Can be used to replace SHA-2"},{"location":"chapter_hashing/hash_algorithm/#hash-values-in-data-structures","title":"Hash Values in Data Structures","text":"<p>We know that the keys in a hash table can be of various data types such as integers, decimals, or strings. Programming languages usually provide built-in hash algorithms for these data types to calculate the bucket indices in the hash table. Taking Python as an example, we can use the <code>hash()</code> function to compute the hash values for various data types.</p> <ul> <li>The hash values of integers and booleans are their own values.</li> <li>The calculation of hash values for floating-point numbers and strings is more complex, and interested readers are encouraged to study this on their own.</li> <li>The hash value of a tuple is a combination of the hash values of each of its elements, resulting in a single hash value.</li> <li>The hash value of an object is generated based on its memory address. By overriding the hash method of an object, hash values can be generated based on content.</li> </ul> <p>Tip</p> <p>Be aware that the definition and methods of the built-in hash value calculation functions in different programming languages vary.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig built_in_hash.py<pre><code>num = 3\nhash_num = hash(num)\n# Hash value of integer 3 is 3\n\nbol = True\nhash_bol = hash(bol)\n# Hash value of boolean True is 1\n\ndec = 3.14159\nhash_dec = hash(dec)\n# Hash value of decimal 3.14159 is 326484311674566659\n\nstr = \"Hello \u7b97\u6cd5\"\nhash_str = hash(str)\n# Hash value of string \"Hello \u7b97\u6cd5\" is 4617003410720528961\n\ntup = (12836, \"\u5c0f\u54c8\")\nhash_tup = hash(tup)\n# Hash value of tuple (12836, '\u5c0f\u54c8') is 1029005403108185979\n\nobj = ListNode(0)\nhash_obj = hash(obj)\n# Hash value of ListNode object at 0x1058fd810 is 274267521\n</code></pre> built_in_hash.cpp<pre><code>int num = 3;\nsize_t hashNum = hash&lt;int&gt;()(num);\n// Hash value of integer 3 is 3\n\nbool bol = true;\nsize_t hashBol = hash&lt;bool&gt;()(bol);\n// Hash value of boolean 1 is 1\n\ndouble dec = 3.14159;\nsize_t hashDec = hash&lt;double&gt;()(dec);\n// Hash value of decimal 3.14159 is 4614256650576692846\n\nstring str = \"Hello \u7b97\u6cd5\";\nsize_t hashStr = hash&lt;string&gt;()(str);\n// Hash value of string \"Hello \u7b97\u6cd5\" is 15466937326284535026\n\n// In C++, built-in std::hash() only provides hash values for basic data types\n// Hash values for arrays and objects need to be implemented separately\n</code></pre> built_in_hash.java<pre><code>int num = 3;\nint hashNum = Integer.hashCode(num);\n// Hash value of integer 3 is 3\n\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// Hash value of boolean true is 1231\n\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// Hash value of decimal 3.14159 is -1340954729\n\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode();\n// Hash value of string \"Hello \u7b97\u6cd5\" is -727081396\n\nObject[] arr = { 12836, \"\u5c0f\u54c8\" };\nint hashTup = Arrays.hashCode(arr);\n// Hash value of array [12836, \u5c0f\u54c8] is 1151158\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// Hash value of ListNode object utils.ListNode@7dc5e7b4 is 2110121908\n</code></pre> built_in_hash.cs<pre><code>int num = 3;\nint hashNum = num.GetHashCode();\n// Hash value of integer 3 is 3;\n\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// Hash value of boolean true is 1;\n\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// Hash value of decimal 3.14159 is -1340954729;\n\nstring str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.GetHashCode();\n// Hash value of string \"Hello \u7b97\u6cd5\" is -586107568;\n\nobject[] arr = [12836, \"\u5c0f\u54c8\"];\nint hashTup = arr.GetHashCode();\n// Hash value of array [12836, \u5c0f\u54c8] is 42931033;\n\nListNode obj = new(0);\nint hashObj = obj.GetHashCode();\n// Hash value of ListNode object 0 is 39053774;\n</code></pre> built_in_hash.go<pre><code>// Go does not provide built-in hash code functions\n</code></pre> built_in_hash.swift<pre><code>let num = 3\nlet hashNum = num.hashValue\n// Hash value of integer 3 is 9047044699613009734\n\nlet bol = true\nlet hashBol = bol.hashValue\n// Hash value of boolean true is -4431640247352757451\n\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// Hash value of decimal 3.14159 is -2465384235396674631\n\nlet str = \"Hello \u7b97\u6cd5\"\nlet hashStr = str.hashValue\n// Hash value of string \"Hello \u7b97\u6cd5\" is -7850626797806988787\n\nlet arr = [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")]\nlet hashTup = arr.hashValue\n// Hash value of array [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")] is -2308633508154532996\n\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// Hash value of ListNode object utils.ListNode is -2434780518035996159\n</code></pre> built_in_hash.js<pre><code>// JavaScript does not provide built-in hash code functions\n</code></pre> built_in_hash.ts<pre><code>// TypeScript does not provide built-in hash code functions\n</code></pre> built_in_hash.dart<pre><code>int num = 3;\nint hashNum = num.hashCode;\n// Hash value of integer 3 is 34803\n\nbool bol = true;\nint hashBol = bol.hashCode;\n// Hash value of boolean true is 1231\n\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// Hash value of decimal 3.14159 is 2570631074981783\n\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode;\n// Hash value of string \"Hello \u7b97\u6cd5\" is 468167534\n\nList arr = [12836, \"\u5c0f\u54c8\"];\nint hashArr = arr.hashCode;\n// Hash value of array [12836, \u5c0f\u54c8] is 976512528\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// Hash value of ListNode object Instance of 'ListNode' is 1033450432\n</code></pre> built_in_hash.rs<pre><code>use std::collections::hash_map::DefaultHasher;\nuse std::hash::{Hash, Hasher};\n\nlet num = 3;\nlet mut num_hasher = DefaultHasher::new();\nnum.hash(&amp;mut num_hasher);\nlet hash_num = num_hasher.finish();\n// Hash value of integer 3 is 568126464209439262\n\nlet bol = true;\nlet mut bol_hasher = DefaultHasher::new();\nbol.hash(&amp;mut bol_hasher);\nlet hash_bol = bol_hasher.finish();\n// Hash value of boolean true is 4952851536318644461\n\nlet dec: f32 = 3.14159;\nlet mut dec_hasher = DefaultHasher::new();\ndec.to_bits().hash(&amp;mut dec_hasher);\nlet hash_dec = dec_hasher.finish();\n// Hash value of decimal 3.14159 is 2566941990314602357\n\nlet str = \"Hello \u7b97\u6cd5\";\nlet mut str_hasher = DefaultHasher::new();\nstr.hash(&amp;mut str_hasher);\nlet hash_str = str_hasher.finish();\n// Hash value of string \"Hello \u7b97\u6cd5\" is 16092673739211250988\n\nlet arr = (&amp;12836, &amp;\"\u5c0f\u54c8\");\nlet mut tup_hasher = DefaultHasher::new();\narr.hash(&amp;mut tup_hasher);\nlet hash_tup = tup_hasher.finish();\n// Hash value of tuple (12836, \"\u5c0f\u54c8\") is 1885128010422702749\n\nlet node = ListNode::new(42);\nlet mut hasher = DefaultHasher::new();\nnode.borrow().val.hash(&amp;mut hasher);\nlet hash = hasher.finish();\n// Hash value of ListNode object RefCell { value: ListNode { val: 42, next: None } } is 15387811073369036852\n</code></pre> built_in_hash.c<pre><code>// C does not provide built-in hash code functions\n</code></pre> built_in_hash.kt<pre><code>\n</code></pre> built_in_hash.zig<pre><code>\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>In many programming languages, only immutable objects can serve as the <code>key</code> in a hash table. If we use a list (dynamic array) as a <code>key</code>, when the contents of the list change, its hash value also changes, and we would no longer be able to find the original <code>value</code> in the hash table.</p> <p>Although the member variables of a custom object (such as a linked list node) are mutable, it is hashable. This is because the hash value of an object is usually generated based on its memory address, and even if the contents of the object change, the memory address remains the same, so the hash value remains unchanged.</p> <p>You might have noticed that the hash values output in different consoles are different. This is because the Python interpreter adds a random salt to the string hash function each time it starts up. This approach effectively prevents HashDoS attacks and enhances the security of the hash algorithm.</p>"},{"location":"chapter_hashing/hash_collision/","title":"6.2 \u00a0 Hash Collision","text":"<p>As mentioned in the previous section, usually the input space of a hash function is much larger than its output space, making hash collisions theoretically inevitable. For example, if the input space consists of all integers and the output space is the size of the array capacity, multiple integers will inevitably map to the same bucket index.</p> <p>Hash collisions can lead to incorrect query results, severely affecting the usability of hash tables. To solve this problem, we expand the hash table whenever a hash collision occurs, until the collision is resolved. This method is simple and effective but inefficient due to the extensive data transfer and hash value computation involved in resizing the hash table. To improve efficiency, we can adopt the following strategies:</p> <ol> <li>Improve the data structure of the hash table, allowing it to function normally in the event of a hash collision.</li> <li>Only perform resizing when necessary, i.e., when hash collisions are severe.</li> </ol> <p>There are mainly two methods for improving the structure of hash tables: \"Separate Chaining\" and \"Open Addressing\".</p>"},{"location":"chapter_hashing/hash_collision/#621-separate-chaining","title":"6.2.1 \u00a0 Separate Chaining","text":"<p>In the original hash table, each bucket can store only one key-value pair. \"Separate chaining\" transforms individual elements into a linked list, with key-value pairs as list nodes, storing all colliding key-value pairs in the same list. The Figure 6-5  shows an example of a hash table with separate chaining.</p> <p></p> <p> Figure 6-5 \u00a0 Separate Chaining Hash Table </p> <p>The operations of a hash table implemented with separate chaining have changed as follows:</p> <ul> <li>Querying Elements: Input <code>key</code>, pass through the hash function to obtain the bucket index, access the head node of the list, then traverse the list and compare <code>key</code> to find the target key-value pair.</li> <li>Adding Elements: First access the list head node via the hash function, then add the node (key-value pair) to the list.</li> <li>Deleting Elements: Access the list head based on the hash function's result, then traverse the list to find and remove the target node.</li> </ul> <p>Separate chaining has the following limitations:</p> <ul> <li>Increased Space Usage: The linked list contains node pointers, which consume more memory space than arrays.</li> <li>Reduced Query Efficiency: Due to the need for linear traversal of the list to find the corresponding element.</li> </ul> <p>The code below provides a simple implementation of a separate chaining hash table, with two things to note:</p> <ul> <li>Lists (dynamic arrays) are used instead of linked lists for simplicity. In this setup, the hash table (array) contains multiple buckets, each of which is a list.</li> <li>This implementation includes a method for resizing the hash table. When the load factor exceeds \\(\\frac{2}{3}\\), we resize the hash table to twice its original size.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map_chaining.py<pre><code>class HashMapChaining:\n    \"\"\"\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\n        self.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\n        self.load_thres = 2.0 / 3.0  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        self.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\n        self.buckets = [[] for _ in range(self.capacity)]  # \u6876\u6570\u7ec4\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\n        return self.size / self.capacity\n\n    def get(self, key: int) -&gt; str | None:\n        \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for pair in bucket:\n            if pair.key == key:\n                return pair.val\n        # \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n        # \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for pair in bucket:\n            if pair.key == key:\n                pair.val = val\n                return\n        # \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        pair = Pair(key, val)\n        bucket.append(pair)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for pair in bucket:\n            if pair.key == key:\n                bucket.remove(pair)\n                self.size -= 1\n                break\n\n    def extend(self):\n        \"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n        # \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        buckets = self.buckets\n        # \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio\n        self.buckets = [[] for _ in range(self.capacity)]\n        self.size = 0\n        # \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for bucket in buckets:\n            for pair in bucket:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n        for bucket in self.buckets:\n            res = []\n            for pair in bucket:\n                res.append(str(pair.key) + \" -&gt; \" + pair.val)\n            print(res)\n</code></pre> hash_map_chaining.cpp<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n  private:\n    int size;                       // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity;                   // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres;               // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio;                // \u6269\u5bb9\u500d\u6570\n    vector&lt;vector&lt;Pair *&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    HashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3.0), extendRatio(2) {\n        buckets.resize(capacity);\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~HashMapChaining() {\n        for (auto &amp;bucket : buckets) {\n            for (Pair *pair : bucket) {\n                // \u91ca\u653e\u5185\u5b58\n                delete pair;\n            }\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double loadFactor() {\n        return (double)size / (double)capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    string get(int key) {\n        int index = hashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                return pair-&gt;val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n        return \"\";\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                pair-&gt;val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        buckets[index].push_back(new Pair(key, val));\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        auto &amp;bucket = buckets[index];\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (int i = 0; i &lt; bucket.size(); i++) {\n            if (bucket[i]-&gt;key == key) {\n                Pair *tmp = bucket[i];\n                bucket.erase(bucket.begin() + i); // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n                delete tmp;                       // \u91ca\u653e\u5185\u5b58\n                size--;\n                return;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        vector&lt;vector&lt;Pair *&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets.clear();\n        buckets.resize(capacity);\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (auto &amp;bucket : bucketsTmp) {\n            for (Pair *pair : bucket) {\n                put(pair-&gt;key, pair-&gt;val);\n                // \u91ca\u653e\u5185\u5b58\n                delete pair;\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (auto &amp;bucket : buckets) {\n            cout &lt;&lt; \"[\";\n            for (Pair *pair : bucket) {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; \", \";\n            }\n            cout &lt;&lt; \"]\\n\";\n        }\n    }\n};\n</code></pre> hash_map_chaining.java<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio; // \u6269\u5bb9\u500d\u6570\n    List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    String get(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, String val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        Pair pair = new Pair(key, val);\n        bucket.add(pair);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (List&lt;Pair&gt; bucket : bucketsTmp) {\n            for (Pair pair : bucket) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (List&lt;Pair&gt; bucket : buckets) {\n            List&lt;String&gt; res = new ArrayList&lt;&gt;();\n            for (Pair pair : bucket) {\n                res.add(pair.key + \" -&gt; \" + pair.val);\n            }\n            System.out.println(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.cs<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio; // \u6269\u5bb9\u500d\u6570\n    List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        int index = HashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        buckets[index].Add(new Pair(key, val));\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        foreach (Pair pair in buckets[index].ToList()) {\n            if (pair.key == key) {\n                buckets[index].Remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void Extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        foreach (List&lt;Pair&gt; bucket in bucketsTmp) {\n            foreach (Pair pair in bucket) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void Print() {\n        foreach (List&lt;Pair&gt; bucket in buckets) {\n            List&lt;string&gt; res = [];\n            foreach (Pair pair in bucket) {\n                res.Add(pair.key + \" -&gt; \" + pair.val);\n            }\n            foreach (string kv in res) {\n                Console.WriteLine(kv);\n            }\n        }\n    }\n}\n</code></pre> hash_map_chaining.go<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntype hashMapChaining struct {\n    size        int      // \u952e\u503c\u5bf9\u6570\u91cf\n    capacity    int      // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    loadThres   float64  // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    extendRatio int      // \u6269\u5bb9\u500d\u6570\n    buckets     [][]pair // \u6876\u6570\u7ec4\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapChaining() *hashMapChaining {\n    buckets := make([][]pair, 4)\n    for i := 0; i &lt; 4; i++ {\n        buckets[i] = make([]pair, 0)\n    }\n    return &amp;hashMapChaining{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     buckets,\n    }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapChaining) hashFunc(key int) int {\n    return key % m.capacity\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapChaining) loadFactor() float64 {\n    return float64(m.size) / float64(m.capacity)\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapChaining) get(key int) string {\n    idx := m.hashFunc(key)\n    bucket := m.buckets[idx]\n    // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    for _, p := range bucket {\n        if p.key == key {\n            return p.val\n        }\n    }\n    // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n    return \"\"\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapChaining) put(key int, val string) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if m.loadFactor() &gt; m.loadThres {\n        m.extend()\n    }\n    idx := m.hashFunc(key)\n    // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n    for i := range m.buckets[idx] {\n        if m.buckets[idx][i].key == key {\n            m.buckets[idx][i].val = val\n            return\n        }\n    }\n    // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n    p := pair{\n        key: key,\n        val: val,\n    }\n    m.buckets[idx] = append(m.buckets[idx], p)\n    m.size += 1\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapChaining) remove(key int) {\n    idx := m.hashFunc(key)\n    // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n    for i, p := range m.buckets[idx] {\n        if p.key == key {\n            // \u5207\u7247\u5220\u9664\n            m.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\n            m.size -= 1\n            break\n        }\n    }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    tmpBuckets := make([][]pair, len(m.buckets))\n    for i := 0; i &lt; len(m.buckets); i++ {\n        tmpBuckets[i] = make([]pair, len(m.buckets[i]))\n        copy(tmpBuckets[i], m.buckets[i])\n    }\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    m.capacity *= m.extendRatio\n    m.buckets = make([][]pair, m.capacity)\n    for i := 0; i &lt; m.capacity; i++ {\n        m.buckets[i] = make([]pair, 0)\n    }\n    m.size = 0\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for _, bucket := range tmpBuckets {\n        for _, p := range bucket {\n            m.put(p.key, p.val)\n        }\n    }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) print() {\n    var builder strings.Builder\n\n    for _, bucket := range m.buckets {\n        builder.WriteString(\"[\")\n        for _, p := range bucket {\n            builder.WriteString(strconv.Itoa(p.key) + \" -&gt; \" + p.val + \" \")\n        }\n        builder.WriteString(\"]\")\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> hash_map_chaining.swift<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n    var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    var loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    var extendRatio: Int // \u6269\u5bb9\u500d\u6570\n    var buckets: [[Pair]] // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: [], count: capacity)\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for pair in bucket {\n            if pair.key == key {\n                return pair.val\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de nil\n        return nil\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val\n                return\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        let pair = Pair(key: key, val: val)\n        buckets[index].append(pair)\n        size += 1\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (pairIndex, pair) in bucket.enumerated() {\n            if pair.key == key {\n                buckets[index].remove(at: pairIndex)\n                size -= 1\n                break\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    func extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        buckets = Array(repeating: [], count: capacity)\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for bucket in bucketsTmp {\n            for pair in bucket {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    func print() {\n        for bucket in buckets {\n            let res = bucket.map { \"\\($0.key) -&gt; \\($0.val)\" }\n            Swift.print(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.js<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    #size; // \u952e\u503c\u5bf9\u6570\u91cf\n    #capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    #loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    #extendRatio; // \u6269\u5bb9\u500d\u6570\n    #buckets; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key) {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key, val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key) {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    #extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print() {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.ts<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    #size: number; // \u952e\u503c\u5bf9\u6570\u91cf\n    #capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    #loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    #extendRatio: number; // \u6269\u5bb9\u500d\u6570\n    #buckets: Pair[][]; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key: number): number {\n        return key % this.#capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    #loadFactor(): number {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key: number): string | null {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key: number, val: string): void {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key: number): void {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    #extend(): void {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print(): void {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.dart<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n  late int size; // \u952e\u503c\u5bf9\u6570\u91cf\n  late int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n  late double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n  late int extendRatio; // \u6269\u5bb9\u500d\u6570\n  late List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  HashMapChaining() {\n    size = 0;\n    capacity = 4;\n    loadThres = 2.0 / 3.0;\n    extendRatio = 2;\n    buckets = List.generate(capacity, (_) =&gt; []);\n  }\n\n  /* \u54c8\u5e0c\u51fd\u6570 */\n  int hashFunc(int key) {\n    return key % capacity;\n  }\n\n  /* \u8d1f\u8f7d\u56e0\u5b50 */\n  double loadFactor() {\n    return size / capacity;\n  }\n\n  /* \u67e5\u8be2\u64cd\u4f5c */\n  String? get(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        return pair.val;\n      }\n    }\n    // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n    return null;\n  }\n\n  /* \u6dfb\u52a0\u64cd\u4f5c */\n  void put(int key, String val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor() &gt; loadThres) {\n      extend();\n    }\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        pair.val = val;\n        return;\n      }\n    }\n    // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n    Pair pair = Pair(key, val);\n    bucket.add(pair);\n    size++;\n  }\n\n  /* \u5220\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        bucket.remove(pair);\n        size--;\n        break;\n      }\n    }\n  }\n\n  /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n  void extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    capacity *= extendRatio;\n    buckets = List.generate(capacity, (_) =&gt; []);\n    size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (List&lt;Pair&gt; bucket in bucketsTmp) {\n      for (Pair pair in bucket) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* \u6253\u5370\u54c8\u5e0c\u8868 */\n  void printHashMap() {\n    for (List&lt;Pair&gt; bucket in buckets) {\n      List&lt;String&gt; res = [];\n      for (Pair pair in bucket) {\n        res.add(\"${pair.key} -&gt; ${pair.val}\");\n      }\n      print(res);\n    }\n  }\n}\n</code></pre> hash_map_chaining.rs<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapChaining {\n    size: i32,\n    capacity: i32,\n    load_thres: f32,\n    extend_ratio: i32,\n    buckets: Vec&lt;Vec&lt;Pair&gt;&gt;,\n}\n\nimpl HashMapChaining {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![vec![]; 4],\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % self.capacity as usize\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fn load_factor(&amp;self) -&gt; f32 {\n        self.size as f32 / self.capacity as f32\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fn remove(&amp;mut self, key: i32) -&gt; Option&lt;String&gt; {\n        let index = self.hash_func(key);\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for i in 0..bucket.len() {\n            if bucket[i].key == key {\n                let pair = bucket.remove(i);\n                self.size -= 1;\n                return Some(pair.val);\n            }\n        }\n\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de None\n        None\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fn extend(&amp;mut self) {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let buckets_tmp = std::mem::replace(&amp;mut self.buckets, vec![]);\n\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![Vec::new(); self.capacity as usize];\n        self.size = 0;\n\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for bucket in buckets_tmp {\n            for pair in bucket {\n                self.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fn print(&amp;self) {\n        for bucket in &amp;self.buckets {\n            let mut res = Vec::new();\n            for pair in bucket {\n                res.push(format!(\"{} -&gt; {}\", pair.key, pair.val));\n            }\n            println!(\"{:?}\", res);\n        }\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n\n        let index = self.hash_func(key);\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val.clone();\n                return;\n            }\n        }\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        let pair = Pair {\n            key,\n            val: val.clone(),\n        };\n        bucket.push(pair);\n        self.size += 1;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        let index = self.hash_func(key);\n        let bucket = &amp;self.buckets[index];\n\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for pair in bucket {\n            if pair.key == key {\n                return Some(&amp;pair.val);\n            }\n        }\n\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de None\n        None\n    }\n}\n</code></pre> hash_map_chaining.c<pre><code>/* \u94fe\u8868\u8282\u70b9 */\ntypedef struct Node {\n    Pair *pair;\n    struct Node *next;\n} Node;\n\n/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\n    int size;         // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity;     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio;  // \u6269\u5bb9\u500d\u6570\n    Node **buckets;   // \u6876\u6570\u7ec4\n} HashMapChaining;\n\n/* \u6784\u9020\u51fd\u6570 */\nHashMapChaining *newHashMapChaining() {\n    HashMapChaining *hashMap = (HashMapChaining *)malloc(sizeof(HashMapChaining));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    return hashMap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapChaining(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        while (cur) {\n            Node *tmp = cur;\n            cur = cur-&gt;next;\n            free(tmp-&gt;pair);\n            free(tmp);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap);\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapChaining *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapChaining *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            return cur-&gt;pair-&gt;val;\n        }\n        cur = cur-&gt;next;\n    }\n    return \"\"; // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapChaining *hashMap, int key, const char *val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    int index = hashFunc(hashMap, key);\n    // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            strcpy(cur-&gt;pair-&gt;val, val); // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n            return;\n        }\n        cur = cur-&gt;next;\n    }\n    // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n    Pair *newPair = (Pair *)malloc(sizeof(Pair));\n    newPair-&gt;key = key;\n    strcpy(newPair-&gt;val, val);\n    Node *newNode = (Node *)malloc(sizeof(Node));\n    newNode-&gt;pair = newPair;\n    newNode-&gt;next = hashMap-&gt;buckets[index];\n    hashMap-&gt;buckets[index] = newNode;\n    hashMap-&gt;size++;\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapChaining *hashMap) {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    int oldCapacity = hashMap-&gt;capacity;\n    Node **oldBuckets = hashMap-&gt;buckets;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    hashMap-&gt;size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Node *cur = oldBuckets[i];\n        while (cur) {\n            put(hashMap, cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            Node *temp = cur;\n            cur = cur-&gt;next;\n            // \u91ca\u653e\u5185\u5b58\n            free(temp-&gt;pair);\n            free(temp);\n        }\n    }\n\n    free(oldBuckets);\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    Node *cur = hashMap-&gt;buckets[index];\n    Node *pre = NULL;\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n            if (pre) {\n                pre-&gt;next = cur-&gt;next;\n            } else {\n                hashMap-&gt;buckets[index] = cur-&gt;next;\n            }\n            // \u91ca\u653e\u5185\u5b58\n            free(cur-&gt;pair);\n            free(cur);\n            hashMap-&gt;size--;\n            return;\n        }\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        printf(\"[\");\n        while (cur) {\n            printf(\"%d -&gt; %s, \", cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            cur = cur-&gt;next;\n        }\n        printf(\"]\\n\");\n    }\n}\n</code></pre> hash_map_chaining.kt<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining() {\n    var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n    var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    val loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    val extendRatio: Int // \u6269\u5bb9\u500d\u6570\n    var buckets: MutableList&lt;MutableList&lt;Pair&gt;&gt; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = ArrayList(capacity)\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (pair in bucket) {\n            if (pair.key == key) return pair.value\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (pair in bucket) {\n            if (pair.key == key) {\n                pair.value = value\n                return\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        val pair = Pair(key, value)\n        bucket.add(pair)\n        size++\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (pair in bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair)\n                size--\n                break\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fun extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        val bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        // mutablelist \u65e0\u56fa\u5b9a\u5927\u5c0f\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (bucket in bucketsTmp) {\n            for (pair in bucket) {\n                put(pair.key, pair.value)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (bucket in buckets) {\n            val res = mutableListOf&lt;String&gt;()\n            for (pair in bucket) {\n                val k = pair.key\n                val v = pair.value\n                res.add(\"$k -&gt; $v\")\n            }\n            println(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.rb<pre><code>[class]{HashMapChaining}-[func]{}\n</code></pre> hash_map_chaining.zig<pre><code>[class]{HashMapChaining}-[func]{}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>It's worth noting that when the list is very long, the query efficiency \\(O(n)\\) is poor. At this point, the list can be converted to an \"AVL tree\" or \"Red-Black tree\" to optimize the time complexity of the query operation to \\(O(\\log n)\\).</p>"},{"location":"chapter_hashing/hash_collision/#622-open-addressing","title":"6.2.2 \u00a0 Open Addressing","text":"<p>\"Open addressing\" does not introduce additional data structures but uses \"multiple probes\" to handle hash collisions. The probing methods mainly include linear probing, quadratic probing, and double hashing.</p> <p>Let's use linear probing as an example to introduce the mechanism of open addressing hash tables.</p>"},{"location":"chapter_hashing/hash_collision/#1-linear-probing","title":"1. \u00a0 Linear Probing","text":"<p>Linear probing uses a fixed-step linear search for probing, differing from ordinary hash tables.</p> <ul> <li>Inserting Elements: Calculate the bucket index using the hash function. If the bucket already contains an element, linearly traverse forward from the conflict position (usually with a step size of \\(1\\)) until an empty bucket is found, then insert the element.</li> <li>Searching for Elements: If a hash collision is found, use the same step size to linearly traverse forward until the corresponding element is found and return <code>value</code>; if an empty bucket is encountered, it means the target element is not in the hash table, so return <code>None</code>.</li> </ul> <p>The Figure 6-6  shows the distribution of key-value pairs in an open addressing (linear probing) hash table. According to this hash function, keys with the same last two digits will be mapped to the same bucket. Through linear probing, they are stored consecutively in that bucket and the buckets below it.</p> <p></p> <p> Figure 6-6 \u00a0 Distribution of Key-Value Pairs in Open Addressing (Linear Probing) Hash Table </p> <p>However, linear probing tends to create \"clustering\". Specifically, the longer a continuous position in the array is occupied, the more likely these positions are to encounter hash collisions, further promoting the growth of these clusters and eventually leading to deterioration in the efficiency of operations.</p> <p>It's important to note that we cannot directly delete elements in an open addressing hash table. Deleting an element creates an empty bucket <code>None</code> in the array. When searching for elements, if linear probing encounters this empty bucket, it will return, making the elements below this bucket inaccessible. The program may incorrectly assume these elements do not exist, as shown in the Figure 6-7 .</p> <p></p> <p> Figure 6-7 \u00a0 Query Issues Caused by Deletion in Open Addressing </p> <p>To solve this problem, we can use a \"lazy deletion\" mechanism: instead of directly removing elements from the hash table, use a constant <code>TOMBSTONE</code> to mark the bucket. In this mechanism, both <code>None</code> and <code>TOMBSTONE</code> represent empty buckets and can hold key-value pairs. However, when linear probing encounters <code>TOMBSTONE</code>, it should continue traversing since there may still be key-value pairs below it.</p> <p>However, lazy deletion may accelerate the degradation of hash table performance. Every deletion operation produces a delete mark, and as <code>TOMBSTONE</code> increases, so does the search time, as linear probing may have to skip multiple <code>TOMBSTONE</code> to find the target element.</p> <p>Therefore, consider recording the index of the first <code>TOMBSTONE</code> encountered during linear probing and swapping the target element found with this <code>TOMBSTONE</code>. The advantage of this is that each time a query or addition is performed, the element is moved to a bucket closer to the ideal position (starting point of probing), thereby optimizing the query efficiency.</p> <p>The code below implements an open addressing (linear probing) hash table with lazy deletion. To make fuller use of the hash table space, we treat the hash table as a \"circular array,\" continuing to traverse from the beginning when the end of the array is passed.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map_open_addressing.py<pre><code>class HashMapOpenAddressing:\n    \"\"\"\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\n        self.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\n        self.load_thres = 2.0 / 3.0  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        self.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\n        self.buckets: list[Pair | None] = [None] * self.capacity  # \u6876\u6570\u7ec4\n        self.TOMBSTONE = Pair(-1, \"-1\")  # \u5220\u9664\u6807\u8bb0\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\n        return self.size / self.capacity\n\n    def find_bucket(self, key: int) -&gt; int:\n        \"\"\"\u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\"\"\"\n        index = self.hash_func(key)\n        first_tombstone = -1\n        # \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while self.buckets[index] is not None:\n            # \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if self.buckets[index].key == key:\n                # \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if first_tombstone != -1:\n                    self.buckets[first_tombstone] = self.buckets[index]\n                    self.buckets[index] = self.TOMBSTONE\n                    return first_tombstone  # \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                return index  # \u8fd4\u56de\u6876\u7d22\u5f15\n            # \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if first_tombstone == -1 and self.buckets[index] is self.TOMBSTONE:\n                first_tombstone = index\n            # \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % self.capacity\n        # \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return index if first_tombstone == -1 else first_tombstone\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n        # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            return self.buckets[index].val\n        # \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n        # \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index].val = val\n            return\n        # \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        self.buckets[index] = Pair(key, val)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n        # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index] = self.TOMBSTONE\n            self.size -= 1\n\n    def extend(self):\n        \"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n        # \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        buckets_tmp = self.buckets\n        # \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio\n        self.buckets = [None] * self.capacity\n        self.size = 0\n        # \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for pair in buckets_tmp:\n            if pair not in [None, self.TOMBSTONE]:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n        for pair in self.buckets:\n            if pair is None:\n                print(\"None\")\n            elif pair is self.TOMBSTONE:\n                print(\"TOMBSTONE\")\n            else:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> hash_map_open_addressing.cpp<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n  private:\n    int size;                             // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity = 4;                     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    const double loadThres = 2.0 / 3.0;     // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    const int extendRatio = 2;            // \u6269\u5bb9\u500d\u6570\n    vector&lt;Pair *&gt; buckets;               // \u6876\u6570\u7ec4\n    Pair *TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    HashMapOpenAddressing() : size(0), buckets(capacity, nullptr) {\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~HashMapOpenAddressing() {\n        for (Pair *pair : buckets) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                delete pair;\n            }\n        }\n        delete TOMBSTONE;\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double loadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != nullptr) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index]-&gt;key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    string get(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]-&gt;val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n        return \"\";\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]-&gt;val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            delete buckets[index];\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        vector&lt;Pair *&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = vector&lt;Pair *&gt;(capacity, nullptr);\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (Pair *pair : bucketsTmp) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                put(pair-&gt;key, pair-&gt;val);\n                delete pair;\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (Pair *pair : buckets) {\n            if (pair == nullptr) {\n                cout &lt;&lt; \"nullptr\" &lt;&lt; endl;\n            } else if (pair == TOMBSTONE) {\n                cout &lt;&lt; \"TOMBSTONE\" &lt;&lt; endl;\n            } else {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; endl;\n            }\n        }\n    }\n};\n</code></pre> hash_map_open_addressing.java<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    private int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    private int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    private final double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    private final int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n    private Pair[] buckets; // \u6876\u6570\u7ec4\n    private final Pair TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    private double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    private int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index].key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public String get(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void put(int key, String val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void remove(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    private void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        Pair[] bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (Pair pair : bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void print() {\n        for (Pair pair : buckets) {\n            if (pair == null) {\n                System.out.println(\"null\");\n            } else if (pair == TOMBSTONE) {\n                System.out.println(\"TOMBSTONE\");\n            } else {\n                System.out.println(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.cs<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n    Pair[] buckets; // \u6876\u6570\u7ec4\n    Pair TOMBSTONE = new(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    int FindBucket(int key) {\n        int index = HashFunc(key);\n        int firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index].key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public string? Get(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void Extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        Pair[] bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        foreach (Pair pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void Print() {\n        foreach (Pair pair in buckets) {\n            if (pair == null) {\n                Console.WriteLine(\"null\");\n            } else if (pair == TOMBSTONE) {\n                Console.WriteLine(\"TOMBSTONE\");\n            } else {\n                Console.WriteLine(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.go<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntype hashMapOpenAddressing struct {\n    size        int     // \u952e\u503c\u5bf9\u6570\u91cf\n    capacity    int     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    loadThres   float64 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    extendRatio int     // \u6269\u5bb9\u500d\u6570\n    buckets     []*pair // \u6876\u6570\u7ec4\n    TOMBSTONE   *pair   // \u5220\u9664\u6807\u8bb0\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\n    return &amp;hashMapOpenAddressing{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     make([]*pair, 4),\n        TOMBSTONE:   &amp;pair{-1, \"-1\"},\n    }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (h *hashMapOpenAddressing) hashFunc(key int) int {\n    return key % h.capacity // \u6839\u636e\u952e\u8ba1\u7b97\u54c8\u5e0c\u503c\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (h *hashMapOpenAddressing) loadFactor() float64 {\n    return float64(h.size) / float64(h.capacity) // \u8ba1\u7b97\u5f53\u524d\u8d1f\u8f7d\u56e0\u5b50\n}\n\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nfunc (h *hashMapOpenAddressing) findBucket(key int) int {\n    index := h.hashFunc(key) // \u83b7\u53d6\u521d\u59cb\u7d22\u5f15\n    firstTombstone := -1     // \u8bb0\u5f55\u9047\u5230\u7684\u7b2c\u4e00\u4e2aTOMBSTONE\u7684\u4f4d\u7f6e\n    for h.buckets[index] != nil {\n        if h.buckets[index].key == key {\n            if firstTombstone != -1 {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                h.buckets[firstTombstone] = h.buckets[index]\n                h.buckets[index] = h.TOMBSTONE\n                return firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n            }\n            return index // \u8fd4\u56de\u627e\u5230\u7684\u7d22\u5f15\n        }\n        if firstTombstone == -1 &amp;&amp; h.buckets[index] == h.TOMBSTONE {\n            firstTombstone = index // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\u7684\u4f4d\u7f6e\n        }\n        index = (index + 1) % h.capacity // \u7ebf\u6027\u63a2\u6d4b\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n    if firstTombstone != -1 {\n        return firstTombstone\n    }\n    return index\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) get(key int) string {\n    index := h.findBucket(key) // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        return h.buckets[index].val // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    }\n    return \"\" // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de \"\"\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) put(key int, val string) {\n    if h.loadFactor() &gt; h.loadThres {\n        h.extend() // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    }\n    index := h.findBucket(key) // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] == nil || h.buckets[index] == h.TOMBSTONE {\n        h.buckets[index] = &amp;pair{key, val} // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        h.size++\n    } else {\n        h.buckets[index].val = val // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val\n    }\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) remove(key int) {\n    index := h.findBucket(key) // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        h.buckets[index] = h.TOMBSTONE // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        h.size--\n    }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (h *hashMapOpenAddressing) extend() {\n    oldBuckets := h.buckets               // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    h.capacity *= h.extendRatio           // \u66f4\u65b0\u5bb9\u91cf\n    h.buckets = make([]*pair, h.capacity) // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    h.size = 0                            // \u91cd\u7f6e\u5927\u5c0f\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for _, pair := range oldBuckets {\n        if pair != nil &amp;&amp; pair != h.TOMBSTONE {\n            h.put(pair.key, pair.val)\n        }\n    }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (h *hashMapOpenAddressing) print() {\n    for _, pair := range h.buckets {\n        if pair == nil {\n            fmt.Println(\"nil\")\n        } else if pair == h.TOMBSTONE {\n            fmt.Println(\"TOMBSTONE\")\n        } else {\n            fmt.Printf(\"%d -&gt; %s\\n\", pair.key, pair.val)\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.swift<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n    var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    var loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    var extendRatio: Int // \u6269\u5bb9\u500d\u6570\n    var buckets: [Pair?] // \u6876\u6570\u7ec4\n    var TOMBSTONE: Pair // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: nil, count: capacity)\n        TOMBSTONE = Pair(key: -1, val: \"-1\")\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    func findBucket(key: Int) -&gt; Int {\n        var index = hashFunc(key: key)\n        var firstTombstone = -1\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while buckets[index] != nil {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if buckets[index]!.key == key {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if firstTombstone != -1 {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE {\n                firstTombstone = index\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            return buckets[index]!.val\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return nil\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index]!.val = val\n            return\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = Pair(key: key, val: val)\n        size += 1\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index] = TOMBSTONE\n            size -= 1\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    func extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        buckets = Array(repeating: nil, count: capacity)\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for pair in bucketsTmp {\n            if let pair, pair != TOMBSTONE {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    func print() {\n        for pair in buckets {\n            if pair == nil {\n                Swift.print(\"null\")\n            } else if pair == TOMBSTONE {\n                Swift.print(\"TOMBSTONE\")\n            } else {\n                Swift.print(\"\\(pair!.key) -&gt; \\(pair!.val)\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.js<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    #size; // \u952e\u503c\u5bf9\u6570\u91cf\n    #capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    #loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    #extendRatio; // \u6269\u5bb9\u500d\u6570\n    #buckets; // \u6876\u6570\u7ec4\n    #TOMBSTONE; // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\n        this.#capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n        this.#loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        this.#extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n        this.#buckets = Array(this.#capacity).fill(null); // \u6876\u6570\u7ec4\n        this.#TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    #findBucket(key) {\n        let index = this.#hashFunc(key);\n        let firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (this.#buckets[index] !== null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (this.#buckets[index].key === key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone !== -1) {\n                    this.#buckets[firstTombstone] = this.#buckets[index];\n                    this.#buckets[index] = this.#TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.#buckets[index] === this.#TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % this.#capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            return this.#buckets[index].val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key, val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        this.#buckets[index] = new Pair(key, val);\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index] = this.#TOMBSTONE;\n            this.#size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    #extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = Array(this.#capacity).fill(null);\n        this.#size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.#TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print() {\n        for (const pair of this.#buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.#TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.ts<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    private size: number; // \u952e\u503c\u5bf9\u6570\u91cf\n    private capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    private loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    private extendRatio: number; // \u6269\u5bb9\u500d\u6570\n    private buckets: Array&lt;Pair | null&gt;; // \u6876\u6570\u7ec4\n    private TOMBSTONE: Pair; // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\n        this.capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n        this.loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        this.extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n        this.buckets = Array(this.capacity).fill(null); // \u6876\u6570\u7ec4\n        this.TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private hashFunc(key: number): number {\n        return key % this.capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    private loadFactor(): number {\n        return this.size / this.capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    private findBucket(key: number): number {\n        let index = this.hashFunc(key);\n        let firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (this.buckets[index] !== null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (this.buckets[index]!.key === key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone !== -1) {\n                    this.buckets[firstTombstone] = this.buckets[index];\n                    this.buckets[index] = this.TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.buckets[index] === this.TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % this.capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key: number): string | null {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            return this.buckets[index]!.val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key: number, val: string): void {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.loadFactor() &gt; this.loadThres) {\n            this.extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index]!.val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        this.buckets[index] = new Pair(key, val);\n        this.size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key: number): void {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index] = this.TOMBSTONE;\n            this.size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    private extend(): void {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.capacity *= this.extendRatio;\n        this.buckets = Array(this.capacity).fill(null);\n        this.size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print(): void {\n        for (const pair of this.buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.dart<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n  late int _size; // \u952e\u503c\u5bf9\u6570\u91cf\n  int _capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n  double _loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n  int _extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n  late List&lt;Pair?&gt; _buckets; // \u6876\u6570\u7ec4\n  Pair _TOMBSTONE = Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  HashMapOpenAddressing() {\n    _size = 0;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n  }\n\n  /* \u54c8\u5e0c\u51fd\u6570 */\n  int hashFunc(int key) {\n    return key % _capacity;\n  }\n\n  /* \u8d1f\u8f7d\u56e0\u5b50 */\n  double loadFactor() {\n    return _size / _capacity;\n  }\n\n  /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n  int findBucket(int key) {\n    int index = hashFunc(key);\n    int firstTombstone = -1;\n    // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n    while (_buckets[index] != null) {\n      // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n      if (_buckets[index]!.key == key) {\n        // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n        if (firstTombstone != -1) {\n          _buckets[firstTombstone] = _buckets[index];\n          _buckets[index] = _TOMBSTONE;\n          return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n        }\n        return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n      }\n      // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n      if (firstTombstone == -1 &amp;&amp; _buckets[index] == _TOMBSTONE) {\n        firstTombstone = index;\n      }\n      // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n      index = (index + 1) % _capacity;\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n    return firstTombstone == -1 ? index : firstTombstone;\n  }\n\n  /* \u67e5\u8be2\u64cd\u4f5c */\n  String? get(int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      return _buckets[index]!.val;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n    return null;\n  }\n\n  /* \u6dfb\u52a0\u64cd\u4f5c */\n  void put(int key, String val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor() &gt; _loadThres) {\n      extend();\n    }\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index]!.val = val;\n      return;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n    _buckets[index] = new Pair(key, val);\n    _size++;\n  }\n\n  /* \u5220\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index] = _TOMBSTONE;\n      _size--;\n    }\n  }\n\n  /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n  void extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    List&lt;Pair?&gt; bucketsTmp = _buckets;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    _capacity *= _extendRatio;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n    _size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (Pair? pair in bucketsTmp) {\n      if (pair != null &amp;&amp; pair != _TOMBSTONE) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* \u6253\u5370\u54c8\u5e0c\u8868 */\n  void printHashMap() {\n    for (Pair? pair in _buckets) {\n      if (pair == null) {\n        print(\"null\");\n      } else if (pair == _TOMBSTONE) {\n        print(\"TOMBSTONE\");\n      } else {\n        print(\"${pair.key} -&gt; ${pair.val}\");\n      }\n    }\n  }\n}\n</code></pre> hash_map_open_addressing.rs<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapOpenAddressing {\n    size: usize,                // \u952e\u503c\u5bf9\u6570\u91cf\n    capacity: usize,            // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    load_thres: f64,            // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    extend_ratio: usize,        // \u6269\u5bb9\u500d\u6570\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;, // \u6876\u6570\u7ec4\n    TOMBSTONE: Option&lt;Pair&gt;,    // \u5220\u9664\u6807\u8bb0\n}\n\nimpl HashMapOpenAddressing {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![None; 4],\n            TOMBSTONE: Some(Pair {\n                key: -1,\n                val: \"-1\".to_string(),\n            }),\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        (key % self.capacity as i32) as usize\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fn load_factor(&amp;self) -&gt; f64 {\n        self.size as f64 / self.capacity as f64\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    fn find_bucket(&amp;mut self, key: i32) -&gt; usize {\n        let mut index = self.hash_func(key);\n        let mut first_tombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while self.buckets[index].is_some() {\n            // \u82e5\u9047\u5230 key\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if self.buckets[index].as_ref().unwrap().key == key {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u5efa\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n                if first_tombstone != -1 {\n                    self.buckets[first_tombstone as usize] = self.buckets[index].take();\n                    self.buckets[index] = self.TOMBSTONE.clone();\n                    return first_tombstone as usize; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if first_tombstone == -1 &amp;&amp; self.buckets[index] == self.TOMBSTONE {\n                first_tombstone = index as i32;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % self.capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        if first_tombstone == -1 {\n            index\n        } else {\n            first_tombstone as usize\n        }\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fn get(&amp;mut self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            return self.buckets[index].as_ref().map(|pair| &amp;pair.val as &amp;str);\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        None\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index].as_mut().unwrap().val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        self.buckets[index] = Some(Pair { key, val });\n        self.size += 1;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fn remove(&amp;mut self, key: i32) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index] = self.TOMBSTONE.clone();\n            self.size -= 1;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fn extend(&amp;mut self) {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let buckets_tmp = self.buckets.clone();\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![None; self.capacity];\n        self.size = 0;\n\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for pair in buckets_tmp {\n            if pair.is_none() || pair == self.TOMBSTONE {\n                continue;\n            }\n            let pair = pair.unwrap();\n\n            self.put(pair.key, pair.val);\n        }\n    }\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fn print(&amp;self) {\n        for pair in &amp;self.buckets {\n            if pair.is_none() {\n                println!(\"null\");\n            } else if pair == &amp;self.TOMBSTONE {\n                println!(\"TOMBSTONE\");\n            } else {\n                let pair = pair.as_ref().unwrap();\n                println!(\"{} -&gt; {}\", pair.key, pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.c<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\n    int size;         // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity;     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio;  // \u6269\u5bb9\u500d\u6570\n    Pair **buckets;   // \u6876\u6570\u7ec4\n    Pair *TOMBSTONE;  // \u5220\u9664\u6807\u8bb0\n} HashMapOpenAddressing;\n\n/* \u6784\u9020\u51fd\u6570 */\nHashMapOpenAddressing *newHashMapOpenAddressing() {\n    HashMapOpenAddressing *hashMap = (HashMapOpenAddressing *)malloc(sizeof(HashMapOpenAddressing));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Pair **)malloc(sizeof(Pair *) * hashMap-&gt;capacity);\n    hashMap-&gt;TOMBSTONE = (Pair *)malloc(sizeof(Pair));\n    hashMap-&gt;TOMBSTONE-&gt;key = -1;\n    hashMap-&gt;TOMBSTONE-&gt;val = \"-1\";\n\n    return hashMap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapOpenAddressing(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapOpenAddressing *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapOpenAddressing *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nint findBucket(HashMapOpenAddressing *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    int firstTombstone = -1;\n    // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n    while (hashMap-&gt;buckets[index] != NULL) {\n        // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        if (hashMap-&gt;buckets[index]-&gt;key == key) {\n            // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n            if (firstTombstone != -1) {\n                hashMap-&gt;buckets[firstTombstone] = hashMap-&gt;buckets[index];\n                hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n                return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n            }\n            return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n        }\n        // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n        if (firstTombstone == -1 &amp;&amp; hashMap-&gt;buckets[index] == hashMap-&gt;TOMBSTONE) {\n            firstTombstone = index;\n        }\n        // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n        index = (index + 1) % hashMap-&gt;capacity;\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n    return firstTombstone == -1 ? index : firstTombstone;\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapOpenAddressing *hashMap, int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        return hashMap-&gt;buckets[index]-&gt;val;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n    return \"\";\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapOpenAddressing *hashMap, int key, char *val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        free(hashMap-&gt;buckets[index]-&gt;val);\n        hashMap-&gt;buckets[index]-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n        strcpy(hashMap-&gt;buckets[index]-&gt;val, val);\n        hashMap-&gt;buckets[index]-&gt;val[strlen(val)] = '\\0';\n        return;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n    Pair *pair = (Pair *)malloc(sizeof(Pair));\n    pair-&gt;key = key;\n    pair-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n    strcpy(pair-&gt;val, val);\n    pair-&gt;val[strlen(val)] = '\\0';\n\n    hashMap-&gt;buckets[index] = pair;\n    hashMap-&gt;size++;\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapOpenAddressing *hashMap, int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        Pair *pair = hashMap-&gt;buckets[index];\n        free(pair-&gt;val);\n        free(pair);\n        hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n        hashMap-&gt;size--;\n    }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapOpenAddressing *hashMap) {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    Pair **bucketsTmp = hashMap-&gt;buckets;\n    int oldCapacity = hashMap-&gt;capacity;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Pair **)malloc(sizeof(Pair *) * hashMap-&gt;capacity);\n    hashMap-&gt;size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Pair *pair = bucketsTmp[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            put(hashMap, pair-&gt;key, pair-&gt;val);\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(bucketsTmp);\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair == NULL) {\n            printf(\"NULL\\n\");\n        } else if (pair == hashMap-&gt;TOMBSTONE) {\n            printf(\"TOMBSTONE\\n\");\n        } else {\n            printf(\"%d -&gt; %s\\n\", pair-&gt;key, pair-&gt;val);\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.kt<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    private var size: Int = 0 // \u952e\u503c\u5bf9\u6570\u91cf\n    private var capacity = 4 // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    private val loadThres: Double = 2.0 / 3.0 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    private val extendRatio = 2 // \u6269\u5bb9\u500d\u6570\n    private var buckets: Array&lt;Pair?&gt; // \u6876\u6570\u7ec4\n    private val TOMBSTONE = Pair(-1, \"-1\") // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init {\n        buckets = arrayOfNulls(capacity)\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    fun findBucket(key: Int): Int {\n        var index = hashFunc(key)\n        var firstTombstone = -1\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index]?.key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return if (firstTombstone == -1) index else firstTombstone\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]?.value\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]!!.value = value\n            return\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = Pair(key, value)\n        size++\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE\n            size--\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fun extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        val bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        buckets = arrayOfNulls(capacity)\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair.value)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (pair in buckets) {\n            if (pair == null) {\n                println(\"null\")\n            } else if (pair == TOMBSTONE) {\n                println(\"TOMESTOME\")\n            } else {\n                println(\"${pair.key} -&gt; ${pair.value}\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.rb<pre><code>[class]{HashMapOpenAddressing}-[func]{}\n</code></pre> hash_map_open_addressing.zig<pre><code>[class]{HashMapOpenAddressing}-[func]{}\n</code></pre>"},{"location":"chapter_hashing/hash_collision/#2-quadratic-probing","title":"2. \u00a0 Quadratic Probing","text":"<p>Quadratic probing is similar to linear probing and is one of the common strategies of open addressing. When a collision occurs, quadratic probing does not simply skip a fixed number of steps but skips \"the square of the number of probes,\" i.e., \\(1, 4, 9, \\dots\\) steps.</p> <p>Quadratic probing has the following advantages:</p> <ul> <li>Quadratic probing attempts to alleviate the clustering effect of linear probing by skipping the distance of the square of the number of probes.</li> <li>Quadratic probing skips larger distances to find empty positions, helping to distribute data more evenly.</li> </ul> <p>However, quadratic probing is not perfect:</p> <ul> <li>Clustering still exists, i.e., some positions are more likely to be occupied than others.</li> <li>Due to the growth of squares, quadratic probing may not probe the entire hash table, meaning it might not access empty buckets even if they exist in the hash table.</li> </ul>"},{"location":"chapter_hashing/hash_collision/#3-double-hashing","title":"3. \u00a0 Double Hashing","text":"<p>As the name suggests, the double hashing method uses multiple hash functions \\(f_1(x)\\), \\(f_2(x)\\), \\(f_3(x)\\), \\(\\dots\\) for probing.</p> <ul> <li>Inserting Elements: If hash function \\(f_1(x)\\) encounters a conflict, try \\(f_2(x)\\), and so on, until an empty position is found and the element is inserted.</li> <li>Searching for Elements: Search in the same order of hash functions until the target element is found and returned; if an empty position is encountered or all hash functions have been tried, it indicates the element is not in the hash table, then return <code>None</code>.</li> </ul> <p>Compared to linear probing, double hashing is less prone to clustering but involves additional computation for multiple hash functions.</p> <p>Tip</p> <p>Please note that open addressing (linear probing, quadratic probing, and double hashing) hash tables all have the issue of \"not being able to directly delete elements.\"</p>"},{"location":"chapter_hashing/hash_collision/#623-choice-of-programming-languages","title":"6.2.3 \u00a0 Choice of Programming Languages","text":"<p>Various programming languages have adopted different hash table implementation strategies, here are a few examples:</p> <ul> <li>Python uses open addressing. The <code>dict</code> dictionary uses pseudo-random numbers for probing.</li> <li>Java uses separate chaining. Since JDK 1.8, when the array length in <code>HashMap</code> reaches 64 and the length of a linked list reaches 8, the linked list is converted to a red-black tree to improve search performance.</li> <li>Go uses separate chaining. Go stipulates that each bucket can store up to 8 key-value pairs, and if the capacity is exceeded, an overflow bucket is connected; when there are too many overflow buckets, a special equal-size expansion operation is performed to ensure performance.</li> </ul>"},{"location":"chapter_hashing/hash_map/","title":"6.1 \u00a0 Hash Table","text":"<p>A \"hash table\", also known as a \"hash map\", achieves efficient element querying by establishing a mapping between keys and values. Specifically, when we input a <code>key</code> into the hash table, we can retrieve the corresponding <code>value</code> in \\(O(1)\\) time.</p> <p>As shown in the Figure 6-1 , given \\(n\\) students, each with two pieces of data: \"name\" and \"student number\". If we want to implement a query feature that returns the corresponding name when given a student number, we can use the hash table shown in the Figure 6-1 .</p> <p></p> <p> Figure 6-1 \u00a0 Abstract representation of a hash table </p> <p>Apart from hash tables, arrays and linked lists can also be used to implement querying functions. Their efficiency is compared in the Table 6-1 .</p> <ul> <li>Adding Elements: Simply add the element to the end of the array (or linked list), using \\(O(1)\\) time.</li> <li>Querying Elements: Since the array (or linked list) is unordered, it requires traversing all the elements, using \\(O(n)\\) time.</li> <li>Deleting Elements: First, locate the element, then delete it from the array (or linked list), using \\(O(n)\\) time.</li> </ul> <p> Table 6-1 \u00a0 Comparison of Element Query Efficiency </p> Array Linked List Hash Table Find Element \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) Add Element \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) Delete Element \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) <p>Observations reveal that the time complexity for adding, deleting, and querying in a hash table is \\(O(1)\\), which is highly efficient.</p>"},{"location":"chapter_hashing/hash_map/#611-common-operations-of-hash-table","title":"6.1.1 \u00a0 Common Operations of Hash Table","text":"<p>Common operations of a hash table include initialization, querying, adding key-value pairs, and deleting key-value pairs, etc. Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig hash_map.py<pre><code># Initialize hash table\nhmap: dict = {}\n\n# Add operation\n# Add key-value pair (key, value) to the hash table\nhmap[12836] = \"Xiao Ha\"\nhmap[15937] = \"Xiao Luo\"\nhmap[16750] = \"Xiao Suan\"\nhmap[13276] = \"Xiao Fa\"\nhmap[10583] = \"Xiao Ya\"\n\n# Query operation\n# Input key into hash table, get value\nname: str = hmap[15937]\n\n# Delete operation\n# Delete key-value pair (key, value) from hash table\nhmap.pop(10583)\n</code></pre> hash_map.cpp<pre><code>/* Initialize hash table */\nunordered_map&lt;int, string&gt; map;\n\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nmap[12836] = \"Xiao Ha\";\nmap[15937] = \"Xiao Luo\";\nmap[16750] = \"Xiao Suan\";\nmap[13276] = \"Xiao Fa\";\nmap[10583] = \"Xiao Ya\";\n\n/* Query operation */\n// Input key into hash table, get value\nstring name = map[15937];\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.erase(10583);\n</code></pre> hash_map.java<pre><code>/* Initialize hash table */\nMap&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nmap.put(12836, \"Xiao Ha\");   \nmap.put(15937, \"Xiao Luo\");   \nmap.put(16750, \"Xiao Suan\");   \nmap.put(13276, \"Xiao Fa\");\nmap.put(10583, \"Xiao Ya\");\n\n/* Query operation */\n// Input key into hash table, get value\nString name = map.get(15937);\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.remove(10583);\n</code></pre> hash_map.cs<pre><code>/* Initialize hash table */\nDictionary&lt;int, string&gt; map = new() {\n    /* Add operation */\n    // Add key-value pair (key, value) to the hash table\n    { 12836, \"Xiao Ha\" },\n    { 15937, \"Xiao Luo\" },\n    { 16750, \"Xiao Suan\" },\n    { 13276, \"Xiao Fa\" },\n    { 10583, \"Xiao Ya\" }\n};\n\n/* Query operation */\n// Input key into hash table, get value\nstring name = map[15937];\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.Remove(10583);\n</code></pre> hash_map_test.go<pre><code>/* Initialize hash table */\nhmap := make(map[int]string)\n\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nhmap[12836] = \"Xiao Ha\"\nhmap[15937] = \"Xiao Luo\"\nhmap[16750] = \"Xiao Suan\"\nhmap[13276] = \"Xiao Fa\"\nhmap[10583] = \"Xiao Ya\"\n\n/* Query operation */\n// Input key into hash table, get value\nname := hmap[15937]\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\ndelete(hmap, 10583)\n</code></pre> hash_map.swift<pre><code>/* Initialize hash table */\nvar map: [Int: String] = [:]\n\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nmap[12836] = \"Xiao Ha\"\nmap[15937] = \"Xiao Luo\"\nmap[16750] = \"Xiao Suan\"\nmap[13276] = \"Xiao Fa\"\nmap[10583] = \"Xiao Ya\"\n\n/* Query operation */\n// Input key into hash table, get value\nlet name = map[15937]!\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.removeValue(forKey: 10583)\n</code></pre> hash_map.js<pre><code>/* Initialize hash table */\nconst map = new Map();\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nmap.set(12836, 'Xiao Ha');\nmap.set(15937, 'Xiao Luo');\nmap.set(16750, 'Xiao Suan');\nmap.set(13276, 'Xiao Fa');\nmap.set(10583, 'Xiao Ya');\n\n/* Query operation */\n// Input key into hash table, get value\nlet name = map.get(15937);\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.delete(10583);\n</code></pre> hash_map.ts<pre><code>/* Initialize hash table */\nconst map = new Map&lt;number, string&gt;();\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nmap.set(12836, 'Xiao Ha');\nmap.set(15937, 'Xiao Luo');\nmap.set(16750, 'Xiao Suan');\nmap.set(13276, 'Xiao Fa');\nmap.set(10583, 'Xiao Ya');\nconsole.info('\\nAfter adding, the hash table is\\nKey -&gt; Value');\nconsole.info(map);\n\n/* Query operation */\n// Input key into hash table, get value\nlet name = map.get(15937);\nconsole.info('\\nInput student number 15937, query name ' + name);\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.delete(10583);\nconsole.info('\\nAfter deleting 10583, the hash table is\\nKey -&gt; Value');\nconsole.info(map);\n</code></pre> hash_map.dart<pre><code>/* Initialize hash table */\nMap&lt;int, String&gt; map = {};\n\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nmap[12836] = \"Xiao Ha\";\nmap[15937] = \"Xiao Luo\";\nmap[16750] = \"Xiao Suan\";\nmap[13276] = \"Xiao Fa\";\nmap[10583] = \"Xiao Ya\";\n\n/* Query operation */\n// Input key into hash table, get value\nString name = map[15937];\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.remove(10583);\n</code></pre> hash_map.rs<pre><code>use std::collections::HashMap;\n\n/* Initialize hash table */\nlet mut map: HashMap&lt;i32, String&gt; = HashMap::new();\n\n/* Add operation */\n// Add key-value pair (key, value) to the hash table\nmap.insert(12836, \"Xiao Ha\".to_string());\nmap.insert(15937, \"Xiao Luo\".to_string());\nmap.insert(16750, \"Xiao Suan\".to_string());\nmap.insert(13279, \"Xiao Fa\".to_string());\nmap.insert(10583, \"Xiao Ya\".to_string());\n\n/* Query operation */\n// Input key into hash table, get value\nlet _name: Option&lt;&amp;String&gt; = map.get(&amp;15937);\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nlet _removed_value: Option&lt;String&gt; = map.remove(&amp;10583);\n</code></pre> hash_map.c<pre><code>// C does not provide a built-in hash table\n</code></pre> hash_map.kt<pre><code>\n</code></pre> hash_map.zig<pre><code>\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>There are three common ways to traverse a hash table: traversing key-value pairs, keys, and values. Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig hash_map.py<pre><code># Traverse hash table\n# Traverse key-value pairs key-&gt;value\nfor key, value in hmap.items():\n    print(key, \"-&gt;\", value)\n# Traverse keys only\nfor key in hmap.keys():\n    print(key)\n# Traverse values only\nfor value in hmap.values():\n    print(value)\n</code></pre> hash_map.cpp<pre><code>/* Traverse hash table */\n// Traverse key-value pairs key-&gt;value\nfor (auto kv: map) {\n    cout &lt;&lt; kv.first &lt;&lt; \" -&gt; \" &lt;&lt; kv.second &lt;&lt; endl;\n}\n// Traverse using iterator key-&gt;value\nfor (auto iter = map.begin(); iter != map.end(); iter++) {\n    cout &lt;&lt; iter-&gt;first &lt;&lt; \"-&gt;\" &lt;&lt; iter-&gt;second &lt;&lt; endl;\n}\n</code></pre> hash_map.java<pre><code>/* Traverse hash table */\n// Traverse key-value pairs key-&gt;value\nfor (Map.Entry&lt;Integer, String&gt; kv: map.entrySet()) {\n    System.out.println(kv.getKey() + \" -&gt; \" + kv.getValue());\n}\n// Traverse keys only\nfor (int key: map.keySet()) {\n    System.out.println(key);\n}\n// Traverse values only\nfor (String val: map.values()) {\n    System.out.println(val);\n}\n</code></pre> hash_map.cs<pre><code>/* Traverse hash table */\n// Traverse key-value pairs Key-&gt;Value\nforeach (var kv in map) {\n    Console.WriteLine(kv.Key + \" -&gt; \" + kv.Value);\n}\n// Traverse keys only\nforeach (int key in map.Keys) {\n    Console.WriteLine(key);\n}\n// Traverse values only\nforeach (string val in map.Values) {\n    Console.WriteLine(val);\n}\n</code></pre> hash_map_test.go<pre><code>/* Traverse hash table */\n// Traverse key-value pairs key-&gt;value\nfor key, value := range hmap {\n    fmt.Println(key, \"-&gt;\", value)\n}\n// Traverse keys only\nfor key := range hmap {\n    fmt.Println(key)\n}\n// Traverse values only\nfor _, value := range hmap {\n    fmt.Println(value)\n}\n</code></pre> hash_map.swift<pre><code>/* Traverse hash table */\n// Traverse key-value pairs Key-&gt;Value\nfor (key, value) in map {\n    print(\"\\(key) -&gt; \\(value)\")\n}\n// Traverse keys only\nfor key in map.keys {\n    print(key)\n}\n// Traverse values only\nfor value in map.values {\n    print(value)\n}\n</code></pre> hash_map.js<pre><code>/* Traverse hash table */\nconsole.info('\\nTraverse key-value pairs Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\nTraverse keys only Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\nTraverse values only Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.ts<pre><code>/* Traverse hash table */\nconsole.info('\\nTraverse key-value pairs Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\nTraverse keys only Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\nTraverse values only Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.dart<pre><code>/* Traverse hash table */\n// Traverse key-value pairs Key-&gt;Value\nmap.forEach((key, value) {\nprint('$key -&gt; $value');\n});\n\n// Traverse keys only Key\nmap.keys.forEach((key) {\nprint(key);\n});\n\n// Traverse values only Value\nmap.values.forEach((value) {\nprint(value);\n});\n</code></pre> hash_map.rs<pre><code>/* Traverse hash table */\n// Traverse key-value pairs Key-&gt;Value\nfor (key, value) in &amp;map {\n    println!(\"{key} -&gt; {value}\");\n}\n\n// Traverse keys only Key\nfor key in map.keys() {\n    println!(\"{key}\"); \n}\n\n// Traverse values only Value\nfor value in map.values() {\n    println!(\"{value}\");\n}\n</code></pre> hash_map.c<pre><code>// C does not provide a built-in hash table\n</code></pre> hash_map.kt<pre><code>\n</code></pre> hash_map.zig<pre><code>// Zig example is not provided\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_hashing/hash_map/#612-simple-implementation-of-hash-table","title":"6.1.2 \u00a0 Simple Implementation of Hash Table","text":"<p>First, let's consider the simplest case: implementing a hash table using just an array. In the hash table, each empty slot in the array is called a \"bucket\", and each bucket can store one key-value pair. Therefore, the query operation involves finding the bucket corresponding to the <code>key</code> and retrieving the <code>value</code> from it.</p> <p>So, how do we locate the appropriate bucket based on the <code>key</code>? This is achieved through a \"hash function\". The role of the hash function is to map a larger input space to a smaller output space. In a hash table, the input space is all possible keys, and the output space is all buckets (array indices). In other words, input a <code>key</code>, and we can use the hash function to determine the storage location of the corresponding key-value pair in the array.</p> <p>The calculation process of the hash function for a given <code>key</code> is divided into the following two steps:</p> <ol> <li>Calculate the hash value using a certain hash algorithm <code>hash()</code>.</li> <li>Take the modulus of the hash value with the number of buckets (array length) <code>capacity</code> to obtain the array index <code>index</code>.</li> </ol> <pre><code>index = hash(key) % capacity\n</code></pre> <p>Afterward, we can use <code>index</code> to access the corresponding bucket in the hash table and thereby retrieve the <code>value</code>.</p> <p>Assuming array length <code>capacity = 100</code> and hash algorithm <code>hash(key) = key</code>, the hash function is <code>key % 100</code>. The Figure 6-2  uses <code>key</code> as the student number and <code>value</code> as the name to demonstrate the working principle of the hash function.</p> <p></p> <p> Figure 6-2 \u00a0 Working principle of hash function </p> <p>The following code implements a simple hash table. Here, we encapsulate <code>key</code> and <code>value</code> into a class <code>Pair</code> to represent the key-value pair.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_hash_map.py<pre><code>class Pair:\n    \"\"\"\u952e\u503c\u5bf9\"\"\"\n\n    def __init__(self, key: int, val: str):\n        self.key = key\n        self.val = val\n\nclass ArrayHashMap:\n    \"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        # \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        self.buckets: list[Pair | None] = [None] * 100\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n        index = key % 100\n        return index\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n        index: int = self.hash_func(key)\n        pair: Pair = self.buckets[index]\n        if pair is None:\n            return None\n        return pair.val\n\n    def put(self, key: int, val: str):\n        \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n        pair = Pair(key, val)\n        index: int = self.hash_func(key)\n        self.buckets[index] = pair\n\n    def remove(self, key: int):\n        \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n        index: int = self.hash_func(key)\n        # \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\n        self.buckets[index] = None\n\n    def entry_set(self) -&gt; list[Pair]:\n        \"\"\"\u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\"\"\"\n        result: list[Pair] = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair)\n        return result\n\n    def key_set(self) -&gt; list[int]:\n        \"\"\"\u83b7\u53d6\u6240\u6709\u952e\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.key)\n        return result\n\n    def value_set(self) -&gt; list[str]:\n        \"\"\"\u83b7\u53d6\u6240\u6709\u503c\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.val)\n        return result\n\n    def print(self):\n        \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n        for pair in self.buckets:\n            if pair is not None:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> array_hash_map.cpp<pre><code>/* \u952e\u503c\u5bf9 */\nstruct Pair {\n  public:\n    int key;\n    string val;\n    Pair(int key, string val) {\n        this-&gt;key = key;\n        this-&gt;val = val;\n    }\n};\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n  private:\n    vector&lt;Pair *&gt; buckets;\n\n  public:\n    ArrayHashMap() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = vector&lt;Pair *&gt;(100);\n    }\n\n    ~ArrayHashMap() {\n        // \u91ca\u653e\u5185\u5b58\n        for (const auto &amp;bucket : buckets) {\n            delete bucket;\n        }\n        buckets.clear();\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    string get(int key) {\n        int index = hashFunc(key);\n        Pair *pair = buckets[index];\n        if (pair == nullptr)\n            return \"\";\n        return pair-&gt;val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, string val) {\n        Pair *pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        // \u91ca\u653e\u5185\u5b58\u5e76\u7f6e\u4e3a nullptr\n        delete buckets[index];\n        buckets[index] = nullptr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    vector&lt;Pair *&gt; pairSet() {\n        vector&lt;Pair *&gt; pairSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                pairSet.push_back(pair);\n            }\n        }\n        return pairSet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    vector&lt;int&gt; keySet() {\n        vector&lt;int&gt; keySet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                keySet.push_back(pair-&gt;key);\n            }\n        }\n        return keySet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    vector&lt;string&gt; valueSet() {\n        vector&lt;string&gt; valueSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                valueSet.push_back(pair-&gt;val);\n            }\n        }\n        return valueSet;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (Pair *kv : pairSet()) {\n            cout &lt;&lt; kv-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; kv-&gt;val &lt;&lt; endl;\n        }\n    }\n};\n</code></pre> array_hash_map.java<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair {\n    public int key;\n    public String val;\n\n    public Pair(int key, String val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private List&lt;Pair&gt; buckets;\n\n    public ArrayHashMap() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = new ArrayList&lt;&gt;();\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.add(null);\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public String get(int key) {\n        int index = hashFunc(key);\n        Pair pair = buckets.get(index);\n        if (pair == null)\n            return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void put(int key, String val) {\n        Pair pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets.set(index, pair);\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void remove(int key) {\n        int index = hashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets.set(index, null);\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    public List&lt;Pair&gt; pairSet() {\n        List&lt;Pair&gt; pairSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                pairSet.add(pair);\n        }\n        return pairSet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    public List&lt;Integer&gt; keySet() {\n        List&lt;Integer&gt; keySet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                keySet.add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    public List&lt;String&gt; valueSet() {\n        List&lt;String&gt; valueSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                valueSet.add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void print() {\n        for (Pair kv : pairSet()) {\n            System.out.println(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.cs<pre><code>/* \u952e\u503c\u5bf9 int-&gt;string */\nclass Pair(int key, string val) {\n    public int key = key;\n    public string val = val;\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    List&lt;Pair?&gt; buckets;\n    public ArrayHashMap() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = [];\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.Add(null);\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int HashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        Pair? pair = buckets[index];\n        if (pair == null) return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        Pair pair = new(key, val);\n        int index = HashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = null;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    public List&lt;Pair&gt; PairSet() {\n        List&lt;Pair&gt; pairSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                pairSet.Add(pair);\n        }\n        return pairSet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    public List&lt;int&gt; KeySet() {\n        List&lt;int&gt; keySet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                keySet.Add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    public List&lt;string&gt; ValueSet() {\n        List&lt;string&gt; valueSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                valueSet.Add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void Print() {\n        foreach (Pair kv in PairSet()) {\n            Console.WriteLine(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.go<pre><code>/* \u952e\u503c\u5bf9 */\ntype pair struct {\n    key int\n    val string\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntype arrayHashMap struct {\n    buckets []*pair\n}\n\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nfunc newArrayHashMap() *arrayHashMap {\n    // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n    buckets := make([]*pair, 100)\n    return &amp;arrayHashMap{buckets: buckets}\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (a *arrayHashMap) hashFunc(key int) int {\n    index := key % 100\n    return index\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (a *arrayHashMap) get(key int) string {\n    index := a.hashFunc(key)\n    pair := a.buckets[index]\n    if pair == nil {\n        return \"Not Found\"\n    }\n    return pair.val\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (a *arrayHashMap) put(key int, val string) {\n    pair := &amp;pair{key: key, val: val}\n    index := a.hashFunc(key)\n    a.buckets[index] = pair\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (a *arrayHashMap) remove(key int) {\n    index := a.hashFunc(key)\n    // \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\n    a.buckets[index] = nil\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e\u5bf9 */\nfunc (a *arrayHashMap) pairSet() []*pair {\n    var pairs []*pair\n    for _, pair := range a.buckets {\n        if pair != nil {\n            pairs = append(pairs, pair)\n        }\n    }\n    return pairs\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc (a *arrayHashMap) keySet() []int {\n    var keys []int\n    for _, pair := range a.buckets {\n        if pair != nil {\n            keys = append(keys, pair.key)\n        }\n    }\n    return keys\n}\n\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc (a *arrayHashMap) valueSet() []string {\n    var values []string\n    for _, pair := range a.buckets {\n        if pair != nil {\n            values = append(values, pair.val)\n        }\n    }\n    return values\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (a *arrayHashMap) print() {\n    for _, pair := range a.buckets {\n        if pair != nil {\n            fmt.Println(pair.key, \"-&gt;\", pair.val)\n        }\n    }\n}\n</code></pre> array_hash_map.swift<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair: Equatable {\n    public var key: Int\n    public var val: String\n\n    public init(key: Int, val: String) {\n        self.key = key\n        self.val = val\n    }\n\n    public static func == (lhs: Pair, rhs: Pair) -&gt; Bool {\n        lhs.key == rhs.key &amp;&amp; lhs.val == rhs.val\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private var buckets: [Pair?]\n\n    init() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = Array(repeating: nil, count: 100)\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private func hashFunc(key: Int) -&gt; Int {\n        let index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let pair = buckets[index]\n        return pair?.val\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        let pair = Pair(key: key, val: val)\n        let index = hashFunc(key: key)\n        buckets[index] = pair\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        // \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = nil\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    func pairSet() -&gt; [Pair] {\n        buckets.compactMap { $0 }\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    func keySet() -&gt; [Int] {\n        buckets.compactMap { $0?.key }\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    func valueSet() -&gt; [String] {\n        buckets.compactMap { $0?.val }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    func print() {\n        for pair in pairSet() {\n            Swift.print(\"\\(pair.key) -&gt; \\(pair.val)\")\n        }\n    }\n}\n</code></pre> array_hash_map.js<pre><code>/* \u952e\u503c\u5bf9 Number -&gt; String */\nclass Pair {\n    constructor(key, val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    #buckets;\n    constructor() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        this.#buckets = new Array(100).fill(null);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key) {\n        return key % 100;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key) {\n        let index = this.#hashFunc(key);\n        let pair = this.#buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    set(key, val) {\n        let index = this.#hashFunc(key);\n        this.#buckets[index] = new Pair(key, val);\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    delete(key) {\n        let index = this.#hashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        this.#buckets[index] = null;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    entries() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    keys() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    values() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.ts<pre><code>/* \u952e\u503c\u5bf9 Number -&gt; String */\nclass Pair {\n    public key: number;\n    public val: string;\n\n    constructor(key: number, val: string) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private readonly buckets: (Pair | null)[];\n\n    constructor() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        this.buckets = new Array(100).fill(null);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private hashFunc(key: number): number {\n        return key % 100;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public get(key: number): string | null {\n        let index = this.hashFunc(key);\n        let pair = this.buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public set(key: number, val: string) {\n        let index = this.hashFunc(key);\n        this.buckets[index] = new Pair(key, val);\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public delete(key: number) {\n        let index = this.hashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        this.buckets[index] = null;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    public entries(): (Pair | null)[] {\n        let arr: (Pair | null)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    public keys(): (number | undefined)[] {\n        let arr: (number | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    public values(): (string | undefined)[] {\n        let arr: (string | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.dart<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair {\n  int key;\n  String val;\n  Pair(this.key, this.val);\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n  late List&lt;Pair?&gt; _buckets;\n\n  ArrayHashMap() {\n    // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n    _buckets = List.filled(100, null);\n  }\n\n  /* \u54c8\u5e0c\u51fd\u6570 */\n  int _hashFunc(int key) {\n    final int index = key % 100;\n    return index;\n  }\n\n  /* \u67e5\u8be2\u64cd\u4f5c */\n  String? get(int key) {\n    final int index = _hashFunc(key);\n    final Pair? pair = _buckets[index];\n    if (pair == null) {\n      return null;\n    }\n    return pair.val;\n  }\n\n  /* \u6dfb\u52a0\u64cd\u4f5c */\n  void put(int key, String val) {\n    final Pair pair = Pair(key, val);\n    final int index = _hashFunc(key);\n    _buckets[index] = pair;\n  }\n\n  /* \u5220\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    final int index = _hashFunc(key);\n    _buckets[index] = null;\n  }\n\n  /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n  List&lt;Pair&gt; pairSet() {\n    List&lt;Pair&gt; pairSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        pairSet.add(pair);\n      }\n    }\n    return pairSet;\n  }\n\n  /* \u83b7\u53d6\u6240\u6709\u952e */\n  List&lt;int&gt; keySet() {\n    List&lt;int&gt; keySet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        keySet.add(pair.key);\n      }\n    }\n    return keySet;\n  }\n\n  /* \u83b7\u53d6\u6240\u6709\u503c */\n  List&lt;String&gt; values() {\n    List&lt;String&gt; valueSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        valueSet.add(pair.val);\n      }\n    }\n    return valueSet;\n  }\n\n  /* \u6253\u5370\u54c8\u5e0c\u8868 */\n  void printHashMap() {\n    for (final Pair kv in pairSet()) {\n      print(\"${kv.key} -&gt; ${kv.val}\");\n    }\n  }\n}\n</code></pre> array_hash_map.rs<pre><code>/* \u952e\u503c\u5bf9 */\n#[derive(Debug, Clone, PartialEq)]\npub struct Pair {\n    pub key: i32,\n    pub val: String,\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\npub struct ArrayHashMap {\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;,\n}\n\nimpl ArrayHashMap {\n    pub fn new() -&gt; ArrayHashMap {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        Self {\n            buckets: vec![None; 100],\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % 100\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    pub fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;String&gt; {\n        let index = self.hash_func(key);\n        self.buckets[index].as_ref().map(|pair| &amp;pair.val)\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    pub fn put(&amp;mut self, key: i32, val: &amp;str) {\n        let index = self.hash_func(key);\n        self.buckets[index] = Some(Pair {\n            key,\n            val: val.to_string(),\n        });\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    pub fn remove(&amp;mut self, key: i32) {\n        let index = self.hash_func(key);\n        // \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\n        self.buckets[index] = None;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    pub fn entry_set(&amp;self) -&gt; Vec&lt;&amp;Pair&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref())\n            .collect()\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    pub fn key_set(&amp;self) -&gt; Vec&lt;&amp;i32&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.key))\n            .collect()\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    pub fn value_set(&amp;self) -&gt; Vec&lt;&amp;String&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.val))\n            .collect()\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    pub fn print(&amp;self) {\n        for pair in self.entry_set() {\n            println!(\"{} -&gt; {}\", pair.key, pair.val);\n        }\n    }\n}\n</code></pre> array_hash_map.c<pre><code>/* \u952e\u503c\u5bf9 int-&gt;string */\ntypedef struct {\n    int key;\n    char *val;\n} Pair;\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntypedef struct {\n    Pair *buckets[HASHTABLE_CAPACITY];\n} ArrayHashMap;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayHashMap *newArrayHashMap() {\n    ArrayHashMap *hmap = malloc(sizeof(ArrayHashMap));\n    return hmap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayHashMap(ArrayHashMap *hmap) {\n    for (int i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            free(hmap-&gt;buckets[i]-&gt;val);\n            free(hmap-&gt;buckets[i]);\n        }\n    }\n    free(hmap);\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(ArrayHashMap *hmap, const int key, const char *val) {\n    Pair *Pair = malloc(sizeof(Pair));\n    Pair-&gt;key = key;\n    Pair-&gt;val = malloc(strlen(val) + 1);\n    strcpy(Pair-&gt;val, val);\n\n    int index = hashFunc(key);\n    hmap-&gt;buckets[index] = Pair;\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(ArrayHashMap *hmap, const int key) {\n    int index = hashFunc(key);\n    free(hmap-&gt;buckets[index]-&gt;val);\n    free(hmap-&gt;buckets[index]);\n    hmap-&gt;buckets[index] = NULL;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nvoid pairSet(ArrayHashMap *hmap, MapSet *set) {\n    Pair *entries;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    entries = malloc(sizeof(Pair) * total);\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            entries[index].key = hmap-&gt;buckets[i]-&gt;key;\n            entries[index].val = malloc(strlen(hmap-&gt;buckets[i]-&gt;val) + 1);\n            strcpy(entries[index].val, hmap-&gt;buckets[i]-&gt;val);\n            index++;\n        }\n    }\n    set-&gt;set = entries;\n    set-&gt;len = total;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e */\nvoid keySet(ArrayHashMap *hmap, MapSet *set) {\n    int *keys;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    keys = malloc(total * sizeof(int));\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            keys[index] = hmap-&gt;buckets[i]-&gt;key;\n            index++;\n        }\n    }\n    set-&gt;set = keys;\n    set-&gt;len = total;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvoid valueSet(ArrayHashMap *hmap, MapSet *set) {\n    char **vals;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    vals = malloc(total * sizeof(char *));\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            vals[index] = hmap-&gt;buckets[i]-&gt;val;\n            index++;\n        }\n    }\n    set-&gt;set = vals;\n    set-&gt;len = total;\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(ArrayHashMap *hmap) {\n    int i;\n    MapSet set;\n    pairSet(hmap, &amp;set);\n    Pair *entries = (Pair *)set.set;\n    for (i = 0; i &lt; set.len; i++) {\n        printf(\"%d -&gt; %s\\n\", entries[i].key, entries[i].val);\n    }\n    free(set.set);\n}\n</code></pre> array_hash_map.kt<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair(\n    var key: Int,\n    var value: String\n)\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private val buckets = arrayOfNulls&lt;Pair&gt;(100)\n\n    init {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        for (i in 0..&lt;100) {\n            buckets[i] = null\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        val index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val pair = buckets[index] ?: return null\n        return pair.value\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        val pair = Pair(key, value)\n        val index = hashFunc(key)\n        buckets[index] = pair\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = null\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    fun pairSet(): MutableList&lt;Pair&gt; {\n        val pairSet = ArrayList&lt;Pair&gt;()\n        for (pair in buckets) {\n            if (pair != null) pairSet.add(pair)\n        }\n        return pairSet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    fun keySet(): MutableList&lt;Int&gt; {\n        val keySet = ArrayList&lt;Int&gt;()\n        for (pair in buckets) {\n            if (pair != null) keySet.add(pair.key)\n        }\n        return keySet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    fun valueSet(): MutableList&lt;String&gt; {\n        val valueSet = ArrayList&lt;String&gt;()\n        for (pair in buckets) {\n            pair?.let { valueSet.add(it.value) }\n        }\n        return valueSet\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (kv in pairSet()) {\n            val key = kv.key\n            val value = kv.value\n            println(\"${key}-&gt;${value}\")\n        }\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private val buckets = arrayOfNulls&lt;Pair&gt;(100)\n\n    init {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        for (i in 0..&lt;100) {\n            buckets[i] = null\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        val index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val pair = buckets[index] ?: return null\n        return pair.value\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        val pair = Pair(key, value)\n        val index = hashFunc(key)\n        buckets[index] = pair\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = null\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    fun pairSet(): MutableList&lt;Pair&gt; {\n        val pairSet = ArrayList&lt;Pair&gt;()\n        for (pair in buckets) {\n            if (pair != null) pairSet.add(pair)\n        }\n        return pairSet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    fun keySet(): MutableList&lt;Int&gt; {\n        val keySet = ArrayList&lt;Int&gt;()\n        for (pair in buckets) {\n            if (pair != null) keySet.add(pair.key)\n        }\n        return keySet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    fun valueSet(): MutableList&lt;String&gt; {\n        val valueSet = ArrayList&lt;String&gt;()\n        for (pair in buckets) {\n            pair?.let { valueSet.add(it.value) }\n        }\n        return valueSet\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (kv in pairSet()) {\n            val key = kv.key\n            val value = kv.value\n            println(\"${key}-&gt;${value}\")\n        }\n    }\n}\n</code></pre> array_hash_map.rb<pre><code>[class]{Pair}-[func]{}\n\n[class]{ArrayHashMap}-[func]{}\n</code></pre> array_hash_map.zig<pre><code>// \u952e\u503c\u5bf9\nconst Pair = struct {\n    key: usize = undefined,\n    val: []const u8 = undefined,\n\n   pub fn init(key: usize, val: []const u8) Pair {\n        return Pair {\n            .key = key,\n            .val = val,\n        };\n    }\n};\n\n// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\nfn ArrayHashMap(comptime T: type) type {\n    return struct {\n        bucket: ?std.ArrayList(?T) = null,\n        mem_allocator: std.mem.Allocator = undefined,\n\n        const Self = @This();\n\n        // \u6784\u9020\u51fd\u6570\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            self.mem_allocator = allocator;\n            // \u521d\u59cb\u5316\u4e00\u4e2a\u957f\u5ea6\u4e3a 100 \u7684\u6876\uff08\u6570\u7ec4\uff09\n            self.bucket = std.ArrayList(?T).init(self.mem_allocator);\n            var i: i32 = 0;\n            while (i &lt; 100) : (i += 1) {\n                try self.bucket.?.append(null);\n            }\n        }\n\n        // \u6790\u6784\u51fd\u6570\n        pub fn deinit(self: *Self) void {\n            if (self.bucket != null) self.bucket.?.deinit();\n        }\n\n        // \u54c8\u5e0c\u51fd\u6570\n        fn hashFunc(key: usize) usize {\n            var index = key % 100;\n            return index;\n        }\n\n        // \u67e5\u8be2\u64cd\u4f5c\n        pub fn get(self: *Self, key: usize) []const u8 {\n            var index = hashFunc(key);\n            var pair = self.bucket.?.items[index];\n            return pair.?.val;\n        }\n\n        // \u6dfb\u52a0\u64cd\u4f5c\n        pub fn put(self: *Self, key: usize, val: []const u8) !void {\n            var pair = Pair.init(key, val);\n            var index = hashFunc(key);\n            self.bucket.?.items[index] = pair;\n        }\n\n        // \u5220\u9664\u64cd\u4f5c\n        pub fn remove(self: *Self, key: usize) !void {\n            var index = hashFunc(key);\n            // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n            self.bucket.?.items[index] = null;\n        }       \n\n        // \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\n        pub fn pairSet(self: *Self) !std.ArrayList(T) {\n            var entry_set = std.ArrayList(T).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try entry_set.append(item.?);\n            }\n            return entry_set;\n        }  \n\n        // \u83b7\u53d6\u6240\u6709\u952e\n        pub fn keySet(self: *Self) !std.ArrayList(usize) {\n            var key_set = std.ArrayList(usize).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try key_set.append(item.?.key);\n            }\n            return key_set;\n        }  \n\n        // \u83b7\u53d6\u6240\u6709\u503c\n        pub fn valueSet(self: *Self) !std.ArrayList([]const u8) {\n            var value_set = std.ArrayList([]const u8).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try value_set.append(item.?.val);\n            }\n            return value_set;\n        }\n\n        // \u6253\u5370\u54c8\u5e0c\u8868\n        pub fn print(self: *Self) !void {\n            var entry_set = try self.pairSet();\n            defer entry_set.deinit();\n            for (entry_set.items) |item| {\n                std.debug.print(\"{} -&gt; {s}\\n\", .{item.key, item.val});\n            }\n        }\n    };\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_hashing/hash_map/#613-hash-collision-and-resizing","title":"6.1.3 \u00a0 Hash Collision and Resizing","text":"<p>Fundamentally, the role of the hash function is to map the entire input space of all keys to the output space of all array indices. However, the input space is often much larger than the output space. Therefore, theoretically, there must be situations where \"multiple inputs correspond to the same output\".</p> <p>For the hash function in the above example, if the last two digits of the input <code>key</code> are the same, the output of the hash function will also be the same. For example, when querying for students with student numbers 12836 and 20336, we find:</p> <pre><code>12836 % 100 = 36\n20336 % 100 = 36\n</code></pre> <p>As shown in the Figure 6-3 , both student numbers point to the same name, which is obviously incorrect. This situation where multiple inputs correspond to the same output is known as \"hash collision\".</p> <p></p> <p> Figure 6-3 \u00a0 Example of hash collision </p> <p>It is easy to understand that the larger the capacity \\(n\\) of the hash table, the lower the probability of multiple keys being allocated to the same bucket, and the fewer the collisions. Therefore, expanding the capacity of the hash table can reduce hash collisions.</p> <p>As shown in the Figure 6-4 , before expansion, key-value pairs <code>(136, A)</code> and <code>(236, D)</code> collided; after expansion, the collision is resolved.</p> <p></p> <p> Figure 6-4 \u00a0 Hash table expansion </p> <p>Similar to array expansion, resizing a hash table requires migrating all key-value pairs from the original hash table to the new one, which is time-consuming. Furthermore, since the capacity <code>capacity</code> of the hash table changes, we need to recalculate the storage positions of all key-value pairs using the hash function, which adds to the computational overhead of the resizing process. Therefore, programming languages often reserve a sufficiently large capacity for the hash table to prevent frequent resizing.</p> <p>The \"load factor\" is an important concept for hash tables. It is defined as the ratio of the number of elements in the hash table to the number of buckets. It is used to measure the severity of hash collisions and is often used as a trigger for resizing the hash table. For example, in Java, when the load factor exceeds \\(0.75\\), the system will resize the hash table to twice its original size.</p>"},{"location":"chapter_hashing/summary/","title":"6.4 \u00a0 Summary","text":""},{"location":"chapter_hashing/summary/#1-key-review","title":"1. \u00a0 Key Review","text":"<ul> <li>Given an input <code>key</code>, a hash table can retrieve the corresponding <code>value</code> in \\(O(1)\\) time, which is highly efficient.</li> <li>Common hash table operations include querying, adding key-value pairs, deleting key-value pairs, and traversing the hash table.</li> <li>The hash function maps a <code>key</code> to an array index, allowing access to the corresponding bucket to retrieve the <code>value</code>.</li> <li>Two different keys may end up with the same array index after hashing, leading to erroneous query results. This phenomenon is known as hash collision.</li> <li>The larger the capacity of the hash table, the lower the probability of hash collisions. Therefore, hash table resizing can mitigate hash collisions. Similar to array resizing, hash table resizing is costly.</li> <li>Load factor, defined as the ratio of the number of elements to the number of buckets in the hash table, reflects the severity of hash collisions and is often used as a trigger for resizing the hash table.</li> <li>Chaining addresses hash collisions by converting each element into a linked list, storing all colliding elements in the same list. However, excessively long lists can reduce query efficiency, which can be improved by converting the lists into red-black trees.</li> <li>Open addressing handles hash collisions through multiple probes. Linear probing uses a fixed step size but cannot delete elements and is prone to clustering. Multiple hashing uses several hash functions for probing, making it less susceptible to clustering but increasing computational load.</li> <li>Different programming languages adopt various hash table implementations. For example, Java's <code>HashMap</code> uses chaining, while Python's <code>dict</code> employs open addressing.</li> <li>In hash tables, we desire hash algorithms with determinism, high efficiency, and uniform distribution. In cryptography, hash algorithms should also possess collision resistance and the avalanche effect.</li> <li>Hash algorithms typically use large prime numbers as moduli to ensure uniform distribution of hash values and reduce hash collisions.</li> <li>Common hash algorithms include MD5, SHA-1, SHA-2, and SHA-3. MD5 is often used for file integrity checks, while SHA-2 is commonly used in secure applications and protocols.</li> <li>Programming languages usually provide built-in hash algorithms for data types to calculate bucket indices in hash tables. Generally, only immutable objects are hashable.</li> </ul>"},{"location":"chapter_hashing/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: When does the time complexity of a hash table degrade to \\(O(n)\\)?</p> <p>The time complexity of a hash table can degrade to \\(O(n)\\) when hash collisions are severe. When the hash function is well-designed, the capacity is set appropriately, and collisions are evenly distributed, the time complexity is \\(O(1)\\). We usually consider the time complexity to be \\(O(1)\\) when using built-in hash tables in programming languages.</p> <p>Q: Why not use the hash function \\(f(x) = x\\)? This would eliminate collisions.</p> <p>Under the hash function \\(f(x) = x\\), each element corresponds to a unique bucket index, which is equivalent to an array. However, the input space is usually much larger than the output space (array length), so the last step of a hash function is often to take the modulo of the array length. In other words, the goal of a hash table is to map a larger state space to a smaller one while providing \\(O(1)\\) query efficiency.</p> <p>Q: Why can hash tables be more efficient than arrays, linked lists, or binary trees, even though they are implemented using these structures?</p> <p>Firstly, hash tables have higher time efficiency but lower space efficiency. A significant portion of memory in hash tables remains unused.</p> <p>Secondly, they are only more efficient in specific use cases. If a feature can be implemented with the same time complexity using an array or a linked list, it's usually faster than using a hash table. This is because the computation of the hash function incurs overhead, making the constant factor in the time complexity larger.</p> <p>Lastly, the time complexity of hash tables can degrade. For example, in chaining, we perform search operations in a linked list or red-black tree, which still risks degrading to \\(O(n)\\) time.</p> <p>Q: Does multiple hashing also have the flaw of not being able to delete elements directly? Can space marked as deleted be reused?</p> <p>Multiple hashing is a form of open addressing, and all open addressing methods have the drawback of not being able to delete elements directly; they require marking elements as deleted. Marked spaces can be reused. When inserting new elements into the hash table, and the hash function points to a position marked as deleted, that position can be used by the new element. This maintains the probing sequence of the hash table while ensuring efficient use of space.</p> <p>Q: Why do hash collisions occur during the search process in linear probing?</p> <p>During the search process, the hash function points to the corresponding bucket and key-value pair. If the <code>key</code> doesn't match, it indicates a hash collision. Therefore, linear probing will search downwards at a predetermined step size until the correct key-value pair is found or the search fails.</p> <p>Q: Why can resizing a hash table alleviate hash collisions?</p> <p>The last step of a hash function often involves taking the modulo of the array length \\(n\\), to keep the output within the array index range. When resizing, the array length \\(n\\) changes, and the indices corresponding to the keys may also change. Keys that were previously mapped to the same bucket might be distributed across multiple buckets after resizing, thereby mitigating hash collisions.</p>"},{"location":"chapter_heap/","title":"Chapter 8. \u00a0 Heap","text":"<p>Abstract</p> <p>The heap is like mountain peaks, stacked and undulating, each with its unique shape.</p> <p>Among these peaks, the highest one always catches the eye first.</p>"},{"location":"chapter_heap/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>8.1 \u00a0 Heap</li> <li>8.2 \u00a0 Building a Heap</li> <li>8.3 \u00a0 Top-k Problem</li> <li>8.4 \u00a0 Summary</li> </ul>"},{"location":"chapter_heap/build_heap/","title":"8.2 \u00a0 Heap construction operation","text":"<p>In some cases, we want to build a heap using all elements of a list, and this process is known as \"heap construction operation.\"</p>"},{"location":"chapter_heap/build_heap/#821-implementing-with-heap-insertion-operation","title":"8.2.1 \u00a0 Implementing with heap insertion operation","text":"<p>First, we create an empty heap and then iterate through the list, performing the \"heap insertion operation\" on each element in turn. This means adding the element to the end of the heap and then \"heapifying\" it from bottom to top.</p> <p>Each time an element is added to the heap, the length of the heap increases by one. Since nodes are added to the binary tree from top to bottom, the heap is constructed \"from top to bottom.\"</p> <p>Let the number of elements be \\(n\\), and each element's insertion operation takes \\(O(\\log{n})\\) time, thus the time complexity of this heap construction method is \\(O(n \\log n)\\).</p>"},{"location":"chapter_heap/build_heap/#822-implementing-by-heapifying-through-traversal","title":"8.2.2 \u00a0 Implementing by heapifying through traversal","text":"<p>In fact, we can implement a more efficient method of heap construction in two steps.</p> <ol> <li>Add all elements of the list as they are into the heap, at this point the properties of the heap are not yet satisfied.</li> <li>Traverse the heap in reverse order (reverse of level-order traversal), and perform \"top to bottom heapify\" on each non-leaf node.</li> </ol> <p>After heapifying a node, the subtree with that node as the root becomes a valid sub-heap. Since the traversal is in reverse order, the heap is built \"from bottom to top.\"</p> <p>The reason for choosing reverse traversal is that it ensures the subtree below the current node is already a valid sub-heap, making the heapification of the current node effective.</p> <p>It's worth mentioning that since leaf nodes have no children, they naturally form valid sub-heaps and do not need to be heapified. As shown in the following code, the last non-leaf node is the parent of the last node; we start from it and traverse in reverse order to perform heapification:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def __init__(self, nums: list[int]):\n    \"\"\"\u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\"\"\"\n    # \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    self.max_heap = nums\n    # \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in range(self.parent(self.size() - 1), -1, -1):\n        self.sift_down(i)\n</code></pre> my_heap.cpp<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(vector&lt;int&gt; nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = nums;\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List&lt;Integer&gt; nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = new ArrayList&lt;&gt;(nums);\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(IEnumerable&lt;int&gt; nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = new List&lt;int&gt;(nums);\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    var size = Parent(this.Size() - 1);\n    for (int i = size; i &gt;= 0; i--) {\n        SiftDown(i);\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nfunc newMaxHeap(nums []any) *maxHeap {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    h := &amp;maxHeap{data: nums}\n    for i := h.parent(len(h.data) - 1); i &gt;= 0; i-- {\n        // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n        h.siftDown(i)\n    }\n    return h\n}\n</code></pre> my_heap.swift<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\ninit(nums: [Int]) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = nums\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in (0 ... parent(i: size() - 1)).reversed() {\n        siftDown(i: i)\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    this.#maxHeap = nums === undefined ? [] : [...nums];\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = this.#parent(this.size() - 1); i &gt;= 0; i--) {\n        this.#siftDown(i);\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums?: number[]) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    this.maxHeap = nums === undefined ? [] : [...nums];\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = this.parent(this.size() - 1); i &gt;= 0; i--) {\n        this.siftDown(i);\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List&lt;int&gt; nums) {\n  // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n  _maxHeap = nums;\n  // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n  for (int i = _parent(size() - 1); i &gt;= 0; i--) {\n    siftDown(i);\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nfn new(nums: Vec&lt;i32&gt;) -&gt; Self {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    let mut heap = MaxHeap { max_heap: nums };\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in (0..=Self::parent(heap.size() - 1)).rev() {\n        heap.sift_down(i);\n    }\n    heap\n}\n</code></pre> my_heap.c<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nMaxHeap *newMaxHeap(int nums[], int size) {\n    // \u6240\u6709\u5143\u7d20\u5165\u5806\n    MaxHeap *maxHeap = (MaxHeap *)malloc(sizeof(MaxHeap));\n    maxHeap-&gt;size = size;\n    memcpy(maxHeap-&gt;data, nums, size * sizeof(int));\n    for (int i = parent(maxHeap, size - 1); i &gt;= 0; i--) {\n        // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n        siftDown(maxHeap, i);\n    }\n    return maxHeap;\n}\n</code></pre> my_heap.kt<pre><code>/* \u5927\u9876\u5806 */\nclass MaxHeap(nums: List&lt;Int&gt;?) {\n    // \u4f7f\u7528\u5217\u8868\u800c\u975e\u6570\u7ec4\uff0c\u8fd9\u6837\u65e0\u987b\u8003\u8651\u6269\u5bb9\u95ee\u9898\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    private val maxHeap = ArrayList(nums!!)\n\n    /* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\n    init {\n        // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n        for (i in parent(size() - 1) downTo 0) {\n            siftDown(i)\n        }\n    }\n\n    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    private fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    private fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    private fun parent(i: Int): Int {\n        return (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u4ea4\u6362\u5143\u7d20 */\n    private fun swap(i: Int, j: Int) {\n        maxHeap[i] = maxHeap[j].also { maxHeap[j] = maxHeap[i] }\n    }\n\n    /* \u83b7\u53d6\u5806\u5927\u5c0f */\n    fun size(): Int {\n        return maxHeap.size\n    }\n\n    /* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        /* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\n        return size() == 0\n    }\n\n    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\n    fun peek(): Int {\n        return maxHeap[0]\n    }\n\n    /* \u5143\u7d20\u5165\u5806 */\n    fun push(value: Int) {\n        // \u6dfb\u52a0\u8282\u70b9\n        maxHeap.add(value)\n        // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n        siftUp(size() - 1)\n    }\n\n    /* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n    private fun siftUp(it: Int) {\n        // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n        var i = it\n        while (true) {\n            // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n            val p = parent(i)\n            // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n            if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n            // \u4ea4\u6362\u4e24\u8282\u70b9\n            swap(i, p)\n            // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n            i = p\n        }\n    }\n\n    /* \u5143\u7d20\u51fa\u5806 */\n    fun pop(): Int {\n        // \u5224\u7a7a\u5904\u7406\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        swap(0, size() - 1)\n        // \u5220\u9664\u8282\u70b9\n        val value = maxHeap.removeAt(size() - 1)\n        // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n        siftDown(0)\n        // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n        return value\n    }\n\n    /* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n    private fun siftDown(it: Int) {\n        // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n        var i = it\n        while (true) {\n            // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n            val l = left(i)\n            val r = right(i)\n            var ma = i\n            if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n            if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n            // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n            if (ma == i) break\n            // \u4ea4\u6362\u4e24\u8282\u70b9\n            swap(i, ma)\n            // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n            i = ma\n        }\n    }\n\n    /* \u6253\u5370\u5806\uff08\u4e8c\u53c9\u6811\uff09 */\n    fun print() {\n        val queue = PriorityQueue { a: Int, b: Int -&gt; b - a }\n        queue.addAll(maxHeap)\n        printHeap(queue)\n    }\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{__init__}\n</code></pre> my_heap.zig<pre><code>// \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\nfn init(self: *Self, allocator: std.mem.Allocator, nums: []const T) !void {\n    if (self.max_heap != null) return;\n    self.max_heap = std.ArrayList(T).init(allocator);\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    try self.max_heap.?.appendSlice(nums);\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    var i: usize = parent(self.size() - 1) + 1;\n    while (i &gt; 0) : (i -= 1) {\n        try self.siftDown(i - 1);\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_heap/build_heap/#823-complexity-analysis","title":"8.2.3 \u00a0 Complexity analysis","text":"<p>Next, let's attempt to calculate the time complexity of this second method of heap construction.</p> <ul> <li>Assuming the number of nodes in the complete binary tree is \\(n\\), then the number of leaf nodes is \\((n + 1) / 2\\), where \\(/\\) is integer division. Therefore, the number of nodes that need to be heapified is \\((n - 1) / 2\\).</li> <li>In the process of \"top to bottom heapification,\" each node is heapified to the leaf nodes at most, so the maximum number of iterations is the height of the binary tree \\(\\log n\\).</li> </ul> <p>Multiplying the two, we get the time complexity of the heap construction process as \\(O(n \\log n)\\). But this estimate is not accurate, because it does not take into account the nature of the binary tree having far more nodes at the lower levels than at the top.</p> <p>Let's perform a more accurate calculation. To simplify the calculation, assume a \"perfect binary tree\" with \\(n\\) nodes and height \\(h\\); this assumption does not affect the correctness of the result.</p> <p></p> <p> Figure 8-5 \u00a0 Node counts at each level of a perfect binary tree </p> <p>As shown in the Figure 8-5 , the maximum number of iterations for a node \"to be heapified from top to bottom\" is equal to the distance from that node to the leaf nodes, which is precisely \"node height.\" Therefore, we can sum the \"number of nodes \\(\\times\\) node height\" at each level, to get the total number of heapification iterations for all nodes.</p> \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{(h-1)}\\times1 \\] <p>To simplify the above equation, we need to use knowledge of sequences from high school, first multiply \\(T(h)\\) by \\(2\\), to get:</p> \\[ \\begin{aligned} T(h) &amp; = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{h-1}\\times1 \\newline 2T(h) &amp; = 2^1h + 2^2(h-1) + 2^3(h-2) + \\dots + 2^h\\times1 \\newline \\end{aligned} \\] <p>By subtracting \\(T(h)\\) from \\(2T(h)\\) using the method of displacement, we get:</p> \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\dots + 2^{h-1} + 2^h \\] <p>Observing the equation, \\(T(h)\\) is an geometric series, which can be directly calculated using the sum formula, resulting in a time complexity of:</p> \\[ \\begin{aligned} T(h) &amp; = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline &amp; = 2^{h+1} - h - 2 \\newline &amp; = O(2^h) \\end{aligned} \\] <p>Further, a perfect binary tree with height \\(h\\) has \\(n = 2^{h+1} - 1\\) nodes, thus the complexity is \\(O(2^h) = O(n)\\). This calculation shows that the time complexity of inputting a list and constructing a heap is \\(O(n)\\), which is very efficient.</p>"},{"location":"chapter_heap/heap/","title":"8.1 \u00a0 Heap","text":"<p>A \"heap\" is a complete binary tree that satisfies specific conditions and can be mainly divided into two types, as shown in the Figure 8-1 .</p> <ul> <li>\"Min heap\": The value of any node \\(\\leq\\) the values of its child nodes.</li> <li>\"Max heap\": The value of any node \\(\\geq\\) the values of its child nodes.</li> </ul> <p></p> <p> Figure 8-1 \u00a0 Min heap and max heap </p> <p>As a special case of a complete binary tree, heaps have the following characteristics:</p> <ul> <li>The bottom layer nodes are filled from left to right, and nodes in other layers are fully filled.</li> <li>The root node of the binary tree is called the \"heap top,\" and the bottom-rightmost node is called the \"heap bottom.\"</li> <li>For max heaps (min heaps), the value of the heap top element (root node) is the largest (smallest).</li> </ul>"},{"location":"chapter_heap/heap/#811-common-operations-on-heaps","title":"8.1.1 \u00a0 Common operations on heaps","text":"<p>It should be noted that many programming languages provide a \"priority queue,\" which is an abstract data structure defined as a queue with priority sorting.</p> <p>In fact, heaps are often used to implement priority queues, with max heaps equivalent to priority queues where elements are dequeued in descending order. From a usage perspective, we can consider \"priority queue\" and \"heap\" as equivalent data structures. Therefore, this book does not make a special distinction between the two, uniformly referring to them as \"heap.\"</p> <p>Common operations on heaps are shown in the Table 8-1 , and the method names depend on the programming language.</p> <p> Table 8-1 \u00a0 Efficiency of Heap Operations </p> Method name Description Time complexity <code>push()</code> Add an element to the heap \\(O(\\log n)\\) <code>pop()</code> Remove the top element from the heap \\(O(\\log n)\\) <code>peek()</code> Access the top element (for max/min heap, the max/min value) \\(O(1)\\) <code>size()</code> Get the number of elements in the heap \\(O(1)\\) <code>isEmpty()</code> Check if the heap is empty \\(O(1)\\) <p>In practice, we can directly use the heap class (or priority queue class) provided by programming languages.</p> <p>Similar to sorting algorithms where we have \"ascending order\" and \"descending order,\" we can switch between \"min heap\" and \"max heap\" by setting a <code>flag</code> or modifying the <code>Comparator</code>. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig heap.py<pre><code># \u521d\u59cb\u5316\u5c0f\u9876\u5806\nmin_heap, flag = [], 1\n# \u521d\u59cb\u5316\u5927\u9876\u5806\nmax_heap, flag = [], -1\n\n# Python \u7684 heapq \u6a21\u5757\u9ed8\u8ba4\u5b9e\u73b0\u5c0f\u9876\u5806\n# \u8003\u8651\u5c06\u201c\u5143\u7d20\u53d6\u8d1f\u201d\u540e\u518d\u5165\u5806\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5c06\u5927\u5c0f\u5173\u7cfb\u98a0\u5012\uff0c\u4ece\u800c\u5b9e\u73b0\u5927\u9876\u5806\n# \u5728\u672c\u793a\u4f8b\u4e2d\uff0cflag = 1 \u65f6\u5bf9\u5e94\u5c0f\u9876\u5806\uff0cflag = -1 \u65f6\u5bf9\u5e94\u5927\u9876\u5806\n\n# \u5143\u7d20\u5165\u5806\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n\n# \u83b7\u53d6\u5806\u9876\u5143\u7d20\npeek: int = flag * max_heap[0] # 5\n\n# \u5806\u9876\u5143\u7d20\u51fa\u5806\n# \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n\n# \u83b7\u53d6\u5806\u5927\u5c0f\nsize: int = len(max_heap)\n\n# \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = not max_heap\n\n# \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\nmin_heap: list[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n</code></pre> heap.cpp<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap;\n// \u521d\u59cb\u5316\u5927\u9876\u5806\npriority_queue&lt;int, vector&lt;int&gt;, less&lt;int&gt;&gt; maxHeap;\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.top(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.empty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nvector&lt;int&gt; input{1, 3, 2, 5, 4};\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap(input.begin(), input.end());\n</code></pre> heap.java<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nQueue&lt;Integer&gt; minHeap = new PriorityQueue&lt;&gt;();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nQueue&lt;Integer&gt; maxHeap = new PriorityQueue&lt;&gt;((a, b) -&gt; b - a);\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.peek(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll(); // 5\npeek = maxHeap.poll(); // 4\npeek = maxHeap.poll(); // 3\npeek = maxHeap.poll(); // 2\npeek = maxHeap.poll(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = maxHeap.isEmpty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue&lt;&gt;(Arrays.asList(1, 3, 2, 5, 4));\n</code></pre> heap.cs<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nPriorityQueue&lt;int, int&gt; minHeap = new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nPriorityQueue&lt;int, int&gt; maxHeap = new(Comparer&lt;int&gt;.Create((x, y) =&gt; y - x));\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.Peek();//5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.Count;\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.Count == 0;\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue&lt;int, int&gt;([(1, 1), (3, 3), (2, 2), (5, 5), (4, 4)]);\n</code></pre> heap.go<pre><code>// Go \u8bed\u8a00\u4e2d\u53ef\u4ee5\u901a\u8fc7\u5b9e\u73b0 heap.Interface \u6765\u6784\u5efa\u6574\u6570\u5927\u9876\u5806\n// \u5b9e\u73b0 heap.Interface \u9700\u8981\u540c\u65f6\u5b9e\u73b0 sort.Interface\ntype intHeap []any\n\n// Push heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u63a8\u5165\u5143\u7d20\u5230\u5806\nfunc (h *intHeap) Push(x any) {\n    // Push \u548c Pop \u4f7f\u7528 pointer receiver \u4f5c\u4e3a\u53c2\u6570\n    // \u56e0\u4e3a\u5b83\u4eec\u4e0d\u4ec5\u4f1a\u5bf9\u5207\u7247\u7684\u5185\u5bb9\u8fdb\u884c\u8c03\u6574\uff0c\u8fd8\u4f1a\u4fee\u6539\u5207\u7247\u7684\u957f\u5ea6\u3002\n    *h = append(*h, x.(int))\n}\n\n// Pop heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u5f39\u51fa\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Pop() any {\n    // \u5f85\u51fa\u5806\u5143\u7d20\u5b58\u653e\u5728\u6700\u540e\n    last := (*h)[len(*h)-1]\n    *h = (*h)[:len(*h)-1]\n    return last\n}\n\n// Len sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Len() int {\n    return len(*h)\n}\n\n// Less sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Less(i, j int) bool {\n    // \u5982\u679c\u5b9e\u73b0\u5c0f\u9876\u5806\uff0c\u5219\u9700\u8981\u8c03\u6574\u4e3a\u5c0f\u4e8e\u53f7\n    return (*h)[i].(int) &gt; (*h)[j].(int)\n}\n\n// Swap sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Swap(i, j int) {\n    (*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n\n// Top \u83b7\u53d6\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Top() any {\n    return (*h)[0]\n}\n\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n    /* \u521d\u59cb\u5316\u5806 */\n    // \u521d\u59cb\u5316\u5927\u9876\u5806\n    maxHeap := &amp;intHeap{}\n    heap.Init(maxHeap)\n    /* \u5143\u7d20\u5165\u5806 */\n    // \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u6dfb\u52a0\u5143\u7d20\n    heap.Push(maxHeap, 1)\n    heap.Push(maxHeap, 3)\n    heap.Push(maxHeap, 2)\n    heap.Push(maxHeap, 4)\n    heap.Push(maxHeap, 5)\n\n    /* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\n    top := maxHeap.Top()\n    fmt.Printf(\"\u5806\u9876\u5143\u7d20\u4e3a %d\\n\", top)\n\n    /* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n    // \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u79fb\u9664\u5143\u7d20\n    heap.Pop(maxHeap) // 5\n    heap.Pop(maxHeap) // 4\n    heap.Pop(maxHeap) // 3\n    heap.Pop(maxHeap) // 2\n    heap.Pop(maxHeap) // 1\n\n    /* \u83b7\u53d6\u5806\u5927\u5c0f */\n    size := len(*maxHeap)\n    fmt.Printf(\"\u5806\u5143\u7d20\u6570\u91cf\u4e3a %d\\n\", size)\n\n    /* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\n    isEmpty := len(*maxHeap) == 0\n    fmt.Printf(\"\u5806\u662f\u5426\u4e3a\u7a7a %t\\n\", isEmpty)\n}\n</code></pre> heap.swift<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// Swift \u7684 Heap \u7c7b\u578b\u540c\u65f6\u652f\u6301\u6700\u5927\u5806\u548c\u6700\u5c0f\u5806\uff0c\u4e14\u9700\u8981\u5f15\u5165 swift-collections\nvar heap = Heap&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u5806 */\nheap.insert(1)\nheap.insert(3)\nheap.insert(2)\nheap.insert(5)\nheap.insert(4)\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nvar peek = heap.max()!\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\npeek = heap.removeMax() // 5\npeek = heap.removeMax() // 4\npeek = heap.removeMax() // 3\npeek = heap.removeMax() // 2\npeek = heap.removeMax() // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nlet size = heap.count\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = heap.isEmpty\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nlet heap2 = Heap([1, 3, 2, 5, 4])\n</code></pre> heap.js<pre><code>// JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.ts<pre><code>// TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.dart<pre><code>// Dart \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.rs<pre><code>use std::collections::BinaryHeap;\nuse std::cmp::Reverse;\n\n/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nlet mut min_heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\nlet mut max_heap = BinaryHeap::new();\n\n/* \u5143\u7d20\u5165\u5806 */\nmax_heap.push(1);\nmax_heap.push(3);\nmax_heap.push(2);\nmax_heap.push(5);\nmax_heap.push(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nlet peek = max_heap.peek().unwrap();  // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nlet peek = max_heap.pop().unwrap();   // 5\nlet peek = max_heap.pop().unwrap();   // 4\nlet peek = max_heap.pop().unwrap();   // 3\nlet peek = max_heap.pop().unwrap();   // 2\nlet peek = max_heap.pop().unwrap();   // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nlet size = max_heap.len();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = max_heap.is_empty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nlet min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]);\n</code></pre> heap.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.kt<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nvar minHeap = PriorityQueue&lt;Int&gt;()\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nval maxHeap = PriorityQueue { a: Int, b: Int -&gt; b - a }\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1)\nmaxHeap.offer(3)\nmaxHeap.offer(2)\nmaxHeap.offer(5)\nmaxHeap.offer(4)\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nvar peek = maxHeap.peek() // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll() // 5\npeek = maxHeap.poll() // 4\npeek = maxHeap.poll() // 3\npeek = maxHeap.poll() // 2\npeek = maxHeap.poll() // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nval size = maxHeap.size\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nval isEmpty = maxHeap.isEmpty()\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = PriorityQueue(mutableListOf(1, 3, 2, 5, 4))\n</code></pre> heap.rb<pre><code>\n</code></pre> heap.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p>https://pythontutor.com/render.html#code=import%20heapq%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%B0%8F%E9%A1%B6%E5%A0%86%0A%20%20%20%20min_heap,%20flag%20%3D%20%5B%5D,%201%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%A4%A7%E9%A1%B6%E5%A0%86%0A%20%20%20%20max_heap,%20flag%20%3D%20%5B%5D,%20-1%0A%20%20%20%20%0A%20%20%20%20%23%20Python%20%E7%9A%84%20heapq%20%E6%A8%A1%E5%9D%97%E9%BB%98%E8%AE%A4%E5%AE%9E%E7%8E%B0%E5%B0%8F%E9%A1%B6%E5%A0%86%0A%20%20%20%20%23%20%E8%80%83%E8%99%91%E5%B0%86%E2%80%9C%E5%85%83%E7%B4%A0%E5%8F%96%E8%B4%9F%E2%80%9D%E5%90%8E%E5%86%8D%E5%85%A5%E5%A0%86%EF%BC%8C%E8%BF%99%E6%A0%B7%E5%B0%B1%E5%8F%AF%E4%BB%A5%E5%B0%86%E5%A4%A7%E5%B0%8F%E5%85%B3%E7%B3%BB%E9%A2%A0%E5%80%92%EF%BC%8C%E4%BB%8E%E8%80%8C%E5%AE%9E%E7%8E%B0%E5%A4%A7%E9%A1%B6%E5%A0%86%0A%20%20%20%20%23%20%E5%9C%A8%E6%9C%AC%E7%A4%BA%E4%BE%8B%E4%B8%AD%EF%BC%8Cflag%20%3D%201%20%E6%97%B6%E5%AF%B9%E5%BA%94%E5%B0%8F%E9%A1%B6%E5%A0%86%EF%BC%8Cflag%20%3D%20-1%20%E6%97%B6%E5%AF%B9%E5%BA%94%E5%A4%A7%E9%A1%B6%E5%A0%86%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%85%A5%E5%A0%86%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%201%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%203%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%202%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%205%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%204%29%0A%20%20%20%20%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%A0%86%E9%A1%B6%E5%85%83%E7%B4%A0%0A%20%20%20%20peek%20%3D%20flag%20*%20max_heap%5B0%5D%20%23%205%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%A0%86%E9%A1%B6%E5%85%83%E7%B4%A0%E5%87%BA%E5%A0%86%0A%20%20%20%20%23%20%E5%87%BA%E5%A0%86%E5%85%83%E7%B4%A0%E4%BC%9A%E5%BD%A2%E6%88%90%E4%B8%80%E4%B8%AA%E4%BB%8E%E5%A4%A7%E5%88%B0%E5%B0%8F%E7%9A%84%E5%BA%8F%E5%88%97%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%205%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%204%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%203%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%202%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%201%0A%20%20%20%20%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%A0%86%E5%A4%A7%E5%B0%8F%0A%20%20%20%20size%20%3D%20len%28max_heap%29%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%88%A4%E6%96%AD%E5%A0%86%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%0A%20%20%20%20is_empty%20%3D%20not%20max_heap%0A%20%20%20%20%0A%20%20%20%20%23%20%E8%BE%93%E5%85%A5%E5%88%97%E8%A1%A8%E5%B9%B6%E5%BB%BA%E5%A0%86%0A%20%20%20%20min_heap%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20heapq.heapify%28min_heap%29&amp;cumulative=false&amp;curInstr=3&amp;heapPrimitives=nevernest&amp;mode=display&amp;origin=opt-frontend.js&amp;py=311&amp;rawInputLstJSON=%5B%5D&amp;textReferences=false</p>"},{"location":"chapter_heap/heap/#812-implementation-of-heaps","title":"8.1.2 \u00a0 Implementation of heaps","text":"<p>The following implementation is of a max heap. To convert it into a min heap, simply invert all size logic comparisons (for example, replace \\(\\geq\\) with \\(\\leq\\)). Interested readers are encouraged to implement it on their own.</p>"},{"location":"chapter_heap/heap/#1-storage-and-representation-of-heaps","title":"1. \u00a0 Storage and representation of heaps","text":"<p>As mentioned in the \"Binary Trees\" section, complete binary trees are well-suited for array representation. Since heaps are a type of complete binary tree, we will use arrays to store heaps.</p> <p>When using an array to represent a binary tree, elements represent node values, and indexes represent node positions in the binary tree. Node pointers are implemented through an index mapping formula.</p> <p>As shown in the Figure 8-2 , given an index \\(i\\), the index of its left child is \\(2i + 1\\), the index of its right child is \\(2i + 2\\), and the index of its parent is \\((i - 1) / 2\\) (floor division). When the index is out of bounds, it signifies a null node or the node does not exist.</p> <p></p> <p> Figure 8-2 \u00a0 Representation and storage of heaps </p> <p>We can encapsulate the index mapping formula into functions for convenient later use:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def left(self, i: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n    return 2 * i + 1\n\ndef right(self, i: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n    return 2 * i + 2\n\ndef parent(self, i: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n    return (i - 1) // 2  # \u5411\u4e0b\u6574\u9664\n</code></pre> my_heap.cpp<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.java<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.cs<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint Left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint Right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint Parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.go<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) left(i int) int {\n    return 2*i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) right(i int) int {\n    return 2*i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) parent(i int) int {\n    // \u5411\u4e0b\u6574\u9664\n    return (i - 1) / 2\n}\n</code></pre> my_heap.swift<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc left(i: Int) -&gt; Int {\n    2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc right(i: Int) -&gt; Int {\n    2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc parent(i: Int) -&gt; Int {\n    (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.js<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n#left(i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n#right(i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n#parent(i) {\n    return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.ts<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nleft(i: number): number {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nright(i: number): number {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nparent(i: number): number {\n    return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.dart<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint _left(int i) {\n  return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint _right(int i) {\n  return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint _parent(int i) {\n  return (i - 1) ~/ 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.rs<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfn left(i: usize) -&gt; usize {\n    2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfn right(i: usize) -&gt; usize {\n    2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfn parent(i: usize) -&gt; usize {\n    (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.c<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint left(MaxHeap *maxHeap, int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint right(MaxHeap *maxHeap, int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint parent(MaxHeap *maxHeap, int i) {\n    return (i - 1) / 2;\n}\n</code></pre> my_heap.kt<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfun left(i: Int): Int {\n    return 2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfun right(i: Int): Int {\n    return 2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfun parent(i: Int): Int {\n    return (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{left}\n\n[class]{MaxHeap}-[func]{right}\n\n[class]{MaxHeap}-[func]{parent}\n</code></pre> my_heap.zig<pre><code>// \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\nfn left(i: usize) usize {\n    return 2 * i + 1;\n}\n\n// \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\nfn right(i: usize) usize {\n    return 2 * i + 2;\n}\n\n// \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15\nfn parent(i: usize) usize {\n    // return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n    return @divFloor(i - 1, 2);\n}\n</code></pre>"},{"location":"chapter_heap/heap/#2-accessing-the-top-element-of-the-heap","title":"2. \u00a0 Accessing the top element of the heap","text":"<p>The top element of the heap is the root node of the binary tree, which is also the first element of the list:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def peek(self) -&gt; int:\n    \"\"\"\u8bbf\u95ee\u5806\u9876\u5143\u7d20\"\"\"\n    return self.max_heap[0]\n</code></pre> my_heap.cpp<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.java<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n    return maxHeap.get(0);\n}\n</code></pre> my_heap.cs<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint Peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.go<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc (h *maxHeap) peek() any {\n    return h.data[0]\n}\n</code></pre> my_heap.swift<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc peek() -&gt; Int {\n    maxHeap[0]\n}\n</code></pre> my_heap.js<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek() {\n    return this.#maxHeap[0];\n}\n</code></pre> my_heap.ts<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek(): number {\n    return this.maxHeap[0];\n}\n</code></pre> my_heap.dart<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n  return _maxHeap[0];\n}\n</code></pre> my_heap.rs<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfn peek(&amp;self) -&gt; Option&lt;i32&gt; {\n    self.max_heap.first().copied()\n}\n</code></pre> my_heap.c<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek(MaxHeap *maxHeap) {\n    return maxHeap-&gt;data[0];\n}\n</code></pre> my_heap.kt<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfun peek(): Int {\n    return maxHeap[0]\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{peek}\n</code></pre> my_heap.zig<pre><code>// \u8bbf\u95ee\u5806\u9876\u5143\u7d20\nfn peek(self: *Self) T {\n    return self.max_heap.?.items[0];\n}  \n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_heap/heap/#3-inserting-an-element-into-the-heap","title":"3. \u00a0 Inserting an element into the heap","text":"<p>Given an element <code>val</code>, we first add it to the bottom of the heap. After addition, since <code>val</code> may be larger than other elements in the heap, the heap's integrity might be compromised, thus it's necessary to repair the path from the inserted node to the root node. This operation is called \"heapifying\".</p> <p>Considering starting from the node inserted, perform heapify from bottom to top. As shown in the Figure 8-3 , we compare the value of the inserted node with its parent node, and if the inserted node is larger, we swap them. Then continue this operation, repairing each node in the heap from bottom to top until passing the root node or encountering a node that does not need to be swapped.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 8-3 \u00a0 Steps of element insertion into the heap </p> <p>Given a total of \\(n\\) nodes, the height of the tree is \\(O(\\log n)\\). Hence, the loop iterations for the heapify operation are at most \\(O(\\log n)\\), making the time complexity of the element insertion operation \\(O(\\log n)\\). The code is as shown:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def push(self, val: int):\n    \"\"\"\u5143\u7d20\u5165\u5806\"\"\"\n    # \u6dfb\u52a0\u8282\u70b9\n    self.max_heap.append(val)\n    # \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    self.sift_up(self.size() - 1)\n\ndef sift_up(self, i: int):\n    \"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\"\"\"\n    while True:\n        # \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        p = self.parent(i)\n        # \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if p &lt; 0 or self.max_heap[i] &lt;= self.max_heap[p]:\n            break\n        # \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, p)\n        # \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.push_back(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(maxHeap[i], maxHeap[p]);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.add(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap.get(i) &lt;= maxHeap.get(p))\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid Push(int val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.Add(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    SiftUp(Size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid SiftUp(int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = Parent(i);\n        // \u82e5\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        Swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunc (h *maxHeap) push(val any) {\n    // \u6dfb\u52a0\u8282\u70b9\n    h.data = append(h.data, val)\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    h.siftUp(len(h.data) - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc (h *maxHeap) siftUp(i int) {\n    for true {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        p := h.parent(i)\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if p &lt; 0 || h.data[i].(int) &lt;= h.data[p].(int) {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        h.swap(i, p)\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunc push(val: Int) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.append(val)\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(i: size() - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc siftUp(i: Int) {\n    var i = i\n    while true {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        let p = parent(i: i)\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if p &lt; 0 || maxHeap[i] &lt;= maxHeap[p] {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i: i, j: p)\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u5165\u5806 */\npush(val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    this.#maxHeap.push(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    this.#siftUp(this.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n#siftUp(i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        const p = this.#parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || this.#maxHeap[i] &lt;= this.#maxHeap[p]) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.#swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u5165\u5806 */\npush(val: number): void {\n    // \u6dfb\u52a0\u8282\u70b9\n    this.maxHeap.push(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    this.siftUp(this.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nsiftUp(i: number): void {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        const p = this.parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || this.maxHeap[i] &lt;= this.maxHeap[p]) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n  // \u6dfb\u52a0\u8282\u70b9\n  _maxHeap.add(val);\n  // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n  siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n  while (true) {\n    // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n    int p = _parent(i);\n    // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n    if (p &lt; 0 || _maxHeap[i] &lt;= _maxHeap[p]) {\n      break;\n    }\n    // \u4ea4\u6362\u4e24\u8282\u70b9\n    _swap(i, p);\n    // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n    i = p;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfn push(&amp;mut self, val: i32) {\n    // \u6dfb\u52a0\u8282\u70b9\n    self.max_heap.push(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    self.sift_up(self.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfn sift_up(&amp;mut self, mut i: usize) {\n    loop {\n        // \u8282\u70b9 i \u5df2\u7ecf\u662f\u5806\u9876\u8282\u70b9\u4e86\uff0c\u7ed3\u675f\u5806\u5316\n        if i == 0 {\n            break;\n        }\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        let p = Self::parent(i);\n        // \u5f53\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if self.max_heap[i] &lt;= self.max_heap[p] {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(MaxHeap *maxHeap, int val) {\n    // \u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c\u4e0d\u5e94\u8be5\u6dfb\u52a0\u8fd9\u4e48\u591a\u8282\u70b9\n    if (maxHeap-&gt;size == MAX_SIZE) {\n        printf(\"heap is full!\");\n        return;\n    }\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap-&gt;data[maxHeap-&gt;size] = val;\n    maxHeap-&gt;size++;\n\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(maxHeap, maxHeap-&gt;size - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = parent(maxHeap, i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap-&gt;data[i] &lt;= maxHeap-&gt;data[p]) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(maxHeap, i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfun push(value: Int) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.add(value)\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(size() - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfun siftUp(it: Int) {\n    // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n    var i = it\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        val p = parent(i)\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, p)\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{push}\n\n[class]{MaxHeap}-[func]{sift_up}\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u5165\u5806\nfn push(self: *Self, val: T) !void {\n    // \u6dfb\u52a0\u8282\u70b9\n    try self.max_heap.?.append(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    try self.siftUp(self.size() - 1);\n}  \n\n// \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\nfn siftUp(self: *Self, i_: usize) !void {\n    var i = i_;\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        var p = parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 or self.max_heap.?.items[i] &lt;= self.max_heap.?.items[p]) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        try self.swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_heap/heap/#4-removing-the-top-element-from-the-heap","title":"4. \u00a0 Removing the top element from the heap","text":"<p>The top element of the heap is the root node of the binary tree, that is, the first element of the list. If we directly remove the first element from the list, all node indexes in the binary tree would change, making it difficult to use heapify for repairs subsequently. To minimize changes in element indexes, we use the following steps.</p> <ol> <li>Swap the top element with the bottom element of the heap (swap the root node with the rightmost leaf node).</li> <li>After swapping, remove the bottom of the heap from the list (note, since it has been swapped, what is actually being removed is the original top element).</li> <li>Starting from the root node, perform heapify from top to bottom.</li> </ol> <p>As shown in the Figure 8-4 , the direction of \"heapify from top to bottom\" is opposite to \"heapify from bottom to top\". We compare the value of the root node with its two children and swap it with the largest child. Then repeat this operation until passing the leaf node or encountering a node that does not need to be swapped.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 8-4 \u00a0 Steps of removing the top element from the heap </p> <p>Similar to the element insertion operation, the time complexity of the top element removal operation is also \\(O(\\log n)\\). The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def pop(self) -&gt; int:\n    \"\"\"\u5143\u7d20\u51fa\u5806\"\"\"\n    # \u5224\u7a7a\u5904\u7406\n    if self.is_empty():\n        raise IndexError(\"\u5806\u4e3a\u7a7a\")\n    # \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    self.swap(0, self.size() - 1)\n    # \u5220\u9664\u8282\u70b9\n    val = self.max_heap.pop()\n    # \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    self.sift_down(0)\n    # \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val\n\ndef sift_down(self, i: int):\n    \"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\n    while True:\n        # \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        l, r, ma = self.left(i), self.right(i), i\n        if l &lt; self.size() and self.max_heap[l] &gt; self.max_heap[ma]:\n            ma = l\n        if r &lt; self.size() and self.max_heap[r] &gt; self.max_heap[ma]:\n            ma = r\n        # \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i:\n            break\n        # \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, ma)\n        # \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nvoid pop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty()) {\n        throw out_of_range(\"\u5806\u4e3a\u7a7a\");\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(maxHeap[0], maxHeap[size() - 1]);\n    // \u5220\u9664\u8282\u70b9\n    maxHeap.pop_back();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(0);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        swap(maxHeap[i], maxHeap[ma]);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty())\n        throw new IndexOutOfBoundsException();\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(0, size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    int val = maxHeap.remove(size() - 1);\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap.get(l) &gt; maxHeap.get(ma))\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap.get(r) &gt; maxHeap.get(ma))\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint Pop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (IsEmpty())\n        throw new IndexOutOfRangeException();\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    Swap(0, Size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    int val = maxHeap.Last();\n    maxHeap.RemoveAt(Size() - 1);\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    SiftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid SiftDown(int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = Left(i), r = Right(i), ma = i;\n        if (l &lt; Size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; Size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // \u82e5\u201c\u8282\u70b9 i \u6700\u5927\u201d\u6216\u201c\u8d8a\u8fc7\u53f6\u8282\u70b9\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\n        if (ma == i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        Swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfunc (h *maxHeap) pop() any {\n    // \u5224\u7a7a\u5904\u7406\n    if h.isEmpty() {\n        fmt.Println(\"error\")\n        return nil\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    h.swap(0, h.size()-1)\n    // \u5220\u9664\u8282\u70b9\n    val := h.data[len(h.data)-1]\n    h.data = h.data[:len(h.data)-1]\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    h.siftDown(0)\n\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc (h *maxHeap) siftDown(i int) {\n    for true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\n        l, r, max := h.left(i), h.right(i), i\n        if l &lt; h.size() &amp;&amp; h.data[l].(int) &gt; h.data[max].(int) {\n            max = l\n        }\n        if r &lt; h.size() &amp;&amp; h.data[r].(int) &gt; h.data[max].(int) {\n            max = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if max == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        h.swap(i, max)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = max\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfunc pop() -&gt; Int {\n    // \u5224\u7a7a\u5904\u7406\n    if isEmpty() {\n        fatalError(\"\u5806\u4e3a\u7a7a\")\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(i: 0, j: size() - 1)\n    // \u5220\u9664\u8282\u70b9\n    let val = maxHeap.remove(at: size() - 1)\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(i: 0)\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(i: Int) {\n    var i = i\n    while true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = left(i: i)\n        let r = right(i: i)\n        var ma = i\n        if l &lt; size(), maxHeap[l] &gt; maxHeap[ma] {\n            ma = l\n        }\n        if r &lt; size(), maxHeap[r] &gt; maxHeap[ma] {\n            ma = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i: i, j: ma)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u51fa\u5806 */\npop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (this.isEmpty()) throw new Error('\u5806\u4e3a\u7a7a');\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    this.#swap(0, this.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    const val = this.#maxHeap.pop();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    this.#siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n#siftDown(i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        const l = this.#left(i),\n            r = this.#right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.#maxHeap[l] &gt; this.#maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.#maxHeap[r] &gt; this.#maxHeap[ma]) ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.#swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u51fa\u5806 */\npop(): number {\n    // \u5224\u7a7a\u5904\u7406\n    if (this.isEmpty()) throw new RangeError('Heap is empty.');\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    this.swap(0, this.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    const val = this.maxHeap.pop();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    this.siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nsiftDown(i: number): void {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        const l = this.left(i),\n            r = this.right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.maxHeap[l] &gt; this.maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.maxHeap[r] &gt; this.maxHeap[ma]) ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n  // \u5224\u7a7a\u5904\u7406\n  if (isEmpty()) throw Exception('\u5806\u4e3a\u7a7a');\n  // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n  _swap(0, size() - 1);\n  // \u5220\u9664\u8282\u70b9\n  int val = _maxHeap.removeLast();\n  // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n  siftDown(0);\n  // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n  return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n  while (true) {\n    // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n    int l = _left(i);\n    int r = _right(i);\n    int ma = i;\n    if (l &lt; size() &amp;&amp; _maxHeap[l] &gt; _maxHeap[ma]) ma = l;\n    if (r &lt; size() &amp;&amp; _maxHeap[r] &gt; _maxHeap[ma]) ma = r;\n    // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n    if (ma == i) break;\n    // \u4ea4\u6362\u4e24\u8282\u70b9\n    _swap(i, ma);\n    // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n    i = ma;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfn pop(&amp;mut self) -&gt; i32 {\n    // \u5224\u7a7a\u5904\u7406\n    if self.is_empty() {\n        panic!(\"index out of bounds\");\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    self.swap(0, self.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    let val = self.max_heap.remove(self.size() - 1);\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    self.sift_down(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfn sift_down(&amp;mut self, mut i: usize) {\n    loop {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let (l, r, mut ma) = (Self::left(i), Self::right(i), i);\n        if l &lt; self.size() &amp;&amp; self.max_heap[l] &gt; self.max_heap[ma] {\n            ma = l;\n        }\n        if r &lt; self.size() &amp;&amp; self.max_heap[r] &gt; self.max_heap[ma] {\n            ma = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop(MaxHeap *maxHeap) {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty(maxHeap)) {\n        printf(\"heap is empty!\");\n        return INT_MAX;\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(maxHeap, 0, size(maxHeap) - 1);\n    // \u5220\u9664\u8282\u70b9\n    int val = maxHeap-&gt;data[maxHeap-&gt;size - 1];\n    maxHeap-&gt;size--;\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(maxHeap, 0);\n\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\n        int l = left(maxHeap, i);\n        int r = right(maxHeap, i);\n        int max = i;\n        if (l &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[l] &gt; maxHeap-&gt;data[max]) {\n            max = l;\n        }\n        if (r &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[r] &gt; maxHeap-&gt;data[max]) {\n            max = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (max == i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(maxHeap, i, max);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = max;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfun pop(): Int {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty()) throw IndexOutOfBoundsException()\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(0, size() - 1)\n    // \u5220\u9664\u8282\u70b9\n    val value = maxHeap.removeAt(size() - 1)\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(0)\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return value\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfun siftDown(it: Int) {\n    // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n    var i = it\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        val l = left(i)\n        val r = right(i)\n        var ma = i\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) break\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, ma)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{pop}\n\n[class]{MaxHeap}-[func]{sift_down}\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u51fa\u5806\nfn pop(self: *Self) !T {\n    // \u5224\u65ad\u5904\u7406\n    if (self.isEmpty()) unreachable;\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    try self.swap(0, self.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    var val = self.max_heap.?.pop();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    try self.siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n} \n\n// \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\nfn siftDown(self: *Self, i_: usize) !void {\n    var i = i_;\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        var l = left(i);\n        var r = right(i);\n        var ma = i;\n        if (l &lt; self.size() and self.max_heap.?.items[l] &gt; self.max_heap.?.items[ma]) ma = l;\n        if (r &lt; self.size() and self.max_heap.?.items[r] &gt; self.max_heap.?.items[ma]) ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        try self.swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_heap/heap/#813-common-applications-of-heaps","title":"8.1.3 \u00a0 Common applications of heaps","text":"<ul> <li>Priority Queue: Heaps are often the preferred data structure for implementing priority queues, with both enqueue and dequeue operations having a time complexity of \\(O(\\log n)\\), and building a queue having a time complexity of \\(O(n)\\), all of which are very efficient.</li> <li>Heap Sort: Given a set of data, we can create a heap from them and then continually perform element removal operations to obtain ordered data. However, we usually use a more elegant method to implement heap sort, as detailed in the \"Heap Sort\" section.</li> <li>Finding the Largest \\(k\\) Elements: This is a classic algorithm problem and also a typical application, such as selecting the top 10 hot news for Weibo hot search, picking the top 10 selling products, etc.</li> </ul>"},{"location":"chapter_heap/summary/","title":"8.4 \u00a0 Summary","text":""},{"location":"chapter_heap/summary/#1-key-review","title":"1. \u00a0 Key review","text":"<ul> <li>A heap is a complete binary tree, which can be divided into a max heap and a min heap based on its property. The top element of a max (min) heap is the largest (smallest).</li> <li>A priority queue is defined as a queue with dequeue priority, usually implemented using a heap.</li> <li>Common operations of a heap and their corresponding time complexities include: element insertion into the heap \\(O(\\log n)\\), removing the top element from the heap \\(O(\\log n)\\), and accessing the top element of the heap \\(O(1)\\).</li> <li>A complete binary tree is well-suited to be represented by an array, thus heaps are commonly stored using arrays.</li> <li>Heapify operations are used to maintain the properties of the heap and are used in both heap insertion and removal operations.</li> <li>The time complexity of inserting \\(n\\) elements into a heap and building the heap can be optimized to \\(O(n)\\), which is highly efficient.</li> <li>Top-k is a classic algorithm problem that can be efficiently solved using the heap data structure, with a time complexity of \\(O(n \\log k)\\).</li> </ul>"},{"location":"chapter_heap/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: Is the \"heap\" in data structures the same concept as the \"heap\" in memory management?</p> <p>The two are not the same concept, even though they are both referred to as \"heap\". The heap in computer system memory is part of dynamic memory allocation, where the program can use it to store data during execution. The program can request a certain amount of heap memory to store complex structures like objects and arrays. When these data are no longer needed, the program needs to release this memory to prevent memory leaks. Compared to stack memory, the management and usage of heap memory need to be more cautious, as improper use may lead to memory leaks and dangling pointers.</p>"},{"location":"chapter_heap/top_k/","title":"8.3 \u00a0 Top-k problem","text":"<p>Question</p> <p>Given an unordered array <code>nums</code> of length \\(n\\), return the largest \\(k\\) elements in the array.</p> <p>For this problem, we will first introduce two straightforward solutions, then explain a more efficient heap-based method.</p>"},{"location":"chapter_heap/top_k/#831-method-1-iterative-selection","title":"8.3.1 \u00a0 Method 1: Iterative selection","text":"<p>We can perform \\(k\\) rounds of iterations as shown in the Figure 8-6 , extracting the \\(1^{st}\\), \\(2^{nd}\\), \\(\\dots\\), \\(k^{th}\\) largest elements in each round, with a time complexity of \\(O(nk)\\).</p> <p>This method is only suitable when \\(k \\ll n\\), as the time complexity approaches \\(O(n^2)\\) when \\(k\\) is close to \\(n\\), which is very time-consuming.</p> <p></p> <p> Figure 8-6 \u00a0 Iteratively finding the largest k elements </p> <p>Tip</p> <p>When \\(k = n\\), we can obtain a complete ordered sequence, which is equivalent to the \"selection sort\" algorithm.</p>"},{"location":"chapter_heap/top_k/#832-method-2-sorting","title":"8.3.2 \u00a0 Method 2: Sorting","text":"<p>As shown in the Figure 8-7 , we can first sort the array <code>nums</code> and then return the last \\(k\\) elements, with a time complexity of \\(O(n \\log n)\\).</p> <p>Clearly, this method \"overachieves\" the task, as we only need to find the largest \\(k\\) elements, without the need to sort the other elements.</p> <p></p> <p> Figure 8-7 \u00a0 Sorting to find the largest k elements </p>"},{"location":"chapter_heap/top_k/#833-method-3-heap","title":"8.3.3 \u00a0 Method 3: Heap","text":"<p>We can solve the Top-k problem more efficiently based on heaps, as shown in the following process.</p> <ol> <li>Initialize a min heap, where the top element is the smallest.</li> <li>First, insert the first \\(k\\) elements of the array into the heap.</li> <li>Starting from the \\(k + 1^{th}\\) element, if the current element is greater than the top element of the heap, remove the top element of the heap and insert the current element into the heap.</li> <li>After completing the traversal, the heap contains the largest \\(k\\) elements.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 8-8 \u00a0 Find the largest k elements based on heap </p> <p>Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig top_k.py<pre><code>def top_k_heap(nums: list[int], k: int) -&gt; list[int]:\n    \"\"\"\u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\"\"\"\n    # \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    heap = []\n    # \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for i in range(k):\n        heapq.heappush(heap, nums[i])\n    # \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for i in range(k, len(nums)):\n        # \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if nums[i] &gt; heap[0]:\n            heapq.heappop(heap)\n            heapq.heappush(heap, nums[i])\n    return heap\n</code></pre> top_k.cpp<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; topKHeap(vector&lt;int&gt; &amp;nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    priority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; heap;\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        heap.push(nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; nums.size(); i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.top()) {\n            heap.pop();\n            heap.push(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.java<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nQueue&lt;Integer&gt; topKHeap(int[] nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    Queue&lt;Integer&gt; heap = new PriorityQueue&lt;Integer&gt;();\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        heap.offer(nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; nums.length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll();\n            heap.offer(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.cs<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nPriorityQueue&lt;int, int&gt; TopKHeap(int[] nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    PriorityQueue&lt;int, int&gt; heap = new();\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        heap.Enqueue(nums[i], nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; nums.Length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.Peek()) {\n            heap.Dequeue();\n            heap.Enqueue(nums[i], nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.go<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums []int, k int) *minHeap {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    h := &amp;minHeap{}\n    heap.Init(h)\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for i := 0; i &lt; k; i++ {\n        heap.Push(h, nums[i])\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for i := k; i &lt; len(nums); i++ {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if nums[i] &gt; h.Top().(int) {\n            heap.Pop(h)\n            heap.Push(h, nums[i])\n        }\n    }\n    return h\n}\n</code></pre> top_k.swift<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums: [Int], k: Int) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5e76\u5c06\u524d k \u4e2a\u5143\u7d20\u5efa\u5806\n    var heap = Heap(nums.prefix(k))\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for i in nums.indices.dropFirst(k) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if nums[i] &gt; heap.min()! {\n            _ = heap.removeMin()\n            heap.insert(nums[i])\n        }\n    }\n    return heap.unordered\n}\n</code></pre> top_k.js<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap, val) {\n    // \u5143\u7d20\u53d6\u53cd\n    maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.pop();\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return maxHeap.getMaxHeap().map((num) =&gt; -num);\n}\n\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums, k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n    const maxHeap = new MaxHeap([]);\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (let i = k; i &lt; nums.length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.ts<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap: MaxHeap, val: number): void {\n    // \u5143\u7d20\u53d6\u53cd\n    maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap: MaxHeap): number {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.pop();\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap: MaxHeap): number {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap: MaxHeap): number[] {\n    // \u5143\u7d20\u53d6\u53cd\n    return maxHeap.getMaxHeap().map((num: number) =&gt; -num);\n}\n\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums: number[], k: number): number[] {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n    const maxHeap = new MaxHeap([]);\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (let i = k; i &lt; nums.length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.dart<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nMinHeap topKHeap(List&lt;int&gt; nums, int k) {\n  // \u521d\u59cb\u5316\u5c0f\u9876\u5806\uff0c\u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n  MinHeap heap = MinHeap(nums.sublist(0, k));\n  // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n  for (int i = k; i &lt; nums.length; i++) {\n    // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n    if (nums[i] &gt; heap.peek()) {\n      heap.pop();\n      heap.push(nums[i]);\n    }\n  }\n  return heap;\n}\n</code></pre> top_k.rs<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfn top_k_heap(nums: Vec&lt;i32&gt;, k: usize) -&gt; BinaryHeap&lt;Reverse&lt;i32&gt;&gt; {\n    // BinaryHeap \u662f\u5927\u9876\u5806\uff0c\u4f7f\u7528 Reverse \u5c06\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u5b9e\u73b0\u5c0f\u9876\u5806\n    let mut heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for &amp;num in nums.iter().take(k) {\n        heap.push(Reverse(num));\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for &amp;num in nums.iter().skip(k) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if num &gt; heap.peek().unwrap().0 {\n            heap.pop();\n            heap.push(Reverse(num));\n        }\n    }\n    heap\n}\n</code></pre> top_k.c<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid pushMinHeap(MaxHeap *maxHeap, int val) {\n    // \u5143\u7d20\u53d6\u53cd\n    push(maxHeap, -val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nint popMinHeap(MaxHeap *maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -pop(maxHeap);\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peekMinHeap(MaxHeap *maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -peek(maxHeap);\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n// \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\u7684\u51fd\u6570\nint *topKHeap(int *nums, int sizeNums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n    int *empty = (int *)malloc(0);\n    MaxHeap *maxHeap = newMaxHeap(empty, 0);\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; sizeNums; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    int *res = getMinHeap(maxHeap);\n    // \u91ca\u653e\u5185\u5b58\n    delMaxHeap(maxHeap);\n    return res;\n}\n</code></pre> top_k.kt<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfun topKHeap(nums: IntArray, k: Int): Queue&lt;Int&gt; {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    val heap = PriorityQueue&lt;Int&gt;()\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (i in 0..&lt;k) {\n        heap.offer(nums[i])\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (i in k..&lt;nums.size) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll()\n            heap.offer(nums[i])\n        }\n    }\n    return heap\n}\n</code></pre> top_k.rb<pre><code>[class]{}-[func]{top_k_heap}\n</code></pre> top_k.zig<pre><code>[class]{}-[func]{topKHeap}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>A total of \\(n\\) rounds of heap insertions and deletions are performed, with the maximum heap size being \\(k\\), hence the time complexity is \\(O(n \\log k)\\). This method is very efficient; when \\(k\\) is small, the time complexity tends towards \\(O(n)\\); when \\(k\\) is large, the time complexity will not exceed \\(O(n \\log n)\\).</p> <p>Additionally, this method is suitable for scenarios with dynamic data streams. By continuously adding data, we can maintain the elements within the heap, thereby achieving dynamic updates of the largest \\(k\\) elements.</p>"},{"location":"chapter_introduction/","title":"Chapter 1. \u00a0 Introduction to Algorithms","text":"<p>Abstract</p> <p>A graceful maiden dances, intertwined with the data, her skirt swaying to the melody of algorithms.</p> <p>She invites you to a dance, follow her steps, and enter the world of algorithms full of logic and beauty.</p>"},{"location":"chapter_introduction/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>1.1 \u00a0 Algorithms are Everywhere</li> <li>1.2 \u00a0 What is an Algorithm</li> <li>1.3 \u00a0 Summary</li> </ul>"},{"location":"chapter_introduction/algorithms_are_everywhere/","title":"1.1 \u00a0 Algorithms are Everywhere","text":"<p>When we hear the word \"algorithm,\" we naturally think of mathematics. However, many algorithms do not involve complex mathematics but rely more on basic logic, which can be seen everywhere in our daily lives.</p> <p>Before formally discussing algorithms, there's an interesting fact worth sharing: you have already unconsciously learned many algorithms and have become accustomed to applying them in your daily life. Here, I will give a few specific examples to prove this point.</p> <p>Example 1: Looking Up a Dictionary. In an English dictionary, words are listed alphabetically. Suppose we're searching for a word that starts with the letter \\(r\\). This is typically done in the following way:</p> <ol> <li>Open the dictionary to about halfway and check the first letter on the page, let's say the letter is \\(m\\).</li> <li>Since \\(r\\) comes after \\(m\\) in the alphabet, we can ignore the first half of the dictionary and focus on the latter half.</li> <li>Repeat steps <code>1.</code> and <code>2.</code> until you find the page where the word starts with \\(r\\).</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 1-1 \u00a0 Process of Looking Up a Dictionary </p> <p>This essential skill for elementary students, looking up a dictionary, is actually the famous \"Binary Search\" algorithm. From a data structure perspective, we can consider the dictionary as a sorted \"array\"; from an algorithmic perspective, the series of actions taken to look up a word in the dictionary can be viewed as \"Binary Search.\"</p> <p>Example 2: Organizing Playing Cards. When playing cards, we need to arrange the cards in our hand in ascending order, as shown in the following process.</p> <ol> <li>Divide the playing cards into \"ordered\" and \"unordered\" sections, assuming initially the leftmost card is already in order.</li> <li>Take out a card from the unordered section and insert it into the correct position in the ordered section; after this, the leftmost two cards are in order.</li> <li>Continue to repeat step <code>2.</code> until all cards are in order.</li> </ol> <p></p> <p> Figure 1-2 \u00a0 Playing Cards Sorting Process </p> <p>The above method of organizing playing cards is essentially the \"Insertion Sort\" algorithm, which is very efficient for small datasets. Many programming languages' sorting functions include the insertion sort.</p> <p>Example 3: Making Change. Suppose we buy goods worth \\(69\\) yuan at a supermarket and give the cashier \\(100\\) yuan, then the cashier needs to give us \\(31\\) yuan in change. They would naturally complete the thought process as shown below.</p> <ol> <li>The options are currencies smaller than \\(31\\), including \\(1\\), \\(5\\), \\(10\\), and \\(20\\).</li> <li>Take out the largest \\(20\\) from the options, leaving \\(31 - 20 = 11\\).</li> <li>Take out the largest \\(10\\) from the remaining options, leaving \\(11 - 10 = 1\\).</li> <li>Take out the largest \\(1\\) from the remaining options, leaving \\(1 - 1 = 0\\).</li> <li>Complete the change-making, with the solution being \\(20 + 10 + 1 = 31\\).</li> </ol> <p></p> <p> Figure 1-3 \u00a0 Change making process </p> <p>In the above steps, we make the best choice at each step (using the largest denomination possible), ultimately resulting in a feasible change-making plan. From the perspective of data structures and algorithms, this method is essentially a \"Greedy\" algorithm.</p> <p>From cooking a meal to interstellar travel, almost all problem-solving involves algorithms. The advent of computers allows us to store data structures in memory and write code to call the CPU and GPU to execute algorithms. In this way, we can transfer real-life problems to computers, solving various complex issues more efficiently.</p> <p>Tip</p> <p>If concepts such as data structures, algorithms, arrays, and binary search still seem somewhat obsecure, I encourage you to continue reading. This book will gently guide you into the realm of understanding data structures and algorithms.</p>"},{"location":"chapter_introduction/summary/","title":"1.3 \u00a0 Summary","text":"<ul> <li>Algorithms are ubiquitous in daily life and are not as inaccessible and complex as they might seem. In fact, we have already unconsciously learned many algorithms to solve various problems in life.</li> <li>The principle of looking up a word in a dictionary is consistent with the binary search algorithm. The binary search algorithm embodies the important algorithmic concept of divide and conquer.</li> <li>The process of organizing playing cards is very similar to the insertion sort algorithm. The insertion sort algorithm is suitable for sorting small datasets.</li> <li>The steps of making change in currency essentially follow the greedy algorithm, where each step involves making the best possible choice at the moment.</li> <li>An algorithm is a set of instructions or steps used to solve a specific problem within a finite amount of time, while a data structure is the way data is organized and stored in a computer.</li> <li>Data structures and algorithms are closely linked. Data structures are the foundation of algorithms, and algorithms are the stage to utilize the functions of data structures.</li> <li>We can liken data structures and algorithms to building blocks. The blocks represent data, the shape and connection method of the blocks represent data structures, and the steps of assembling the blocks correspond to algorithms.</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/","title":"1.2 \u00a0 What is an Algorithm","text":""},{"location":"chapter_introduction/what_is_dsa/#121-definition-of-an-algorithm","title":"1.2.1 \u00a0 Definition of an Algorithm","text":"<p>An \"algorithm\" is a set of instructions or steps to solve a specific problem within a finite amount of time. It has the following characteristics:</p> <ul> <li>The problem is clearly defined, including unambiguous definitions of input and output.</li> <li>The algorithm is feasible, meaning it can be completed within a finite number of steps, time, and memory space.</li> <li>Each step has a definitive meaning. The output is consistently the same under the same inputs and conditions.</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#122-definition-of-a-data-structure","title":"1.2.2 \u00a0 Definition of a Data Structure","text":"<p>A \"data structure\" is a way of organizing and storing data in a computer, with the following design goals:</p> <ul> <li>Minimize space occupancy to save computer memory.</li> <li>Make data operations as fast as possible, covering data access, addition, deletion, updating, etc.</li> <li>Provide concise data representation and logical information to enable efficient algorithm execution.</li> </ul> <p>Designing data structures is a balancing act, often requiring trade-offs. If you want to improve in one aspect, you often need to compromise in another. Here are two examples:</p> <ul> <li>Compared to arrays, linked lists offer more convenience in data addition and deletion but sacrifice data access speed.</li> <li>Graphs, compared to linked lists, provide richer logical information but require more memory space.</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#123-relationship-between-data-structures-and-algorithms","title":"1.2.3 \u00a0 Relationship Between Data Structures and Algorithms","text":"<p>As shown in the Figure 1-4 , data structures and algorithms are highly related and closely integrated, specifically in the following three aspects:</p> <ul> <li>Data structures are the foundation of algorithms. They provide structured data storage and methods for manipulating data for algorithms.</li> <li>Algorithms are the stage where data structures come into play. The data structure alone only stores data information; it is through the application of algorithms that specific problems can be solved.</li> <li>Algorithms can often be implemented based on different data structures, but their execution efficiency can vary greatly. Choosing the right data structure is key.</li> </ul> <p></p> <p> Figure 1-4 \u00a0 Relationship between data structures and algorithms </p> <p>Data structures and algorithms can be likened to a set of building blocks, as illustrated in the Figure 1-5 . A building block set includes numerous pieces, accompanied by detailed assembly instructions. Following these instructions step by step allows us to construct an intricate block model.</p> <p></p> <p> Figure 1-5 \u00a0 Assembling blocks </p> <p>The detailed correspondence between the two is shown in the Table 1-1 .</p> <p> Table 1-1 \u00a0 Comparing Data Structures and Algorithms to Building Blocks </p> Data Structures and Algorithms Building Blocks Input data Unassembled blocks Data structure Organization of blocks, including shape, size, connections, etc Algorithm A series of steps to assemble the blocks into the desired shape Output data Completed Block model <p>It's worth noting that data structures and algorithms are independent of programming languages. For this reason, this book is able to provide implementations in multiple programming languages.</p> <p>Conventional Abbreviation</p> <p>In real-life discussions, we often refer to \"Data Structures and Algorithms\" simply as \"Algorithms\". For example, the well-known LeetCode algorithm problems actually test both data structure and algorithm knowledge.</p>"},{"location":"chapter_preface/","title":"Chapter 0. \u00a0 Preface","text":"<p>Abstract</p> <p>Algorithms are like a beautiful symphony, with each line of code flowing like a rhythm.</p> <p>May this book ring softly in your mind, leaving a unique and profound melody.</p>"},{"location":"chapter_preface/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>0.1 \u00a0 About This Book</li> <li>0.2 \u00a0 How to Read</li> <li>0.3 \u00a0 Summary</li> </ul>"},{"location":"chapter_preface/about_the_book/","title":"0.1 \u00a0 About This Book","text":"<p>This open-source project aims to create a free, and beginner-friendly crash course on data structures and algorithms.</p> <ul> <li>Using animated illustrations, it delivers structured insights into data structures and algorithmic concepts, ensuring comprehensibility and a smooth learning curve.</li> <li>Run code with just one click, supporting Java, C++, Python, Go, JS, TS, C#, Swift, Rust, Dart, Zig and other languages.</li> <li>Readers are encouraged to engage with each other in the discussion area for each section, questions and comments are usually answered within two days.</li> </ul>"},{"location":"chapter_preface/about_the_book/#011-target-audience","title":"0.1.1 \u00a0 Target Audience","text":"<p>If you are new to algorithms with limited exposure, or you have accumulated some experience in algorithms, but you only have a vague understanding of data structures and algorithms, and you are constantly jumping between \"yep\" and \"hmm\", then this book is for you!</p> <p>If you have already accumulated a certain amount of problem-solving experience, and are familiar with most types of problems, then this book can help you review and organize your algorithm knowledge system. The repository's source code can be used as a \"problem-solving toolkit\" or an \"algorithm cheat sheet\".</p> <p>If you are an algorithm expert, we look forward to receiving your valuable suggestions, or join us and collaborate.</p> <p>Prerequisites</p> <p>You should know how to write and read simple code in at least one programming language.</p>"},{"location":"chapter_preface/about_the_book/#012-content-structure","title":"0.1.2 \u00a0 Content Structure","text":"<p>The main content of the book is shown in the following figure.</p> <ul> <li>Complexity Analysis: explores aspects and methods for evaluating data structures and algorithms. Covers methods of deriving time complexity and space complexity, along with common types and examples.</li> <li>Data Structures: focuses on fundamental data types, classification methods, definitions, pros and cons, common operations, types, applications, and implementation methods of data structures such as array, linked list, stack, queue, hash table, tree, heap, graph, etc.</li> <li>Algorithms: defines algorithms, discusses their pros and cons, efficiency, application scenarios, problem-solving steps, and includes sample questions for various algorithms such as search, sorting, divide and conquer, backtracking, dynamic programming, greedy algorithms, and more.</li> </ul> <p></p> <p> Figure 0-1 \u00a0 Main Content of the Book </p>"},{"location":"chapter_preface/about_the_book/#013-acknowledgements","title":"0.1.3 \u00a0 Acknowledgements","text":"<p>This book is continuously improved with the joint efforts of many contributors from the open-source community. Thanks to each writer who invested their time and energy, listed in the order generated by GitHub: krahets, codingonion, nuomi1, Gonglja, Reanon, justin-tse, danielsss, hpstory, S-N-O-R-L-A-X, night-cruise, msk397, gvenusleo, RiverTwilight, gyt95, zhuoqinyue, Zuoxun, Xia-Sang, mingXta, FangYuan33, GN-Yu, IsChristina, xBLACKICEx, guowei-gong, Cathay-Chen, mgisr, JoseHung, qualifier1024, pengchzn, Guanngxu, longsizhuo, L-Super, what-is-me, yuan0221, lhxsm, Slone123c, WSL0809, longranger2, theNefelibatas, xiongsp, JeffersonHuang, hongyun-robot, K3v123, yuelinxin, a16su, gaofer, malone6, Wonderdch, xjr7670, DullSword, Horbin-Magician, NI-SW, reeswell, XC-Zero, XiaChuerwu, yd-j, iron-irax, huawuque404, MolDuM, Nigh, KorsChen, foursevenlove, 52coder, bubble9um, youshaoXG, curly210102, gltianwen, fanchenggang, Transmigration-zhou, FloranceYeh, FreddieLi, ShiMaRing, lipusheng, Javesun99, JackYang-hellobobo, shanghai-Jerry, 0130w, Keynman, psychelzh, logan-qiu, ZnYang2018, MwumLi, 1ch0, Phoenix0415, qingpeng9802, Richard-Zhang1019, QiLOL, Suremotoo, Turing-1024-Lee, Evilrabbit520, GaochaoZhu, ZJKung, linzeyan, hezhizhen, ZongYangL, beintentional, czruby, coderlef, dshlstarr, szu17dmy, fbigm, gledfish, hts0000, boloboloda, iStig, jiaxianhua, wenjianmin, keshida, kilikilikid, lclc6, lwbaptx, liuxjerry, lucaswangdev, lyl625760, chadyi, noobcodemaker, selear, siqyka, syd168, 4yDX3906, tao363, wangwang105, weibk, yabo083, yi427, yishangzhang, zhouLion, baagod, ElaBosak233, xb534, luluxia, yanedie, thomasq0, YangXuanyi and th1nk3r-ing.</p> <p>The code review work for this book was completed by codingonion, Gonglja, gvenusleo, hpstory, justin\u2010tse, krahets, night-cruise, nuomi1, and Reanon (listed in alphabetical order). Thanks to them for their time and effort, ensuring the standardization and uniformity of the code in various languages.</p> <p>Throughout the creation of this book, numerous individuals provided invaluable assistance, including but not limited to:</p> <ul> <li>Thanks to my mentor at the company, Dr. Xi Li, who encouraged me in a conversation to \"get moving fast,\" which solidified my determination to write this book;</li> <li>Thanks to my girlfriend Bubble, as the first reader of this book, for offering many valuable suggestions from the perspective of a beginner in algorithms, making this book more suitable for newbies;</li> <li>Thanks to Tengbao, Qibao, and Feibao for coming up with a creative name for this book, evoking everyone's fond memories of writing their first line of code \"Hello World!\";</li> <li>Thanks to Xiaoquan for providing professional help in intellectual property, which has played a significant role in the development of this open-source book;</li> <li>Thanks to Sutong for designing a beautiful cover and logo for this book, and for patiently making multiple revisions under my insistence;</li> <li>Thanks to @squidfunk for providing writing and typesetting suggestions, as well as his developed open-source documentation theme Material-for-MkDocs.</li> </ul> <p>Throughout the writing journey, I delved into numerous textbooks and articles on data structures and algorithms. These works served as exemplary models, ensuring the accuracy and quality of this book's content. I extend my gratitude to all who preceded me for their invaluable contributions!</p> <p>This book advocates a combination of hands-on and minds-on learning, inspired in this regard by \"Dive into Deep Learning\". I highly recommend this excellent book to all readers.</p> <p>Heartfelt thanks to my parents, whose ongoing support and encouragement have allowed me to do this interesting work.</p>"},{"location":"chapter_preface/suggestions/","title":"0.2 \u00a0 How to Read","text":"<p>Tip</p> <p>For the best reading experience, it is recommended that you read through this section.</p>"},{"location":"chapter_preface/suggestions/#021-writing-conventions","title":"0.2.1 \u00a0 Writing Conventions","text":"<ul> <li>Chapters marked with '*' after the title are optional and contain relatively challenging content. If you are short on time, it is advisable to skip them.</li> <li>Technical terms will be in boldface (in the print and PDF versions) or underlined (in the web version), for instance, array. It's advisable to familiarize yourself with these for better comprehension of technical texts.</li> <li>Bolded text indicates key content or summary statements, which deserve special attention.</li> <li>Words and phrases with specific meanings are indicated with \u201cquotation marks\u201d to avoid ambiguity.</li> <li>When it comes to terms that are inconsistent between programming languages, this book follows Python, for example using <code>None</code> to mean <code>null</code>.</li> <li>This book partially ignores the comment conventions for programming languages in exchange for a more compact layout of the content. The comments primarily consist of three types: title comments, content comments, and multi-line comments.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig <pre><code>\"\"\"Header comments for labeling functions, classes, test samples, etc\"\"\"\"\n\n# Comments for explaining details\n\n\"\"\"\nMultiline\ncomments\n\"\"\"\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>/* Header comments for labeling functions, classes, test samples, etc */\n\n// Comments for explaining details.\n\n/**\n * Multiline\n * comments\n */\n</code></pre> <pre><code>// Header comments for labeling functions, classes, test samples, etc\n\n// Comments for explaining details.\n\n// Multiline\n// comments\n</code></pre>"},{"location":"chapter_preface/suggestions/#022-efficient-learning-via-animated-illustrations","title":"0.2.2 \u00a0 Efficient Learning via Animated Illustrations","text":"<p>Compared with text, videos and pictures have a higher density of information and are more structured, making them easier to understand. In this book, key and difficult concepts are mainly presented through animations and illustrations, with text serving as explanations and supplements.</p> <p>When encountering content with animations or illustrations as shown in the Figure 0-2 , prioritize understanding the figure, with text as supplementary, integrating both for a comprehensive understanding.</p> <p></p> <p> Figure 0-2 \u00a0 Animated Illustration Example </p>"},{"location":"chapter_preface/suggestions/#023-deepen-understanding-through-coding-practice","title":"0.2.3 \u00a0 Deepen Understanding through Coding Practice","text":"<p>The source code of this book is hosted on the GitHub Repository. As shown in the Figure 0-3 , the source code comes with test examples and can be executed with just a single click.</p> <p>If time permits, it's recommended to type out the code yourself. If pressed for time, at least read and run all the codes.</p> <p>Compared to just reading code, writing code often yields more learning. Learning by doing is the real way to learn.</p> <p></p> <p> Figure 0-3 \u00a0 Running Code Example </p> <p>Setting up to run the code involves three main steps.</p> <p>Step 1: Install a local programming environment. Follow the tutorial in the appendix for installation, or skip this step if already installed.</p> <p>Step 2: Clone or download the code repository. Visit the GitHub Repository.</p> <p>If Git is installed, use the following command to clone the repository:</p> <pre><code>git clone https://github.com/krahets/hello-algo.git\n</code></pre> <p>Alternatively, you can also click the \"Download ZIP\" button at the location shown in the Figure 0-4  to directly download the code as a compressed ZIP file. Then, you can simply extract it locally.</p> <p></p> <p> Figure 0-4 \u00a0 Cloning Repository and Downloading Code </p> <p>Step 3: Run the source code. As shown in the Figure 0-5 , for the code block labeled with the file name at the top, we can find the corresponding source code file in the <code>codes</code> folder of the repository. These files can be executed with a single click, which will help you save unnecessary debugging time and allow you to focus on learning.</p> <p></p> <p> Figure 0-5 \u00a0 Code Block and Corresponding Source Code File </p>"},{"location":"chapter_preface/suggestions/#024-learning-together-in-discussion","title":"0.2.4 \u00a0 Learning Together in Discussion","text":"<p>While reading this book, please don't skip over the points that you didn't learn. Feel free to post your questions in the comment section. We will be happy to answer them and can usually respond within two days.</p> <p>As illustrated in the Figure 0-6 , each chapter features a comment section at the bottom. I encourage you to pay attention to these comments. They not only expose you to others' encountered problems, aiding in identifying knowledge gaps and sparking deeper contemplation, but also invite you to generously contribute by answering fellow readers' inquiries, sharing insights, and fostering mutual improvement.</p> <p></p> <p> Figure 0-6 \u00a0 Comment Section Example </p>"},{"location":"chapter_preface/suggestions/#025-algorithm-learning-path","title":"0.2.5 \u00a0 Algorithm Learning Path","text":"<p>Overall, the journey of mastering data structures and algorithms can be divided into three stages:</p> <ol> <li>Stage 1: Introduction to algorithms. We need to familiarize ourselves with the characteristics and usage of various data structures and learn about the principles, processes, uses, and efficiency of different algorithms.</li> <li>Stage 2: Practicing algorithm problems. It is recommended to start from popular problems, such as Sword for Offer and LeetCode Hot 100, and accumulate at least 100 questions to familiarize yourself with mainstream algorithmic problems. Forgetfulness can be a challenge when you start practicing, but rest assured that this is normal. We can follow the \"Ebbinghaus Forgetting Curve\" to review the questions, and usually after 3~5 rounds of repetitions, we will be able to memorize them.</li> <li>Stage 3: Building the knowledge system. In terms of learning, we can read algorithm column articles, solution frameworks, and algorithm textbooks to continuously enrich the knowledge system. In terms of practicing, we can try advanced strategies, such as categorizing by topic, multiple solutions for a single problem,  and one solution for multiple problems, etc. Insights on these strategies can be found in various communities.</li> </ol> <p>As shown in the Figure 0-7 , this book mainly covers \u201cStage 1,\u201d aiming to help you more efficiently embark on Stages 2 and 3.</p> <p></p> <p> Figure 0-7 \u00a0 Algorithm Learning Path </p>"},{"location":"chapter_preface/summary/","title":"0.3 \u00a0 Summary","text":"<ul> <li>The main audience of this book is beginners in algorithm. If you already have some basic knowledge, this book can help you systematically review your algorithm knowledge, and the source code in this book can also be used as a \"Coding Toolkit\".</li> <li>The book consists of three main sections, Complexity Analysis, Data Structures, and Algorithms, covering most of the topics in the field.</li> <li>For newcomers to algorithms, it is crucial to read an introductory book in the beginning stages to avoid many detours or common pitfalls.</li> <li>Animations and figures within the book are usually used to introduce key points and difficult knowledge. These should be given more attention when reading the book.</li> <li>Practice is the best way to learn programming. It is highly recommended that you run the source code and type in the code yourself.</li> <li>Each chapter in the web version of this book features a discussion section, and you are welcome to share your questions and insights at any time.</li> </ul>"},{"location":"chapter_stack_and_queue/","title":"Chapter 5. \u00a0 Stack and Queue","text":"<p>Abstract</p> <p>A stack is like cats placed on top of each other, while a queue is like cats lined up one by one.</p> <p>They represent the logical relationships of Last-In-First-Out (LIFO) and First-In-First-Out (FIFO), respectively.</p>"},{"location":"chapter_stack_and_queue/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>5.1 \u00a0 Stack</li> <li>5.2 \u00a0 Queue</li> <li>5.3 \u00a0 Double-ended Queue</li> <li>5.4 \u00a0 Summary</li> </ul>"},{"location":"chapter_stack_and_queue/deque/","title":"5.3 \u00a0 Double-Ended Queue","text":"<p>In a queue, we can only delete elements from the head or add elements to the tail. As shown in the following diagram, a \"double-ended queue (deque)\" offers more flexibility, allowing the addition or removal of elements at both the head and the tail.</p> <p></p> <p> Figure 5-7 \u00a0 Operations in Double-Ended Queue </p>"},{"location":"chapter_stack_and_queue/deque/#531-common-operations-in-double-ended-queue","title":"5.3.1 \u00a0 Common Operations in Double-Ended Queue","text":"<p>The common operations in a double-ended queue are listed below, and the names of specific methods depend on the programming language used.</p> <p> Table 5-3 \u00a0 Efficiency of Double-Ended Queue Operations </p> Method Name Description Time Complexity <code>pushFirst()</code> Add an element to the head \\(O(1)\\) <code>pushLast()</code> Add an element to the tail \\(O(1)\\) <code>popFirst()</code> Remove the first element \\(O(1)\\) <code>popLast()</code> Remove the last element \\(O(1)\\) <code>peekFirst()</code> Access the first element \\(O(1)\\) <code>peekLast()</code> Access the last element \\(O(1)\\) <p>Similarly, we can directly use the double-ended queue classes implemented in programming languages:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig deque.py<pre><code>from collections import deque\n\n# Initialize the deque\ndeque: deque[int] = deque()\n\n# Enqueue elements\ndeque.append(2)      # Add to the tail\ndeque.append(5)\ndeque.append(4)\ndeque.appendleft(3)  # Add to the head\ndeque.appendleft(1)\n\n# Access elements\nfront: int = deque[0]  # The first element\nrear: int = deque[-1]  # The last element\n\n# Dequeue elements\npop_front: int = deque.popleft()  # The first element dequeued\npop_rear: int = deque.pop()       # The last element dequeued\n\n# Get the length of the deque\nsize: int = len(deque)\n\n# Check if the deque is empty\nis_empty: bool = len(deque) == 0\n</code></pre> deque.cpp<pre><code>/* Initialize the deque */\ndeque&lt;int&gt; deque;\n\n/* Enqueue elements */\ndeque.push_back(2);   // Add to the tail\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // Add to the head\ndeque.push_front(1);\n\n/* Access elements */\nint front = deque.front(); // The first element\nint back = deque.back();   // The last element\n\n/* Dequeue elements */\ndeque.pop_front();  // The first element dequeued\ndeque.pop_back();   // The last element dequeued\n\n/* Get the length of the deque */\nint size = deque.size();\n\n/* Check if the deque is empty */\nbool empty = deque.empty();\n</code></pre> deque.java<pre><code>/* Initialize the deque */\nDeque&lt;Integer&gt; deque = new LinkedList&lt;&gt;();\n\n/* Enqueue elements */\ndeque.offerLast(2);   // Add to the tail\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // Add to the head\ndeque.offerFirst(1);\n\n/* Access elements */\nint peekFirst = deque.peekFirst();  // The first element\nint peekLast = deque.peekLast();    // The last element\n\n/* Dequeue elements */\nint popFirst = deque.pollFirst();  // The first element dequeued\nint popLast = deque.pollLast();    // The last element dequeued\n\n/* Get the length of the deque */\nint size = deque.size();\n\n/* Check if the deque is empty */\nboolean isEmpty = deque.isEmpty();\n</code></pre> deque.cs<pre><code>/* Initialize the deque */\n// In C#, LinkedList is used as a deque\nLinkedList&lt;int&gt; deque = new();\n\n/* Enqueue elements */\ndeque.AddLast(2);   // Add to the tail\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // Add to the head\ndeque.AddFirst(1);\n\n/* Access elements */\nint peekFirst = deque.First.Value;  // The first element\nint peekLast = deque.Last.Value;    // The last element\n\n/* Dequeue elements */\ndeque.RemoveFirst();  // The first element dequeued\ndeque.RemoveLast();   // The last element dequeued\n\n/* Get the length of the deque */\nint size = deque.Count;\n\n/* Check if the deque is empty */\nbool isEmpty = deque.Count == 0;\n</code></pre> deque_test.go<pre><code>/* Initialize the deque */\n// In Go, use list as a deque\ndeque := list.New()\n\n/* Enqueue elements */\ndeque.PushBack(2)      // Add to the tail\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // Add to the head\ndeque.PushFront(1)\n\n/* Access elements */\nfront := deque.Front() // The first element\nrear := deque.Back()   // The last element\n\n/* Dequeue elements */\ndeque.Remove(front)    // The first element dequeued\ndeque.Remove(rear)     // The last element dequeued\n\n/* Get the length of the deque */\nsize := deque.Len()\n\n/* Check if the deque is empty */\nisEmpty := deque.Len() == 0\n</code></pre> deque.swift<pre><code>/* Initialize the deque */\n// Swift does not have a built-in deque class, so Array can be used as a deque\nvar deque: [Int] = []\n\n/* Enqueue elements */\ndeque.append(2) // Add to the tail\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // Add to the head\ndeque.insert(1, at: 0)\n\n/* Access elements */\nlet peekFirst = deque.first! // The first element\nlet peekLast = deque.last!   // The last element\n\n/* Dequeue elements */\n// Using Array, popFirst has a complexity of O(n)\nlet popFirst = deque.removeFirst() // The first element dequeued\nlet popLast = deque.removeLast()   // The last element dequeued\n\n/* Get the length of the deque */\nlet size = deque.count\n\n/* Check if the deque is empty */\nlet isEmpty = deque.isEmpty\n</code></pre> deque.js<pre><code>/* Initialize the deque */\n// JavaScript does not have a built-in deque, so Array is used as a deque\nconst deque = [];\n\n/* Enqueue elements */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// Note that unshift() has a time complexity of O(n) as it's an array\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* Access elements */\nconst peekFirst = deque[0]; // The first element\nconst peekLast = deque[deque.length - 1]; // The last element\n\n/* Dequeue elements */\n// Note that shift() has a time complexity of O(n) as it's an array\nconst popFront = deque.shift(); // The first element dequeued\nconst popBack = deque.pop();    // The last element dequeued\n\n/* Get the length of the deque */\nconst size = deque.length;\n\n/* Check if the deque is empty */\nconst isEmpty = size === 0;\n</code></pre> deque.ts<pre><code>/* Initialize the deque */\n// TypeScript does not have a built-in deque, so Array is used as a deque\nconst deque: number[] = [];\n\n/* Enqueue elements */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// Note that unshift() has a time complexity of O(n) as it's an array\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* Access elements */\nconst peekFirst: number = deque[0]; // The first element\nconst peekLast: number = deque[deque.length - 1]; // The last element\n\n/* Dequeue elements */\n// Note that shift() has a time complexity of O(n) as it's an array\nconst popFront: number = deque.shift() as number; // The first element dequeued\nconst popBack: number = deque.pop() as number;    // The last element dequeued\n\n/* Get the length of the deque */\nconst size: number = deque.length;\n\n/* Check if the deque is empty */\nconst isEmpty: boolean = size === 0;\n</code></pre> deque.dart<pre><code>/* Initialize the deque */\n// In Dart, Queue is defined as a deque\nQueue&lt;int&gt; deque = Queue&lt;int&gt;();\n\n/* Enqueue elements */\ndeque.addLast(2);  // Add to the tail\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // Add to the head\ndeque.addFirst(1);\n\n/* Access elements */\nint peekFirst = deque.first; // The first element\nint peekLast = deque.last;   // The last element\n\n/* Dequeue elements */\nint popFirst = deque.removeFirst(); // The first element dequeued\nint popLast = deque.removeLast();   // The last element dequeued\n\n/* Get the length of the deque */\nint size = deque.length;\n\n/* Check if the deque is empty */\nbool isEmpty = deque.isEmpty;\n</code></pre> deque.rs<pre><code>/* Initialize the deque */\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* Enqueue elements */\ndeque.push_back(2);  // Add to the tail\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // Add to the head\ndeque.push_front(1);\n\n/* Access elements */\nif let Some(front) = deque.front() { // The first element\n}\nif let Some(rear) = deque.back() {   // The last element\n}\n\n/* Dequeue elements */\nif let Some(pop_front) = deque.pop_front() { // The first element dequeued\n}\nif let Some(pop_rear) = deque.pop_back() {   // The last element dequeued\n}\n\n/* Get the length of the deque */\nlet size = deque.len();\n\n/* Check if the deque is empty */\nlet is_empty = deque.is_empty();\n</code></pre> deque.c<pre><code>// C does not provide a built-in deque\n</code></pre> deque.kt<pre><code>\n</code></pre> deque.zig<pre><code>\n</code></pre> Visualizing Code <p>https://pythontutor.com/render.html#code=from%20collections%20import%20deque%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%0A%20%20%20%20deq%20%3D%20deque%28%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%85%A5%E9%98%9F%0A%20%20%20%20deq.append%282%29%20%20%23%20%E6%B7%BB%E5%8A%A0%E8%87%B3%E9%98%9F%E5%B0%BE%0A%20%20%20%20deq.append%285%29%0A%20%20%20%20deq.append%284%29%0A%20%20%20%20deq.appendleft%283%29%20%20%23%20%E6%B7%BB%E5%8A%A0%E8%87%B3%E9%98%9F%E9%A6%96%0A%20%20%20%20deq.appendleft%281%29%0A%20%20%20%20print%28%22%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%20deque%20%3D%22,%20deq%29%0A%0A%20%20%20%20%23%20%E8%AE%BF%E9%97%AE%E5%85%83%E7%B4%A0%0A%20%20%20%20front%20%3D%20deq%5B0%5D%20%20%23%20%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%20front%20%3D%22,%20front%29%0A%20%20%20%20rear%20%3D%20deq%5B-1%5D%20%20%23%20%E9%98%9F%E5%B0%BE%E5%85%83%E7%B4%A0%0A%20%20%20%20print%28%22%E9%98%9F%E5%B0%BE%E5%85%83%E7%B4%A0%20rear%20%3D%22,%20rear%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20pop_front%20%3D%20deq.popleft%28%29%20%20%23%20%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%87%BA%E9%98%9F%E5%85%83%E7%B4%A0%20%20pop_front%20%3D%22,%20pop_front%29%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%87%BA%E9%98%9F%E5%90%8E%20deque%20%3D%22,%20deq%29%0A%20%20%20%20pop_rear%20%3D%20deq.pop%28%29%20%20%23%20%E9%98%9F%E5%B0%BE%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20print%28%22%E9%98%9F%E5%B0%BE%E5%87%BA%E9%98%9F%E5%85%83%E7%B4%A0%20%20pop_rear%20%3D%22,%20pop_rear%29%0A%20%20%20%20print%28%22%E9%98%9F%E5%B0%BE%E5%87%BA%E9%98%9F%E5%90%8E%20deque%20%3D%22,%20deq%29%0A%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E7%9A%84%E9%95%BF%E5%BA%A6%0A%20%20%20%20size%20%3D%20len%28deq%29%0A%20%20%20%20print%28%22%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E9%95%BF%E5%BA%A6%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20%E5%88%A4%E6%96%AD%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%0A%20%20%20%20is_empty%20%3D%20len%28deq%29%20%3D%3D%200%0A%20%20%20%20print%28%22%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%20%3D%22,%20is_empty%29&amp;cumulative=false&amp;curInstr=3&amp;heapPrimitives=nevernest&amp;mode=display&amp;origin=opt-frontend.js&amp;py=311&amp;rawInputLstJSON=%5B%5D&amp;textReferences=false</p>"},{"location":"chapter_stack_and_queue/deque/#532-implementing-a-double-ended-queue","title":"5.3.2 \u00a0 Implementing a Double-Ended Queue *","text":"<p>The implementation of a double-ended queue is similar to that of a regular queue, it can be based on either a linked list or an array as the underlying data structure.</p>"},{"location":"chapter_stack_and_queue/deque/#1-implementation-based-on-doubly-linked-list","title":"1. \u00a0 Implementation Based on Doubly Linked List","text":"<p>Recall from the previous section that we used a regular singly linked list to implement a queue, as it conveniently allows for deleting from the head (corresponding to the dequeue operation) and adding new elements after the tail (corresponding to the enqueue operation).</p> <p>For a double-ended queue, both the head and the tail can perform enqueue and dequeue operations. In other words, a double-ended queue needs to implement operations in the opposite direction as well. For this, we use a \"doubly linked list\" as the underlying data structure of the double-ended queue.</p> <p>As shown in the Figure 5-8 , we treat the head and tail nodes of the doubly linked list as the front and rear of the double-ended queue, respectively, and implement the functionality to add and remove nodes at both ends.</p> LinkedListDequepushLast()pushFirst()popLast()popFirst() <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 5-8 \u00a0 Implementing Double-Ended Queue with Doubly Linked List for Enqueue and Dequeue Operations </p> <p>The implementation code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_deque.py<pre><code>class ListNode:\n    \"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\"\"\"\n\n    def __init__(self, val: int):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self.val: int = val\n        self.next: ListNode | None = None  # \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n        self.prev: ListNode | None = None  # \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\nclass LinkedListDeque:\n    \"\"\"\u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._front: ListNode | None = None  # \u5934\u8282\u70b9 front\n        self._rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\n        self._size: int = 0  # \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self.size() == 0\n\n    def push(self, num: int, is_front: bool):\n        \"\"\"\u5165\u961f\u64cd\u4f5c\"\"\"\n        node = ListNode(num)\n        # \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if self.is_empty():\n            self._front = self._rear = node\n        # \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        elif is_front:\n            # \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            self._front.prev = node\n            node.next = self._front\n            self._front = node  # \u66f4\u65b0\u5934\u8282\u70b9\n        # \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else:\n            # \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            self._rear.next = node\n            node.prev = self._rear\n            self._rear = node  # \u66f4\u65b0\u5c3e\u8282\u70b9\n        self._size += 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n\n    def push_first(self, num: int):\n        \"\"\"\u961f\u9996\u5165\u961f\"\"\"\n        self.push(num, True)\n\n    def push_last(self, num: int):\n        \"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\n        self.push(num, False)\n\n    def pop(self, is_front: bool) -&gt; int:\n        \"\"\"\u51fa\u961f\u64cd\u4f5c\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        # \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if is_front:\n            val: int = self._front.val  # \u6682\u5b58\u5934\u8282\u70b9\u503c\n            # \u5220\u9664\u5934\u8282\u70b9\n            fnext: ListNode | None = self._front.next\n            if fnext != None:\n                fnext.prev = None\n                self._front.next = None\n            self._front = fnext  # \u66f4\u65b0\u5934\u8282\u70b9\n        # \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else:\n            val: int = self._rear.val  # \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            # \u5220\u9664\u5c3e\u8282\u70b9\n            rprev: ListNode | None = self._rear.prev\n            if rprev != None:\n                rprev.next = None\n                self._rear.prev = None\n            self._rear = rprev  # \u66f4\u65b0\u5c3e\u8282\u70b9\n        self._size -= 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val\n\n    def pop_first(self) -&gt; int:\n        \"\"\"\u961f\u9996\u51fa\u961f\"\"\"\n        return self.pop(True)\n\n    def pop_last(self) -&gt; int:\n        \"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\n        return self.pop(False)\n\n    def peek_first(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        return self._front.val\n\n    def peek_last(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        return self._rear.val\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n        node = self._front\n        res = [0] * self.size()\n        for i in range(self.size()):\n            res[i] = node.val\n            node = node.next\n        return res\n</code></pre> linkedlist_deque.cpp<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nstruct DoublyListNode {\n    int val;              // \u8282\u70b9\u503c\n    DoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n    DoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\u6307\u9488\n    DoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n    }\n};\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n  private:\n    DoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    int queSize = 0;              // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    LinkedListDeque() : front(nullptr), rear(nullptr) {\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~LinkedListDeque() {\n        // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n        DoublyListNode *pre, *cur = front;\n        while (cur != nullptr) {\n            pre = cur;\n            cur = cur-&gt;next;\n            delete pre;\n        }\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    void push(int num, bool isFront) {\n        DoublyListNode *node = new DoublyListNode(num);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty())\n            front = rear = node;\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front-&gt;prev = node;\n            node-&gt;next = front;\n            front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear-&gt;next = node;\n            node-&gt;prev = rear;\n            rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    int pop(bool isFront) {\n        if (isEmpty())\n            throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n        int val;\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            val = front-&gt;val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            DoublyListNode *fNext = front-&gt;next;\n            if (fNext != nullptr) {\n                fNext-&gt;prev = nullptr;\n                front-&gt;next = nullptr;\n            }\n            delete front;\n            front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        } else {\n            val = rear-&gt;val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            DoublyListNode *rPrev = rear-&gt;prev;\n            if (rPrev != nullptr) {\n                rPrev-&gt;next = nullptr;\n                rear-&gt;prev = nullptr;\n            }\n            delete rear;\n            rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    int popFirst() {\n        return pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    int popLast() {\n        return pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        return front-&gt;val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        return rear-&gt;val;\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    vector&lt;int&gt; toVector() {\n        DoublyListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_deque.java<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    int val; // \u8282\u70b9\u503c\n    ListNode next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    ListNode prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n    ListNode(int val) {\n        this.val = val;\n        prev = next = null;\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private int queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    public LinkedListDeque() {\n        front = rear = null;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    private void push(int num, boolean isFront) {\n        ListNode node = new ListNode(num);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty())\n            front = rear = node;\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front.prev = node;\n            node.next = front;\n            front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear.next = node;\n            node.prev = rear;\n            rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    private int pop(boolean isFront) {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        int val;\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            val = front.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            ListNode fNext = front.next;\n            if (fNext != null) {\n                fNext.prev = null;\n                front.next = null;\n            }\n            front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        } else {\n            val = rear.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            ListNode rPrev = rear.prev;\n            if (rPrev != null) {\n                rPrev.next = null;\n                rear.prev = null;\n            }\n            rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int popFirst() {\n        return pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int popLast() {\n        return pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return rear.val;\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.cs<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode(int val) {\n    public int val = val;       // \u8282\u70b9\u503c\n    public ListNode? next = null; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    public ListNode? prev = null; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    ListNode? front, rear; // \u5934\u8282\u70b9 front, \u5c3e\u8282\u70b9 rear\n    int queSize = 0;      // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    public LinkedListDeque() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    void Push(int num, bool isFront) {\n        ListNode node = new(num);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (IsEmpty()) {\n            front = node;\n            rear = node;\n        }\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front!.prev = node;\n            node.next = front;\n            front = node; // \u66f4\u65b0\u5934\u8282\u70b9                           \n        }\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear!.next = node;\n            node.prev = rear;\n            rear = node;  // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n\n        queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void PushFirst(int num) {\n        Push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void PushLast(int num) {\n        Push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    int? Pop(bool isFront) {\n        if (IsEmpty())\n            throw new Exception();\n        int? val;\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            val = front?.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            ListNode? fNext = front?.next;\n            if (fNext != null) {\n                fNext.prev = null;\n                front!.next = null;\n            }\n            front = fNext;   // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else {\n            val = rear?.val;  // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            ListNode? rPrev = rear?.prev;\n            if (rPrev != null) {\n                rPrev.next = null;\n                rear!.prev = null;\n            }\n            rear = rPrev;    // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n\n        queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int? PopFirst() {\n        return Pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int? PopLast() {\n        return Pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int? PeekFirst() {\n        if (IsEmpty())\n            throw new Exception();\n        return front?.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int? PeekLast() {\n        if (IsEmpty())\n            throw new Exception();\n        return rear?.val;\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int?[] ToArray() {\n        ListNode? node = front;\n        int?[] res = new int?[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node?.val;\n            node = node?.next;\n        }\n\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.go<pre><code>/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype linkedListDeque struct {\n    // \u4f7f\u7528\u5185\u7f6e\u5305 list\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u53cc\u7aef\u961f\u5217 */\nfunc newLinkedListDeque() *linkedListDeque {\n    return &amp;linkedListDeque{\n        data: list.New(),\n    }\n}\n\n/* \u961f\u9996\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushFirst(value any) {\n    s.data.PushFront(value)\n}\n\n/* \u961f\u5c3e\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushLast(value any) {\n    s.data.PushBack(value)\n}\n\n/* \u961f\u9996\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u961f\u5c3e\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListDeque) peekFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (s *linkedListDeque) peekLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListDeque) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListDeque) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListDeque) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_deque.swift<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    var val: Int // \u8282\u70b9\u503c\n    var next: ListNode? // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    weak var prev: ListNode? // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n    init(val: Int) {\n        self.val = val\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private var front: ListNode? // \u5934\u8282\u70b9 front\n    private var rear: ListNode? // \u5c3e\u8282\u70b9 rear\n    private var _size: Int // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    init() {\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    private func push(num: Int, isFront: Bool) {\n        let node = ListNode(val: num)\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if isEmpty() {\n            front = node\n            rear = node\n        }\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if isFront {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front?.prev = node\n            node.next = front\n            front = node // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear?.next = node\n            node.prev = rear\n            rear = node // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        _size += 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    func pushFirst(num: Int) {\n        push(num: num, isFront: true)\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    func pushLast(num: Int) {\n        push(num: num, isFront: false)\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    private func pop(isFront: Bool) -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        let val: Int\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if isFront {\n            val = front!.val // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            let fNext = front?.next\n            if fNext != nil {\n                fNext?.prev = nil\n                front?.next = nil\n            }\n            front = fNext // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else {\n            val = rear!.val // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            let rPrev = rear?.prev\n            if rPrev != nil {\n                rPrev?.next = nil\n                rear?.prev = nil\n            }\n            rear = rPrev // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        _size -= 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    func popFirst() -&gt; Int {\n        pop(isFront: true)\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    func popLast() -&gt; Int {\n        pop(isFront: false)\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return front!.val\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return rear!.val\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.js<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    val; // \u8282\u70b9\u503c\n\n    constructor(val) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    #front; // \u5934\u8282\u70b9 front\n    #rear; // \u5c3e\u8282\u70b9 rear\n    #queSize; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n        this.#queSize = 0;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\n    pushLast(val) {\n        const node = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            this.#rear.next = node;\n            node.prev = this.#rear;\n            this.#rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        this.#queSize++;\n    }\n\n    /* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\n    pushFirst(val) {\n        const node = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            this.#front.prev = node;\n            node.next = this.#front;\n            this.#front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        this.#queSize++;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\n    popLast() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5c3e\u8282\u70b9\n        let temp = this.#rear.prev;\n        if (temp !== null) {\n            temp.next = null;\n            this.#rear.prev = null;\n        }\n        this.#rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        this.#queSize--;\n        return value;\n    }\n\n    /* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\n    popFirst() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5934\u8282\u70b9\n        let temp = this.#front.next;\n        if (temp !== null) {\n            temp.prev = null;\n            this.#front.next = null;\n        }\n        this.#front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\n        this.#queSize--;\n        return value;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast() {\n        return this.#queSize === 0 ? null : this.#rear.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst() {\n        return this.#queSize === 0 ? null : this.#front.val;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u6253\u5370\u53cc\u5411\u961f\u5217 */\n    print() {\n        const arr = [];\n        let temp = this.#front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.ts<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    prev: ListNode; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    next: ListNode; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    val: number; // \u8282\u70b9\u503c\n\n    constructor(val: number) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private front: ListNode; // \u5934\u8282\u70b9 front\n    private rear: ListNode; // \u5c3e\u8282\u70b9 rear\n    private queSize: number; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n        this.queSize = 0;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\n    pushLast(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            this.rear.next = node;\n            node.prev = this.rear;\n            this.rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        this.queSize++;\n    }\n\n    /* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\n    pushFirst(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            this.front.prev = node;\n            node.next = this.front;\n            this.front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        this.queSize++;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\n    popLast(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5c3e\u8282\u70b9\n        let temp: ListNode = this.rear.prev;\n        if (temp !== null) {\n            temp.next = null;\n            this.rear.prev = null;\n        }\n        this.rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        this.queSize--;\n        return value;\n    }\n\n    /* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\n    popFirst(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5934\u8282\u70b9\n        let temp: ListNode = this.front.next;\n        if (temp !== null) {\n            temp.prev = null;\n            this.front.next = null;\n        }\n        this.front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\n        this.queSize--;\n        return value;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast(): number {\n        return this.queSize === 0 ? null : this.rear.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst(): number {\n        return this.queSize === 0 ? null : this.front.val;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u6253\u5370\u53cc\u5411\u961f\u5217 */\n    print(): void {\n        const arr: number[] = [];\n        let temp: ListNode = this.front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.dart<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n  int val; // \u8282\u70b9\u503c\n  ListNode? next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n  ListNode? prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n  ListNode(this.val, {this.next, this.prev});\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u5bf9\u5217 */\nclass LinkedListDeque {\n  late ListNode? _front; // \u5934\u8282\u70b9 _front\n  late ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\n  int _queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n  LinkedListDeque() {\n    this._front = null;\n    this._rear = null;\n  }\n\n  /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u957f\u5ea6 */\n  int size() {\n    return this._queSize;\n  }\n\n  /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return size() == 0;\n  }\n\n  /* \u5165\u961f\u64cd\u4f5c */\n  void push(int _num, bool isFront) {\n    final ListNode node = ListNode(_num);\n    if (isEmpty()) {\n      // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 _front \u548c _rear \u90fd\u6307\u5411 node\n      _front = _rear = node;\n    } else if (isFront) {\n      // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n      // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n      _front!.prev = node;\n      node.next = _front;\n      _front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n    } else {\n      // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n      // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n      _rear!.next = node;\n      node.prev = _rear;\n      _rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n    }\n    _queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n  }\n\n  /* \u961f\u9996\u5165\u961f */\n  void pushFirst(int _num) {\n    push(_num, true);\n  }\n\n  /* \u961f\u5c3e\u5165\u961f */\n  void pushLast(int _num) {\n    push(_num, false);\n  }\n\n  /* \u51fa\u961f\u64cd\u4f5c */\n  int? pop(bool isFront) {\n    // \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\n    if (isEmpty()) {\n      return null;\n    }\n    final int val;\n    if (isFront) {\n      // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n      val = _front!.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n      // \u5220\u9664\u5934\u8282\u70b9\n      ListNode? fNext = _front!.next;\n      if (fNext != null) {\n        fNext.prev = null;\n        _front!.next = null;\n      }\n      _front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n    } else {\n      // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n      val = _rear!.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n      // \u5220\u9664\u5c3e\u8282\u70b9\n      ListNode? rPrev = _rear!.prev;\n      if (rPrev != null) {\n        rPrev.next = null;\n        _rear!.prev = null;\n      }\n      _rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n    }\n    _queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    return val;\n  }\n\n  /* \u961f\u9996\u51fa\u961f */\n  int? popFirst() {\n    return pop(true);\n  }\n\n  /* \u961f\u5c3e\u51fa\u961f */\n  int? popLast() {\n    return pop(false);\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int? peekFirst() {\n    return _front?.val;\n  }\n\n  /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n  int? peekLast() {\n    return _rear?.val;\n  }\n\n  /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; _queSize; i++) {\n      res.add(node!.val);\n      node = node.next;\n    }\n    return res;\n  }\n}\n</code></pre> linkedlist_deque.rs<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\npub struct ListNode&lt;T&gt; {\n    pub val: T,                                 // \u8282\u70b9\u503c\n    pub next: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n    pub prev: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u524d\u9a71\u8282\u70b9\u6307\u9488\n}\n\nimpl&lt;T&gt; ListNode&lt;T&gt; {\n    pub fn new(val: T) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n        Rc::new(RefCell::new(ListNode {\n            val,\n            next: None,\n            prev: None,\n        }))\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListDeque&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5934\u8282\u70b9 front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // \u5c3e\u8282\u70b9 rear\n    que_size: usize,                         // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListDeque&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    pub fn push(&amp;mut self, num: T, is_front: bool) {\n        let node = ListNode::new(num);\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        if is_front {\n            match self.front.take() {\n                // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n                None =&gt; {\n                    self.rear = Some(node.clone());\n                    self.front = Some(node);\n                }\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n                Some(old_front) =&gt; {\n                    old_front.borrow_mut().prev = Some(node.clone());\n                    node.borrow_mut().next = Some(old_front);\n                    self.front = Some(node); // \u66f4\u65b0\u5934\u8282\u70b9\n                }\n            }\n        }\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else {\n            match self.rear.take() {\n                // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n                None =&gt; {\n                    self.front = Some(node.clone());\n                    self.rear = Some(node);\n                }\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n                Some(old_rear) =&gt; {\n                    old_rear.borrow_mut().next = Some(node.clone());\n                    node.borrow_mut().prev = Some(old_rear);\n                    self.rear = Some(node); // \u66f4\u65b0\u5c3e\u8282\u70b9\n                }\n            }\n        }\n        self.que_size += 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pub fn push_first(&amp;mut self, num: T) {\n        self.push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pub fn push_last(&amp;mut self, num: T) {\n        self.push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    pub fn pop(&amp;mut self, is_front: bool) -&gt; Option&lt;T&gt; {\n        // \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de None\n        if self.is_empty() {\n            return None;\n        };\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if is_front {\n            self.front.take().map(|old_front| {\n                match old_front.borrow_mut().next.take() {\n                    Some(new_front) =&gt; {\n                        new_front.borrow_mut().prev.take();\n                        self.front = Some(new_front); // \u66f4\u65b0\u5934\u8282\u70b9\n                    }\n                    None =&gt; {\n                        self.rear.take();\n                    }\n                }\n                self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n                Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n            })\n        }\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else {\n            self.rear.take().map(|old_rear| {\n                match old_rear.borrow_mut().prev.take() {\n                    Some(new_rear) =&gt; {\n                        new_rear.borrow_mut().next.take();\n                        self.rear = Some(new_rear); // \u66f4\u65b0\u5c3e\u8282\u70b9\n                    }\n                    None =&gt; {\n                        self.front.take();\n                    }\n                }\n                self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n                Rc::try_unwrap(old_rear).ok().unwrap().into_inner().val\n            })\n        }\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    pub fn pop_first(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    pub fn pop_last(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    pub fn peek_first(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    pub fn peek_last(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.rear.as_ref()\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.insert(0, node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_deque.c<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\ntypedef struct DoublyListNode {\n    int val;                     // \u8282\u70b9\u503c\n    struct DoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\n    struct DoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\n} DoublyListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nDoublyListNode *newDoublyListNode(int num) {\n    DoublyListNode *new = (DoublyListNode *)malloc(sizeof(DoublyListNode));\n    new-&gt;val = num;\n    new-&gt;next = NULL;\n    new-&gt;prev = NULL;\n    return new;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delDoublyListNode(DoublyListNode *node) {\n    free(node);\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\n    DoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    int queSize;                  // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n} LinkedListDeque;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListDeque *newLinkedListDeque() {\n    LinkedListDeque *deque = (LinkedListDeque *)malloc(sizeof(LinkedListDeque));\n    deque-&gt;front = NULL;\n    deque-&gt;rear = NULL;\n    deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListdeque(LinkedListDeque *deque) {\n    // \u91ca\u653e\u6240\u6709\u8282\u70b9\n    for (int i = 0; i &lt; deque-&gt;queSize &amp;&amp; deque-&gt;front != NULL; i++) {\n        DoublyListNode *tmp = deque-&gt;front;\n        deque-&gt;front = deque-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // \u91ca\u653e deque \u7ed3\u6784\u4f53\n    free(deque);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListDeque *deque) {\n    return (size(deque) == 0);\n}\n\n/* \u5165\u961f */\nvoid push(LinkedListDeque *deque, int num, bool isFront) {\n    DoublyListNode *node = newDoublyListNode(num);\n    // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411node\n    if (empty(deque)) {\n        deque-&gt;front = deque-&gt;rear = node;\n    }\n    // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n    else if (isFront) {\n        // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n        deque-&gt;front-&gt;prev = node;\n        node-&gt;next = deque-&gt;front;\n        deque-&gt;front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n    }\n    // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n    else {\n        // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n        deque-&gt;rear-&gt;next = node;\n        node-&gt;prev = deque-&gt;rear;\n        deque-&gt;rear = node;\n    }\n    deque-&gt;queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(LinkedListDeque *deque, int num) {\n    push(deque, num, true);\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(LinkedListDeque *deque, int num) {\n    push(deque, num, false);\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;front);\n    return deque-&gt;front-&gt;val;\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;rear);\n    return deque-&gt;rear-&gt;val;\n}\n\n/* \u51fa\u961f */\nint pop(LinkedListDeque *deque, bool isFront) {\n    if (empty(deque))\n        return -1;\n    int val;\n    // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n    if (isFront) {\n        val = peekFirst(deque); // \u6682\u5b58\u5934\u8282\u70b9\u503c\n        DoublyListNode *fNext = deque-&gt;front-&gt;next;\n        if (fNext) {\n            fNext-&gt;prev = NULL;\n            deque-&gt;front-&gt;next = NULL;\n            delDoublyListNode(deque-&gt;front);\n        }\n        deque-&gt;front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n    }\n    // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n    else {\n        val = peekLast(deque); // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n        DoublyListNode *rPrev = deque-&gt;rear-&gt;prev;\n        if (rPrev) {\n            rPrev-&gt;next = NULL;\n            deque-&gt;rear-&gt;prev = NULL;\n            delDoublyListNode(deque-&gt;rear);\n        }\n        deque-&gt;rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n    }\n    deque-&gt;queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    return val;\n}\n\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(LinkedListDeque *deque) {\n    return pop(deque, true);\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(LinkedListDeque *deque) {\n    return pop(deque, false);\n}\n\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListDeque(LinkedListDeque *deque) {\n    int *arr = malloc(sizeof(int) * deque-&gt;queSize);\n    // \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\n    int i;\n    DoublyListNode *node;\n    for (i = 0, node = deque-&gt;front; i &lt; deque-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, deque-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_deque.kt<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode(var value: Int) {\n    // \u8282\u70b9\u503c\n    var next: ListNode? = null // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    var prev: ListNode? = null // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private var front: ListNode? = null // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private var rear: ListNode? = null\n    private var queSize = 0 // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    fun push(num: Int, isFront: Boolean) {\n        val node = ListNode(num)\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty()) {\n            rear = node\n            front = rear\n            // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        } else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front?.prev = node\n            node.next = front\n            front = node // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear?.next = node\n            node.prev = rear\n            rear = node // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize++ // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    fun pushFirst(num: Int) {\n        push(num, true)\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    fun pushLast(num: Int) {\n        push(num, false)\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    fun pop(isFront: Boolean): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n\n        val value: Int\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            value = front!!.value // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            val fNext = front!!.next\n            if (fNext != null) {\n                fNext.prev = null\n                front!!.next = null\n            }\n            front = fNext // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        } else {\n            value = rear!!.value // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            val rPrev = rear!!.prev\n            if (rPrev != null) {\n                rPrev.next = null\n                rear!!.prev = null\n            }\n            rear = rPrev // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize-- // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return value\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    fun popFirst(): Int {\n        return pop(true)\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    fun popLast(): Int {\n        return pop(false)\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peekFirst(): Int {\n        if (isEmpty()) {\n            throw IndexOutOfBoundsException()\n\n        }\n        return front!!.value\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return rear!!.value\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!.value\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.rb<pre><code>[class]{ListNode}-[func]{}\n\n[class]{LinkedListDeque}-[func]{}\n</code></pre> linkedlist_deque.zig<pre><code>// \u53cc\u5411\u94fe\u8868\u8282\u70b9\nfn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = undefined,     // \u8282\u70b9\u503c\n        next: ?*Self = null,    // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n        prev: ?*Self = null,    // \u524d\u9a71\u8282\u70b9\u6307\u9488\n\n        // Initialize a list node with specific value\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n            self.prev = null;\n        }\n    };\n}\n\n// \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\nfn LinkedListDeque(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        front: ?*ListNode(T) = null,                    // \u5934\u8282\u70b9 front\n        rear: ?*ListNode(T) = null,                     // \u5c3e\u8282\u70b9 rear\n        que_size: usize = 0,                             // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.front = null;\n            self.rear = null;\n            self.que_size = 0;\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.que_size;\n        }\n\n        // \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u5165\u961f\u64cd\u4f5c\n        pub fn push(self: *Self, num: T, is_front: bool) !void {\n            var node = try self.mem_allocator.create(ListNode(T));\n            node.init(num);\n            // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n            if (self.isEmpty()) {\n                self.front = node;\n                self.rear = node;\n            // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n            } else if (is_front) {\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n                self.front.?.prev = node;\n                node.next = self.front;\n                self.front = node;  // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n            } else {\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n                self.rear.?.next = node;\n                node.prev = self.rear;\n                self.rear = node;   // \u66f4\u65b0\u5c3e\u8282\u70b9\n            }\n            self.que_size += 1;      // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        } \n\n        // \u961f\u9996\u5165\u961f\n        pub fn pushFirst(self: *Self, num: T) !void {\n            try self.push(num, true);\n        } \n\n        // \u961f\u5c3e\u5165\u961f\n        pub fn pushLast(self: *Self, num: T) !void {\n            try self.push(num, false);\n        } \n\n        // \u51fa\u961f\u64cd\u4f5c\n        pub fn pop(self: *Self, is_front: bool) T {\n            if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n            var val: T = undefined;\n            // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n            if (is_front) {\n                val = self.front.?.val;     // \u6682\u5b58\u5934\u8282\u70b9\u503c\n                // \u5220\u9664\u5934\u8282\u70b9\n                var fNext = self.front.?.next;\n                if (fNext != null) {\n                    fNext.?.prev = null;\n                    self.front.?.next = null;\n                }\n                self.front = fNext;         // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n            } else {\n                val = self.rear.?.val;      // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n                // \u5220\u9664\u5c3e\u8282\u70b9\n                var rPrev = self.rear.?.prev;\n                if (rPrev != null) {\n                    rPrev.?.next = null;\n                    self.rear.?.prev = null;\n                }\n                self.rear = rPrev;          // \u66f4\u65b0\u5c3e\u8282\u70b9\n            }\n            self.que_size -= 1;              // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n            return val;\n        } \n\n        // \u961f\u9996\u51fa\u961f\n        pub fn popFirst(self: *Self) T {\n            return self.pop(true);\n        } \n\n        // \u961f\u5c3e\u51fa\u961f\n        pub fn popLast(self: *Self) T {\n            return self.pop(false);\n        } \n\n        // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n        pub fn peekFirst(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n            return self.front.?.val;\n        }  \n\n        // \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\n        pub fn peekLast(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n            return self.rear.?.val;\n        }\n\n        // \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.front;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[i] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#2-implementation-based-on-array","title":"2. \u00a0 Implementation Based on Array","text":"<p>As shown in the Figure 5-9 , similar to implementing a queue with an array, we can also use a circular array to implement a double-ended queue.</p> ArrayDequepushLast()pushFirst()popLast()popFirst() <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 5-9 \u00a0 Implementing Double-Ended Queue with Array for Enqueue and Dequeue Operations </p> <p>The implementation only needs to add methods for \"front enqueue\" and \"rear dequeue\":</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_deque.py<pre><code>class ArrayDeque:\n    \"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\n\n    def __init__(self, capacity: int):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._nums: list[int] = [0] * capacity\n        self._front: int = 0\n        self._size: int = 0\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self._size == 0\n\n    def index(self, i: int) -&gt; int:\n        \"\"\"\u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15\"\"\"\n        # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        # \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        # \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + self.capacity()) % self.capacity()\n\n    def push_first(self, num: int):\n        \"\"\"\u961f\u9996\u5165\u961f\"\"\"\n        if self._size == self.capacity():\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        # \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        self._front = self.index(self._front - 1)\n        # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        self._nums[self._front] = num\n        self._size += 1\n\n    def push_last(self, num: int):\n        \"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\n        if self._size == self.capacity():\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        # \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        rear = self.index(self._front + self._size)\n        # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop_first(self) -&gt; int:\n        \"\"\"\u961f\u9996\u51fa\u961f\"\"\"\n        num = self.peek_first()\n        # \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        self._front = self.index(self._front + 1)\n        self._size -= 1\n        return num\n\n    def pop_last(self) -&gt; int:\n        \"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\n        num = self.peek_last()\n        self._size -= 1\n        return num\n\n    def peek_first(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        return self._nums[self._front]\n\n    def peek_last(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        # \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        last = self.index(self._front + self._size - 1)\n        return self._nums[last]\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n        # \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        res = []\n        for i in range(self._size):\n            res.append(self._nums[self.index(self._front + i)])\n        return res\n</code></pre> array_deque.cpp<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n  private:\n    vector&lt;int&gt; nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;        // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;      // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    ArrayDeque(int capacity) {\n        nums.resize(capacity);\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    int capacity() {\n        return nums.size();\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    int index(int i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    void pushFirst(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    void pushLast(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        int rear = index(front + queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    int popFirst() {\n        int num = peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        return nums[front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    vector&lt;int&gt; toVector() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        vector&lt;int&gt; res(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n};\n</code></pre> array_deque.java<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    private int[] nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public ArrayDeque(int capacity) {\n        this.nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    private int index(int i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void pushFirst(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void pushLast(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        int rear = index(front + queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int popFirst() {\n        int num = peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int[] toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.cs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public ArrayDeque(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    int Index(int i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + Capacity()) % Capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void PushFirst(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = Index(front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void PushLast(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        int rear = Index(front + queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int PopFirst() {\n        int num = PeekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = Index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int PopLast() {\n        int num = PeekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int PeekFirst() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        return nums[front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int PeekLast() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = Index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int[] ToArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[Index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.go<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype arrayDeque struct {\n    nums        []int // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    queSize     int   // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n    queCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\n    return &amp;arrayDeque{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayDeque) size() int {\n    return q.queSize\n}\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayDeque) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nfunc (q *arrayDeque) index(i int) int {\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n    return (i + q.queCapacity) % q.queCapacity\n}\n\n/* \u961f\u9996\u5165\u961f */\nfunc (q *arrayDeque) pushFirst(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n        return\n    }\n    // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n    q.front = q.index(q.front - 1)\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n    q.nums[q.front] = num\n    q.queSize++\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nfunc (q *arrayDeque) pushLast(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n        return\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    rear := q.index(q.front + q.queSize)\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* \u961f\u9996\u51fa\u961f */\nfunc (q *arrayDeque) popFirst() any {\n    num := q.peekFirst()\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    q.front = q.index(q.front + 1)\n    q.queSize--\n    return num\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nfunc (q *arrayDeque) popLast() any {\n    num := q.peekLast()\n    q.queSize--\n    return num\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayDeque) peekFirst() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (q *arrayDeque) peekLast() any {\n    if q.isEmpty() {\n        return nil\n    }\n    // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    last := q.index(q.front + q.queSize - 1)\n    return q.nums[last]\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayDeque) toSlice() []int {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    res := make([]int, q.queSize)\n    for i, j := 0, q.front; i &lt; q.queSize; i++ {\n        res[i] = q.nums[q.index(j)]\n        j++\n    }\n    return res\n}\n</code></pre> array_deque.swift<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    private var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var _size: Int // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init(capacity: Int) {\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    private func index(i: Int) -&gt; Int {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        (i + capacity()) % capacity()\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    func pushFirst(num: Int) {\n        if size() == capacity() {\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(i: front - 1)\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num\n        _size += 1\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    func pushLast(num: Int) {\n        if size() == capacity() {\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        let rear = index(i: front + size())\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    func popFirst() -&gt; Int {\n        let num = peekFirst()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(i: front + 1)\n        _size -= 1\n        return num\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    func popLast() -&gt; Int {\n        let num = peekLast()\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return nums[front]\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        let last = index(i: front + size() - 1)\n        return nums[last]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    func toArray() -&gt; [Int] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        (front ..&lt; front + size()).map { nums[index(i: $0)] }\n    }\n}\n</code></pre> array_deque.js<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    #nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    #front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    #queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n        this.#front = 0;\n        this.#queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    capacity() {\n        return this.#nums.length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    index(i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pushFirst(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        this.#front = this.index(this.#front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        this.#nums[this.#front] = num;\n        this.#queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pushLast(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        const rear = this.index(this.#front + this.#queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    popFirst() {\n        const num = this.peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        this.#front = this.index(this.#front + 1);\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    popLast() {\n        const num = this.peekLast();\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.#nums[this.#front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        const last = this.index(this.#front + this.#queSize - 1);\n        return this.#nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const res = [];\n        for (let i = 0, j = this.#front; i &lt; this.#queSize; i++, j++) {\n            res[i] = this.#nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.ts<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    private nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private queSize: number; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = 0;\n        this.queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    capacity(): number {\n        return this.nums.length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    index(i: number): number {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pushFirst(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        this.front = this.index(this.front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        this.nums[this.front] = num;\n        this.queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pushLast(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        const rear: number = this.index(this.front + this.queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    popFirst(): number {\n        const num: number = this.peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        this.front = this.index(this.front + 1);\n        this.queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    popLast(): number {\n        const num: number = this.peekLast();\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.nums[this.front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        const last = this.index(this.front + this.queSize - 1);\n        return this.nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    toArray(): number[] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const res: number[] = [];\n        for (let i = 0, j = this.front; i &lt; this.queSize; i++, j++) {\n            res[i] = this.nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.dart<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n  late List&lt;int&gt; _nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n  late int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n  late int _queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  ArrayDeque(int capacity) {\n    this._nums = List.filled(capacity, 0);\n    this._front = this._queSize = 0;\n  }\n\n  /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n  int capacity() {\n    return _nums.length;\n  }\n\n  /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n  int index(int i) {\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n    return (i + capacity()) % capacity();\n  }\n\n  /* \u961f\u9996\u5165\u961f */\n  void pushFirst(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n    }\n    // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 _front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n    _front = index(_front - 1);\n    // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u9996\n    _nums[_front] = _num;\n    _queSize++;\n  }\n\n  /* \u961f\u5c3e\u5165\u961f */\n  void pushLast(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    int rear = index(_front + _queSize);\n    // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* \u961f\u9996\u51fa\u961f */\n  int popFirst() {\n    int _num = peekFirst();\n    // \u961f\u9996\u6307\u9488\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n    _front = index(_front + 1);\n    _queSize--;\n    return _num;\n  }\n\n  /* \u961f\u5c3e\u51fa\u961f */\n  int popLast() {\n    int _num = peekLast();\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int peekFirst() {\n    if (isEmpty()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n    }\n    return _nums[_front];\n  }\n\n  /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n  int peekLast() {\n    if (isEmpty()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n    }\n    // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    int last = index(_front + _queSize - 1);\n    return _nums[last];\n  }\n\n  /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n  List&lt;int&gt; toArray() {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[index(j)];\n    }\n    return res;\n  }\n}\n</code></pre> array_deque.rs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nstruct ArrayDeque {\n    nums: Vec&lt;i32&gt;,  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front: usize,    // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    que_size: usize, // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n}\n\nimpl ArrayDeque {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(capacity: usize) -&gt; Self {\n        Self {\n            nums: vec![0; capacity],\n            front: 0,\n            que_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    pub fn capacity(&amp;self) -&gt; usize {\n        self.nums.len()\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        self.que_size\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    fn index(&amp;self, i: i32) -&gt; usize {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return ((i + self.capacity() as i32) % self.capacity() as i32) as usize;\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pub fn push_first(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        self.front = self.index(self.front as i32 - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        self.nums[self.front] = num;\n        self.que_size += 1;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pub fn push_last(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        let rear = self.index(self.front as i32 + self.que_size as i32);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self.nums[rear] = num;\n        self.que_size += 1;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    fn pop_first(&amp;mut self) -&gt; i32 {\n        let num = self.peek_first();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        self.front = self.index(self.front as i32 + 1);\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    fn pop_last(&amp;mut self) -&gt; i32 {\n        let num = self.peek_last();\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fn peek_first(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        };\n        self.nums[self.front]\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fn peek_last(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        };\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        let last = self.index(self.front as i32 + self.que_size as i32 - 1);\n        self.nums[last]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    fn to_array(&amp;self) -&gt; Vec&lt;i32&gt; {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        let mut res = vec![0; self.que_size];\n        let mut j = self.front;\n        for i in 0..self.que_size {\n            res[i] = self.nums[self.index(j as i32)];\n            j += 1;\n        }\n        res\n    }\n}\n</code></pre> array_deque.c<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\n    int *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n    int queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayDeque;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayDeque *newArrayDeque(int capacity) {\n    ArrayDeque *deque = (ArrayDeque *)malloc(sizeof(ArrayDeque));\n    // \u521d\u59cb\u5316\u6570\u7ec4\n    deque-&gt;queCapacity = capacity;\n    deque-&gt;nums = (int *)malloc(sizeof(int) * deque-&gt;queCapacity);\n    deque-&gt;front = deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayDeque(ArrayDeque *deque) {\n    free(deque-&gt;nums);\n    free(deque);\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayDeque *deque) {\n    return deque-&gt;queCapacity;\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayDeque *deque) {\n    return deque-&gt;queSize == 0;\n}\n\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint dequeIndex(ArrayDeque *deque, int i) {\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n    return ((i + capacity(deque)) % capacity(deque));\n}\n\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\n        return;\n    }\n    // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u56de\u5230\u5c3e\u90e8\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front - 1);\n    // \u5c06 num \u6dfb\u52a0\u5230\u961f\u9996\n    deque-&gt;nums[deque-&gt;front] = num;\n    deque-&gt;queSize++;\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\n        return;\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    int rear = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize);\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    deque-&gt;nums[rear] = num;\n    deque-&gt;queSize++;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(ArrayDeque *deque) {\n    // \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\n    assert(empty(deque) == 0);\n    return deque-&gt;nums[deque-&gt;front];\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(ArrayDeque *deque) {\n    // \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\n    assert(empty(deque) == 0);\n    int last = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize - 1);\n    return deque-&gt;nums[last];\n}\n\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(ArrayDeque *deque) {\n    int num = peekFirst(deque);\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front + 1);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(ArrayDeque *deque) {\n    int num = peekLast(deque);\n    deque-&gt;queSize--;\n    return num;\n}\n</code></pre> array_deque.kt<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque(capacity: Int) {\n    private var nums = IntArray(capacity) // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var queSize = 0 // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    private fun index(i: Int): Int {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity()\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    fun pushFirst(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1)\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num\n        queSize++\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    fun pushLast(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        val rear = index(front + queSize)\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        queSize++\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    fun popFirst(): Int {\n        val num = peekFirst()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(front + 1)\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fun popLast(): Int {\n        val num = peekLast()\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        val last = index(front + queSize - 1)\n        return nums[last]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    fun toArray(): IntArray {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[index(j)]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_deque.rb<pre><code>[class]{ArrayDeque}-[func]{}\n</code></pre> array_deque.zig<pre><code>[class]{ArrayDeque}-[func]{}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#533-applications-of-double-ended-queue","title":"5.3.3 \u00a0 Applications of Double-Ended Queue","text":"<p>The double-ended queue combines the logic of both stacks and queues, thus, it can implement all their respective use cases while offering greater flexibility.</p> <p>We know that software's \"undo\" feature is typically implemented using a stack: the system <code>pushes</code> each change operation onto the stack and then <code>pops</code> to implement undoing. However, considering the limitations of system resources, software often restricts the number of undo steps (for example, only allowing the last 50 steps). When the stack length exceeds 50, the software needs to perform a deletion operation at the bottom of the stack (the front of the queue). But a regular stack cannot perform this function, where a double-ended queue becomes necessary. Note that the core logic of \"undo\" still follows the Last-In-First-Out principle of a stack, but a double-ended queue can more flexibly implement some additional logic.</p>"},{"location":"chapter_stack_and_queue/queue/","title":"5.2 \u00a0 Queue","text":"<p>\"Queue\" is a linear data structure that follows the First-In-First-Out (FIFO) rule. As the name suggests, a queue simulates the phenomenon of lining up, where newcomers join the queue at the rear, and the person at the front leaves the queue first.</p> <p>As shown in the Figure 5-4 , we call the front of the queue the \"head\" and the back the \"tail.\" The operation of adding elements to the rear of the queue is termed \"enqueue,\" and the operation of removing elements from the front is termed \"dequeue.\"</p> <p></p> <p> Figure 5-4 \u00a0 Queue's First-In-First-Out Rule </p>"},{"location":"chapter_stack_and_queue/queue/#521-common-operations-on-queue","title":"5.2.1 \u00a0 Common Operations on Queue","text":"<p>The common operations on a queue are shown in the Table 5-2 . Note that method names may vary across different programming languages. Here, we use the same naming convention as that used for stacks.</p> <p> Table 5-2 \u00a0 Efficiency of Queue Operations </p> Method Name Description Time Complexity <code>push()</code> Enqueue an element, add it to the tail \\(O(1)\\) <code>pop()</code> Dequeue the head element \\(O(1)\\) <code>peek()</code> Access the head element \\(O(1)\\) <p>We can directly use the ready-made queue classes in programming languages:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig queue.py<pre><code>from collections import deque\n\n# Initialize the queue\n# In Python, we generally use the deque class as a queue\n# Although queue.Queue() is a pure queue class, it's not very user-friendly, so it's not recommended\nque: deque[int] = deque()\n\n# Enqueue elements\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n\n# Access the first element\nfront: int = que[0]\n\n# Dequeue an element\npop: int = que.popleft()\n\n# Get the length of the queue\nsize: int = len(que)\n\n# Check if the queue is empty\nis_empty: bool = len(que) == 0\n</code></pre> queue.cpp<pre><code>/* Initialize the queue */\nqueue&lt;int&gt; queue;\n\n/* Enqueue elements */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* Access the first element*/\nint front = queue.front();\n\n/* Dequeue an element */\nqueue.pop();\n\n/* Get the length of the queue */\nint size = queue.size();\n\n/* Check if the queue is empty */\nbool empty = queue.empty();\n</code></pre> queue.java<pre><code>/* Initialize the queue */\nQueue&lt;Integer&gt; queue = new LinkedList&lt;&gt;();\n\n/* Enqueue elements */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n\n/* Access the first element */\nint peek = queue.peek();\n\n/* Dequeue an element */\nint pop = queue.poll();\n\n/* Get the length of the queue */\nint size = queue.size();\n\n/* Check if the queue is empty */\nboolean isEmpty = queue.isEmpty();\n</code></pre> queue.cs<pre><code>/* Initialize the queue */\nQueue&lt;int&gt; queue = new();\n\n/* Enqueue elements */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n\n/* Access the first element */\nint peek = queue.Peek();\n\n/* Dequeue an element */\nint pop = queue.Dequeue();\n\n/* Get the length of the queue */\nint size = queue.Count;\n\n/* Check if the queue is empty */\nbool isEmpty = queue.Count == 0;\n</code></pre> queue_test.go<pre><code>/* Initialize the queue */\n// In Go, use list as a queue\nqueue := list.New()\n\n/* Enqueue elements */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n\n/* Access the first element */\npeek := queue.Front()\n\n/* Dequeue an element */\npop := queue.Front()\nqueue.Remove(pop)\n\n/* Get the length of the queue */\nsize := queue.Len()\n\n/* Check if the queue is empty */\nisEmpty := queue.Len() == 0\n</code></pre> queue.swift<pre><code>/* Initialize the queue */\n// Swift does not have a built-in queue class, so Array can be used as a queue\nvar queue: [Int] = []\n\n/* Enqueue elements */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n\n/* Access the first element */\nlet peek = queue.first!\n\n/* Dequeue an element */\n// Since it's an array, removeFirst has a complexity of O(n)\nlet pool = queue.removeFirst()\n\n/* Get the length of the queue */\nlet size = queue.count\n\n/* Check if the queue is empty */\nlet isEmpty = queue.isEmpty\n</code></pre> queue.js<pre><code>/* Initialize the queue */\n// JavaScript does not have a built-in queue, so Array can be used as a queue\nconst queue = [];\n\n/* Enqueue elements */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* Access the first element */\nconst peek = queue[0];\n\n/* Dequeue an element */\n// Since the underlying structure is an array, shift() method has a time complexity of O(n)\nconst pop = queue.shift();\n\n/* Get the length of the queue */\nconst size = queue.length;\n\n/* Check if the queue is empty */\nconst empty = queue.length === 0;\n</code></pre> queue.ts<pre><code>/* Initialize the queue */\n// TypeScript does not have a built-in queue, so Array can be used as a queue \nconst queue: number[] = [];\n\n/* Enqueue elements */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* Access the first element */\nconst peek = queue[0];\n\n/* Dequeue an element */\n// Since the underlying structure is an array, shift() method has a time complexity of O(n)\nconst pop = queue.shift();\n\n/* Get the length of the queue */\nconst size = queue.length;\n\n/* Check if the queue is empty */\nconst empty = queue.length === 0;\n</code></pre> queue.dart<pre><code>/* Initialize the queue */\n// In Dart, the Queue class is a double-ended queue but can be used as a queue\nQueue&lt;int&gt; queue = Queue();\n\n/* Enqueue elements */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n\n/* Access the first element */\nint peek = queue.first;\n\n/* Dequeue an element */\nint pop = queue.removeFirst();\n\n/* Get the length of the queue */\nint size = queue.length;\n\n/* Check if the queue is empty */\nbool isEmpty = queue.isEmpty;\n</code></pre> queue.rs<pre><code>/* Initialize the double-ended queue */\n// In Rust, use a double-ended queue as a regular queue\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* Enqueue elements */\ndeque.push_back(1);\ndeque.push_back(3);\ndeque.push_back(2);\ndeque.push_back(5);\ndeque.push_back(4);\n\n/* Access the first element */\nif let Some(front) = deque.front() {\n}\n\n/* Dequeue an element */\nif let Some(pop) = deque.pop_front() {\n}\n\n/* Get the length of the queue */\nlet size = deque.len();\n\n/* Check if the queue is empty */\nlet is_empty = deque.is_empty();\n</code></pre> queue.c<pre><code>// C does not provide a built-in queue\n</code></pre> queue.kt<pre><code>\n</code></pre> queue.zig<pre><code>\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_stack_and_queue/queue/#522-implementing-a-queue","title":"5.2.2 \u00a0 Implementing a Queue","text":"<p>To implement a queue, we need a data structure that allows adding elements at one end and removing them at the other. Both linked lists and arrays meet this requirement.</p>"},{"location":"chapter_stack_and_queue/queue/#1-implementation-based-on-a-linked-list","title":"1. \u00a0 Implementation Based on a Linked List","text":"<p>As shown in the Figure 5-5 , we can consider the \"head node\" and \"tail node\" of a linked list as the \"front\" and \"rear\" of the queue, respectively. It is stipulated that nodes can only be added at the rear and removed at the front.</p> LinkedListQueuepush()pop() <p></p> <p></p> <p></p> <p> Figure 5-5 \u00a0 Implementing Queue with Linked List for Enqueue and Dequeue Operations </p> <p>Below is the code for implementing a queue using a linked list:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_queue.py<pre><code>class LinkedListQueue:\n    \"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._front: ListNode | None = None  # \u5934\u8282\u70b9 front\n        self._rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return not self._front\n\n    def push(self, num: int):\n        \"\"\"\u5165\u961f\"\"\"\n        # \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        node = ListNode(num)\n        # \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if self._front is None:\n            self._front = node\n            self._rear = node\n        # \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        else:\n            self._rear.next = node\n            self._rear = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u961f\"\"\"\n        num = self.peek()\n        # \u5220\u9664\u5934\u8282\u70b9\n        self._front = self._front.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\n        return self._front.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        queue = []\n        temp = self._front\n        while temp:\n            queue.append(temp.val)\n            temp = temp.next\n        return queue\n</code></pre> linkedlist_queue.cpp<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n  private:\n    ListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    int queSize;\n\n  public:\n    LinkedListQueue() {\n        front = nullptr;\n        rear = nullptr;\n        queSize = 0;\n    }\n\n    ~LinkedListQueue() {\n        // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n        freeMemoryLinkedList(front);\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u961f */\n    void push(int num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        ListNode *node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == nullptr) {\n            front = node;\n            rear = node;\n        }\n        // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        else {\n            rear-&gt;next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    int pop() {\n        int num = peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        ListNode *tmp = front;\n        front = front-&gt;next;\n        // \u91ca\u653e\u5185\u5b58\n        delete tmp;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peek() {\n        if (size() == 0)\n            throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n        return front-&gt;val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_queue.java<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    private ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f */\n    public void push(int num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        ListNode node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == null) {\n            front = node;\n            rear = node;\n        // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int pop() {\n        int num = peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front.next;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.cs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    ListNode? front, rear;  // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear \n    int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u961f */\n    public void Push(int num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        ListNode node = new(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == null) {\n            front = node;\n            rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else if (rear != null) {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int Pop() {\n        int num = Peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front?.next;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return front!.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] ToArray() {\n        if (front == null)\n            return [];\n\n        ListNode? node = front;\n        int[] res = new int[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.go<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntype linkedListQueue struct {\n    // \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u961f\u5217\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newLinkedListQueue() *linkedListQueue {\n    return &amp;linkedListQueue{\n        data: list.New(),\n    }\n}\n\n/* \u5165\u961f */\nfunc (s *linkedListQueue) push(value any) {\n    s.data.PushBack(value)\n}\n\n/* \u51fa\u961f */\nfunc (s *linkedListQueue) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListQueue) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListQueue) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListQueue) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListQueue) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_queue.swift<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    private var front: ListNode? // \u5934\u8282\u70b9\n    private var rear: ListNode? // \u5c3e\u8282\u70b9\n    private var _size: Int\n\n    init() {\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u961f */\n    func push(num: Int) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        let node = ListNode(x: num)\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if front == nil {\n            front = node\n            rear = node\n        }\n        // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        else {\n            rear?.next = node\n            rear = node\n        }\n        _size += 1\n    }\n\n    /* \u51fa\u961f */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front?.next\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return front!.val\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.js<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    #front; // \u5934\u8282\u70b9 #front\n    #rear; // \u5c3e\u8282\u70b9 #rear\n    #queSize = 0;\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        const node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (!this.#front) {\n            this.#front = node;\n            this.#rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            this.#rear.next = node;\n            this.#rear = node;\n        }\n        this.#queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop() {\n        const num = this.peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        this.#front = this.#front.next;\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek() {\n        if (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.#front.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray() {\n        let node = this.#front;\n        const res = new Array(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.ts<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    private front: ListNode | null; // \u5934\u8282\u70b9 front\n    private rear: ListNode | null; // \u5c3e\u8282\u70b9 rear\n    private queSize: number = 0;\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num: number): void {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        const node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (!this.front) {\n            this.front = node;\n            this.rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            this.rear!.next = node;\n            this.rear = node;\n        }\n        this.queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop(): number {\n        const num = this.peek();\n        if (!this.front) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        // \u5220\u9664\u5934\u8282\u70b9\n        this.front = this.front.next;\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek(): number {\n        if (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.front!.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray(): number[] {\n        let node = this.front;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.dart<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n  ListNode? _front; // \u5934\u8282\u70b9 _front\n  ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\n  int _queSize = 0; // \u961f\u5217\u957f\u5ea6\n\n  LinkedListQueue() {\n    _front = null;\n    _rear = null;\n  }\n\n  /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u5165\u961f */\n  void push(int _num) {\n    // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 _num\n    final node = ListNode(_num);\n    // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n    if (_front == null) {\n      _front = node;\n      _rear = node;\n    } else {\n      // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n      _rear!.next = node;\n      _rear = node;\n    }\n    _queSize++;\n  }\n\n  /* \u51fa\u961f */\n  int pop() {\n    final int _num = peek();\n    // \u5220\u9664\u5934\u8282\u70b9\n    _front = _front!.next;\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int peek() {\n    if (_queSize == 0) {\n      throw Exception('\u961f\u5217\u4e3a\u7a7a');\n    }\n    return _front!.val;\n  }\n\n  /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; queue = [];\n    while (node != null) {\n      queue.add(node.val);\n      node = node.next;\n    }\n    return queue;\n  }\n}\n</code></pre> linkedlist_queue.rs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListQueue&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5934\u8282\u70b9 front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // \u5c3e\u8282\u70b9 rear\n    que_size: usize,                         // \u961f\u5217\u7684\u957f\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListQueue&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u961f */\n    pub fn push(&amp;mut self, num: T) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        let new_rear = ListNode::new(num);\n        match self.rear.take() {\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n            Some(old_rear) =&gt; {\n                old_rear.borrow_mut().next = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n            // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n            None =&gt; {\n                self.front = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n        }\n        self.que_size += 1;\n    }\n\n    /* \u51fa\u961f */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.front.take().map(|old_front| {\n            match old_front.borrow_mut().next.take() {\n                Some(new_front) =&gt; {\n                    self.front = Some(new_front);\n                }\n                None =&gt; {\n                    self.rear.take();\n                }\n            }\n            self.que_size -= 1;\n            Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n        })\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.insert(0, node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_queue.c<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\n    ListNode *front, *rear;\n    int queSize;\n} LinkedListQueue;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListQueue *newLinkedListQueue() {\n    LinkedListQueue *queue = (LinkedListQueue *)malloc(sizeof(LinkedListQueue));\n    queue-&gt;front = NULL;\n    queue-&gt;rear = NULL;\n    queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListQueue(LinkedListQueue *queue) {\n    // \u91ca\u653e\u6240\u6709\u8282\u70b9\n    while (queue-&gt;front != NULL) {\n        ListNode *tmp = queue-&gt;front;\n        queue-&gt;front = queue-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // \u91ca\u653e queue \u7ed3\u6784\u4f53\n    free(queue);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListQueue *queue) {\n    return (size(queue) == 0);\n}\n\n/* \u5165\u961f */\nvoid push(LinkedListQueue *queue, int num) {\n    // \u5c3e\u8282\u70b9\u5904\u6dfb\u52a0 node\n    ListNode *node = newListNode(num);\n    // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n    if (queue-&gt;front == NULL) {\n        queue-&gt;front = node;\n        queue-&gt;rear = node;\n    }\n    // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n    else {\n        queue-&gt;rear-&gt;next = node;\n        queue-&gt;rear = node;\n    }\n    queue-&gt;queSize++;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(LinkedListQueue *queue) {\n    assert(size(queue) &amp;&amp; queue-&gt;front);\n    return queue-&gt;front-&gt;val;\n}\n\n/* \u51fa\u961f */\nint pop(LinkedListQueue *queue) {\n    int num = peek(queue);\n    ListNode *tmp = queue-&gt;front;\n    queue-&gt;front = queue-&gt;front-&gt;next;\n    free(tmp);\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListQueue(LinkedListQueue *queue) {\n    int *arr = malloc(sizeof(int) * queue-&gt;queSize);\n    // \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\n    int i;\n    ListNode *node;\n    for (i = 0, node = queue-&gt;front; i &lt; queue-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, queue-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_queue.kt<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue(\n    // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private var front: ListNode? = null,\n    private var rear: ListNode? = null,\n    private var queSize: Int = 0\n) {\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u961f */\n    fun push(num: Int) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        val node = ListNode(num)\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == null) {\n            front = node\n            rear = node\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            rear?.next = node\n            rear = node\n        }\n        queSize++\n    }\n\n    /* \u51fa\u961f */\n    fun pop(): Int {\n        val num = peek()\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front?.next\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!.value\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!.value\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.rb<pre><code>[class]{LinkedListQueue}-[func]{}\n</code></pre> linkedlist_queue.zig<pre><code>// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\nfn LinkedListQueue(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        front: ?*inc.ListNode(T) = null,                // \u5934\u8282\u70b9 front\n        rear: ?*inc.ListNode(T) = null,                 // \u5c3e\u8282\u70b9 rear\n        que_size: usize = 0,                            // \u961f\u5217\u7684\u957f\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.front = null;\n            self.rear = null;\n            self.que_size = 0;\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.que_size;\n        }\n\n        // \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.size() == 0) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\n            return self.front.?.val;\n        }  \n\n        // \u5165\u961f\n        pub fn push(self: *Self, num: T) !void {\n            // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n            var node = try self.mem_allocator.create(inc.ListNode(T));\n            node.init(num);\n            // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n            if (self.front == null) {\n                self.front = node;\n                self.rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n            } else {\n                self.rear.?.next = node;\n                self.rear = node;\n            }\n            self.que_size += 1;\n        } \n\n        // \u51fa\u961f\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            // \u5220\u9664\u5934\u8282\u70b9\n            self.front = self.front.?.next;\n            self.que_size -= 1;\n            return num;\n        } \n\n        // \u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.front;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[i] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_stack_and_queue/queue/#2-implementation-based-on-an-array","title":"2. \u00a0 Implementation Based on an Array","text":"<p>Deleting the first element in an array has a time complexity of \\(O(n)\\), which would make the dequeue operation inefficient. However, this problem can be cleverly avoided as follows.</p> <p>We use a variable <code>front</code> to indicate the index of the front element and maintain a variable <code>size</code> to record the queue's length. Define <code>rear = front + size</code>, which points to the position immediately following the tail element.</p> <p>With this design, the effective interval of elements in the array is <code>[front, rear - 1]</code>. The implementation methods for various operations are shown in the Figure 5-6 .</p> <ul> <li>Enqueue operation: Assign the input element to the <code>rear</code> index and increase <code>size</code> by 1.</li> <li>Dequeue operation: Simply increase <code>front</code> by 1 and decrease <code>size</code> by 1.</li> </ul> <p>Both enqueue and dequeue operations only require a single operation, each with a time complexity of \\(O(1)\\).</p> ArrayQueuepush()pop() <p></p> <p></p> <p></p> <p> Figure 5-6 \u00a0 Implementing Queue with Array for Enqueue and Dequeue Operations </p> <p>You might notice a problem: as enqueue and dequeue operations are continuously performed, both <code>front</code> and <code>rear</code> move to the right and will eventually reach the end of the array and can't move further. To resolve this, we can treat the array as a \"circular array\" where connecting the end of the array back to its beginning.</p> <p>In a circular array, <code>front</code> or <code>rear</code> needs to loop back to the start of the array upon reaching the end. This cyclical pattern can be achieved with a \"modulo operation\" as shown in the code below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_queue.py<pre><code>class ArrayQueue:\n    \"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\n\n    def __init__(self, size: int):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._nums: list[int] = [0] * size  # \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n        self._front: int = 0  # \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n        self._size: int = 0  # \u961f\u5217\u957f\u5ea6\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"\u5165\u961f\"\"\"\n        if self._size == self.capacity():\n            raise IndexError(\"\u961f\u5217\u5df2\u6ee1\")\n        # \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        rear: int = (self._front + self._size) % self.capacity()\n        # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u961f\"\"\"\n        num: int = self.peek()\n        # \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        self._front = (self._front + 1) % self.capacity()\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\n        return self._nums[self._front]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        res = [0] * self.size()\n        j: int = self._front\n        for i in range(self.size()):\n            res[i] = self._nums[(j % self.capacity())]\n            j += 1\n        return res\n</code></pre> array_queue.cpp<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n  private:\n    int *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;     // \u961f\u5217\u957f\u5ea6\n    int queCapacity; // \u961f\u5217\u5bb9\u91cf\n\n  public:\n    ArrayQueue(int capacity) {\n        // \u521d\u59cb\u5316\u6570\u7ec4\n        nums = new int[capacity];\n        queCapacity = capacity;\n        front = queSize = 0;\n    }\n\n    ~ArrayQueue() {\n        delete[] nums;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    int capacity() {\n        return queCapacity;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f */\n    void push(int num) {\n        if (queSize == queCapacity) {\n            cout &lt;&lt; \"\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        int rear = (front + queSize) % queCapacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    int pop() {\n        int num = peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % queCapacity;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peek() {\n        if (isEmpty())\n            throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n        return nums[front];\n    }\n\n    /* \u5c06\u6570\u7ec4\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        vector&lt;int&gt; arr(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            arr[i] = nums[j % queCapacity];\n        }\n        return arr;\n    }\n};\n</code></pre> array_queue.java<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    private int[] nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private int queSize; // \u961f\u5217\u957f\u5ea6\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u961f */\n    public void push(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        int rear = (front + queSize) % capacity();\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int pop() {\n        int num = peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % capacity();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    public int[] toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.cs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize; // \u961f\u5217\u957f\u5ea6\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u961f */\n    public void Push(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        int rear = (front + queSize) % Capacity();\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int Pop() {\n        int num = Peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % Capacity();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return nums[front];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    public int[] ToArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % this.Capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.go<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntype arrayQueue struct {\n    nums        []int // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    queSize     int   // \u961f\u5217\u957f\u5ea6\n    queCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\n    return &amp;arrayQueue{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayQueue) size() int {\n    return q.queSize\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayQueue) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* \u5165\u961f */\nfunc (q *arrayQueue) push(num int) {\n    // \u5f53 rear == queCapacity \u8868\u793a\u961f\u5217\u5df2\u6ee1\n    if q.queSize == q.queCapacity {\n        return\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n    rear := (q.front + q.queSize) % q.queCapacity\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* \u51fa\u961f */\nfunc (q *arrayQueue) pop() any {\n    num := q.peek()\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n    q.front = (q.front + 1) % q.queCapacity\n    q.queSize--\n    return num\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayQueue) peek() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayQueue) toSlice() []int {\n    rear := (q.front + q.queSize)\n    if rear &gt;= q.queCapacity {\n        rear %= q.queCapacity\n        return append(q.nums[q.front:], q.nums[:rear]...)\n    }\n    return q.nums[q.front:rear]\n}\n</code></pre> array_queue.swift<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    private var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var _size: Int // \u961f\u5217\u957f\u5ea6\n\n    init(capacity: Int) {\n        // \u521d\u59cb\u5316\u6570\u7ec4\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u961f */\n    func push(num: Int) {\n        if size() == capacity() {\n            print(\"\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        let rear = (front + size()) % capacity()\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* \u51fa\u961f */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % capacity()\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return nums[front]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    func toArray() -&gt; [Int] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        (front ..&lt; front + size()).map { nums[$0 % capacity()] }\n    }\n}\n</code></pre> array_queue.js<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    #nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    #front = 0; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    #queSize = 0; // \u961f\u5217\u957f\u5ea6\n\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    get capacity() {\n        return this.#nums.length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num) {\n        if (this.size === this.capacity) {\n            console.log('\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        const rear = (this.#front + this.size) % this.capacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop() {\n        const num = this.peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        this.#front = (this.#front + 1) % this.capacity;\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek() {\n        if (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.#nums[this.#front];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.#front; i &lt; this.size; i++, j++) {\n            arr[i] = this.#nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.ts<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    private nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private queSize: number; // \u961f\u5217\u957f\u5ea6\n\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = this.queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    get capacity(): number {\n        return this.nums.length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num: number): void {\n        if (this.size === this.capacity) {\n            console.log('\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        const rear = (this.front + this.queSize) % this.capacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop(): number {\n        const num = this.peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        this.front = (this.front + 1) % this.capacity;\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek(): number {\n        if (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.nums[this.front];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray(): number[] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.front; i &lt; this.size; i++, j++) {\n            arr[i] = this.nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.dart<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n  late List&lt;int&gt; _nums; // \u7528\u4e8e\u50a8\u5b58\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n  late int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n  late int _queSize; // \u961f\u5217\u957f\u5ea6\n\n  ArrayQueue(int capacity) {\n    _nums = List.filled(capacity, 0);\n    _front = _queSize = 0;\n  }\n\n  /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n  int capaCity() {\n    return _nums.length;\n  }\n\n  /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u5165\u961f */\n  void push(int _num) {\n    if (_queSize == capaCity()) {\n      throw Exception(\"\u961f\u5217\u5df2\u6ee1\");\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n    int rear = (_front + _queSize) % capaCity();\n    // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* \u51fa\u961f */\n  int pop() {\n    int _num = peek();\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n    _front = (_front + 1) % capaCity();\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"\u961f\u5217\u4e3a\u7a7a\");\n    }\n    return _nums[_front];\n  }\n\n  /* \u8fd4\u56de Array */\n  List&lt;int&gt; toArray() {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    final List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[j % capaCity()];\n    }\n    return res;\n  }\n}\n</code></pre> array_queue.rs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nstruct ArrayQueue {\n    nums: Vec&lt;i32&gt;,    // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front: i32,        // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    que_size: i32,     // \u961f\u5217\u957f\u5ea6\n    que_capacity: i32, // \u961f\u5217\u5bb9\u91cf\n}\n\nimpl ArrayQueue {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(capacity: i32) -&gt; ArrayQueue {\n        ArrayQueue {\n            nums: vec![0; capacity as usize],\n            front: 0,\n            que_size: 0,\n            que_capacity: capacity,\n        }\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    fn capacity(&amp;self) -&gt; i32 {\n        self.que_capacity\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    fn size(&amp;self) -&gt; i32 {\n        self.que_size\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* \u5165\u961f */\n    fn push(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        let rear = (self.front + self.que_size) % self.que_capacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self.nums[rear as usize] = num;\n        self.que_size += 1;\n    }\n\n    /* \u51fa\u961f */\n    fn pop(&amp;mut self) -&gt; i32 {\n        let num = self.peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        self.front = (self.front + 1) % self.que_capacity;\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fn peek(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"index out of bounds\");\n        }\n        self.nums[self.front as usize]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    fn to_vector(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let cap = self.que_capacity;\n        let mut j = self.front;\n        let mut arr = vec![0; self.que_size as usize];\n        for i in 0..self.que_size {\n            arr[i as usize] = self.nums[(j % cap) as usize];\n            j += 1;\n        }\n        arr\n    }\n}\n</code></pre> array_queue.c<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\n    int *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n    int queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayQueue;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayQueue *newArrayQueue(int capacity) {\n    ArrayQueue *queue = (ArrayQueue *)malloc(sizeof(ArrayQueue));\n    // \u521d\u59cb\u5316\u6570\u7ec4\n    queue-&gt;queCapacity = capacity;\n    queue-&gt;nums = (int *)malloc(sizeof(int) * queue-&gt;queCapacity);\n    queue-&gt;front = queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayQueue(ArrayQueue *queue) {\n    free(queue-&gt;nums);\n    free(queue);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayQueue *queue) {\n    return queue-&gt;queCapacity;\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayQueue *queue) {\n    return queue-&gt;queSize == 0;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(ArrayQueue *queue) {\n    assert(size(queue) != 0);\n    return queue-&gt;nums[queue-&gt;front];\n}\n\n/* \u5165\u961f */\nvoid push(ArrayQueue *queue, int num) {\n    if (size(queue) == capacity(queue)) {\n        printf(\"\u961f\u5217\u5df2\u6ee1\\r\\n\");\n        return;\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n    int rear = (queue-&gt;front + queue-&gt;queSize) % queue-&gt;queCapacity;\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    queue-&gt;nums[rear] = num;\n    queue-&gt;queSize++;\n}\n\n/* \u51fa\u961f */\nint pop(ArrayQueue *queue) {\n    int num = peek(queue);\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n    queue-&gt;front = (queue-&gt;front + 1) % queue-&gt;queCapacity;\n    queue-&gt;queSize--;\n    return num;\n}\n</code></pre> array_queue.kt<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue(capacity: Int) {\n    private val nums = IntArray(capacity) // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var queSize = 0 // \u961f\u5217\u957f\u5ea6\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* \u5165\u961f */\n    fun push(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        val rear = (front + queSize) % capacity()\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        queSize++\n    }\n\n    /* \u51fa\u961f */\n    fun pop(): Int {\n        val num = peek()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % capacity()\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    fun toArray(): IntArray {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[j % capacity()]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_queue.rb<pre><code>[class]{ArrayQueue}-[func]{}\n</code></pre> array_queue.zig<pre><code>// \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\nfn ArrayQueue(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        nums: []T = undefined,                          // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4     \n        cap: usize = 0,                                 // \u961f\u5217\u5bb9\u91cf\n        front: usize = 0,                               // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n        queSize: usize = 0,                             // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6570\u7ec4\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator, cap: usize) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.cap = cap;\n            self.nums = try self.mem_allocator.alloc(T, self.cap);\n            @memset(self.nums, @as(T, 0));\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\n        pub fn capacity(self: *Self) usize {\n            return self.cap;\n        }\n\n        // \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.queSize;\n        }\n\n        // \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.queSize == 0;\n        }\n\n        // \u5165\u961f\n        pub fn push(self: *Self, num: T) !void {\n            if (self.size() == self.capacity()) {\n                std.debug.print(\"\u961f\u5217\u5df2\u6ee1\\n\", .{});\n                return;\n            }\n            // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n            // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n            var rear = (self.front + self.queSize) % self.capacity();\n            // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n            self.nums[rear] = num;\n            self.queSize += 1;\n        } \n\n        // \u51fa\u961f\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n            self.front = (self.front + 1) % self.capacity();\n            self.queSize -= 1;\n            return num;\n        } \n\n        // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\n            return self.nums[self.front];\n        } \n\n        // \u8fd4\u56de\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            var j: usize = self.front;\n            while (i &lt; self.size()) : ({ i += 1; j += 1; }) {\n                res[i] = self.nums[j % self.capacity()];\n            }\n            return res;\n        }\n    };\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>The above implementation of the queue still has its limitations: its length is fixed. However, this issue is not difficult to resolve. We can replace the array with a dynamic array that can expand itself if needed. Interested readers can try to implement this themselves.</p> <p>The comparison of the two implementations is consistent with that of the stack and is not repeated here.</p>"},{"location":"chapter_stack_and_queue/queue/#523-typical-applications-of-queue","title":"5.2.3 \u00a0 Typical Applications of Queue","text":"<ul> <li>Amazon Orders. After shoppers place orders, these orders join a queue, and the system processes them in order. During events like Singles' Day, a massive number of orders are generated in a short time, making high concurrency a key challenge for engineers.</li> <li>Various To-Do Lists. Any scenario requiring a \"first-come, first-served\" functionality, such as a printer's task queue or a restaurant's food delivery queue, can effectively maintain the order of processing with a queue.</li> </ul>"},{"location":"chapter_stack_and_queue/stack/","title":"5.1 \u00a0 Stack","text":"<p>A \"Stack\" is a linear data structure that follows the principle of Last-In-First-Out (LIFO).</p> <p>We can compare a stack to a pile of plates on a table. To access the bottom plate, one must first remove the plates on top. By replacing the plates with various types of elements (such as integers, characters, objects, etc.), we obtain the data structure known as a stack.</p> <p>As shown in the Figure 5-1 , we refer to the top of the pile of elements as the \"top of the stack\" and the bottom as the \"bottom of the stack.\" The operation of adding elements to the top of the stack is called \"push,\" and the operation of removing the top element is called \"pop.\"</p> <p></p> <p> Figure 5-1 \u00a0 Stack's Last-In-First-Out Rule </p>"},{"location":"chapter_stack_and_queue/stack/#511-common-operations-on-stack","title":"5.1.1 \u00a0 Common Operations on Stack","text":"<p>The common operations on a stack are shown in the Table 5-1 . The specific method names depend on the programming language used. Here, we use <code>push()</code>, <code>pop()</code>, and <code>peek()</code> as examples.</p> <p> Table 5-1 \u00a0 Efficiency of Stack Operations </p> Method Description Time Complexity <code>push()</code> Push an element onto the stack (add to the top) \\(O(1)\\) <code>pop()</code> Pop the top element from the stack \\(O(1)\\) <code>peek()</code> Access the top element of the stack \\(O(1)\\) <p>Typically, we can directly use the stack class built into the programming language. However, some languages may not specifically provide a stack class. In these cases, we can use the language's \"array\" or \"linked list\" as a stack and ignore operations that are not related to stack logic in the program.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinZig stack.py<pre><code># Initialize the stack\n# Python does not have a built-in stack class, so a list can be used as a stack\nstack: list[int] = []\n\n# Push elements onto the stack\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n# Access the top element of the stack\npeek: int = stack[-1]\n\n# Pop an element from the stack\npop: int = stack.pop()\n\n# Get the length of the stack\nsize: int = len(stack)\n\n# Check if the stack is empty\nis_empty: bool = len(stack) == 0\n</code></pre> stack.cpp<pre><code>/* Initialize the stack */\nstack&lt;int&gt; stack;\n\n/* Push elements onto the stack */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access the top element of the stack */\nint top = stack.top();\n\n/* Pop an element from the stack */\nstack.pop(); // No return value\n\n/* Get the length of the stack */\nint size = stack.size();\n\n/* Check if the stack is empty */\nbool empty = stack.empty();\n</code></pre> stack.java<pre><code>/* Initialize the stack */\nStack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n\n/* Push elements onto the stack */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access the top element of the stack */\nint peek = stack.peek();\n\n/* Pop an element from the stack */\nint pop = stack.pop();\n\n/* Get the length of the stack */\nint size = stack.size();\n\n/* Check if the stack is empty */\nboolean isEmpty = stack.isEmpty();\n</code></pre> stack.cs<pre><code>/* Initialize the stack */\nStack&lt;int&gt; stack = new();\n\n/* Push elements onto the stack */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n\n/* Access the top element of the stack */\nint peek = stack.Peek();\n\n/* Pop an element from the stack */\nint pop = stack.Pop();\n\n/* Get the length of the stack */\nint size = stack.Count;\n\n/* Check if the stack is empty */\nbool isEmpty = stack.Count == 0;\n</code></pre> stack_test.go<pre><code>/* Initialize the stack */\n// In Go, it is recommended to use a Slice as a stack\nvar stack []int\n\n/* Push elements onto the stack */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n\n/* Access the top element of the stack */\npeek := stack[len(stack)-1]\n\n/* Pop an element from the stack */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n\n/* Get the length of the stack */\nsize := len(stack)\n\n/* Check if the stack is empty */\nisEmpty := len(stack) == 0\n</code></pre> stack.swift<pre><code>/* Initialize the stack */\n// Swift does not have a built-in stack class, so Array can be used as a stack\nvar stack: [Int] = []\n\n/* Push elements onto the stack */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n/* Access the top element of the stack */\nlet peek = stack.last!\n\n/* Pop an element from the stack */\nlet pop = stack.removeLast()\n\n/* Get the length of the stack */\nlet size = stack.count\n\n/* Check if the stack is empty */\nlet isEmpty = stack.isEmpty\n</code></pre> stack.js<pre><code>/* Initialize the stack */\n// JavaScript does not have a built-in stack class, so Array can be used as a stack\nconst stack = [];\n\n/* Push elements onto the stack */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access the top element of the stack */\nconst peek = stack[stack.length-1];\n\n/* Pop an element from the stack */\nconst pop = stack.pop();\n\n/* Get the length of the stack */\nconst size = stack.length;\n\n/* Check if the stack is empty */\nconst is_empty = stack.length === 0;\n</code></pre> stack.ts<pre><code>/* Initialize the stack */\n// TypeScript does not have a built-in stack class, so Array can be used as a stack\nconst stack: number[] = [];\n\n/* Push elements onto the stack */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access the top element of the stack */\nconst peek = stack[stack.length - 1];\n\n/* Pop an element from the stack */\nconst pop = stack.pop();\n\n/* Get the length of the stack */\nconst size = stack.length;\n\n/* Check if the stack is empty */\nconst is_empty = stack.length === 0;\n</code></pre> stack.dart<pre><code>/* Initialize the stack */\n// Dart does not have a built-in stack class, so List can be used as a stack\nList&lt;int&gt; stack = [];\n\n/* Push elements onto the stack */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n\n/* Access the top element of the stack */\nint peek = stack.last;\n\n/* Pop an element from the stack */\nint pop = stack.removeLast();\n\n/* Get the length of the stack */\nint size = stack.length;\n\n/* Check if the stack is empty */\nbool isEmpty = stack.isEmpty;\n</code></pre> stack.rs<pre><code>/* Initialize the stack */\n// Use Vec as a stack\nlet mut stack: Vec&lt;i32&gt; = Vec::new();\n\n/* Push elements onto the stack */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access the top element of the stack */\nlet top = stack.last().unwrap();\n\n/* Pop an element from the stack */\nlet pop = stack.pop().unwrap();\n\n/* Get the length of the stack */\nlet size = stack.len();\n\n/* Check if the stack is empty */\nlet is_empty = stack.is_empty();\n</code></pre> stack.c<pre><code>// C does not provide a built-in stack\n</code></pre> stack.kt<pre><code>\n</code></pre> stack.zig<pre><code>\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#512-implementing-a-stack","title":"5.1.2 \u00a0 Implementing a Stack","text":"<p>To gain a deeper understanding of how a stack operates, let's try implementing a stack class ourselves.</p> <p>A stack follows the principle of Last-In-First-Out, which means we can only add or remove elements at the top of the stack. However, both arrays and linked lists allow adding and removing elements at any position, therefore a stack can be seen as a restricted array or linked list. In other words, we can \"shield\" certain irrelevant operations of an array or linked list, aligning their external behavior with the characteristics of a stack.</p>"},{"location":"chapter_stack_and_queue/stack/#1-implementation-based-on-linked-list","title":"1. \u00a0 Implementation Based on Linked List","text":"<p>When implementing a stack using a linked list, we can consider the head node of the list as the top of the stack and the tail node as the bottom of the stack.</p> <p>As shown in the Figure 5-2 , for the push operation, we simply insert elements at the head of the linked list. This method of node insertion is known as \"head insertion.\" For the pop operation, we just need to remove the head node from the list.</p> LinkedListStackpush()pop() <p></p> <p></p> <p></p> <p> Figure 5-2 \u00a0 Implementing Stack with Linked List for Push and Pop Operations </p> <p>Below is an example code for implementing a stack based on a linked list:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_stack.py<pre><code>class LinkedListStack:\n    \"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._peek: ListNode | None = None\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return not self._peek\n\n    def push(self, val: int):\n        \"\"\"\u5165\u6808\"\"\"\n        node = ListNode(val)\n        node.next = self._peek\n        self._peek = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u6808\"\"\"\n        num = self.peek()\n        self._peek = self._peek.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u6808\u4e3a\u7a7a\")\n        return self._peek.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        arr = []\n        node = self._peek\n        while node:\n            arr.append(node.val)\n            node = node.next\n        arr.reverse()\n        return arr\n</code></pre> linkedlist_stack.cpp<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n  private:\n    ListNode *stackTop; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    int stkSize;        // \u6808\u7684\u957f\u5ea6\n\n  public:\n    LinkedListStack() {\n        stackTop = nullptr;\n        stkSize = 0;\n    }\n\n    ~LinkedListStack() {\n        // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n        freeMemoryLinkedList(stackTop);\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    int size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    void push(int num) {\n        ListNode *node = new ListNode(num);\n        node-&gt;next = stackTop;\n        stackTop = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    int pop() {\n        int num = top();\n        ListNode *tmp = stackTop;\n        stackTop = stackTop-&gt;next;\n        // \u91ca\u653e\u5185\u5b58\n        delete tmp;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"\u6808\u4e3a\u7a7a\");\n        return stackTop-&gt;val;\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = stackTop;\n        vector&lt;int&gt; res(size());\n        for (int i = res.size() - 1; i &gt;= 0; i--) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_stack.java<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    private ListNode stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private int stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void push(int num) {\n        ListNode node = new ListNode(num);\n        node.next = stackPeek;\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    public int pop() {\n        int num = peek();\n        stackPeek = stackPeek.next;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stackPeek.val;\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] toArray() {\n        ListNode node = stackPeek;\n        int[] res = new int[size()];\n        for (int i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.cs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    ListNode? stackPeek;  // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    int stkSize = 0;   // \u6808\u7684\u957f\u5ea6\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int Size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void Push(int num) {\n        ListNode node = new(num) {\n            next = stackPeek\n        };\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    public int Pop() {\n        int num = Peek();\n        stackPeek = stackPeek!.next;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stackPeek!.val;\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] ToArray() {\n        if (stackPeek == null)\n            return [];\n\n        ListNode? node = stackPeek;\n        int[] res = new int[Size()];\n        for (int i = res.Length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.go<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntype linkedListStack struct {\n    // \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u6808\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u6808 */\nfunc newLinkedListStack() *linkedListStack {\n    return &amp;linkedListStack{\n        data: list.New(),\n    }\n}\n\n/* \u5165\u6808 */\nfunc (s *linkedListStack) push(value int) {\n    s.data.PushBack(value)\n}\n\n/* \u51fa\u6808 */\nfunc (s *linkedListStack) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc (s *linkedListStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc (s *linkedListStack) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListStack) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListStack) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_stack.swift<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    private var _peek: ListNode? // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private var _size: Int // \u6808\u7684\u957f\u5ea6\n\n    init() {\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u6808 */\n    func push(num: Int) {\n        let node = ListNode(x: num)\n        node.next = _peek\n        _peek = node\n        _size += 1\n    }\n\n    /* \u51fa\u6808 */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        _peek = _peek?.next\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u6808\u4e3a\u7a7a\")\n        }\n        return _peek!.val\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        var node = _peek\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices.reversed() {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.js<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    #stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    #stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n    constructor() {\n        this.#stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num) {\n        const node = new ListNode(num);\n        node.next = this.#stackPeek;\n        this.#stackPeek = node;\n        this.#stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    pop() {\n        const num = this.peek();\n        this.#stackPeek = this.#stackPeek.next;\n        this.#stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    peek() {\n        if (!this.#stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.#stackPeek.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray() {\n        let node = this.#stackPeek;\n        const res = new Array(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.ts<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    private stackPeek: ListNode | null; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private stkSize: number = 0; // \u6808\u7684\u957f\u5ea6\n\n    constructor() {\n        this.stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num: number): void {\n        const node = new ListNode(num);\n        node.next = this.stackPeek;\n        this.stackPeek = node;\n        this.stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    pop(): number {\n        const num = this.peek();\n        if (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n        this.stackPeek = this.stackPeek.next;\n        this.stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    peek(): number {\n        if (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.stackPeek.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray(): number[] {\n        let node = this.stackPeek;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.dart<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u7c7b\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n  ListNode? _stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n  int _stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n  LinkedListStack() {\n    _stackPeek = null;\n  }\n\n  /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n  int size() {\n    return _stkSize;\n  }\n\n  /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _stkSize == 0;\n  }\n\n  /* \u5165\u6808 */\n  void push(int _num) {\n    final ListNode node = ListNode(_num);\n    node.next = _stackPeek;\n    _stackPeek = node;\n    _stkSize++;\n  }\n\n  /* \u51fa\u6808 */\n  int pop() {\n    final int _num = peek();\n    _stackPeek = _stackPeek!.next;\n    _stkSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n  int peek() {\n    if (_stackPeek == null) {\n      throw Exception(\"\u6808\u4e3a\u7a7a\");\n    }\n    return _stackPeek!.val;\n  }\n\n  /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a List \u5e76\u8fd4\u56de */\n  List&lt;int&gt; toList() {\n    ListNode? node = _stackPeek;\n    List&lt;int&gt; list = [];\n    while (node != null) {\n      list.add(node.val);\n      node = node.next;\n    }\n    list = list.reversed.toList();\n    return list;\n  }\n}\n</code></pre> linkedlist_stack.rs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\n#[allow(dead_code)]\npub struct LinkedListStack&lt;T&gt; {\n    stack_peek: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    stk_size: usize,                              // \u6808\u7684\u957f\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListStack&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            stack_peek: None,\n            stk_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.stk_size;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    pub fn push(&amp;mut self, num: T) {\n        let node = ListNode::new(num);\n        node.borrow_mut().next = self.stack_peek.take();\n        self.stack_peek = Some(node);\n        self.stk_size += 1;\n    }\n\n    /* \u51fa\u6808 */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack_peek.take().map(|old_head| {\n            match old_head.borrow_mut().next.take() {\n                Some(new_head) =&gt; {\n                    self.stack_peek = Some(new_head);\n                }\n                None =&gt; {\n                    self.stack_peek = None;\n                }\n            }\n            self.stk_size -= 1;\n            Rc::try_unwrap(old_head).ok().unwrap().into_inner().val\n        })\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.stack_peek.as_ref()\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.push(node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_stack.c<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\n    ListNode *top; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    int size;      // \u6808\u7684\u957f\u5ea6\n} LinkedListStack;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListStack *newLinkedListStack() {\n    LinkedListStack *s = malloc(sizeof(LinkedListStack));\n    s-&gt;top = NULL;\n    s-&gt;size = 0;\n    return s;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListStack(LinkedListStack *s) {\n    while (s-&gt;top) {\n        ListNode *n = s-&gt;top-&gt;next;\n        free(s-&gt;top);\n        s-&gt;top = n;\n    }\n    free(s);\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(LinkedListStack *s) {\n    return s-&gt;size;\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(LinkedListStack *s) {\n    return size(s) == 0;\n}\n\n/* \u5165\u6808 */\nvoid push(LinkedListStack *s, int num) {\n    ListNode *node = (ListNode *)malloc(sizeof(ListNode));\n    node-&gt;next = s-&gt;top; // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6307\u9488\u57df\n    node-&gt;val = num;     // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6570\u636e\u57df\n    s-&gt;top = node;       // \u66f4\u65b0\u6808\u9876\n    s-&gt;size++;           // \u66f4\u65b0\u6808\u5927\u5c0f\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(LinkedListStack *s) {\n    if (s-&gt;size == 0) {\n        printf(\"\u6808\u4e3a\u7a7a\\n\");\n        return INT_MAX;\n    }\n    return s-&gt;top-&gt;val;\n}\n\n/* \u51fa\u6808 */\nint pop(LinkedListStack *s) {\n    int val = peek(s);\n    ListNode *tmp = s-&gt;top;\n    s-&gt;top = s-&gt;top-&gt;next;\n    // \u91ca\u653e\u5185\u5b58\n    free(tmp);\n    s-&gt;size--;\n    return val;\n}\n</code></pre> linkedlist_stack.kt<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack(\n    private var stackPeek: ListNode? = null, // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private var stkSize: Int = 0 // \u6808\u7684\u957f\u5ea6\n) {\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return stkSize\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u6808 */\n    fun push(num: Int) {\n        val node = ListNode(num)\n        node.next = stackPeek\n        stackPeek = node\n        stkSize++\n    }\n\n    /* \u51fa\u6808 */\n    fun pop(): Int? {\n        val num = peek()\n        stackPeek = stackPeek?.next\n        stkSize--;\n        return num\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    fun peek(): Int? {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stackPeek?.value\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    fun toArray(): IntArray {\n        var node = stackPeek\n        val res = IntArray(size())\n        for (i in res.size - 1 downTo 0) {\n            res[i] = node?.value!!\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.rb<pre><code>[class]{LinkedListStack}-[func]{}\n</code></pre> linkedlist_stack.zig<pre><code>// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\nfn LinkedListStack(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        stack_top: ?*inc.ListNode(T) = null,             // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n        stk_size: usize = 0,                             // \u6808\u7684\u957f\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,    // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.stack_top = null;\n            self.stk_size = 0;\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u6808\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.stk_size;\n        }\n\n        // \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8bbf\u95ee\u6808\u9876\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.size() == 0) @panic(\"\u6808\u4e3a\u7a7a\");\n            return self.stack_top.?.val;\n        }  \n\n        // \u5165\u6808\n        pub fn push(self: *Self, num: T) !void {\n            var node = try self.mem_allocator.create(inc.ListNode(T));\n            node.init(num);\n            node.next = self.stack_top;\n            self.stack_top = node;\n            self.stk_size += 1;\n        } \n\n        // \u51fa\u6808\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            self.stack_top = self.stack_top.?.next;\n            self.stk_size -= 1;\n            return num;\n        } \n\n        // \u5c06\u6808\u8f6c\u6362\u4e3a\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.stack_top;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[res.len - i - 1] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#2-implementation-based-on-array","title":"2. \u00a0 Implementation Based on Array","text":"<p>When implementing a stack using an array, we can consider the end of the array as the top of the stack. As shown in the Figure 5-3 , push and pop operations correspond to adding and removing elements at the end of the array, respectively, both with a time complexity of \\(O(1)\\).</p> ArrayStackpush()pop() <p></p> <p></p> <p></p> <p> Figure 5-3 \u00a0 Implementing Stack with Array for Push and Pop Operations </p> <p>Since the elements to be pushed onto the stack may continuously increase, we can use a dynamic array, thus avoiding the need to handle array expansion ourselves. Here is an example code:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_stack.py<pre><code>class ArrayStack:\n    \"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._stack: list[int] = []\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\n        return len(self._stack)\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self._stack == []\n\n    def push(self, item: int):\n        \"\"\"\u5165\u6808\"\"\"\n        self._stack.append(item)\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u6808\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u6808\u4e3a\u7a7a\")\n        return self._stack.pop()\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u6808\u4e3a\u7a7a\")\n        return self._stack[-1]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        return self._stack\n</code></pre> array_stack.cpp<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n  private:\n    vector&lt;int&gt; stack;\n\n  public:\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    int size() {\n        return stack.size();\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return stack.size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    void push(int num) {\n        stack.push_back(num);\n    }\n\n    /* \u51fa\u6808 */\n    int pop() {\n        int num = top();\n        stack.pop_back();\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"\u6808\u4e3a\u7a7a\");\n        return stack.back();\n    }\n\n    /* \u8fd4\u56de Vector */\n    vector&lt;int&gt; toVector() {\n        return stack;\n    }\n};\n</code></pre> array_stack.java<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    private ArrayList&lt;Integer&gt; stack;\n\n    public ArrayStack() {\n        // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n        stack = new ArrayList&lt;&gt;();\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int size() {\n        return stack.size();\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void push(int num) {\n        stack.add(num);\n    }\n\n    /* \u51fa\u6808 */\n    public int pop() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.remove(size() - 1);\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.get(size() - 1);\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public Object[] toArray() {\n        return stack.toArray();\n    }\n}\n</code></pre> array_stack.cs<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    List&lt;int&gt; stack;\n    public ArrayStack() {\n        // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n        stack = [];\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int Size() {\n        return stack.Count;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void Push(int num) {\n        stack.Add(num);\n    }\n\n    /* \u51fa\u6808 */\n    public int Pop() {\n        if (IsEmpty())\n            throw new Exception();\n        var val = Peek();\n        stack.RemoveAt(Size() - 1);\n        return val;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stack[Size() - 1];\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] ToArray() {\n        return [.. stack];\n    }\n}\n</code></pre> array_stack.go<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntype arrayStack struct {\n    data []int // \u6570\u636e\n}\n\n/* \u521d\u59cb\u5316\u6808 */\nfunc newArrayStack() *arrayStack {\n    return &amp;arrayStack{\n        // \u8bbe\u7f6e\u6808\u7684\u957f\u5ea6\u4e3a 0\uff0c\u5bb9\u91cf\u4e3a 16\n        data: make([]int, 0, 16),\n    }\n}\n\n/* \u6808\u7684\u957f\u5ea6 */\nfunc (s *arrayStack) size() int {\n    return len(s.data)\n}\n\n/* \u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *arrayStack) isEmpty() bool {\n    return s.size() == 0\n}\n\n/* \u5165\u6808 */\nfunc (s *arrayStack) push(v int) {\n    // \u5207\u7247\u4f1a\u81ea\u52a8\u6269\u5bb9\n    s.data = append(s.data, v)\n}\n\n/* \u51fa\u6808 */\nfunc (s *arrayStack) pop() any {\n    val := s.peek()\n    s.data = s.data[:len(s.data)-1]\n    return val\n}\n\n/* \u83b7\u53d6\u6808\u9876\u5143\u7d20 */\nfunc (s *arrayStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    val := s.data[len(s.data)-1]\n    return val\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (s *arrayStack) toSlice() []int {\n    return s.data\n}\n</code></pre> array_stack.swift<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    private var stack: [Int]\n\n    init() {\n        // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n        stack = []\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        stack.count\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        stack.isEmpty\n    }\n\n    /* \u5165\u6808 */\n    func push(num: Int) {\n        stack.append(num)\n    }\n\n    /* \u51fa\u6808 */\n    @discardableResult\n    func pop() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u6808\u4e3a\u7a7a\")\n        }\n        return stack.removeLast()\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u6808\u4e3a\u7a7a\")\n        }\n        return stack.last!\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        stack\n    }\n}\n</code></pre> array_stack.js<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    #stack;\n    constructor() {\n        this.#stack = [];\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#stack.length;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#stack.length === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num) {\n        this.#stack.push(num);\n    }\n\n    /* \u51fa\u6808 */\n    pop() {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.#stack.pop();\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    top() {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.#stack[this.#stack.length - 1];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        return this.#stack;\n    }\n}\n</code></pre> array_stack.ts<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    private stack: number[];\n    constructor() {\n        this.stack = [];\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.stack.length;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.stack.length === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num: number): void {\n        this.stack.push(num);\n    }\n\n    /* \u51fa\u6808 */\n    pop(): number | undefined {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.stack.pop();\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    top(): number | undefined {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.stack[this.stack.length - 1];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        return this.stack;\n    }\n}\n</code></pre> array_stack.dart<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n  late List&lt;int&gt; _stack;\n  ArrayStack() {\n    _stack = [];\n  }\n\n  /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n  int size() {\n    return _stack.length;\n  }\n\n  /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _stack.isEmpty;\n  }\n\n  /* \u5165\u6808 */\n  void push(int _num) {\n    _stack.add(_num);\n  }\n\n  /* \u51fa\u6808 */\n  int pop() {\n    if (isEmpty()) {\n      throw Exception(\"\u6808\u4e3a\u7a7a\");\n    }\n    return _stack.removeLast();\n  }\n\n  /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"\u6808\u4e3a\u7a7a\");\n    }\n    return _stack.last;\n  }\n\n  /* \u5c06\u6808\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n  List&lt;int&gt; toArray() =&gt; _stack;\n}\n</code></pre> array_stack.rs<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nstruct ArrayStack&lt;T&gt; {\n    stack: Vec&lt;T&gt;,\n}\n\nimpl&lt;T&gt; ArrayStack&lt;T&gt; {\n    /* \u521d\u59cb\u5316\u6808 */\n    fn new() -&gt; ArrayStack&lt;T&gt; {\n        ArrayStack::&lt;T&gt; {\n            stack: Vec::&lt;T&gt;::new(),\n        }\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    fn size(&amp;self) -&gt; usize {\n        self.stack.len()\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.size() == 0\n    }\n\n    /* \u5165\u6808 */\n    fn push(&amp;mut self, num: T) {\n        self.stack.push(num);\n    }\n\n    /* \u51fa\u6808 */\n    fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack.pop()\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    fn peek(&amp;self) -&gt; Option&lt;&amp;T&gt; {\n        if self.is_empty() {\n            panic!(\"\u6808\u4e3a\u7a7a\")\n        };\n        self.stack.last()\n    }\n\n    /* \u8fd4\u56de &amp;Vec */\n    fn to_array(&amp;self) -&gt; &amp;Vec&lt;T&gt; {\n        &amp;self.stack\n    }\n}\n</code></pre> array_stack.c<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\n    int *data;\n    int size;\n} ArrayStack;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayStack *newArrayStack() {\n    ArrayStack *stack = malloc(sizeof(ArrayStack));\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5927\u5bb9\u91cf\uff0c\u907f\u514d\u6269\u5bb9\n    stack-&gt;data = malloc(sizeof(int) * MAX_SIZE);\n    stack-&gt;size = 0;\n    return stack;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayStack(ArrayStack *stack) {\n    free(stack-&gt;data);\n    free(stack);\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(ArrayStack *stack) {\n    return stack-&gt;size;\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(ArrayStack *stack) {\n    return stack-&gt;size == 0;\n}\n\n/* \u5165\u6808 */\nvoid push(ArrayStack *stack, int num) {\n    if (stack-&gt;size == MAX_SIZE) {\n        printf(\"\u6808\u5df2\u6ee1\\n\");\n        return;\n    }\n    stack-&gt;data[stack-&gt;size] = num;\n    stack-&gt;size++;\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(ArrayStack *stack) {\n    if (stack-&gt;size == 0) {\n        printf(\"\u6808\u4e3a\u7a7a\\n\");\n        return INT_MAX;\n    }\n    return stack-&gt;data[stack-&gt;size - 1];\n}\n\n/* \u51fa\u6808 */\nint pop(ArrayStack *stack) {\n    int val = peek(stack);\n    stack-&gt;size--;\n    return val;\n}\n</code></pre> array_stack.kt<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n    private val stack = ArrayList&lt;Int&gt;()\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return stack.size\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u6808 */\n    fun push(num: Int) {\n        stack.add(num)\n    }\n\n    /* \u51fa\u6808 */\n    fun pop(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack.removeAt(size() - 1)\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack[size() - 1]\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    fun toArray(): Array&lt;Any&gt; {\n        return stack.toArray()\n    }\n}\n</code></pre> array_stack.rb<pre><code>[class]{ArrayStack}-[func]{}\n</code></pre> array_stack.zig<pre><code>// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\nfn ArrayStack(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        stack: ?std.ArrayList(T) = null,     \n\n        // \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) void {\n            if (self.stack == null) {\n                self.stack = std.ArrayList(T).init(allocator);\n            }\n        }\n\n        // \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.stack == null) return;\n            self.stack.?.deinit();\n        }\n\n        // \u83b7\u53d6\u6808\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.stack.?.items.len;\n        }\n\n        // \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8bbf\u95ee\u6808\u9876\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u6808\u4e3a\u7a7a\");\n            return self.stack.?.items[self.size() - 1];\n        }  \n\n        // \u5165\u6808\n        pub fn push(self: *Self, num: T) !void {\n            try self.stack.?.append(num);\n        } \n\n        // \u51fa\u6808\n        pub fn pop(self: *Self) T {\n            var num = self.stack.?.pop();\n            return num;\n        } \n\n        // \u8fd4\u56de ArrayList\n        pub fn toList(self: *Self) std.ArrayList(T) {\n            return self.stack.?;\n        }\n    };\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#513-comparison-of-the-two-implementations","title":"5.1.3 \u00a0 Comparison of the Two Implementations","text":"<p>Supported Operations</p> <p>Both implementations support all the operations defined in a stack. The array implementation additionally supports random access, but this is beyond the scope of a stack definition and is generally not used.</p> <p>Time Efficiency</p> <p>In the array-based implementation, both push and pop operations occur in pre-allocated contiguous memory, which has good cache locality and therefore higher efficiency. However, if the push operation exceeds the array capacity, it triggers a resizing mechanism, making the time complexity of that push operation \\(O(n)\\).</p> <p>In the linked list implementation, list expansion is very flexible, and there is no efficiency decrease issue as in array expansion. However, the push operation requires initializing a node object and modifying pointers, so its efficiency is relatively lower. If the elements being pushed are already node objects, then the initialization step can be skipped, improving efficiency.</p> <p>Thus, when the elements for push and pop operations are basic data types like <code>int</code> or <code>double</code>, we can draw the following conclusions:</p> <ul> <li>The array-based stack implementation's efficiency decreases during expansion, but since expansion is a low-frequency operation, its average efficiency is higher.</li> <li>The linked list-based stack implementation provides more stable efficiency performance.</li> </ul> <p>Space Efficiency</p> <p>When initializing a list, the system allocates an \"initial capacity,\" which might exceed the actual need; moreover, the expansion mechanism usually increases capacity by a specific factor (like doubling), which may also exceed the actual need. Therefore, the array-based stack might waste some space.</p> <p>However, since linked list nodes require extra space for storing pointers, the space occupied by linked list nodes is relatively larger.</p> <p>In summary, we cannot simply determine which implementation is more memory-efficient. It requires analysis based on specific circumstances.</p>"},{"location":"chapter_stack_and_queue/stack/#514-typical-applications-of-stack","title":"5.1.4 \u00a0 Typical Applications of Stack","text":"<ul> <li>Back and forward in browsers, undo and redo in software. Every time we open a new webpage, the browser pushes the previous page onto the stack, allowing us to go back to the previous page through the back operation, which is essentially a pop operation. To support both back and forward, two stacks are needed to work together.</li> <li>Memory management in programs. Each time a function is called, the system adds a stack frame at the top of the stack to record the function's context information. In recursive functions, the downward recursion phase keeps pushing onto the stack, while the upward backtracking phase keeps popping from the stack.</li> </ul>"},{"location":"chapter_stack_and_queue/summary/","title":"5.4 \u00a0 Summary","text":""},{"location":"chapter_stack_and_queue/summary/#1-key-review","title":"1. \u00a0 Key Review","text":"<ul> <li>Stack is a data structure that follows the Last-In-First-Out (LIFO) principle and can be implemented using arrays or linked lists.</li> <li>In terms of time efficiency, the array implementation of the stack has a higher average efficiency. However, during expansion, the time complexity for a single push operation can degrade to \\(O(n)\\). In contrast, the linked list implementation of a stack offers more stable efficiency.</li> <li>Regarding space efficiency, the array implementation of the stack may lead to a certain degree of space wastage. However, it's important to note that the memory space occupied by nodes in a linked list is generally larger than that for elements in an array.</li> <li>A queue is a data structure that follows the First-In-First-Out (FIFO) principle, and it can also be implemented using arrays or linked lists. The conclusions regarding time and space efficiency for queues are similar to those for stacks.</li> <li>A double-ended queue (deque) is a more flexible type of queue that allows adding and removing elements at both ends.</li> </ul>"},{"location":"chapter_stack_and_queue/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: Is the browser's forward and backward functionality implemented with a doubly linked list?</p> <p>A browser's forward and backward navigation is essentially a manifestation of the \"stack\" concept. When a user visits a new page, the page is added to the top of the stack; when they click the back button, the page is popped from the top of the stack. A double-ended queue (deque) can conveniently implement some additional operations, as mentioned in the \"Double-Ended Queue\" section.</p> <p>Q: After popping from a stack, is it necessary to free the memory of the popped node?</p> <p>If the popped node will still be used later, it's not necessary to free its memory. In languages like Java and Python that have automatic garbage collection, manual memory release is not necessary; in C and C++, manual memory release is required.</p> <p>Q: A double-ended queue seems like two stacks joined together. What are its uses?</p> <p>A double-ended queue, which is a combination of a stack and a queue or two stacks joined together, exhibits both stack and queue logic. Thus, it can implement all applications of stacks and queues while offering more flexibility.</p> <p>Q: How exactly are undo and redo implemented?</p> <p>Undo and redo operations are implemented using two stacks: Stack A for undo and Stack B for redo.</p> <ol> <li>Each time a user performs an operation, it is pushed onto Stack A, and Stack B is cleared.</li> <li>When the user executes an \"undo\", the most recent operation is popped from Stack A and pushed onto Stack B.</li> <li>When the user executes a \"redo\", the most recent operation is popped from Stack B and pushed back onto Stack A.</li> </ol>"},{"location":"chapter_tree/","title":"Chapter 7. \u00a0 Tree","text":"<p>Abstract</p> <p>The towering tree, full of vitality with its roots deep and leaves lush, branches spreading wide.</p> <p>It vividly demonstrates the form of data divide-and-conquer.</p>"},{"location":"chapter_tree/#chapter-contents","title":"Chapter Contents","text":"<ul> <li>7.1 \u00a0 Binary Tree</li> <li>7.2 \u00a0 Binary Tree Traversal</li> <li>7.3 \u00a0 Array Representation of Tree</li> <li>7.4 \u00a0 Binary Search Tree</li> <li>7.5 \u00a0 AVL Tree *</li> <li>7.6 \u00a0 Summary</li> </ul>"},{"location":"chapter_tree/array_representation_of_tree/","title":"7.3 \u00a0 Array representation of binary trees","text":"<p>Under the linked list representation, the storage unit of a binary tree is a node <code>TreeNode</code>, with nodes connected by pointers. The basic operations of binary trees under the linked list representation were introduced in the previous section.</p> <p>So, can we use an array to represent a binary tree? The answer is yes.</p>"},{"location":"chapter_tree/array_representation_of_tree/#731-representing-perfect-binary-trees","title":"7.3.1 \u00a0 Representing perfect binary trees","text":"<p>Let's analyze a simple case first. Given a perfect binary tree, we store all nodes in an array according to the order of level-order traversal, where each node corresponds to a unique array index.</p> <p>Based on the characteristics of level-order traversal, we can deduce a \"mapping formula\" between the index of a parent node and its children: If a node's index is \\(i\\), then the index of its left child is \\(2i + 1\\) and the right child is \\(2i + 2\\). The Figure 7-12  shows the mapping relationship between the indices of various nodes.</p> <p></p> <p> Figure 7-12 \u00a0 Array representation of a perfect binary tree </p> <p>The mapping formula plays a role similar to the node references (pointers) in linked lists. Given any node in the array, we can access its left (right) child node using the mapping formula.</p>"},{"location":"chapter_tree/array_representation_of_tree/#732-representing-any-binary-tree","title":"7.3.2 \u00a0 Representing any binary tree","text":"<p>Perfect binary trees are a special case; there are often many <code>None</code> values in the middle levels of a binary tree. Since the sequence of level-order traversal does not include these <code>None</code> values, we cannot solely rely on this sequence to deduce the number and distribution of <code>None</code> values. This means that multiple binary tree structures can match the same level-order traversal sequence.</p> <p>As shown in the Figure 7-13 , given a non-perfect binary tree, the above method of array representation fails.</p> <p></p> <p> Figure 7-13 \u00a0 Level-order traversal sequence corresponds to multiple binary tree possibilities </p> <p>To solve this problem, we can consider explicitly writing out all <code>None</code> values in the level-order traversal sequence. As shown in the following figure, after this treatment, the level-order traversal sequence can uniquely represent a binary tree. Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># Array representation of a binary tree\n# Using None to represent empty slots\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using the maximum integer value INT_MAX to mark empty slots\nvector&lt;int&gt; tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using the Integer wrapper class allows for using null to mark empty slots\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using nullable int (int?) allows for using null to mark empty slots\nint?[] tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using an any type slice, allowing for nil to mark empty slots\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using optional Int (Int?) allows for using nil to mark empty slots\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using null to represent empty slots\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using null to represent empty slots\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using nullable int (int?) allows for using null to mark empty slots\nList&lt;int?&gt; tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using None to mark empty slots\nlet tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using the maximum int value to mark empty slots, therefore, node values must not be INT_MAX\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using null to represent empty slots\nval tree = mutableListOf( 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 )\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p></p> <p> Figure 7-14 \u00a0 Array representation of any type of binary tree </p> <p>It's worth noting that complete binary trees are very suitable for array representation. Recalling the definition of a complete binary tree, <code>None</code> appears only at the bottom level and towards the right, meaning all <code>None</code> values definitely appear at the end of the level-order traversal sequence.</p> <p>This means that when using an array to represent a complete binary tree, it's possible to omit storing all <code>None</code> values, which is very convenient. The Figure 7-15  gives an example.</p> <p></p> <p> Figure 7-15 \u00a0 Array representation of a complete binary tree </p> <p>The following code implements a binary tree based on array representation, including the following operations:</p> <ul> <li>Given a node, obtain its value, left (right) child node, and parent node.</li> <li>Obtain the preorder, inorder, postorder, and level-order traversal sequences.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_binary_tree.py<pre><code>class ArrayBinaryTree:\n    \"\"\"\u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b\"\"\"\n\n    def __init__(self, arr: list[int | None]):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._tree = list(arr)\n\n    def size(self):\n        \"\"\"\u5217\u8868\u5bb9\u91cf\"\"\"\n        return len(self._tree)\n\n    def val(self, i: int) -&gt; int:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c\"\"\"\n        # \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 or i &gt;= self.size():\n            return None\n        return self._tree[i]\n\n    def left(self, i: int) -&gt; int | None:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n        return 2 * i + 1\n\n    def right(self, i: int) -&gt; int | None:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n        return 2 * i + 2\n\n    def parent(self, i: int) -&gt; int | None:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n        return (i - 1) // 2\n\n    def level_order(self) -&gt; list[int]:\n        \"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for i in range(self.size()):\n            if self.val(i) is not None:\n                self.res.append(self.val(i))\n        return self.res\n\n    def dfs(self, i: int, order: str):\n        \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\n        if self.val(i) is None:\n            return\n        # \u524d\u5e8f\u904d\u5386\n        if order == \"pre\":\n            self.res.append(self.val(i))\n        self.dfs(self.left(i), order)\n        # \u4e2d\u5e8f\u904d\u5386\n        if order == \"in\":\n            self.res.append(self.val(i))\n        self.dfs(self.right(i), order)\n        # \u540e\u5e8f\u904d\u5386\n        if order == \"post\":\n            self.res.append(self.val(i))\n\n    def pre_order(self) -&gt; list[int]:\n        \"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        self.dfs(0, order=\"pre\")\n        return self.res\n\n    def in_order(self) -&gt; list[int]:\n        \"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        self.dfs(0, order=\"in\")\n        return self.res\n\n    def post_order(self) -&gt; list[int]:\n        \"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        self.dfs(0, order=\"post\")\n        return self.res\n</code></pre> array_binary_tree.cpp<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    ArrayBinaryTree(vector&lt;int&gt; arr) {\n        tree = arr;\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    int size() {\n        return tree.size();\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    int val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size())\n            return INT_MAX;\n        return tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    int left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    int right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    int parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; levelOrder() {\n        vector&lt;int&gt; res;\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != INT_MAX)\n                res.push_back(val(i));\n        }\n        return res;\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; preOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; inOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; postOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"post\", res);\n        return res;\n    }\n\n  private:\n    vector&lt;int&gt; tree;\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    void dfs(int i, string order, vector&lt;int&gt; &amp;res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (val(i) == INT_MAX)\n            return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order == \"pre\")\n            res.push_back(val(i));\n        dfs(left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order == \"in\")\n            res.push_back(val(i));\n        dfs(right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order == \"post\")\n            res.push_back(val(i));\n    }\n};\n</code></pre> array_binary_tree.java<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    private List&lt;Integer&gt; tree;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public ArrayBinaryTree(List&lt;Integer&gt; arr) {\n        tree = new ArrayList&lt;&gt;(arr);\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    public int size() {\n        return tree.size();\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    public Integer val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size())\n            return null;\n        return tree.get(i);\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public Integer left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public Integer right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    public Integer parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; levelOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != null)\n                res.add(val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    private void dfs(Integer i, String order, List&lt;Integer&gt; res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (val(i) == null)\n            return;\n        // \u524d\u5e8f\u904d\u5386\n        if (\"pre\".equals(order))\n            res.add(val(i));\n        dfs(left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (\"in\".equals(order))\n            res.add(val(i));\n        dfs(right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (\"post\".equals(order))\n            res.add(val(i));\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; preOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; inOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; postOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.cs<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree(List&lt;int?&gt; arr) {\n    List&lt;int?&gt; tree = new(arr);\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    public int Size() {\n        return tree.Count;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    public int? Val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= Size())\n            return null;\n        return tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public int Left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public int Right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    public int Parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; LevelOrder() {\n        List&lt;int&gt; res = [];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (int i = 0; i &lt; Size(); i++) {\n            if (Val(i).HasValue)\n                res.Add(Val(i)!.Value);\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    void DFS(int i, string order, List&lt;int&gt; res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (!Val(i).HasValue)\n            return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order == \"pre\")\n            res.Add(Val(i)!.Value);\n        DFS(Left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order == \"in\")\n            res.Add(Val(i)!.Value);\n        DFS(Right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order == \"post\")\n            res.Add(Val(i)!.Value);\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; PreOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; InOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"in\", res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; PostOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.go<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\ntype arrayBinaryTree struct {\n    tree []any\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newArrayBinaryTree(arr []any) *arrayBinaryTree {\n    return &amp;arrayBinaryTree{\n        tree: arr,\n    }\n}\n\n/* \u5217\u8868\u5bb9\u91cf */\nfunc (abt *arrayBinaryTree) size() int {\n    return len(abt.tree)\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nfunc (abt *arrayBinaryTree) val(i int) any {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if i &lt; 0 || i &gt;= abt.size() {\n        return nil\n    }\n    return abt.tree[i]\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) left(i int) int {\n    return 2*i + 1\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) right(i int) int {\n    return 2*i + 2\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) parent(i int) int {\n    return (i - 1) / 2\n}\n\n/* \u5c42\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) levelOrder() []any {\n    var res []any\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n    for i := 0; i &lt; abt.size(); i++ {\n        if abt.val(i) != nil {\n            res = append(res, abt.val(i))\n        }\n    }\n    return res\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nfunc (abt *arrayBinaryTree) dfs(i int, order string, res *[]any) {\n    // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n    if abt.val(i) == nil {\n        return\n    }\n    // \u524d\u5e8f\u904d\u5386\n    if order == \"pre\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.left(i), order, res)\n    // \u4e2d\u5e8f\u904d\u5386\n    if order == \"in\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.right(i), order, res)\n    // \u540e\u5e8f\u904d\u5386\n    if order == \"post\" {\n        *res = append(*res, abt.val(i))\n    }\n}\n\n/* \u524d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) preOrder() []any {\n    var res []any\n    abt.dfs(0, \"pre\", &amp;res)\n    return res\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) inOrder() []any {\n    var res []any\n    abt.dfs(0, \"in\", &amp;res)\n    return res\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) postOrder() []any {\n    var res []any\n    abt.dfs(0, \"post\", &amp;res)\n    return res\n}\n</code></pre> array_binary_tree.swift<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    private var tree: [Int?]\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init(arr: [Int?]) {\n        tree = arr\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    func size() -&gt; Int {\n        tree.count\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    func val(i: Int) -&gt; Int? {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 || i &gt;= size() {\n            return nil\n        }\n        return tree[i]\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    func left(i: Int) -&gt; Int {\n        2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    func right(i: Int) -&gt; Int {\n        2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    func parent(i: Int) -&gt; Int {\n        (i - 1) / 2\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    func levelOrder() -&gt; [Int] {\n        var res: [Int] = []\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for i in 0 ..&lt; size() {\n            if let val = val(i: i) {\n                res.append(val)\n            }\n        }\n        return res\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    private func dfs(i: Int, order: String, res: inout [Int]) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        guard let val = val(i: i) else {\n            return\n        }\n        // \u524d\u5e8f\u904d\u5386\n        if order == \"pre\" {\n            res.append(val)\n        }\n        dfs(i: left(i: i), order: order, res: &amp;res)\n        // \u4e2d\u5e8f\u904d\u5386\n        if order == \"in\" {\n            res.append(val)\n        }\n        dfs(i: right(i: i), order: order, res: &amp;res)\n        // \u540e\u5e8f\u904d\u5386\n        if order == \"post\" {\n            res.append(val)\n        }\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    func preOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"pre\", res: &amp;res)\n        return res\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    func inOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"in\", res: &amp;res)\n        return res\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    func postOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"post\", res: &amp;res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.js<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    #tree;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(arr) {\n        this.#tree = arr;\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    size() {\n        return this.#tree.length;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    val(i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    left(i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    right(i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    parent(i) {\n        return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    levelOrder() {\n        let res = [];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    #dfs(i, order, res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (this.val(i) === null) return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    preOrder() {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    inOrder() {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    postOrder() {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.ts<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    #tree: (number | null)[];\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(arr: (number | null)[]) {\n        this.#tree = arr;\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    size(): number {\n        return this.#tree.length;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    val(i: number): number | null {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    left(i: number): number {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    right(i: number): number {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    parent(i: number): number {\n        return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    levelOrder(): number[] {\n        let res = [];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    #dfs(i: number, order: Order, res: (number | null)[]): void {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (this.val(i) === null) return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    preOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    inOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    postOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.dart<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n  late List&lt;int?&gt; _tree;\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  ArrayBinaryTree(this._tree);\n\n  /* \u5217\u8868\u5bb9\u91cf */\n  int size() {\n    return _tree.length;\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n  int? val(int i) {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if (i &lt; 0 || i &gt;= size()) {\n      return null;\n    }\n    return _tree[i];\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n  int? left(int i) {\n    return 2 * i + 1;\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n  int? right(int i) {\n    return 2 * i + 2;\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n  int? parent(int i) {\n    return (i - 1) ~/ 2;\n  }\n\n  /* \u5c42\u5e8f\u904d\u5386 */\n  List&lt;int&gt; levelOrder() {\n    List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; size(); i++) {\n      if (val(i) != null) {\n        res.add(val(i)!);\n      }\n    }\n    return res;\n  }\n\n  /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n  void dfs(int i, String order, List&lt;int?&gt; res) {\n    // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n    if (val(i) == null) {\n      return;\n    }\n    // \u524d\u5e8f\u904d\u5386\n    if (order == 'pre') {\n      res.add(val(i));\n    }\n    dfs(left(i)!, order, res);\n    // \u4e2d\u5e8f\u904d\u5386\n    if (order == 'in') {\n      res.add(val(i));\n    }\n    dfs(right(i)!, order, res);\n    // \u540e\u5e8f\u904d\u5386\n    if (order == 'post') {\n      res.add(val(i));\n    }\n  }\n\n  /* \u524d\u5e8f\u904d\u5386 */\n  List&lt;int?&gt; preOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'pre', res);\n    return res;\n  }\n\n  /* \u4e2d\u5e8f\u904d\u5386 */\n  List&lt;int?&gt; inOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'in', res);\n    return res;\n  }\n\n  /* \u540e\u5e8f\u904d\u5386 */\n  List&lt;int?&gt; postOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'post', res);\n    return res;\n  }\n}\n</code></pre> array_binary_tree.rs<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nstruct ArrayBinaryTree {\n    tree: Vec&lt;Option&lt;i32&gt;&gt;,\n}\n\nimpl ArrayBinaryTree {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(arr: Vec&lt;Option&lt;i32&gt;&gt;) -&gt; Self {\n        Self { tree: arr }\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    fn size(&amp;self) -&gt; i32 {\n        self.tree.len() as i32\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    fn val(&amp;self, i: i32) -&gt; Option&lt;i32&gt; {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 || i &gt;= self.size() {\n            None\n        } else {\n            self.tree[i as usize]\n        }\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fn left(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fn right(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    fn parent(&amp;self, i: i32) -&gt; i32 {\n        (i - 1) / 2\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    fn level_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for i in 0..self.size() {\n            if let Some(val) = self.val(i) {\n                res.push(val)\n            }\n        }\n        res\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    fn dfs(&amp;self, i: i32, order: &amp;str, res: &amp;mut Vec&lt;i32&gt;) {\n        if self.val(i).is_none() {\n            return;\n        }\n        let val = self.val(i).unwrap();\n        // \u524d\u5e8f\u904d\u5386\n        if order == \"pre\" {\n            res.push(val);\n        }\n        self.dfs(self.left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if order == \"in\" {\n            res.push(val);\n        }\n        self.dfs(self.right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if order == \"post\" {\n            res.push(val);\n        }\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    fn pre_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"pre\", &amp;mut res);\n        res\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    fn in_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"in\", &amp;mut res);\n        res\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    fn post_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"post\", &amp;mut res);\n        res\n    }\n}\n</code></pre> array_binary_tree.c<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7ed3\u6784\u4f53 */\ntypedef struct {\n    int *tree;\n    int size;\n} ArrayBinaryTree;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayBinaryTree *newArrayBinaryTree(int *arr, int arrSize) {\n    ArrayBinaryTree *abt = (ArrayBinaryTree *)malloc(sizeof(ArrayBinaryTree));\n    abt-&gt;tree = malloc(sizeof(int) * arrSize);\n    memcpy(abt-&gt;tree, arr, sizeof(int) * arrSize);\n    abt-&gt;size = arrSize;\n    return abt;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayBinaryTree(ArrayBinaryTree *abt) {\n    free(abt-&gt;tree);\n    free(abt);\n}\n\n/* \u5217\u8868\u5bb9\u91cf */\nint size(ArrayBinaryTree *abt) {\n    return abt-&gt;size;\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nint val(ArrayBinaryTree *abt, int i) {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if (i &lt; 0 || i &gt;= size(abt))\n        return INT_MAX;\n    return abt-&gt;tree[i];\n}\n\n/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; size(abt); i++) {\n        if (val(abt, i) != INT_MAX)\n            res[index++] = val(abt, i);\n    }\n    *returnSize = index;\n    return res;\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nvoid dfs(ArrayBinaryTree *abt, int i, char *order, int *res, int *index) {\n    // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n    if (val(abt, i) == INT_MAX)\n        return;\n    // \u524d\u5e8f\u904d\u5386\n    if (strcmp(order, \"pre\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, left(i), order, res, index);\n    // \u4e2d\u5e8f\u904d\u5386\n    if (strcmp(order, \"in\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, right(i), order, res, index);\n    // \u540e\u5e8f\u904d\u5386\n    if (strcmp(order, \"post\") == 0)\n        res[(*index)++] = val(abt, i);\n}\n\n/* \u524d\u5e8f\u904d\u5386 */\nint *preOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"pre\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nint *inOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"in\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nint *postOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"post\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n</code></pre> array_binary_tree.kt<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree(val tree: List&lt;Int?&gt;) {\n    /* \u5217\u8868\u5bb9\u91cf */\n    fun size(): Int {\n        return tree.size\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    fun value(i: Int): Int? {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size()) return null\n        return tree[i]\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    fun parent(i: Int): Int {\n        return (i - 1) / 2\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    fun levelOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (i in 0..&lt;size()) {\n            if (value(i) != null) res.add(value(i))\n        }\n        return res\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    fun dfs(i: Int, order: String, res: MutableList&lt;Int?&gt;) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (value(i) == null) return\n        // \u524d\u5e8f\u904d\u5386\n        if (\"pre\" == order) res.add(value(i))\n        dfs(left(i), order, res)\n        // \u4e2d\u5e8f\u904d\u5386\n        if (\"in\" == order) res.add(value(i))\n        dfs(right(i), order, res)\n        // \u540e\u5e8f\u904d\u5386\n        if (\"post\" == order) res.add(value(i))\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    fun preOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        dfs(0, \"pre\", res)\n        return res\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    fun inOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        dfs(0, \"in\", res)\n        return res\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    fun postOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        dfs(0, \"post\", res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.rb<pre><code>[class]{ArrayBinaryTree}-[func]{}\n</code></pre> array_binary_tree.zig<pre><code>[class]{ArrayBinaryTree}-[func]{}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_tree/array_representation_of_tree/#733-advantages-and-limitations","title":"7.3.3 \u00a0 Advantages and limitations","text":"<p>The array representation of binary trees has the following advantages:</p> <ul> <li>Arrays are stored in contiguous memory spaces, which is cache-friendly and allows for faster access and traversal.</li> <li>It does not require storing pointers, which saves space.</li> <li>It allows random access to nodes.</li> </ul> <p>However, the array representation also has some limitations:</p> <ul> <li>Array storage requires contiguous memory space, so it is not suitable for storing trees with a large amount of data.</li> <li>Adding or deleting nodes requires array insertion and deletion operations, which are less efficient.</li> <li>When there are many <code>None</code> values in the binary tree, the proportion of node data contained in the array is low, leading to lower space utilization.</li> </ul>"},{"location":"chapter_tree/avl_tree/","title":"7.5 \u00a0 AVL tree *","text":"<p>In the \"Binary Search Tree\" section, we mentioned that after multiple insertions and removals, a binary search tree might degrade to a linked list. In such cases, the time complexity of all operations degrades from \\(O(\\log n)\\) to \\(O(n)\\).</p> <p>As shown in the Figure 7-24 , after two node removal operations, this binary search tree will degrade into a linked list.</p> <p></p> <p> Figure 7-24 \u00a0 Degradation of an AVL tree after removing nodes </p> <p>For example, in the perfect binary tree shown in the Figure 7-25 , after inserting two nodes, the tree will lean heavily to the left, and the time complexity of search operations will also degrade.</p> <p></p> <p> Figure 7-25 \u00a0 Degradation of an AVL tree after inserting nodes </p> <p>In 1962, G. M. Adelson-Velsky and E. M. Landis proposed the \"AVL Tree\" in their paper \"An algorithm for the organization of information\". The paper detailed a series of operations to ensure that after continuously adding and removing nodes, the AVL tree would not degrade, thus maintaining the time complexity of various operations at \\(O(\\log n)\\) level. In other words, in scenarios where frequent additions, removals, searches, and modifications are needed, the AVL tree can always maintain efficient data operation performance, which has great application value.</p>"},{"location":"chapter_tree/avl_tree/#751-common-terminology-in-avl-trees","title":"7.5.1 \u00a0 Common terminology in AVL trees","text":"<p>An AVL tree is both a binary search tree and a balanced binary tree, satisfying all properties of these two types of binary trees, hence it is a \"balanced binary search tree\".</p>"},{"location":"chapter_tree/avl_tree/#1-node-height","title":"1. \u00a0 Node height","text":"<p>Since the operations related to AVL trees require obtaining node heights, we need to add a <code>height</code> variable to the node class:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class TreeNode:\n    \"\"\"AVL tree node\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                 # Node value\n        self.height: int = 0                # Node height\n        self.left: TreeNode | None = None   # Left child reference\n        self.right: TreeNode | None = None  # Right child reference\n</code></pre> <pre><code>/* AVL tree node */\nstruct TreeNode {\n    int val{};          // Node value\n    int height = 0;     // Node height\n    TreeNode *left{};   // Left child\n    TreeNode *right{};  // Right child\n    TreeNode() = default;\n    explicit TreeNode(int x) : val(x){}\n};\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode {\n    public int val;        // Node value\n    public int height;     // Node height\n    public TreeNode left;  // Left child\n    public TreeNode right; // Right child\n    public TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode(int? x) {\n    public int? val = x;    // Node value\n    public int height;      // Node height\n    public TreeNode? left;  // Left child reference\n    public TreeNode? right; // Right child reference\n}\n</code></pre> <pre><code>/* AVL tree node */\ntype TreeNode struct {\n    Val    int       // Node value\n    Height int       // Node height\n    Left   *TreeNode // Left child reference\n    Right  *TreeNode // Right child reference\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode {\n    var val: Int // Node value\n    var height: Int // Node height\n    var left: TreeNode? // Left child\n    var right: TreeNode? // Right child\n\n    init(x: Int) {\n        val = x\n        height = 0\n    }\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode {\n    val; // Node value\n    height; // Node height\n    left; // Left child pointer\n    right; // Right child pointer\n    constructor(val, left, right, height) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode {\n    val: number;            // Node value\n    height: number;         // Node height\n    left: TreeNode | null;  // Left child pointer\n    right: TreeNode | null; // Right child pointer\n    constructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height; \n        this.left = left === undefined ? null : left; \n        this.right = right === undefined ? null : right; \n    }\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode {\n  int val;         // Node value\n  int height;      // Node height\n  TreeNode? left;  // Left child\n  TreeNode? right; // Right child\n  TreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* AVL tree node */\nstruct TreeNode {\n    val: i32,                               // Node value\n    height: i32,                            // Node height\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // Left child\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // Right child\n}\n\nimpl TreeNode {\n    /* Constructor */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            height: 0,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* AVL tree node */\nTreeNode struct TreeNode {\n    int val;\n    int height;\n    struct TreeNode *left;\n    struct TreeNode *right;\n} TreeNode;\n\n/* Constructor */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode(val _val: Int) {  // Node value\n    val height: Int = 0          // Node height\n    val left: TreeNode? = null   // Left child\n    val right: TreeNode? = null  // Right child\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>The \"node height\" refers to the distance from that node to its farthest leaf node, i.e., the number of \"edges\" passed. It is important to note that the height of a leaf node is \\(0\\), and the height of a null node is \\(-1\\). We will create two utility functions for getting and updating the height of a node:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def height(self, node: TreeNode | None) -&gt; int:\n    \"\"\"\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\"\"\"\n    # \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    if node is not None:\n        return node.height\n    return -1\n\ndef update_height(self, node: TreeNode | None):\n    \"\"\"\u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\"\"\"\n    # \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height = max([self.height(node.left), self.height(node.right)]) + 1\n</code></pre> avl_tree.cpp<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node == nullptr ? -1 : node-&gt;height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node-&gt;height = max(height(node-&gt;left), height(node-&gt;right)) + 1;\n}\n</code></pre> avl_tree.java<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint Height(TreeNode? node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid UpdateHeight(TreeNode node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height = Math.Max(Height(node.left), Height(node.right)) + 1;\n}\n</code></pre> avl_tree.go<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    if node != nil {\n        return node.Height\n    }\n    return -1\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\n    lh := t.height(node.Left)\n    rh := t.height(node.Right)\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    if lh &gt; rh {\n        node.Height = lh + 1\n    } else {\n        node.Height = rh + 1\n    }\n}\n</code></pre> avl_tree.swift<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc height(node: TreeNode?) -&gt; Int {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    node?.height ?? -1\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc updateHeight(node: TreeNode?) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n</code></pre> avl_tree.js<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\n#updateHeight(node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node: TreeNode): number {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nupdateHeight(node: TreeNode): void {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode? node) {\n  // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n  return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode? node) {\n  // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n  node!.height = max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfn height(node: OptionTreeNodeRc) -&gt; i32 {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    match node {\n        Some(node) =&gt; node.borrow().height,\n        None =&gt; -1,\n    }\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfn update_height(node: OptionTreeNodeRc) {\n    if let Some(node) = node {\n        let left = node.borrow().left.clone();\n        let right = node.borrow().right.clone();\n        // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n        node.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    if (node != NULL) {\n        return node-&gt;height;\n    }\n    return -1;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n    int lh = height(node-&gt;left);\n    int rh = height(node-&gt;right);\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    if (lh &gt; rh) {\n        node-&gt;height = lh + 1;\n    } else {\n        node-&gt;height = rh + 1;\n    }\n}\n</code></pre> avl_tree.kt<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfun height(node: TreeNode?): Int {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node?.height ?: -1\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfun updateHeight(node: TreeNode?) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node?.height = (max(height(node?.left).toDouble(), height(node?.right).toDouble()) + 1).toInt()\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{height}\n\n[class]{AVLTree}-[func]{update_height}\n</code></pre> avl_tree.zig<pre><code>// \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\nfn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n    _ = self;\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return if (node == null) -1 else node.?.height;\n}\n\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nfn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.?.height = @max(self.height(node.?.left), self.height(node.?.right)) + 1;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2-node-balance-factor","title":"2. \u00a0 Node balance factor","text":"<p>The \"balance factor\" of a node is defined as the height of the node's left subtree minus the height of its right subtree, with the balance factor of a null node defined as \\(0\\). We will also encapsulate the functionality of obtaining the node balance factor into a function for easy use later on:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def balance_factor(self, node: TreeNode | None) -&gt; int:\n    \"\"\"\u83b7\u53d6\u5e73\u8861\u56e0\u5b50\"\"\"\n    # \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if node is None:\n        return 0\n    # \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return self.height(node.left) - self.height(node.right)\n</code></pre> avl_tree.cpp<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == nullptr)\n        return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.java<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null)\n        return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.cs<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint BalanceFactor(TreeNode? node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return Height(node.left) - Height(node.right);\n}\n</code></pre> avl_tree.go<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if node == nil {\n        return 0\n    }\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return t.height(node.Left) - t.height(node.Right)\n}\n</code></pre> avl_tree.swift<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc balanceFactor(node: TreeNode?) -&gt; Int {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    guard let node = node else { return 0 }\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node: node.left) - height(node: node.right)\n}\n</code></pre> avl_tree.js<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node === null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.ts<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node: TreeNode): number {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node === null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.dart<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node) {\n  // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n  if (node == null) return 0;\n  // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n  return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.rs<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfn balance_factor(node: OptionTreeNodeRc) -&gt; i32 {\n    match node {\n        // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n        None =&gt; 0,\n        // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n        Some(node) =&gt; {\n            Self::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())\n        }\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == NULL) {\n        return 0;\n    }\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.kt<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfun balanceFactor(node: TreeNode?): Int {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null) return 0\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node.left) - height(node.right)\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{balance_factor}\n</code></pre> avl_tree.zig<pre><code>// \u83b7\u53d6\u5e73\u8861\u56e0\u5b50\nfn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return self.height(node.?.left) - self.height(node.?.right);\n}\n</code></pre> <p>Note</p> <p>Let the balance factor be \\(f\\), then the balance factor of any node in an AVL tree satisfies \\(-1 \\le f \\le 1\\).</p>"},{"location":"chapter_tree/avl_tree/#752-rotations-in-avl-trees","title":"7.5.2 \u00a0 Rotations in AVL trees","text":"<p>The characteristic feature of an AVL tree is the \"rotation\" operation, which can restore balance to an unbalanced node without affecting the in-order traversal sequence of the binary tree. In other words, the rotation operation can maintain the property of a \"binary search tree\" while also turning the tree back into a \"balanced binary tree\".</p> <p>We call nodes with an absolute balance factor \\(&gt; 1\\) \"unbalanced nodes\". Depending on the type of imbalance, there are four kinds of rotations: right rotation, left rotation, right-left rotation, and left-right rotation. Below, we detail these rotation operations.</p>"},{"location":"chapter_tree/avl_tree/#1-right-rotation","title":"1. \u00a0 Right rotation","text":"<p>As shown in the Figure 7-26 , the first unbalanced node from the bottom up in the binary tree is \"node 3\". Focusing on the subtree with this unbalanced node as the root, denoted as <code>node</code>, and its left child as <code>child</code>, perform a \"right rotation\". After the right rotation, the subtree is balanced again while still maintaining the properties of a binary search tree.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 7-26 \u00a0 Steps of right rotation </p> <p>As shown in the Figure 7-27 , when the <code>child</code> node has a right child (denoted as <code>grand_child</code>), a step needs to be added in the right rotation: set <code>grand_child</code> as the left child of <code>node</code>.</p> <p></p> <p> Figure 7-27 \u00a0 Right rotation with grand_child </p> <p>\"Right rotation\" is a figurative term; in practice, it is achieved by modifying node pointers, as shown in the following code:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def right_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u53f3\u65cb\u64cd\u4f5c\"\"\"\n    child = node.left\n    grand_child = child.right\n    # \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node\n    node.left = grand_child\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    self.update_height(child)\n    # \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;left;\n    TreeNode *grandChild = child-&gt;right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode rightRotate(TreeNode node) {\n    TreeNode child = node.left;\n    TreeNode grandChild = child.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? RightRotate(TreeNode? node) {\n    TreeNode? child = node?.left;\n    TreeNode? grandChild = child?.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\n    child := node.Left\n    grandChild := child.Right\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.Right = node\n    node.Left = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc rightRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.left\n    let grandChild = child?.right\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child?.right = node\n    node?.left = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\n#rightRotate(node) {\n    const child = node.left;\n    const grandChild = child.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nrightRotate(node: TreeNode): TreeNode {\n    const child = node.left;\n    const grandChild = child.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node) {\n  TreeNode? child = node!.left;\n  TreeNode? grandChild = child!.right;\n  // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n  child.right = node;\n  node.left = grandChild;\n  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  updateHeight(node);\n  updateHeight(child);\n  // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfn right_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().left.clone().unwrap();\n            let grand_child = child.borrow().right.clone();\n            // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n            child.borrow_mut().right = Some(node.clone());\n            node.borrow_mut().left = grand_child;\n            // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;left;\n    grandChild = child-&gt;right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfun rightRotate(node: TreeNode?): TreeNode {\n    val child = node!!.left\n    val grandChild = child!!.right\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node\n    node.left = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node)\n    updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{right_rotate}\n</code></pre> avl_tree.zig<pre><code>// \u53f3\u65cb\u64cd\u4f5c\nfn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    var child = node.?.left;\n    var grandChild = child.?.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.?.right = node;\n    node.?.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.updateHeight(node);\n    self.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2-left-rotation","title":"2. \u00a0 Left rotation","text":"<p>Correspondingly, if considering the \"mirror\" of the above unbalanced binary tree, the \"left rotation\" operation shown in the Figure 7-28  needs to be performed.</p> <p></p> <p> Figure 7-28 \u00a0 Left rotation operation </p> <p>Similarly, as shown in the Figure 7-29 , when the <code>child</code> node has a left child (denoted as <code>grand_child</code>), a step needs to be added in the left rotation: set <code>grand_child</code> as the right child of <code>node</code>.</p> <p></p> <p> Figure 7-29 \u00a0 Left rotation with grand_child </p> <p>It can be observed that the right and left rotation operations are logically symmetrical, and they solve two symmetrical types of imbalance. Based on symmetry, by replacing all <code>left</code> with <code>right</code>, and all <code>right</code> with <code>left</code> in the implementation code of right rotation, we can get the implementation code for left rotation:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def left_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u5de6\u65cb\u64cd\u4f5c\"\"\"\n    child = node.right\n    grand_child = child.left\n    # \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node\n    node.right = grand_child\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    self.update_height(child)\n    # \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;right;\n    TreeNode *grandChild = child-&gt;left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode leftRotate(TreeNode node) {\n    TreeNode child = node.right;\n    TreeNode grandChild = child.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? LeftRotate(TreeNode? node) {\n    TreeNode? child = node?.right;\n    TreeNode? grandChild = child?.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\n    child := node.Right\n    grandChild := child.Left\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.Left = node\n    node.Right = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc leftRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.right\n    let grandChild = child?.left\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child?.left = node\n    node?.right = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\n#leftRotate(node) {\n    const child = node.right;\n    const grandChild = child.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nleftRotate(node: TreeNode): TreeNode {\n    const child = node.right;\n    const grandChild = child.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node) {\n  TreeNode? child = node!.right;\n  TreeNode? grandChild = child!.left;\n  // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n  child.left = node;\n  node.right = grandChild;\n  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  updateHeight(node);\n  updateHeight(child);\n  // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfn left_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().right.clone().unwrap();\n            let grand_child = child.borrow().left.clone();\n            // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n            child.borrow_mut().left = Some(node.clone());\n            node.borrow_mut().right = grand_child;\n            // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;right;\n    grandChild = child-&gt;left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfun leftRotate(node: TreeNode?): TreeNode {\n    val child = node!!.right\n    val grandChild = child!!.left\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node\n    node.right = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node)\n    updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{left_rotate}\n</code></pre> avl_tree.zig<pre><code>// \u5de6\u65cb\u64cd\u4f5c\nfn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    var child = node.?.right;\n    var grandChild = child.?.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.?.left = node;\n    node.?.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.updateHeight(node);\n    self.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3-right-left-rotation","title":"3. \u00a0 Right-left rotation","text":"<p>For the unbalanced node 3 shown in the Figure 7-30 , using either left or right rotation alone cannot restore balance to the subtree. In this case, a \"left rotation\" needs to be performed on <code>child</code> first, followed by a \"right rotation\" on <code>node</code>.</p> <p></p> <p> Figure 7-30 \u00a0 Right-left rotation </p>"},{"location":"chapter_tree/avl_tree/#4-left-right-rotation","title":"4. \u00a0 Left-right rotation","text":"<p>As shown in the Figure 7-31 , for the mirror case of the above unbalanced binary tree, a \"right rotation\" needs to be performed on <code>child</code> first, followed by a \"left rotation\" on <code>node</code>.</p> <p></p> <p> Figure 7-31 \u00a0 Left-right rotation </p>"},{"location":"chapter_tree/avl_tree/#5-choice-of-rotation","title":"5. \u00a0 Choice of rotation","text":"<p>The four kinds of imbalances shown in the Figure 7-32  correspond to the cases described above, respectively requiring right rotation, left-right rotation, right-left rotation, and left rotation.</p> <p></p> <p> Figure 7-32 \u00a0 The four rotation cases of AVL tree </p> <p>As shown in the Table 7-3 , we determine which of the above cases an unbalanced node belongs to by judging the sign of the balance factor of the unbalanced node and its higher-side child's balance factor.</p> <p> Table 7-3 \u00a0 Conditions for Choosing Among the Four Rotation Cases </p> Balance factor of unbalanced node Balance factor of child node Rotation method to use \\(&gt; 1\\) (Left-leaning tree) \\(\\geq 0\\) Right rotation \\(&gt; 1\\) (Left-leaning tree) \\(&lt;0\\) Left rotation then right rotation \\(&lt; -1\\) (Right-leaning tree) \\(\\leq 0\\) Left rotation \\(&lt; -1\\) (Right-leaning tree) \\(&gt;0\\) Right rotation then left rotation <p>For convenience, we encapsulate the rotation operations into a function. With this function, we can perform rotations on various kinds of imbalances, restoring balance to unbalanced nodes. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\"\"\"\n    # \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    balance_factor = self.balance_factor(node)\n    # \u5de6\u504f\u6811\n    if balance_factor &gt; 1:\n        if self.balance_factor(node.left) &gt;= 0:\n            # \u53f3\u65cb\n            return self.right_rotate(node)\n        else:\n            # \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = self.left_rotate(node.left)\n            return self.right_rotate(node)\n    # \u53f3\u504f\u6811\n    elif balance_factor &lt; -1:\n        if self.balance_factor(node.right) &lt;= 0:\n            # \u5de6\u65cb\n            return self.left_rotate(node)\n        else:\n            # \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = self.right_rotate(node.right)\n            return self.left_rotate(node)\n    # \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n</code></pre> avl_tree.cpp<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int _balanceFactor = balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (_balanceFactor &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (_balanceFactor &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode rotate(TreeNode node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int balanceFactor = balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = leftRotate(node.left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = rightRotate(node.right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? Rotate(TreeNode? node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int balanceFactorInt = BalanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactorInt &gt; 1) {\n        if (BalanceFactor(node?.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return RightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node!.left = LeftRotate(node!.left);\n            return RightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactorInt &lt; -1) {\n        if (BalanceFactor(node?.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return LeftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node!.right = RightRotate(node!.right);\n            return LeftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    // Go \u63a8\u8350\u77ed\u53d8\u91cf\uff0c\u8fd9\u91cc bf \u6307\u4ee3 t.balanceFactor\n    bf := t.balanceFactor(node)\n    // \u5de6\u504f\u6811\n    if bf &gt; 1 {\n        if t.balanceFactor(node.Left) &gt;= 0 {\n            // \u53f3\u65cb\n            return t.rightRotate(node)\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.Left = t.leftRotate(node.Left)\n            return t.rightRotate(node)\n        }\n    }\n    // \u53f3\u504f\u6811\n    if bf &lt; -1 {\n        if t.balanceFactor(node.Right) &lt;= 0 {\n            // \u5de6\u65cb\n            return t.leftRotate(node)\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.Right = t.rightRotate(node.Right)\n            return t.leftRotate(node)\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc rotate(node: TreeNode?) -&gt; TreeNode? {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    let balanceFactor = balanceFactor(node: node)\n    // \u5de6\u504f\u6811\n    if balanceFactor &gt; 1 {\n        if self.balanceFactor(node: node?.left) &gt;= 0 {\n            // \u53f3\u65cb\n            return rightRotate(node: node)\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node?.left = leftRotate(node: node?.left)\n            return rightRotate(node: node)\n        }\n    }\n    // \u53f3\u504f\u6811\n    if balanceFactor &lt; -1 {\n        if self.balanceFactor(node: node?.right) &lt;= 0 {\n            // \u5de6\u65cb\n            return leftRotate(node: node)\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node?.right = rightRotate(node: node?.right)\n            return leftRotate(node: node)\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n#rotate(node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    const balanceFactor = this.balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return this.#rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = this.#leftRotate(node.left);\n            return this.#rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return this.#leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = this.#rightRotate(node.right);\n            return this.#leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nrotate(node: TreeNode): TreeNode {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    const balanceFactor = this.balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return this.rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = this.leftRotate(node.left);\n            return this.rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return this.leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = this.rightRotate(node.right);\n            return this.leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node) {\n  // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n  int factor = balanceFactor(node);\n  // \u5de6\u504f\u6811\n  if (factor &gt; 1) {\n    if (balanceFactor(node!.left) &gt;= 0) {\n      // \u53f3\u65cb\n      return rightRotate(node);\n    } else {\n      // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n      node.left = leftRotate(node.left);\n      return rightRotate(node);\n    }\n  }\n  // \u53f3\u504f\u6811\n  if (factor &lt; -1) {\n    if (balanceFactor(node!.right) &lt;= 0) {\n      // \u5de6\u65cb\n      return leftRotate(node);\n    } else {\n      // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n      node.right = rightRotate(node.right);\n      return leftRotate(node);\n    }\n  }\n  // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfn rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    let balance_factor = Self::balance_factor(node.clone());\n    // \u5de6\u504f\u6811\n    if balance_factor &gt; 1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().left.clone()) &gt;= 0 {\n            // \u53f3\u65cb\n            Self::right_rotate(Some(node))\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            let left = node.borrow().left.clone();\n            node.borrow_mut().left = Self::left_rotate(left);\n            Self::right_rotate(Some(node))\n        }\n    }\n    // \u53f3\u504f\u6811\n    else if balance_factor &lt; -1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().right.clone()) &lt;= 0 {\n            // \u5de6\u65cb\n            Self::left_rotate(Some(node))\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            let right = node.borrow().right.clone();\n            node.borrow_mut().right = Self::right_rotate(right);\n            Self::left_rotate(Some(node))\n        }\n    } else {\n        // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n        node\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int bf = balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (bf &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (bf &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfun rotate(node: TreeNode): TreeNode {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    val balanceFactor = balanceFactor(node)\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node)\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = leftRotate(node.left)\n            return rightRotate(node)\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node)\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = rightRotate(node.right)\n            return leftRotate(node)\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{rotate}\n</code></pre> avl_tree.zig<pre><code>// \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nfn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    var balance_factor = self.balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balance_factor &gt; 1) {\n        if (self.balanceFactor(node.?.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return self.rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.?.left = self.leftRotate(node.?.left);\n            return self.rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balance_factor &lt; -1) {\n        if (self.balanceFactor(node.?.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return self.leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.?.right = self.rightRotate(node.?.right);\n            return self.leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#753-common-operations-in-avl-trees","title":"7.5.3 \u00a0 Common operations in AVL trees","text":""},{"location":"chapter_tree/avl_tree/#1-node-insertion","title":"1. \u00a0 Node insertion","text":"<p>The node insertion operation in AVL trees is similar to that in binary search trees. The only difference is that after inserting a node in an AVL tree, a series of unbalanced nodes may appear along the path from that node to the root node. Therefore, we need to start from this node and perform rotation operations upwards to restore balance to all unbalanced nodes. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def insert(self, val):\n    \"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n    self._root = self.insert_helper(self._root, val)\n\ndef insert_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode:\n    \"\"\"\u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\n    if node is None:\n        return TreeNode(val)\n    # 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9\n    if val &lt; node.val:\n        node.left = self.insert_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.insert_helper(node.right, val)\n    else:\n        # \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    # 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    else\n        return node;    // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode insertHelper(TreeNode node, int val) {\n    if (node == null)\n        return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val)\n        node.left = insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = insertHelper(node.right, val);\n    else\n        return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int val) {\n    root = InsertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? InsertHelper(TreeNode? node, int val) {\n    if (node == null) return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val)\n        node.left = InsertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = InsertHelper(node.right, val);\n    else\n        return node;     // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    UpdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = Rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc (t *aVLTree) insert(val int) {\n    t.root = t.insertHelper(t.root, val)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return NewTreeNode(val)\n    }\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if val &lt; node.Val.(int) {\n        node.Left = t.insertHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.insertHelper(node.Right, val)\n    } else {\n        // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = t.rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(val: Int) {\n    root = insertHelper(node: root, val: val)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc insertHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return TreeNode(x: val)\n    }\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if val &lt; node!.val {\n        node?.left = insertHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = insertHelper(node: node?.right, val: val)\n    } else {\n        return node // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    updateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node: node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(val) {\n    this.root = this.#insertHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#insertHelper(node, val) {\n    if (node === null) return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val) node.left = this.#insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#insertHelper(node.right, val);\n    else return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    this.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.#rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(val: number): void {\n    this.root = this.insertHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\ninsertHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val) {\n        node.left = this.insertHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.insertHelper(node.right, val);\n    } else {\n        return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    this.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n  root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val) {\n  if (node == null) return TreeNode(val);\n  /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n  if (val &lt; node.val)\n    node.left = insertHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = insertHelper(node.right, val);\n  else\n    return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n  updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n  node = rotate(node);\n  // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfn insert(&amp;mut self, val: i32) {\n    self.root = Self::insert_helper(self.root.clone(), val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn insert_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n            match {\n                let node_val = node.borrow().val;\n                node_val\n            }\n            .cmp(&amp;val)\n            {\n                Ordering::Greater =&gt; {\n                    let left = node.borrow().left.clone();\n                    node.borrow_mut().left = Self::insert_helper(left, val);\n                }\n                Ordering::Less =&gt; {\n                    let right = node.borrow().right.clone();\n                    node.borrow_mut().right = Self::insert_helper(right, val);\n                }\n                Ordering::Equal =&gt; {\n                    return Some(node); // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n                }\n            }\n            Self::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n\n            /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n            node = Self::rotate(Some(node)).unwrap();\n            // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(node)\n        }\n        None =&gt; Some(TreeNode::new(val)),\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(AVLTree *tree, int val) {\n    tree-&gt;root = insertHelper(tree-&gt;root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == NULL) {\n        return newTreeNode(val);\n    }\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    } else {\n        // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node;\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfun insert(value: Int) {\n    root = insertHelper(root, value)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfun insertHelper(n: TreeNode?, value: Int): TreeNode {\n    if (n == null)\n        return TreeNode(value)\n    var node = n\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (value &lt; node.value) node.left = insertHelper(node.left, value)\n    else if (value &gt; node.value) node.right = insertHelper(node.right, value)\n    else return node // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n\n    updateHeight(node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{insert}\n\n[class]{AVLTree}-[func]{insert_helper}\n</code></pre> avl_tree.zig<pre><code>// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, val: T) !void {\n    self.root = (try self.insertHelper(self.root, val)).?;\n}\n\n// \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {\n    var node = node_;\n    if (node == null) {\n        var tmp_node = try self.mem_allocator.create(inc.TreeNode(T));\n        tmp_node.init(val);\n        return tmp_node;\n    }\n    // 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9\n    if (val &lt; node.?.val) {\n        node.?.left = try self.insertHelper(node.?.left, val);\n    } else if (val &gt; node.?.val) {\n        node.?.right = try self.insertHelper(node.?.right, val);\n    } else {\n        return node;            // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    self.updateHeight(node);    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    // 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    node = self.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2-node-removal","title":"2. \u00a0 Node removal","text":"<p>Similarly, based on the method of removing nodes in binary search trees, rotation operations need to be performed from the bottom up to restore balance to all unbalanced nodes. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def remove(self, val: int):\n    \"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n    self._root = self.remove_helper(self._root, val)\n\ndef remove_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode | None:\n    \"\"\"\u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\n    if node is None:\n        return None\n    # 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664\n    if val &lt; node.val:\n        node.left = self.remove_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.remove_helper(node.right, val)\n    else:\n        if node.left is None or node.right is None:\n            child = node.left or node.right\n            # \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if child is None:\n                return None\n            # \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else:\n                node = child\n        else:\n            # \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            temp = node.right\n            while temp.left is not None:\n                temp = temp.left\n            node.right = self.remove_helper(node.right, temp.val)\n            node.val = temp.val\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    # 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return nullptr;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    else {\n        if (node-&gt;left == nullptr || node-&gt;right == nullptr) {\n            TreeNode *child = node-&gt;left != nullptr ? node-&gt;left : node-&gt;right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == nullptr) {\n                delete node;\n                return nullptr;\n            }\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else {\n                delete node;\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != nullptr) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode removeHelper(TreeNode node, int val) {\n    if (node == null)\n        return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val)\n        node.left = removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = removeHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode child = node.left != null ? node.left : node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null)\n                return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else\n                node = child;\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int val) {\n    root = RemoveHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? RemoveHelper(TreeNode? node, int val) {\n    if (node == null) return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val)\n        node.left = RemoveHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = RemoveHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode? child = node.left ?? node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null)\n                return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else\n                node = child;\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode? temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = RemoveHelper(node.right, temp.val!.Value);\n            node.val = temp.val;\n        }\n    }\n    UpdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = Rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc (t *aVLTree) remove(val int) {\n    t.root = t.removeHelper(t.root, val)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return nil\n    }\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if val &lt; node.Val.(int) {\n        node.Left = t.removeHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.removeHelper(node.Right, val)\n    } else {\n        if node.Left == nil || node.Right == nil {\n            child := node.Left\n            if node.Right != nil {\n                child = node.Right\n            }\n            if child == nil {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n                return nil\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                node = child\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            temp := node.Right\n            for temp.Left != nil {\n                temp = temp.Left\n            }\n            node.Right = t.removeHelper(node.Right, temp.Val.(int))\n            node.Val = temp.Val\n        }\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = t.rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc remove(val: Int) {\n    root = removeHelper(node: root, val: val)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc removeHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return nil\n    }\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if val &lt; node!.val {\n        node?.left = removeHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = removeHelper(node: node?.right, val: val)\n    } else {\n        if node?.left == nil || node?.right == nil {\n            let child = node?.left ?? node?.right\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if child == nil {\n                return nil\n            }\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else {\n                node = child\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            var temp = node?.right\n            while temp?.left != nil {\n                temp = temp?.left\n            }\n            node?.right = removeHelper(node: node?.right, val: temp!.val)\n            node?.val = temp!.val\n        }\n    }\n    updateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node: node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(val) {\n    this.root = this.#removeHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#removeHelper(node, val) {\n    if (node === null) return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val) node.left = this.#removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#removeHelper(node.right, val);\n    else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child === null) return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else node = child;\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.#removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.#rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(val: number): void {\n    this.root = this.removeHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nremoveHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val) {\n        node.left = this.removeHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.removeHelper(node.right, val);\n    } else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child === null) {\n                return null;\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n  root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val) {\n  if (node == null) return null;\n  /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n  if (val &lt; node.val)\n    node.left = removeHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = removeHelper(node.right, val);\n  else {\n    if (node.left == null || node.right == null) {\n      TreeNode? child = node.left ?? node.right;\n      // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n      if (child == null)\n        return null;\n      // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n      else\n        node = child;\n    } else {\n      // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n      TreeNode? temp = node.right;\n      while (temp!.left != null) {\n        temp = temp.left;\n      }\n      node.right = removeHelper(node.right, temp.val);\n      node.val = temp.val;\n    }\n  }\n  updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n  node = rotate(node);\n  // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfn remove(&amp;self, val: i32) {\n    Self::remove_helper(self.root.clone(), val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn remove_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n            if val &lt; node.borrow().val {\n                let left = node.borrow().left.clone();\n                node.borrow_mut().left = Self::remove_helper(left, val);\n            } else if val &gt; node.borrow().val {\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, val);\n            } else if node.borrow().left.is_none() || node.borrow().right.is_none() {\n                let child = if node.borrow().left.is_some() {\n                    node.borrow().left.clone()\n                } else {\n                    node.borrow().right.clone()\n                };\n                match child {\n                    // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n                    None =&gt; {\n                        return None;\n                    }\n                    // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                    Some(child) =&gt; node = child,\n                }\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n                let mut temp = node.borrow().right.clone().unwrap();\n                loop {\n                    let temp_left = temp.borrow().left.clone();\n                    if temp_left.is_none() {\n                        break;\n                    }\n                    temp = temp_left.unwrap();\n                }\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);\n                node.borrow_mut().val = temp.borrow().val;\n            }\n            Self::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n\n            /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n            node = Self::rotate(Some(node)).unwrap();\n            // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(node)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(AVLTree *tree, int val) {\n    TreeNode *root = removeHelper(tree-&gt;root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    TreeNode *child, *grandChild;\n    if (node == NULL) {\n        return NULL;\n    }\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    } else {\n        if (node-&gt;left == NULL || node-&gt;right == NULL) {\n            child = node-&gt;left;\n            if (node-&gt;right != NULL) {\n                child = node-&gt;right;\n            }\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == NULL) {\n                return NULL;\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != NULL) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfun remove(value: Int) {\n    root = removeHelper(root, value)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfun removeHelper(n: TreeNode?, value: Int): TreeNode? {\n    var node = n ?: return null\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (value &lt; node.value) node.left = removeHelper(node.left, value)\n    else if (value &gt; node.value) node.right = removeHelper(node.right, value)\n    else {\n        if (node.left == null || node.right == null) {\n            val child = if (node.left != null) node.left else node.right\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null) return null\n            else node = child\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            var temp = node.right\n            while (temp!!.left != null) {\n                temp = temp.left\n            }\n            node.right = removeHelper(node.right, temp.value)\n            node.value = temp.value\n        }\n    }\n    updateHeight(node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{remove}\n\n[class]{AVLTree}-[func]{remove_helper}\n</code></pre> avl_tree.zig<pre><code>// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, val: T) void {\n   self.root = self.removeHelper(self.root, val).?;\n}\n\n// \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {\n    var node = node_;\n    if (node == null) return null;\n    // 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664\n    if (val &lt; node.?.val) {\n        node.?.left = self.removeHelper(node.?.left, val);\n    } else if (val &gt; node.?.val) {\n        node.?.right = self.removeHelper(node.?.right, val);\n    } else {\n        if (node.?.left == null or node.?.right == null) {\n            var child = if (node.?.left != null) node.?.left else node.?.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null) {\n                return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            } else {\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            var temp = node.?.right;\n            while (temp.?.left != null) {\n                temp = temp.?.left;\n            }\n            node.?.right = self.removeHelper(node.?.right, temp.?.val);\n            node.?.val = temp.?.val;\n        }\n    }\n    self.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    // 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    node = self.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3-node-search","title":"3. \u00a0 Node search","text":"<p>The node search operation in AVL trees is consistent with that in binary search trees and will not be detailed here.</p>"},{"location":"chapter_tree/avl_tree/#754-typical-applications-of-avl-trees","title":"7.5.4 \u00a0 Typical applications of AVL trees","text":"<ul> <li>Organizing and storing large amounts of data, suitable for scenarios with high-frequency searches and low-frequency intertions and removals.</li> <li>Used to build index systems in databases.</li> <li>Red-black trees are also a common type of balanced binary search tree. Compared to AVL trees, red-black trees have more relaxed balancing conditions, require fewer rotations for node insertion and removal, and have a higher average efficiency for node addition and removal operations.</li> </ul>"},{"location":"chapter_tree/binary_search_tree/","title":"7.4 \u00a0 Binary search tree","text":"<p>As shown in the Figure 7-16 , a \"binary search tree\" satisfies the following conditions.</p> <ol> <li>For the root node, the value of all nodes in the left subtree &lt; the value of the root node &lt; the value of all nodes in the right subtree.</li> <li>The left and right subtrees of any node are also binary search trees, i.e., they satisfy condition <code>1.</code> as well.</li> </ol> <p></p> <p> Figure 7-16 \u00a0 Binary search tree </p>"},{"location":"chapter_tree/binary_search_tree/#741-operations-on-a-binary-search-tree","title":"7.4.1 \u00a0 Operations on a binary search tree","text":"<p>We encapsulate the binary search tree as a class <code>BinarySearchTree</code> and declare a member variable <code>root</code>, pointing to the tree's root node.</p>"},{"location":"chapter_tree/binary_search_tree/#1-searching-for-a-node","title":"1. \u00a0 Searching for a node","text":"<p>Given a target node value <code>num</code>, one can search according to the properties of the binary search tree. As shown in the Figure 7-17 , we declare a node <code>cur</code> and start from the binary tree's root node <code>root</code>, looping to compare the size relationship between the node value <code>cur.val</code> and <code>num</code>.</p> <ul> <li>If <code>cur.val &lt; num</code>, it means the target node is in <code>cur</code>'s right subtree, thus execute <code>cur = cur.right</code>.</li> <li>If <code>cur.val &gt; num</code>, it means the target node is in <code>cur</code>'s left subtree, thus execute <code>cur = cur.left</code>.</li> <li>If <code>cur.val = num</code>, it means the target node is found, exit the loop and return the node.</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 7-17 \u00a0 Example of searching for a node in a binary search tree </p> <p>The search operation in a binary search tree works on the same principle as the binary search algorithm, eliminating half of the possibilities in each round. The number of loops is at most the height of the binary tree. When the binary tree is balanced, it uses \\(O(\\log n)\\) time. Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def search(self, num: int) -&gt; TreeNode | None:\n    \"\"\"\u67e5\u627e\u8282\u70b9\"\"\"\n    cur = self._root\n    # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur is not None:\n        # \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        elif cur.val &gt; num:\n            cur = cur.left\n        # \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else:\n            break\n    return cur\n</code></pre> binary_search_tree.cpp<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(int num) {\n    TreeNode *cur = root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur-&gt;val &gt; num)\n            cur = cur-&gt;left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode search(int num) {\n    TreeNode cur = root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? Search(int num) {\n    TreeNode? cur = root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur =\n            cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\n    node := bst.root\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    for node != nil {\n        if node.Val.(int) &lt; num {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            node = node.Right\n        } else if node.Val.(int) &gt; num {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            node = node.Left\n        } else {\n            // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            break\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return node\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfunc search(num: Int) -&gt; TreeNode? {\n    var cur = root\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur != nil {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if cur!.val &gt; num {\n            cur = cur?.left\n        }\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else {\n            break\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nsearch(num) {\n    let cur = this.root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num) cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nsearch(num: number): TreeNode | null {\n    let cur = this.root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num) cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? search(int _num) {\n  TreeNode? cur = _root;\n  // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n  while (cur != null) {\n    // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n    else if (cur.val &gt; _num)\n      cur = cur.left;\n    // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n    else\n      break;\n  }\n  // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n  return cur;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u67e5\u627e\u8282\u70b9 */\npub fn search(&amp;self, num: i32) -&gt; OptionTreeNodeRc {\n    let mut cur = self.root.clone();\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            Ordering::Greater =&gt; cur = node.borrow().right.clone(),\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            Ordering::Less =&gt; cur = node.borrow().left.clone(),\n            // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            Ordering::Equal =&gt; break,\n        }\n    }\n\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    cur\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(BinarySearchTree *bst, int num) {\n    TreeNode *cur = bst-&gt;root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != NULL) {\n        if (cur-&gt;val &lt; num) {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            cur = cur-&gt;right;\n        } else if (cur-&gt;val &gt; num) {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            cur = cur-&gt;left;\n        } else {\n            // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfun search(num: Int): TreeNode? {\n    var cur = root\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        cur = if (cur.value &lt; num) cur.right\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.value &gt; num) cur.left\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else break\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur\n}\n</code></pre> binary_search_tree.rb<pre><code>[class]{BinarySearchTree}-[func]{search}\n</code></pre> binary_search_tree.zig<pre><code>// \u67e5\u627e\u8282\u70b9\nfn search(self: *Self, num: T) ?*inc.TreeNode(T) {\n    var cur = self.root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        } else if (cur.?.val &gt; num) {\n            cur = cur.?.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        } else {\n            break;\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_tree/binary_search_tree/#2-inserting-a-node","title":"2. \u00a0 Inserting a node","text":"<p>Given an element <code>num</code> to be inserted, to maintain the property of the binary search tree \"left subtree &lt; root node &lt; right subtree,\" the insertion operation proceeds as shown in the Figure 7-18 .</p> <ol> <li>Finding the insertion position: Similar to the search operation, start from the root node and loop downwards according to the size relationship between the current node value and <code>num</code> until passing through the leaf node (traversing to <code>None</code>) then exit the loop.</li> <li>Insert the node at that position: Initialize the node <code>num</code> and place it where <code>None</code> was.</li> </ol> <p></p> <p> Figure 7-18 \u00a0 Inserting a node into a binary search tree </p> <p>In the code implementation, note the following two points.</p> <ul> <li>The binary search tree does not allow duplicate nodes; otherwise, it will violate its definition. Therefore, if the node to be inserted already exists in the tree, the insertion is not performed, and it directly returns.</li> <li>To perform the insertion operation, we need to use the node <code>pre</code> to save the node from the last loop. This way, when traversing to <code>None</code>, we can get its parent node, thus completing the node insertion operation.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def insert(self, num: int):\n    \"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n    # \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if self._root is None:\n        self._root = TreeNode(num)\n        return\n    # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    cur, pre = self._root, None\n    while cur is not None:\n        # \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if cur.val == num:\n            return\n        pre = cur\n        # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else:\n            cur = cur.left\n    # \u63d2\u5165\u8282\u70b9\n    node = TreeNode(num)\n    if pre.val &lt; num:\n        pre.right = node\n    else:\n        pre.left = node\n</code></pre> binary_search_tree.cpp<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == nullptr) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode *cur = root, *pre = nullptr;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur-&gt;val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur-&gt;left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode *node = new TreeNode(num);\n    if (pre-&gt;val &lt; num)\n        pre-&gt;right = node;\n    else\n        pre-&gt;left = node;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode node = new TreeNode(num);\n    if (pre.val &lt; num)\n        pre.right = node;\n    else\n        pre.left = node;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode? cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode node = new(num);\n    if (pre != null) {\n        if (pre.val &lt; num)\n            pre.right = node;\n        else\n            pre.left = node;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc (bst *binarySearchTree) insert(num int) {\n    cur := bst.root\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if cur == nil {\n        bst.root = NewTreeNode(num)\n        return\n    }\n    // \u5f85\u63d2\u5165\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\n    var pre *TreeNode = nil\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    for cur != nil {\n        if cur.Val == num {\n            return\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            cur = cur.Right\n        } else {\n            cur = cur.Left\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    node := NewTreeNode(num)\n    if pre.Val.(int) &lt; num {\n        pre.Right = node\n    } else {\n        pre.Left = node\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if root == nil {\n        root = TreeNode(x: num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur != nil {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if cur!.val == num {\n            return\n        }\n        pre = cur\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else {\n            cur = cur?.left\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    let node = TreeNode(x: num)\n    if pre!.val &lt; num {\n        pre?.right = node\n    } else {\n        pre?.left = node\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur = this.root,\n        pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val === num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    const node = new TreeNode(num);\n    if (pre.val &lt; num) pre.right = node;\n    else pre.left = node;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(num: number): void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val === num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    const node = new TreeNode(num);\n    if (pre!.val &lt; num) pre!.right = node;\n    else pre!.left = node;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int _num) {\n  // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n  if (_root == null) {\n    _root = TreeNode(_num);\n    return;\n  }\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n  while (cur != null) {\n    // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n    if (cur.val == _num) return;\n    pre = cur;\n    // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n    else\n      cur = cur.left;\n  }\n  // \u63d2\u5165\u8282\u70b9\n  TreeNode? node = TreeNode(_num);\n  if (pre!.val &lt; _num)\n    pre.right = node;\n  else\n    pre.left = node;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\npub fn insert(&amp;mut self, num: i32) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if self.root.is_none() {\n        self.root = Some(TreeNode::new(num));\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n            Ordering::Equal =&gt; return,\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    let pre = pre.unwrap();\n    let node = Some(TreeNode::new(num));\n    if num &gt; pre.borrow().val {\n        pre.borrow_mut().right = node;\n    } else {\n        pre.borrow_mut().left = node;\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(BinarySearchTree *bst, int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (bst-&gt;root == NULL) {\n        bst-&gt;root = newTreeNode(num);\n        return;\n    }\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != NULL) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur-&gt;val == num) {\n            return;\n        }\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            cur = cur-&gt;right;\n        } else {\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            cur = cur-&gt;left;\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode *node = newTreeNode(num);\n    if (pre-&gt;val &lt; num) {\n        pre-&gt;right = node;\n    } else {\n        pre-&gt;left = node;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfun insert(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == null) {\n        root = TreeNode(num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode? = null\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.value == num) return\n        pre = cur\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        cur = if (cur.value &lt; num) cur.right\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur.left\n    }\n    // \u63d2\u5165\u8282\u70b9\n    val node = TreeNode(num)\n    if (pre?.value!! &lt; num) pre.right = node\n    else pre.left = node\n}\n</code></pre> binary_search_tree.rb<pre><code>[class]{BinarySearchTree}-[func]{insert}\n</code></pre> binary_search_tree.zig<pre><code>// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, num: T) !void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (self.root == null) {\n        self.root = try self.mem_allocator.create(inc.TreeNode(T));\n        return;\n    }\n    var cur = self.root;\n    var pre: ?*inc.TreeNode(T) = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.?.val == num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        } else {\n            cur = cur.?.left;\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    var node = try self.mem_allocator.create(inc.TreeNode(T));\n    node.init(num);\n    if (pre.?.val &lt; num) {\n        pre.?.right = node;\n    } else {\n        pre.?.left = node;\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Similar to searching for a node, inserting a node uses \\(O(\\log n)\\) time.</p>"},{"location":"chapter_tree/binary_search_tree/#3-removing-a-node","title":"3. \u00a0 Removing a node","text":"<p>First, find the target node in the binary tree, then remove it. Similar to inserting a node, we need to ensure that after the removal operation is completed, the property of the binary search tree \"left subtree &lt; root node &lt; right subtree\" is still satisfied. Therefore, based on the number of child nodes of the target node, we divide it into 0, 1, and 2 cases, performing the corresponding node removal operations.</p> <p>As shown in the Figure 7-19 , when the degree of the node to be removed is \\(0\\), it means the node is a leaf node, and it can be directly removed.</p> <p></p> <p> Figure 7-19 \u00a0 Removing a node in a binary search tree (degree 0) </p> <p>As shown in the Figure 7-20 , when the degree of the node to be removed is \\(1\\), replacing the node to be removed with its child node is sufficient.</p> <p></p> <p> Figure 7-20 \u00a0 Removing a node in a binary search tree (degree 1) </p> <p>When the degree of the node to be removed is \\(2\\), we cannot remove it directly, but need to use a node to replace it. To maintain the property of the binary search tree \"left subtree &lt; root node &lt; right subtree,\" this node can be either the smallest node of the right subtree or the largest node of the left subtree.</p> <p>Assuming we choose the smallest node of the right subtree (the next node in in-order traversal), then the removal operation proceeds as shown in the Figure 7-21 .</p> <ol> <li>Find the next node in the \"in-order traversal sequence\" of the node to be removed, denoted as <code>tmp</code>.</li> <li>Replace the value of the node to be removed with <code>tmp</code>'s value, and recursively remove the node <code>tmp</code> in the tree.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 7-21 \u00a0 Removing a node in a binary search tree (degree 2) </p> <p>The operation of removing a node also uses \\(O(\\log n)\\) time, where finding the node to be removed requires \\(O(\\log n)\\) time, and obtaining the in-order traversal successor node requires \\(O(\\log n)\\) time. Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def remove(self, num: int):\n    \"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n    # \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if self._root is None:\n        return\n    # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    cur, pre = self._root, None\n    while cur is not None:\n        # \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if cur.val == num:\n            break\n        pre = cur\n        # \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else:\n            cur = cur.left\n    # \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur is None:\n        return\n\n    # \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if cur.left is None or cur.right is None:\n        # \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        child = cur.left or cur.right\n        # \u5220\u9664\u8282\u70b9 cur\n        if cur != self._root:\n            if pre.left == cur:\n                pre.left = child\n            else:\n                pre.right = child\n        else:\n            # \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            self._root = child\n    # \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else:\n        # \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        tmp: TreeNode = cur.right\n        while tmp.left is not None:\n            tmp = tmp.left\n        # \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        self.remove(tmp.val)\n        # \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val\n</code></pre> binary_search_tree.cpp<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == nullptr)\n        return;\n    TreeNode *cur = root, *pre = nullptr;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur-&gt;left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == nullptr)\n        return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur-&gt;left == nullptr || cur-&gt;right == nullptr) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n        TreeNode *child = cur-&gt;left != nullptr ? cur-&gt;left : cur-&gt;right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre-&gt;left == cur)\n                pre-&gt;left = child;\n            else\n                pre-&gt;right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child;\n        }\n        // \u91ca\u653e\u5185\u5b58\n        delete cur;\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != nullptr) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(tmp-&gt;val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return;\n    TreeNode cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        TreeNode child = cur.left != null ? cur.left : cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(tmp.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return;\n    TreeNode? cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        TreeNode? child = cur.left ?? cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre!.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode? tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        Remove(tmp.val!.Value);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc (bst *binarySearchTree) remove(num int) {\n    cur := bst.root\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5f85\u5220\u9664\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\n    var pre *TreeNode = nil\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    for cur != nil {\n        if cur.Val == num {\n            break\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728\u53f3\u5b50\u6811\u4e2d\n            cur = cur.Right\n        } else {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728\u5de6\u5b50\u6811\u4e2d\n            cur = cur.Left\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5b50\u8282\u70b9\u6570\u4e3a 0 \u6216 1\n    if cur.Left == nil || cur.Right == nil {\n        var child *TreeNode = nil\n        // \u53d6\u51fa\u5f85\u5220\u9664\u8282\u70b9\u7684\u5b50\u8282\u70b9\n        if cur.Left != nil {\n            child = cur.Left\n        } else {\n            child = cur.Right\n        }\n        // \u5220\u9664\u8282\u70b9 cur\n        if cur != bst.root {\n            if pre.Left == cur {\n                pre.Left = child\n            } else {\n                pre.Right = child\n            }\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            bst.root = child\n        }\n        // \u5b50\u8282\u70b9\u6570\u4e3a 2\n    } else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u5f85\u5220\u9664\u8282\u70b9 cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        tmp := cur.Right\n        for tmp.Left != nil {\n            tmp = tmp.Left\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        bst.remove(tmp.Val.(int))\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.Val = tmp.Val\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc remove(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if root == nil {\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur != nil {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if cur!.val == num {\n            break\n        }\n        pre = cur\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else {\n            cur = cur?.left\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if cur?.left == nil || cur?.right == nil {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        let child = cur?.left ?? cur?.right\n        // \u5220\u9664\u8282\u70b9 cur\n        if cur !== root {\n            if pre?.left === cur {\n                pre?.left = child\n            } else {\n                pre?.right = child\n            }\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        var tmp = cur?.right\n        while tmp?.left != nil {\n            tmp = tmp?.left\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(num: tmp!.val)\n        // \u7528 tmp \u8986\u76d6 cur\n        cur?.val = tmp!.val\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (this.root === null) return;\n    let cur = this.root,\n        pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val === num) break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur === null) return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        const child = cur.left !== null ? cur.left : cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur !== this.root) {\n            if (pre.left === cur) pre.left = child;\n            else pre.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            this.root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        let tmp = cur.right;\n        while (tmp.left !== null) {\n            tmp = tmp.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        this.remove(tmp.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(num: number): void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (this.root === null) return;\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val === num) break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur === null) return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        const child: TreeNode | null =\n            cur.left !== null ? cur.left : cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur !== this.root) {\n            if (pre!.left === cur) pre!.left = child;\n            else pre!.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            this.root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        let tmp: TreeNode | null = cur.right;\n        while (tmp!.left !== null) {\n            tmp = tmp!.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        this.remove(tmp!.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp!.val;\n    }\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int _num) {\n  // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n  if (_root == null) return;\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n  while (cur != null) {\n    // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n    if (cur.val == _num) break;\n    pre = cur;\n    // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n    else\n      cur = cur.left;\n  }\n  // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n  if (cur == null) return;\n  // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n  if (cur.left == null || cur.right == null) {\n    // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n    TreeNode? child = cur.left ?? cur.right;\n    // \u5220\u9664\u8282\u70b9 cur\n    if (cur != _root) {\n      if (pre!.left == cur)\n        pre.left = child;\n      else\n        pre.right = child;\n    } else {\n      // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n      _root = child;\n    }\n  } else {\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n    TreeNode? tmp = cur.right;\n    while (tmp!.left != null) {\n      tmp = tmp.left;\n    }\n    // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n    remove(tmp.val);\n    // \u7528 tmp \u8986\u76d6 cur\n    cur.val = tmp.val;\n  }\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u5220\u9664\u8282\u70b9 */\npub fn remove(&amp;mut self, num: i32) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if self.root.is_none() {\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            Ordering::Equal =&gt; break,\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur.is_none() {\n        return;\n    }\n    let cur = cur.unwrap();\n    let (left_child, right_child) = (cur.borrow().left.clone(), cur.borrow().right.clone());\n    match (left_child.clone(), right_child.clone()) {\n        // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n        (None, None) | (Some(_), None) | (None, Some(_)) =&gt; {\n            // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n            let child = left_child.or(right_child);\n            let pre = pre.unwrap();\n            // \u5220\u9664\u8282\u70b9 cur\n            if !Rc::ptr_eq(&amp;cur, self.root.as_ref().unwrap()) {\n                let left = pre.borrow().left.clone();\n                if left.is_some() &amp;&amp; Rc::ptr_eq(&amp;left.as_ref().unwrap(), &amp;cur) {\n                    pre.borrow_mut().left = child;\n                } else {\n                    pre.borrow_mut().right = child;\n                }\n            } else {\n                // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n                self.root = child;\n            }\n        }\n        // \u5b50\u8282\u70b9\u6570\u91cf = 2\n        (Some(_), Some(_)) =&gt; {\n            // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n            let mut tmp = cur.borrow().right.clone();\n            while let Some(node) = tmp.clone() {\n                if node.borrow().left.is_some() {\n                    tmp = node.borrow().left.clone();\n                } else {\n                    break;\n                }\n            }\n            let tmpval = tmp.unwrap().borrow().val;\n            // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n            self.remove(tmpval);\n            // \u7528 tmp \u8986\u76d6 cur\n            cur.borrow_mut().val = tmpval;\n        }\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(BinarySearchTree *bst, int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (bst-&gt;root == NULL)\n        return;\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != NULL) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u53f3\u5b50\u6811\u4e2d\n            cur = cur-&gt;right;\n        } else {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u5de6\u5b50\u6811\u4e2d\n            cur = cur-&gt;left;\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == NULL)\n        return;\n    // \u5224\u65ad\u5f85\u5220\u9664\u8282\u70b9\u662f\u5426\u5b58\u5728\u5b50\u8282\u70b9\n    if (cur-&gt;left == NULL || cur-&gt;right == NULL) {\n        /* \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1 */\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n        TreeNode *child = cur-&gt;left != NULL ? cur-&gt;left : cur-&gt;right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (pre-&gt;left == cur) {\n            pre-&gt;left = child;\n        } else {\n            pre-&gt;right = child;\n        }\n        // \u91ca\u653e\u5185\u5b58\n        free(cur);\n    } else {\n        /* \u5b50\u8282\u70b9\u6570\u91cf = 2 */\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != NULL) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        removeItem(bst, tmp-&gt;val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfun remove(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null) return\n    var cur = root\n    var pre: TreeNode? = null\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.value == num) break\n        pre = cur\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        cur = if (cur.value &lt; num) cur.right\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur.left\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null) return\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        val child = if (cur.left != null) cur.left else cur.right\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre!!.left == cur) pre.left = child\n            else pre.right = child\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child\n        }\n        // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    } else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        var tmp = cur.right\n        while (tmp!!.left != null) {\n            tmp = tmp.left\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(tmp.value)\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.value = tmp.value\n    }\n}\n</code></pre> binary_search_tree.rb<pre><code>[class]{BinarySearchTree}-[func]{remove}\n</code></pre> binary_search_tree.zig<pre><code>// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, num: T) void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (self.root == null) return;\n    var cur = self.root;\n    var pre: ?*inc.TreeNode(T) = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.?.val == num) break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        } else {\n            cur = cur.?.left;\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null) return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.?.left == null or cur.?.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        var child = if (cur.?.left != null) cur.?.left else cur.?.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (pre.?.left == cur) {\n            pre.?.left = child;\n        } else {\n            pre.?.right = child;\n        }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    } else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        var tmp = cur.?.right;\n        while (tmp.?.left != null) {\n            tmp = tmp.?.left;\n        }\n        var tmp_val = tmp.?.val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        self.remove(tmp.?.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.?.val = tmp_val;\n    }\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_tree/binary_search_tree/#4-in-order-traversal-is-ordered","title":"4. \u00a0 In-order traversal is ordered","text":"<p>As shown in the Figure 7-22 , the in-order traversal of a binary tree follows the \"left \\(\\rightarrow\\) root \\(\\rightarrow\\) right\" traversal order, and a binary search tree satisfies the size relationship \"left child node &lt; root node &lt; right child node\".</p> <p>This means that in-order traversal in a binary search tree always traverses the next smallest node first, thus deriving an important property: The in-order traversal sequence of a binary search tree is ascending.</p> <p>Using the ascending property of in-order traversal, obtaining ordered data in a binary search tree requires only \\(O(n)\\) time, without the need for additional sorting operations, which is very efficient.</p> <p></p> <p> Figure 7-22 \u00a0 In-order traversal sequence of a binary search tree </p>"},{"location":"chapter_tree/binary_search_tree/#742-efficiency-of-binary-search-trees","title":"7.4.2 \u00a0 Efficiency of binary search trees","text":"<p>Given a set of data, we consider using an array or a binary search tree for storage. Observing the Table 7-2 , the operations on a binary search tree all have logarithmic time complexity, which is stable and efficient. Only in scenarios of high-frequency addition and low-frequency search and removal, arrays are more efficient than binary search trees.</p> <p> Table 7-2 \u00a0 Efficiency comparison between arrays and search trees </p> Unsorted array Binary search tree Search element \\(O(n)\\) \\(O(\\log n)\\) Insert element \\(O(1)\\) \\(O(\\log n)\\) Remove element \\(O(n)\\) \\(O(\\log n)\\) <p>In ideal conditions, the binary search tree is \"balanced,\" thus any node can be found within \\(\\log n\\) loops.</p> <p>However, continuously inserting and removing nodes in a binary search tree may lead to the binary tree degenerating into a chain list as shown in the Figure 7-23 , at which point the time complexity of various operations also degrades to \\(O(n)\\).</p> <p></p> <p> Figure 7-23 \u00a0 Degradation of a binary search tree </p>"},{"location":"chapter_tree/binary_search_tree/#743-common-applications-of-binary-search-trees","title":"7.4.3 \u00a0 Common applications of binary search trees","text":"<ul> <li>Used as multi-level indexes in systems to implement efficient search, insertion, and removal operations.</li> <li>Serves as the underlying data structure for certain search algorithms.</li> <li>Used to store data streams to maintain their ordered state.</li> </ul>"},{"location":"chapter_tree/binary_tree/","title":"7.1 \u00a0 Binary tree","text":"<p>A \"binary tree\" is a non-linear data structure that represents the ancestral and descendent relationships, embodying the \"divide and conquer\" logic. Similar to a linked list, the basic unit of a binary tree is a node, each containing a value, a reference to the left child node, and a reference to the right child node.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class TreeNode:\n    \"\"\"Binary tree node\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # Node value\n        self.left: TreeNode | None = None  # Reference to left child node\n        self.right: TreeNode | None = None # Reference to right child node\n</code></pre> <pre><code>/* Binary tree node */\nstruct TreeNode {\n    int val;          // Node value\n    TreeNode *left;   // Pointer to left child node\n    TreeNode *right;  // Pointer to right child node\n    TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode {\n    int val;         // Node value\n    TreeNode left;   // Reference to left child node\n    TreeNode right;  // Reference to right child node\n    TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode(int? x) {\n    public int? val = x;    // Node value\n    public TreeNode? left;  // Reference to left child node\n    public TreeNode? right; // Reference to right child node\n}\n</code></pre> <pre><code>/* Binary tree node */\ntype TreeNode struct {\n    Val   int\n    Left  *TreeNode\n    Right *TreeNode\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc NewTreeNode(v int) *TreeNode {\n    return &amp;TreeNode{\n        Left:  nil, // Pointer to left child node\n        Right: nil, // Pointer to right child node\n        Val:   v,   // Node value\n    }\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode {\n    var val: Int // Node value\n    var left: TreeNode? // Reference to left child node\n    var right: TreeNode? // Reference to right child node\n\n    init(x: Int) {\n        val = x\n    }\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode {\n    val; // Node value\n    left; // Pointer to left child node\n    right; // Pointer to right child node\n    constructor(val, left, right) {\n        this.val = val === undefined ? 0 : val;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode {\n    val: number;\n    left: TreeNode | null;\n    right: TreeNode | null;\n\n    constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val; // Node value\n        this.left = left === undefined ? null : left; // Reference to left child node\n        this.right = right === undefined ? null : right; // Reference to right child node\n    }\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode {\n  int val;         // Node value\n  TreeNode? left;  // Reference to left child node\n  TreeNode? right; // Reference to right child node\n  TreeNode(this.val, [this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* Binary tree node */\nstruct TreeNode {\n    val: i32,                               // Node value\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // Reference to left child node\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // Reference to right child node\n}\n\nimpl TreeNode {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* Binary tree node */\ntypedef struct TreeNode {\n    int val;                // Node value\n    int height;             // \u8282\u70b9\u9ad8\u5ea6\n    struct TreeNode *left;  // Pointer to left child node\n    struct TreeNode *right; // Pointer to right child node\n} TreeNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode(val _val: Int) {  // Node value\n    val left: TreeNode? = null   // Reference to left child node\n    val right: TreeNode? = null  // Reference to right child node\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>Each node has two references (pointers), pointing to the \"left-child node\" and \"right-child node,\" respectively. This node is called the \"parent node\" of these two child nodes. When given a node of a binary tree, we call the tree formed by this node's left child and all nodes under it the \"left subtree\" of this node. Similarly, the \"right subtree\" can be defined.</p> <p>In a binary tree, except for leaf nodes, all other nodes contain child nodes and non-empty subtrees. As shown in the Figure 7-1 , if \"Node 2\" is considered as the parent node, then its left and right child nodes are \"Node 4\" and \"Node 5,\" respectively. The left subtree is \"the tree formed by Node 4 and all nodes under it,\" and the right subtree is \"the tree formed by Node 5 and all nodes under it.\"</p> <p></p> <p> Figure 7-1 \u00a0 Parent Node, child Node, subtree </p>"},{"location":"chapter_tree/binary_tree/#711-common-terminology-of-binary-trees","title":"7.1.1 \u00a0 Common terminology of binary trees","text":"<p>The commonly used terminology of binary trees is shown in the following figure.</p> <ul> <li>\"Root node\": The node at the top level of the binary tree, which has no parent node.</li> <li>\"Leaf node\": A node with no children, both of its pointers point to <code>None</code>.</li> <li>\"Edge\": The line segment connecting two nodes, i.e., node reference (pointer).</li> <li>The \"level\" of a node: Incrementing from top to bottom, with the root node's level being 1.</li> <li>The \"degree\" of a node: The number of a node's children. In a binary tree, the degree can be 0, 1, or 2.</li> <li>The \"height\" of a binary tree: The number of edges passed from the root node to the farthest leaf node.</li> <li>The \"depth\" of a node: The number of edges passed from the root node to the node.</li> <li>The \"height\" of a node: The number of edges from the farthest leaf node to the node.</li> </ul> <p></p> <p> Figure 7-2 \u00a0 Common Terminology of Binary Trees </p> <p>Tip</p> <p>Please note that we usually define \"height\" and \"depth\" as \"the number of edges passed,\" but some problems or textbooks may define them as \"the number of nodes passed.\" In this case, both height and depth need to be incremented by 1.</p>"},{"location":"chapter_tree/binary_tree/#712-basic-operations-of-binary-trees","title":"7.1.2 \u00a0 Basic operations of binary trees","text":""},{"location":"chapter_tree/binary_tree/#1-initializing-a-binary-tree","title":"1. \u00a0 Initializing a binary tree","text":"<p>Similar to a linked list, initialize nodes first, then construct references (pointers).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree.py<pre><code># Initializing a binary tree\n# Initializing nodes\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# Linking references (pointers) between nodes\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.cpp<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// Linking references (pointers) between nodes\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.java<pre><code>// Initializing nodes\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// Linking references (pointers) between nodes\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.cs<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nTreeNode n1 = new(1);\nTreeNode n2 = new(2);\nTreeNode n3 = new(3);\nTreeNode n4 = new(4);\nTreeNode n5 = new(5);\n// Linking references (pointers) between nodes\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.go<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// Linking references (pointers) between nodes\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n</code></pre> binary_tree.swift<pre><code>// Initializing nodes\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// Linking references (pointers) between nodes\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.js<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// Linking references (pointers) between nodes\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.ts<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// Linking references (pointers) between nodes\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.dart<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// Linking references (pointers) between nodes\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.rs<pre><code>// Initializing nodes\nlet n1 = TreeNode::new(1);\nlet n2 = TreeNode::new(2);\nlet n3 = TreeNode::new(3);\nlet n4 = TreeNode::new(4);\nlet n5 = TreeNode::new(5);\n// Linking references (pointers) between nodes\nn1.borrow_mut().left = Some(n2.clone());\nn1.borrow_mut().right = Some(n3);\nn2.borrow_mut().left = Some(n4);\nn2.borrow_mut().right = Some(n5);\n</code></pre> binary_tree.c<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// Linking references (pointers) between nodes\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.kt<pre><code>// Initializing nodes\nval n1 = TreeNode(1)\nval n2 = TreeNode(2)\nval n3 = TreeNode(3)\nval n4 = TreeNode(4)\nval n5 = TreeNode(5)\n// Linking references (pointers) between nodes\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.rb<pre><code>\n</code></pre> binary_tree.zig<pre><code>\n</code></pre> Code visualization <p>https://pythontutor.com/render.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%8F%89%E6%A0%91%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E8%8A%82%E7%82%B9%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E9%92%88%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5&amp;cumulative=false&amp;curInstr=3&amp;heapPrimitives=nevernest&amp;mode=display&amp;origin=opt-frontend.js&amp;py=311&amp;rawInputLstJSON=%5B%5D&amp;textReferences=false</p>"},{"location":"chapter_tree/binary_tree/#2-inserting-and-removing-nodes","title":"2. \u00a0 Inserting and removing nodes","text":"<p>Similar to a linked list, inserting and removing nodes in a binary tree can be achieved by modifying pointers. The Figure 7-3  provides an example.</p> <p></p> <p> Figure 7-3 \u00a0 Inserting and removing nodes in a binary tree </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree.py<pre><code># Inserting and removing nodes\np = TreeNode(0)\n# Inserting node P between n1 -&gt; n2\nn1.left = p\np.left = n2\n# Removing node P\nn1.left = n2\n</code></pre> binary_tree.cpp<pre><code>/* Inserting and removing nodes */\nTreeNode* P = new TreeNode(0);\n// Inserting node P between n1 and n2\nn1-&gt;left = P;\nP-&gt;left = n2;\n// Removing node P\nn1-&gt;left = n2;\n</code></pre> binary_tree.java<pre><code>TreeNode P = new TreeNode(0);\n// Inserting node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Removing node P\nn1.left = n2;\n</code></pre> binary_tree.cs<pre><code>/* Inserting and removing nodes */\nTreeNode P = new(0);\n// Inserting node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Removing node P\nn1.left = n2;\n</code></pre> binary_tree.go<pre><code>/* Inserting and removing nodes */\n// Inserting node P between n1 and n2\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// Removing node P\nn1.Left = n2\n</code></pre> binary_tree.swift<pre><code>let P = TreeNode(x: 0)\n// Inserting node P between n1 and n2\nn1.left = P\nP.left = n2\n// Removing node P\nn1.left = n2\n</code></pre> binary_tree.js<pre><code>/* Inserting and removing nodes */\nlet P = new TreeNode(0);\n// Inserting node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Removing node P\nn1.left = n2;\n</code></pre> binary_tree.ts<pre><code>/* Inserting and removing nodes */\nconst P = new TreeNode(0);\n// Inserting node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Removing node P\nn1.left = n2;\n</code></pre> binary_tree.dart<pre><code>/* Inserting and removing nodes */\nTreeNode P = new TreeNode(0);\n// Inserting node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Removing node P\nn1.left = n2;\n</code></pre> binary_tree.rs<pre><code>let p = TreeNode::new(0);\n// Inserting node P between n1 and n2\nn1.borrow_mut().left = Some(p.clone());\np.borrow_mut().left = Some(n2.clone());\n// Removing node P\nn1.borrow_mut().left = Some(n2);\n</code></pre> binary_tree.c<pre><code>/* Inserting and removing nodes */\nTreeNode *P = newTreeNode(0);\n// Inserting node P between n1 and n2\nn1-&gt;left = P;\nP-&gt;left = n2;\n// Removing node P\nn1-&gt;left = n2;\n</code></pre> binary_tree.kt<pre><code>val P = TreeNode(0)\n// Inserting node P between n1 and n2\nn1.left = P\nP.left = n2\n// Removing node P\nn1.left = n2\n</code></pre> binary_tree.rb<pre><code>\n</code></pre> binary_tree.zig<pre><code>\n</code></pre> Code visualization <p>https://pythontutor.com/render.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%8F%89%E6%A0%91%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E8%8A%82%E7%82%B9%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E9%92%88%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E4%B8%8E%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%0A%20%20%20%20p%20%3D%20TreeNode%280%29%0A%20%20%20%20%23%20%E5%9C%A8%20n1%20-%3E%20n2%20%E4%B8%AD%E9%97%B4%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20p%0A%20%20%20%20p.left%20%3D%20n2%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20n2&amp;cumulative=false&amp;curInstr=37&amp;heapPrimitives=nevernest&amp;mode=display&amp;origin=opt-frontend.js&amp;py=311&amp;rawInputLstJSON=%5B%5D&amp;textReferences=false</p> <p>Note</p> <p>It's important to note that inserting nodes may change the original logical structure of the binary tree, while removing nodes usually means removing the node and all its subtrees. Therefore, in a binary tree, insertion and removal are usually performed through a set of operations to achieve meaningful actions.</p>"},{"location":"chapter_tree/binary_tree/#713-common-types-of-binary-trees","title":"7.1.3 \u00a0 Common types of binary trees","text":""},{"location":"chapter_tree/binary_tree/#1-perfect-binary-tree","title":"1. \u00a0 Perfect binary tree","text":"<p>As shown in the Figure 7-4 , in a \"perfect binary tree,\" all levels of nodes are fully filled. In a perfect binary tree, the degree of leaf nodes is \\(0\\), and the degree of all other nodes is \\(2\\); if the tree's height is \\(h\\), then the total number of nodes is \\(2^{h+1} - 1\\), showing a standard exponential relationship, reflecting the common phenomenon of cell division in nature.</p> <p>Tip</p> <p>Please note that in the Chinese community, a perfect binary tree is often referred to as a \"full binary tree.\"</p> <p></p> <p> Figure 7-4 \u00a0 Perfect binary tree </p>"},{"location":"chapter_tree/binary_tree/#2-complete-binary-tree","title":"2. \u00a0 Complete binary tree","text":"<p>As shown in the Figure 7-5 , a \"complete binary tree\" has only the bottom level nodes not fully filled, and the bottom level nodes are filled as far left as possible.</p> <p></p> <p> Figure 7-5 \u00a0 Complete binary tree </p>"},{"location":"chapter_tree/binary_tree/#3-full-binary-tree","title":"3. \u00a0 Full binary tree","text":"<p>As shown in the Figure 7-6 , a \"full binary tree\" has all nodes except leaf nodes having two children.</p> <p></p> <p> Figure 7-6 \u00a0 Full binary tree </p>"},{"location":"chapter_tree/binary_tree/#4-balanced-binary-tree","title":"4. \u00a0 Balanced binary tree","text":"<p>As shown in the Figure 7-7 , in a \"balanced binary tree,\" the absolute difference in height between the left and right subtrees of any node does not exceed 1.</p> <p></p> <p> Figure 7-7 \u00a0 Balanced binary tree </p>"},{"location":"chapter_tree/binary_tree/#714-degeneration-of-binary-trees","title":"7.1.4 \u00a0 Degeneration of binary trees","text":"<p>The Figure 7-8  shows the ideal and degenerate structures of binary trees. When every level of a binary tree is filled, it reaches the \"perfect binary tree\"; when all nodes are biased towards one side, the binary tree degenerates into a \"linked list\".</p> <ul> <li>The perfect binary tree is the ideal situation, fully leveraging the \"divide and conquer\" advantage of binary trees.</li> <li>A linked list is another extreme, where operations become linear, degrading the time complexity to \\(O(n)\\).</li> </ul> <p></p> <p> Figure 7-8 \u00a0 The Best and Worst Structures of Binary Trees </p> <p>As shown in the Table 7-1 , in the best and worst structures, the number of leaf nodes, total number of nodes, and height of the binary tree reach their maximum or minimum values.</p> <p> Table 7-1 \u00a0 The Best and Worst Structures of Binary Trees </p> Perfect binary tree Linked list Number of nodes at level \\(i\\) \\(2^{i-1}\\) \\(1\\) Number of leaf nodes in a tree with height \\(h\\) \\(2^h\\) \\(1\\) Total number of nodes in a tree with height \\(h\\) \\(2^{h+1} - 1\\) \\(h + 1\\) Height of a tree with \\(n\\) total nodes \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)"},{"location":"chapter_tree/binary_tree_traversal/","title":"7.2 \u00a0 Binary tree traversal","text":"<p>From the perspective of physical structure, a tree is a data structure based on linked lists, hence its traversal method involves accessing nodes one by one through pointers. However, a tree is a non-linear data structure, which makes traversing a tree more complex than traversing a linked list, requiring the assistance of search algorithms to achieve.</p> <p>Common traversal methods for binary trees include level-order traversal, preorder traversal, inorder traversal, and postorder traversal, among others.</p>"},{"location":"chapter_tree/binary_tree_traversal/#721-level-order-traversal","title":"7.2.1 \u00a0 Level-order traversal","text":"<p>As shown in the Figure 7-9 , \"level-order traversal\" traverses the binary tree from top to bottom, layer by layer, and accesses nodes in each layer in a left-to-right order.</p> <p>Level-order traversal essentially belongs to \"breadth-first traversal\", also known as \"breadth-first search (BFS)\", which embodies a \"circumferentially outward expanding\" layer-by-layer traversal method.</p> <p></p> <p> Figure 7-9 \u00a0 Level-order traversal of a binary tree </p>"},{"location":"chapter_tree/binary_tree_traversal/#1-code-implementation","title":"1. \u00a0 Code implementation","text":"<p>Breadth-first traversal is usually implemented with the help of a \"queue\". The queue follows the \"first in, first out\" rule, while breadth-first traversal follows the \"layer-by-layer progression\" rule, the underlying ideas of the two are consistent. The implementation code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree_bfs.py<pre><code>def level_order(root: TreeNode | None) -&gt; list[int]:\n    \"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n    # \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    queue: deque[TreeNode] = deque()\n    queue.append(root)\n    # \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    res = []\n    while queue:\n        node: TreeNode = queue.popleft()  # \u961f\u5217\u51fa\u961f\n        res.append(node.val)  # \u4fdd\u5b58\u8282\u70b9\u503c\n        if node.left is not None:\n            queue.append(node.left)  # \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if node.right is not None:\n            queue.append(node.right)  # \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    return res\n</code></pre> binary_tree_bfs.cpp<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nvector&lt;int&gt; levelOrder(TreeNode *root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    queue&lt;TreeNode *&gt; queue;\n    queue.push(root);\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    vector&lt;int&gt; vec;\n    while (!queue.empty()) {\n        TreeNode *node = queue.front();\n        queue.pop();              // \u961f\u5217\u51fa\u961f\n        vec.push_back(node-&gt;val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node-&gt;left != nullptr)\n            queue.push(node-&gt;left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node-&gt;right != nullptr)\n            queue.push(node-&gt;right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return vec;\n}\n</code></pre> binary_tree_bfs.java<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;Integer&gt; levelOrder(TreeNode root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    Queue&lt;TreeNode&gt; queue = new LinkedList&lt;&gt;();\n    queue.add(root);\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    List&lt;Integer&gt; list = new ArrayList&lt;&gt;();\n    while (!queue.isEmpty()) {\n        TreeNode node = queue.poll(); // \u961f\u5217\u51fa\u961f\n        list.add(node.val);           // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null)\n            queue.offer(node.left);   // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node.right != null)\n            queue.offer(node.right);  // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.cs<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;int&gt; LevelOrder(TreeNode root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    Queue&lt;TreeNode&gt; queue = new();\n    queue.Enqueue(root);\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    List&lt;int&gt; list = [];\n    while (queue.Count != 0) {\n        TreeNode node = queue.Dequeue(); // \u961f\u5217\u51fa\u961f\n        list.Add(node.val!.Value);       // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null)\n            queue.Enqueue(node.left);    // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node.right != null)\n            queue.Enqueue(node.right);   // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.go<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root *TreeNode) []any {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    queue := list.New()\n    queue.PushBack(root)\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5207\u7247\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    nums := make([]any, 0)\n    for queue.Len() &gt; 0 {\n        // \u961f\u5217\u51fa\u961f\n        node := queue.Remove(queue.Front()).(*TreeNode)\n        // \u4fdd\u5b58\u8282\u70b9\u503c\n        nums = append(nums, node.Val)\n        if node.Left != nil {\n            // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n            queue.PushBack(node.Left)\n        }\n        if node.Right != nil {\n            // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n            queue.PushBack(node.Right)\n        }\n    }\n    return nums\n}\n</code></pre> binary_tree_bfs.swift<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root: TreeNode) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    var queue: [TreeNode] = [root]\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    var list: [Int] = []\n    while !queue.isEmpty {\n        let node = queue.removeFirst() // \u961f\u5217\u51fa\u961f\n        list.append(node.val) // \u4fdd\u5b58\u8282\u70b9\u503c\n        if let left = node.left {\n            queue.append(left) // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if let right = node.right {\n            queue.append(right) // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        }\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.js<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    const queue = [root];\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    const list = [];\n    while (queue.length) {\n        let node = queue.shift(); // \u961f\u5217\u51fa\u961f\n        list.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left) queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node.right) queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.ts<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root: TreeNode | null): number[] {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    const queue = [root];\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    const list: number[] = [];\n    while (queue.length) {\n        let node = queue.shift() as TreeNode; // \u961f\u5217\u51fa\u961f\n        list.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left) {\n            queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if (node.right) {\n            queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        }\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.dart<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;int&gt; levelOrder(TreeNode? root) {\n  // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n  Queue&lt;TreeNode?&gt; queue = Queue();\n  queue.add(root);\n  // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n  List&lt;int&gt; res = [];\n  while (queue.isNotEmpty) {\n    TreeNode? node = queue.removeFirst(); // \u961f\u5217\u51fa\u961f\n    res.add(node!.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n    if (node.left != null) queue.add(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n    if (node.right != null) queue.add(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n  }\n  return res;\n}\n</code></pre> binary_tree_bfs.rs<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfn level_order(root: &amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    let mut que = VecDeque::new();\n    que.push_back(Rc::clone(&amp;root));\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    let mut vec = Vec::new();\n\n    while let Some(node) = que.pop_front() {\n        // \u961f\u5217\u51fa\u961f\n        vec.push(node.borrow().val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if let Some(left) = node.borrow().left.as_ref() {\n            que.push_back(Rc::clone(left)); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if let Some(right) = node.borrow().right.as_ref() {\n            que.push_back(Rc::clone(right)); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        };\n    }\n    vec\n}\n</code></pre> binary_tree_bfs.c<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(TreeNode *root, int *size) {\n    /* \u8f85\u52a9\u961f\u5217 */\n    int front, rear;\n    int index, *arr;\n    TreeNode *node;\n    TreeNode **queue;\n\n    /* \u8f85\u52a9\u961f\u5217 */\n    queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);\n    // \u961f\u5217\u6307\u9488\n    front = 0, rear = 0;\n    // \u52a0\u5165\u6839\u8282\u70b9\n    queue[rear++] = root;\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    /* \u8f85\u52a9\u6570\u7ec4 */\n    arr = (int *)malloc(sizeof(int) * MAX_SIZE);\n    // \u6570\u7ec4\u6307\u9488\n    index = 0;\n    while (front &lt; rear) {\n        // \u961f\u5217\u51fa\u961f\n        node = queue[front++];\n        // \u4fdd\u5b58\u8282\u70b9\u503c\n        arr[index++] = node-&gt;val;\n        if (node-&gt;left != NULL) {\n            // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n            queue[rear++] = node-&gt;left;\n        }\n        if (node-&gt;right != NULL) {\n            // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n            queue[rear++] = node-&gt;right;\n        }\n    }\n    // \u66f4\u65b0\u6570\u7ec4\u957f\u5ea6\u7684\u503c\n    *size = index;\n    arr = realloc(arr, sizeof(int) * (*size));\n\n    // \u91ca\u653e\u8f85\u52a9\u6570\u7ec4\u7a7a\u95f4\n    free(queue);\n    return arr;\n}\n</code></pre> binary_tree_bfs.kt<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfun levelOrder(root: TreeNode?): MutableList&lt;Int&gt; {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    val queue = LinkedList&lt;TreeNode?&gt;()\n    queue.add(root)\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    val list = ArrayList&lt;Int&gt;()\n    while (!queue.isEmpty()) {\n        val node = queue.poll() // \u961f\u5217\u51fa\u961f\n        list.add(node?.value!!) // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null) queue.offer(node.left) // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n\n        if (node.right != null) queue.offer(node.right) // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.rb<pre><code>[class]{}-[func]{level_order}\n</code></pre> binary_tree_bfs.zig<pre><code>// \u5c42\u5e8f\u904d\u5386\nfn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    const L = std.TailQueue(*inc.TreeNode(T));\n    var queue = L{};\n    var root_node = try mem_allocator.create(L.Node);\n    root_node.data = root;\n    queue.append(root_node); \n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    var list = std.ArrayList(T).init(std.heap.page_allocator);\n    while (queue.len &gt; 0) {\n        var queue_node = queue.popFirst().?;    // \u961f\u5217\u51fa\u961f\n        var node = queue_node.data;\n        try list.append(node.val);              // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null) {\n            var tmp_node = try mem_allocator.create(L.Node);\n            tmp_node.data = node.left.?;\n            queue.append(tmp_node);             // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if (node.right != null) {\n            var tmp_node = try mem_allocator.create(L.Node);\n            tmp_node.data = node.right.?;\n            queue.append(tmp_node);             // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        }        \n    }\n    return list;\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p>"},{"location":"chapter_tree/binary_tree_traversal/#2-complexity-analysis","title":"2. \u00a0 Complexity analysis","text":"<ul> <li>Time complexity is \\(O(n)\\): All nodes are visited once, using \\(O(n)\\) time, where \\(n\\) is the number of nodes.</li> <li>Space complexity is \\(O(n)\\): In the worst case, i.e., a full binary tree, before traversing to the lowest level, the queue can contain at most \\((n + 1) / 2\\) nodes at the same time, occupying \\(O(n)\\) space.</li> </ul>"},{"location":"chapter_tree/binary_tree_traversal/#722-preorder-inorder-and-postorder-traversal","title":"7.2.2 \u00a0 Preorder, inorder, and postorder traversal","text":"<p>Correspondingly, preorder, inorder, and postorder traversal all belong to \"depth-first traversal\", also known as \"depth-first search (DFS)\", which embodies a \"proceed to the end first, then backtrack and continue\" traversal method.</p> <p>The Figure 7-10  shows the working principle of performing a depth-first traversal on a binary tree. Depth-first traversal is like walking around the perimeter of the entire binary tree, encountering three positions at each node, corresponding to preorder traversal, inorder traversal, and postorder traversal.</p> <p></p> <p> Figure 7-10 \u00a0 Preorder, inorder, and postorder traversal of a binary search tree </p>"},{"location":"chapter_tree/binary_tree_traversal/#1-code-implementation_1","title":"1. \u00a0 Code implementation","text":"<p>Depth-first search is usually implemented based on recursion:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree_dfs.py<pre><code>def pre_order(root: TreeNode | None):\n    \"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\n    if root is None:\n        return\n    # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    res.append(root.val)\n    pre_order(root=root.left)\n    pre_order(root=root.right)\n\ndef in_order(root: TreeNode | None):\n    \"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\n    if root is None:\n        return\n    # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    in_order(root=root.left)\n    res.append(root.val)\n    in_order(root=root.right)\n\ndef post_order(root: TreeNode | None):\n    \"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\n    if root is None:\n        return\n    # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    post_order(root=root.left)\n    post_order(root=root.right)\n    res.append(root.val)\n</code></pre> binary_tree_dfs.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    vec.push_back(root-&gt;val);\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root-&gt;left);\n    vec.push_back(root-&gt;val);\n    inOrder(root-&gt;right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root-&gt;left);\n    postOrder(root-&gt;right);\n    vec.push_back(root-&gt;val);\n}\n</code></pre> binary_tree_dfs.java<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.add(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left);\n    list.add(root.val);\n    inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left);\n    postOrder(root.right);\n    list.add(root.val);\n}\n</code></pre> binary_tree_dfs.cs<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.Add(root.val!.Value);\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid InOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    InOrder(root.left);\n    list.Add(root.val!.Value);\n    InOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid PostOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    PostOrder(root.left);\n    PostOrder(root.right);\n    list.Add(root.val!.Value);\n}\n</code></pre> binary_tree_dfs.go<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    nums = append(nums, node.Val)\n    preOrder(node.Left)\n    preOrder(node.Right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(node.Left)\n    nums = append(nums, node.Val)\n    inOrder(node.Right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(node.Left)\n    postOrder(node.Right)\n    nums = append(nums, node.Val)\n}\n</code></pre> binary_tree_dfs.swift<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.append(root.val)\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root: root.left)\n    list.append(root.val)\n    inOrder(root: root.right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root: root.left)\n    postOrder(root: root.right)\n    list.append(root.val)\n}\n</code></pre> binary_tree_dfs.js<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root) {\n    if (root === null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root) {\n    if (root === null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root) {\n    if (root === null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.ts<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.dart<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n  list.add(node.val);\n  preOrder(node.left);\n  preOrder(node.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n  inOrder(node.left);\n  list.add(node.val);\n  inOrder(node.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n  postOrder(node.left);\n  postOrder(node.right);\n  list.add(node.val);\n}\n</code></pre> binary_tree_dfs.rs<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfn pre_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    if let Some(node) = root {\n        // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n        result.push(node.borrow().val);\n        result.append(&amp;mut pre_order(node.borrow().left.as_ref()));\n        result.append(&amp;mut pre_order(node.borrow().right.as_ref()));\n    }\n    result\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfn in_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    if let Some(node) = root {\n        // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n        result.append(&amp;mut in_order(node.borrow().left.as_ref()));\n        result.push(node.borrow().val);\n        result.append(&amp;mut in_order(node.borrow().right.as_ref()));\n    }\n    result\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfn post_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    if let Some(node) = root {\n        // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n        result.append(&amp;mut post_order(node.borrow().left.as_ref()));\n        result.append(&amp;mut post_order(node.borrow().right.as_ref()));\n        result.push(node.borrow().val);\n    }\n    result\n}\n</code></pre> binary_tree_dfs.c<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    arr[(*size)++] = root-&gt;val;\n    preOrder(root-&gt;left, size);\n    preOrder(root-&gt;right, size);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root-&gt;left, size);\n    arr[(*size)++] = root-&gt;val;\n    inOrder(root-&gt;right, size);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root-&gt;left, size);\n    postOrder(root-&gt;right, size);\n    arr[(*size)++] = root-&gt;val;\n}\n</code></pre> binary_tree_dfs.kt<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.add(root.value)\n    preOrder(root.left)\n    preOrder(root.right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfun inOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left)\n    list.add(root.value)\n    inOrder(root.right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfun postOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left)\n    postOrder(root.right)\n    list.add(root.value)\n}\n</code></pre> binary_tree_dfs.rb<pre><code>[class]{}-[func]{pre_order}\n\n[class]{}-[func]{in_order}\n\n[class]{}-[func]{post_order}\n</code></pre> binary_tree_dfs.zig<pre><code>// \u524d\u5e8f\u904d\u5386\nfn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    try list.append(root.?.val);\n    try preOrder(T, root.?.left);\n    try preOrder(T, root.?.right);\n}\n\n// \u4e2d\u5e8f\u904d\u5386\nfn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    try inOrder(T, root.?.left);\n    try list.append(root.?.val);\n    try inOrder(T, root.?.right);\n}\n\n// \u540e\u5e8f\u904d\u5386\nfn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    try postOrder(T, root.?.left);\n    try postOrder(T, root.?.right);\n    try list.append(root.?.val);\n}\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Tip</p> <p>Depth-first search can also be implemented based on iteration, interested readers can study this on their own.</p> <p>The Figure 7-11  shows the recursive process of preorder traversal of a binary tree, which can be divided into two opposite parts: \"recursion\" and \"return\".</p> <ol> <li>\"Recursion\" means starting a new method, the program accesses the next node in this process.</li> <li>\"Return\" means the function returns, indicating the current node has been fully accessed.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 7-11 \u00a0 The recursive process of preorder traversal </p>"},{"location":"chapter_tree/binary_tree_traversal/#2-complexity-analysis_1","title":"2. \u00a0 Complexity analysis","text":"<ul> <li>Time complexity is \\(O(n)\\): All nodes are visited once, using \\(O(n)\\) time.</li> <li>Space complexity is \\(O(n)\\): In the worst case, i.e., the tree degrades into a linked list, the recursion depth reaches \\(n\\), the system occupies \\(O(n)\\) stack frame space.</li> </ul>"},{"location":"chapter_tree/summary/","title":"7.6 \u00a0 Summary","text":""},{"location":"chapter_tree/summary/#1-key-review","title":"1. \u00a0 Key review","text":"<ul> <li>A binary tree is a non-linear data structure that reflects the \"divide and conquer\" logic of splitting one into two. Each binary tree node contains a value and two pointers, which point to its left and right child nodes, respectively.</li> <li>For a node in a binary tree, the tree formed by its left (right) child node and all nodes under it is called the node's left (right) subtree.</li> <li>Related terminology of binary trees includes root node, leaf node, level, degree, edge, height, and depth, among others.</li> <li>The operations of initializing a binary tree, inserting nodes, and removing nodes are similar to those of linked list operations.</li> <li>Common types of binary trees include perfect binary trees, complete binary trees, full binary trees, and balanced binary trees. The perfect binary tree represents the ideal state, while the linked list is the worst state after degradation.</li> <li>A binary tree can be represented using an array by arranging the node values and empty slots in a level-order traversal sequence and implementing pointers based on the index mapping relationship between parent nodes and child nodes.</li> <li>The level-order traversal of a binary tree is a breadth-first search method, which reflects a layer-by-layer traversal manner of \"expanding circle by circle.\" It is usually implemented using a queue.</li> <li>Pre-order, in-order, and post-order traversals are all depth-first search methods, reflecting the traversal manner of \"going to the end first, then backtracking to continue.\" They are usually implemented using recursion.</li> <li>A binary search tree is an efficient data structure for element searching, with the time complexity of search, insert, and remove operations all being \\(O(\\log n)\\). When a binary search tree degrades into a linked list, these time complexities deteriorate to \\(O(n)\\).</li> <li>An AVL tree, also known as a balanced binary search tree, ensures that the tree remains balanced after continuous node insertions and removals through rotation operations.</li> <li>Rotation operations in an AVL tree include right rotation, left rotation, right-then-left rotation, and left-then-right rotation. After inserting or removing nodes, an AVL tree performs rotation operations from bottom to top to rebalance the tree.</li> </ul>"},{"location":"chapter_tree/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q: For a binary tree with only one node, are both the height of the tree and the depth of the root node \\(0\\)?</p> <p>Yes, because height and depth are typically defined as \"the number of edges passed.\"</p> <p>Q: The insertion and removal in a binary tree are generally completed by a set of operations. What does \"a set of operations\" refer to here? Can it be understood as the release of resources of the child nodes?</p> <p>Taking the binary search tree as an example, the operation of removing a node needs to be handled in three different scenarios, each requiring multiple steps of node operations.</p> <p>Q: Why are there three sequences: pre-order, in-order, and post-order for DFS traversal of a binary tree, and what are their uses?</p> <p>Similar to sequential and reverse traversal of arrays, pre-order, in-order, and post-order traversals are three methods of traversing a binary tree, allowing us to obtain a traversal result in a specific order. For example, in a binary search tree, since the node sizes satisfy <code>left child node value &lt; root node value &lt; right child node value</code>, we can obtain an ordered node sequence by traversing the tree in the \"left \u2192 root \u2192 right\" priority.</p> <p>Q: In a right rotation operation that deals with the relationship between the imbalance nodes <code>node</code>, <code>child</code>, <code>grand_child</code>, isn't the connection between <code>node</code> and its parent node and the original link of <code>node</code> lost after the right rotation?</p> <p>We need to view this problem from a recursive perspective. The <code>right_rotate(root)</code> operation passes the root node of the subtree and eventually returns the root node of the rotated subtree with <code>return child</code>. The connection between the subtree's root node and its parent node is established after this function returns, which is outside the scope of the right rotation operation's maintenance.</p> <p>Q: In C++, functions are divided into <code>private</code> and <code>public</code> sections. What considerations are there for this? Why are the <code>height()</code> function and the <code>updateHeight()</code> function placed in <code>public</code> and <code>private</code>, respectively?</p> <p>It depends on the scope of the method's use. If a method is only used within the class, then it is designed to be <code>private</code>. For example, it makes no sense for users to call <code>updateHeight()</code> on their own, as it is just a step in the insertion or removal operations. However, <code>height()</code> is for accessing node height, similar to <code>vector.size()</code>, thus it is set to <code>public</code> for use.</p> <p>Q: How do you build a binary search tree from a set of input data? Is the choice of root node very important?</p> <p>Yes, the method for building the tree is provided in the <code>build_tree()</code> method in the binary search tree code. As for the choice of the root node, we usually sort the input data and then select the middle element as the root node, recursively building the left and right subtrees. This approach maximizes the balance of the tree.</p> <p>Q: In Java, do you always have to use the <code>equals()</code> method for string comparison?</p> <p>In Java, for primitive data types, <code>==</code> is used to compare whether the values of two variables are equal. For reference types, the working principles of the two symbols are different.</p> <ul> <li><code>==</code>: Used to compare whether two variables point to the same object, i.e., whether their positions in memory are the same.</li> <li><code>equals()</code>: Used to compare whether the values of two objects are equal.</li> </ul> <p>Therefore, to compare values, we should use <code>equals()</code>. However, strings initialized with <code>String a = \"hi\"; String b = \"hi\";</code> are stored in the string constant pool and point to the same object, so <code>a == b</code> can also be used to compare the contents of two strings.</p> <p>Q: Before reaching the bottom level, is the number of nodes in the queue \\(2^h\\) in breadth-first traversal?</p> <p>Yes, for example, a full binary tree with height \\(h = 2\\) has a total of \\(n = 7\\) nodes, then the bottom level has \\(4 = 2^h = (n + 1) / 2\\) nodes.</p>"}]}
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+                      <div class="md-nav__link md-nav__container">
+                        <a class="md-nav__link" href="chapter_hello_algo/">
+                          <svg viewbox="0 0 24 24" xmlns="http://www.w3.org/2000/svg">
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+                          </svg>
+                          <span class="md-ellipsis">
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+                          </span>
+                        </a>
+                      </div>
+                      <nav aria-expanded="false" aria-labelledby="__nav_1_label" class="md-nav" data-md-level="1">
+                        <label class="md-nav__title" for="__nav_1">
+                          <span class="md-nav__icon md-icon">
+                          </span>
+                          序
+                        </label>
+                        <ul class="md-nav__list" data-md-scrollfix="">
+                        </ul>
+                      </nav>
+                    </li>
+                    <li class="md-nav__item md-nav__item--nested">
+                      <input class="md-nav__toggle md-toggle" id="__nav_2" type="checkbox"/>
                       <div class="md-nav__link md-nav__container">
                         <a class="md-nav__link" href="chapter_preface/">
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                             第 0 章   前言
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-                        <label class="md-nav__title" for="__nav_1">
+                      <nav aria-expanded="false" aria-labelledby="__nav_2_label" class="md-nav" data-md-level="1">
+                        <label class="md-nav__title" for="__nav_2">
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                           第 0 章   前言
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-                      <input class="md-nav__toggle md-toggle" id="__nav_2" type="checkbox"/>
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                         <a class="md-nav__link" href="chapter_introduction/">
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                             第 1 章   初识算法
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+                        <label class="md-nav__title" for="__nav_3">
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                             第 2 章   复杂度分析
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+                        <label class="md-nav__title" for="__nav_4">
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                             第 3 章   数据结构
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+                        <label class="md-nav__title" for="__nav_5">
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-                      <input class="md-nav__toggle md-toggle" id="__nav_5" type="checkbox"/>
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                         <a class="md-nav__link" href="chapter_array_and_linkedlist/">
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                             第 4 章   数组与链表
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-                        <label class="md-nav__title" for="__nav_5">
+                      <nav aria-expanded="false" aria-labelledby="__nav_6_label" class="md-nav" data-md-level="1">
+                        <label class="md-nav__title" for="__nav_6">
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                           第 4 章   数组与链表
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-                      <input class="md-nav__toggle md-toggle" id="__nav_6" type="checkbox"/>
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                         <a class="md-nav__link" href="chapter_stack_and_queue/">
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                             第 5 章   栈与队列
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-                        <label class="md-nav__link" for="__nav_6" id="__nav_6_label" tabindex="0">
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-                        <label class="md-nav__title" for="__nav_6">
+                      <nav aria-expanded="false" aria-labelledby="__nav_7_label" class="md-nav" data-md-level="1">
+                        <label class="md-nav__title" for="__nav_7">
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                           第 5 章   栈与队列
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                     </li>
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-                      <input class="md-nav__toggle md-toggle" id="__nav_7" type="checkbox"/>
+                      <input class="md-nav__toggle md-toggle" id="__nav_8" type="checkbox"/>
                       <div class="md-nav__link md-nav__container">
                         <a class="md-nav__link" href="chapter_hashing/">
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                             第 6 章   哈希表
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-                        <label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
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-                      <nav aria-expanded="false" aria-labelledby="__nav_7_label" class="md-nav" data-md-level="1">
-                        <label class="md-nav__title" for="__nav_7">
+                      <nav aria-expanded="false" aria-labelledby="__nav_8_label" class="md-nav" data-md-level="1">
+                        <label class="md-nav__title" for="__nav_8">
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                           第 6 章   哈希表
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-                      <input class="md-nav__toggle md-toggle" id="__nav_8" type="checkbox"/>
+                      <input class="md-nav__toggle md-toggle" id="__nav_9" type="checkbox"/>
                       <div class="md-nav__link md-nav__container">
                         <a class="md-nav__link" href="chapter_tree/">
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                             第 7 章   树
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-                        <label class="md-nav__title" for="__nav_8">
+                      <nav aria-expanded="false" aria-labelledby="__nav_9_label" class="md-nav" data-md-level="1">
+                        <label class="md-nav__title" for="__nav_9">
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-                      <input class="md-nav__toggle md-toggle" id="__nav_9" type="checkbox"/>
+                      <input class="md-nav__toggle md-toggle" id="__nav_10" type="checkbox"/>
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                             第 8 章   堆
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-                        <label class="md-nav__title" for="__nav_9">
+                      <nav aria-expanded="false" aria-labelledby="__nav_10_label" class="md-nav" data-md-level="1">
+                        <label class="md-nav__title" for="__nav_10">
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@@ -1180,7 +1194,7 @@
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                     </li>
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-                      <input class="md-nav__toggle md-toggle" id="__nav_10" type="checkbox"/>
+                      <input class="md-nav__toggle md-toggle" id="__nav_11" type="checkbox"/>
                       <div class="md-nav__link md-nav__container">
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@@ -1191,13 +1205,13 @@
                             第 9 章   图
                           </span>
                         </a>
-                        <label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
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-                      <nav aria-expanded="false" aria-labelledby="__nav_10_label" class="md-nav" data-md-level="1">
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+                      <nav aria-expanded="false" aria-labelledby="__nav_11_label" class="md-nav" data-md-level="1">
+                        <label class="md-nav__title" for="__nav_11">
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-                      <input class="md-nav__toggle md-toggle" id="__nav_11" type="checkbox"/>
+                      <input class="md-nav__toggle md-toggle" id="__nav_12" type="checkbox"/>
                       <div class="md-nav__link md-nav__container">
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                             第 10 章   搜索
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-                        <label class="md-nav__title" for="__nav_11">
+                      <nav aria-expanded="false" aria-labelledby="__nav_12_label" class="md-nav" data-md-level="1">
+                        <label class="md-nav__title" for="__nav_12">
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                     </li>
                     <li class="md-nav__item md-nav__item--nested">
-                      <input class="md-nav__toggle md-toggle" id="__nav_12" type="checkbox"/>
+                      <input class="md-nav__toggle md-toggle" id="__nav_13" type="checkbox"/>
                       <div class="md-nav__link md-nav__container">
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                             第 11 章   排序
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                         <a class="md-nav__link" href="chapter_divide_and_conquer/">
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@@ -1419,13 +1433,13 @@
                             第 12 章   分治
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-                        <label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
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                           </span>
                           第 12 章   分治
@@ -1470,7 +1484,7 @@
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-                      <input class="md-nav__toggle md-toggle" id="__nav_14" type="checkbox"/>
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                             第 13 章   回溯
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-                        <label class="md-nav__link" for="__nav_14" id="__nav_14_label" tabindex="0">
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                           第 13 章   回溯
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@@ -1543,13 +1557,13 @@
                             第 14 章   动态规划
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                           第 14 章   动态规划
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                         <a class="md-nav__link" href="chapter_greedy/">
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                             第 15 章   贪心
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-                        <label class="md-nav__link" for="__nav_16" id="__nav_16_label" tabindex="0">
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                           第 15 章   贪心
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                             第 16 章   附录
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+                        <label class="md-nav__title" for="__nav_18">
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                           第 16 章   附录
@@ -1718,7 +1732,7 @@
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                     <li class="md-nav__item md-nav__item--nested">
-                      <input class="md-nav__toggle md-toggle" id="__nav_18" type="checkbox"/>
+                      <input class="md-nav__toggle md-toggle" id="__nav_19" type="checkbox"/>
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                         <a class="md-nav__link" href="chapter_reference/">
                           <svg viewbox="0 0 24 24" xmlns="http://www.w3.org/2000/svg">
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                           参考文献
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-                      <input class="md-nav__toggle md-toggle" id="__nav_19" type="checkbox"/>
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                           纸质书
@@ -1805,7 +1818,9 @@
           <div class="md-footer-meta__inner md-grid">
             <div class="md-copyright">
               <div class="md-copyright__highlight">
-                Copyright © 2022-2024 krahetsThe website content is licensed under
+                Copyright © 2022-2024 krahets
+                <br/>
+                The website content is licensed under
                 <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">
                   CC BY-NC-SA 4.0
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diff --git a/search/search_index.json b/search/search_index.json
index debbc0beb..30a2d9353 100644
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+++ b/search/search_index.json
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<p>\u7eb8\u8d28\u56fe\u4e66\u7684\u4e24\u6b21\u5370\u5237\u7684\u95f4\u9694\u65f6\u95f4\u5f80\u5f80\u8f83\u4e45\uff0c\u5185\u5bb9\u66f4\u65b0\u975e\u5e38\u4e0d\u65b9\u4fbf\u3002</p> <p>\u800c\u5728\u672c\u5f00\u6e90\u4e66\u4e2d\uff0c\u5185\u5bb9\u66f4\u8fed\u7684\u65f6\u95f4\u88ab\u7f29\u77ed\u81f3\u6570\u65e5\u751a\u81f3\u51e0\u4e2a\u5c0f\u65f6\u3002</p>"},{"location":"chapter_appendix/contribution/#1","title":"1. \u00a0 \u5185\u5bb9\u5fae\u8c03","text":"<p>\u5982\u56fe 16-3 \u6240\u793a\uff0c\u6bcf\u4e2a\u9875\u9762\u7684\u53f3\u4e0a\u89d2\u90fd\u6709\u201c\u7f16\u8f91\u56fe\u6807\u201d\u3002\u60a8\u53ef\u4ee5\u6309\u7167\u4ee5\u4e0b\u6b65\u9aa4\u4fee\u6539\u6587\u672c\u6216\u4ee3\u7801\u3002</p> <ol> <li>\u70b9\u51fb\u201c\u7f16\u8f91\u56fe\u6807\u201d\uff0c\u5982\u679c\u9047\u5230\u201c\u9700\u8981 Fork \u6b64\u4ed3\u5e93\u201d\u7684\u63d0\u793a\uff0c\u8bf7\u540c\u610f\u8be5\u64cd\u4f5c\u3002</li> <li>\u4fee\u6539 Markdown \u6e90\u6587\u4ef6\u5185\u5bb9\uff0c\u68c0\u67e5\u5185\u5bb9\u7684\u6b63\u786e\u6027\uff0c\u5e76\u5c3d\u91cf\u4fdd\u6301\u6392\u7248\u683c\u5f0f\u7684\u7edf\u4e00\u3002</li> <li>\u5728\u9875\u9762\u5e95\u90e8\u586b\u5199\u4fee\u6539\u8bf4\u660e\uff0c\u7136\u540e\u70b9\u51fb\u201cPropose file change\u201d\u6309\u94ae\u3002\u9875\u9762\u8df3\u8f6c\u540e\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u3002</li> </ol> <p></p> <p> \u56fe 16-3 \u00a0 \u9875\u9762\u7f16\u8f91\u6309\u952e </p> <p>\u56fe\u7247\u65e0\u6cd5\u76f4\u63a5\u4fee\u6539\uff0c\u9700\u8981\u901a\u8fc7\u65b0\u5efa Issue \u6216\u8bc4\u8bba\u7559\u8a00\u6765\u63cf\u8ff0\u95ee\u9898\uff0c\u6211\u4eec\u4f1a\u5c3d\u5feb\u91cd\u65b0\u7ed8\u5236\u5e76\u66ff\u6362\u56fe\u7247\u3002</p>"},{"location":"chapter_appendix/contribution/#2","title":"2. \u00a0 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<li>\u5237\u65b0\u4ed3\u5e93\u7f51\u9875\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u3002</li> </ol>"},{"location":"chapter_appendix/contribution/#3-docker","title":"3. \u00a0 Docker \u90e8\u7f72","text":"<p>\u5728 <code>hello-algo</code> \u6839\u76ee\u5f55\u4e0b\uff0c\u6267\u884c\u4ee5\u4e0b Docker \u811a\u672c\uff0c\u5373\u53ef\u5728 <code>http://localhost:8000</code> \u8bbf\u95ee\u672c\u9879\u76ee\uff1a</p> <pre><code>docker-compose up -d\n</code></pre> <p>\u4f7f\u7528\u4ee5\u4e0b\u547d\u4ee4\u5373\u53ef\u5220\u9664\u90e8\u7f72\uff1a</p> <pre><code>docker-compose down\n</code></pre>"},{"location":"chapter_appendix/installation/","title":"16.1 \u00a0 \u7f16\u7a0b\u73af\u5883\u5b89\u88c5","text":""},{"location":"chapter_appendix/installation/#1611-ide","title":"16.1.1 \u00a0 \u5b89\u88c5 IDE","text":"<p>\u63a8\u8350\u4f7f\u7528\u5f00\u6e90\u3001\u8f7b\u91cf\u7684 VS Code \u4f5c\u4e3a\u672c\u5730\u96c6\u6210\u5f00\u53d1\u73af\u5883\uff08IDE\uff09\u3002\u8bbf\u95ee VS Code \u5b98\u7f51\uff0c\u6839\u636e\u64cd\u4f5c\u7cfb\u7edf\u9009\u62e9\u76f8\u5e94\u7248\u672c\u7684 VS Code \u8fdb\u884c\u4e0b\u8f7d\u548c\u5b89\u88c5\u3002</p> <p></p> <p> \u56fe 16-1 \u00a0 \u4ece\u5b98\u7f51\u4e0b\u8f7d VS Code </p> <p>VS Code \u62e5\u6709\u5f3a\u5927\u7684\u6269\u5c55\u5305\u751f\u6001\u7cfb\u7edf\uff0c\u652f\u6301\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u7684\u8fd0\u884c\u548c\u8c03\u8bd5\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u5b89\u88c5\u201cPython Extension Pack\u201d\u6269\u5c55\u5305\u4e4b\u540e\uff0c\u5373\u53ef\u8fdb\u884c Python \u4ee3\u7801\u8c03\u8bd5\u3002\u5b89\u88c5\u6b65\u9aa4\u5982\u56fe 16-2 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 16-2 \u00a0 \u5b89\u88c5 VS Code \u6269\u5c55\u5305 </p>"},{"location":"chapter_appendix/installation/#1612","title":"16.1.2 \u00a0 \u5b89\u88c5\u8bed\u8a00\u73af\u5883","text":""},{"location":"chapter_appendix/installation/#1-python","title":"1. \u00a0 Python \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Miniconda3 \uff0c\u9700\u8981 Python 3.10 \u6216\u66f4\u65b0\u7248\u672c\u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>python</code> \uff0c\u5b89\u88c5 Python Extension Pack \u3002</li> <li>\uff08\u53ef\u9009\uff09\u5728\u547d\u4ee4\u884c\u8f93\u5165 <code>pip install black</code> \uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#2-cc","title":"2. \u00a0 C/C++ \u73af\u5883","text":"<ol> <li>Windows \u7cfb\u7edf\u9700\u8981\u5b89\u88c5 MinGW\uff08\u914d\u7f6e\u6559\u7a0b\uff09\uff1bMacOS \u81ea\u5e26 Clang \uff0c\u65e0\u987b\u5b89\u88c5\u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>c++</code> \uff0c\u5b89\u88c5 C/C++ Extension Pack \u3002</li> <li>\uff08\u53ef\u9009\uff09\u6253\u5f00 Settings \u9875\u9762\uff0c\u641c\u7d22 <code>Clang_format_fallback Style</code> \u4ee3\u7801\u683c\u5f0f\u5316\u9009\u9879\uff0c\u8bbe\u7f6e\u4e3a <code>{ BasedOnStyle: Microsoft, BreakBeforeBraces: Attach }</code> \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#3-java","title":"3. \u00a0 Java \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 OpenJDK\uff08\u7248\u672c\u9700\u6ee1\u8db3 &gt; JDK 9\uff09\u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>java</code> \uff0c\u5b89\u88c5 Extension Pack for Java \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#4-c","title":"4. \u00a0 C# \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 .Net 8.0 \u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>C# Dev Kit</code> \uff0c\u5b89\u88c5 C# Dev Kit \uff08\u914d\u7f6e\u6559\u7a0b\uff09\u3002</li> <li>\u4e5f\u53ef\u4f7f\u7528 Visual Studio\uff08\u5b89\u88c5\u6559\u7a0b\uff09\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#5-go","title":"5. \u00a0 Go \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 go \u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>go</code> \uff0c\u5b89\u88c5 Go \u3002</li> <li>\u6309\u5feb\u6377\u952e <code>Ctrl + Shift + P</code> \u547c\u51fa\u547d\u4ee4\u680f\uff0c\u8f93\u5165 go \uff0c\u9009\u62e9 <code>Go: Install/Update Tools</code> \uff0c\u5168\u90e8\u52fe\u9009\u5e76\u5b89\u88c5\u5373\u53ef\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#6-swift","title":"6. \u00a0 Swift \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Swift \u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>swift</code> \uff0c\u5b89\u88c5 Swift for Visual Studio Code \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#7-javascript","title":"7. \u00a0 JavaScript \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 node.js \u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>javascript</code> \uff0c\u5b89\u88c5 JavaScript (ES6) code snippets \u3002</li> <li>\uff08\u53ef\u9009\uff09\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>Prettier</code> \uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#8-dart","title":"8. \u00a0 Dart \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Dart \u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>dart</code> \uff0c\u5b89\u88c5 Dart \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#9-rust","title":"9. \u00a0 Rust \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Rust \u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>rust</code> \uff0c\u5b89\u88c5 rust-analyzer \u3002</li> </ol>"},{"location":"chapter_appendix/terminology/","title":"16.3 \u00a0 \u672f\u8bed\u8868","text":"<p>\u8868 16-1 \u5217\u51fa\u4e86\u4e66\u4e2d\u51fa\u73b0\u7684\u91cd\u8981\u672f\u8bed\uff0c\u503c\u5f97\u6ce8\u610f\u4ee5\u4e0b\u51e0\u70b9\u3002</p> <ul> <li>\u5efa\u8bae\u8bb0\u4f4f\u540d\u8bcd\u7684\u82f1\u6587\u53eb\u6cd5\uff0c\u4ee5\u4fbf\u9605\u8bfb\u82f1\u6587\u6587\u732e\u3002</li> <li>\u90e8\u5206\u540d\u8bcd\u5728\u7b80\u4f53\u4e2d\u6587\u548c\u7e41\u4f53\u4e2d\u6587\u4e0b\u7684\u53eb\u6cd5\u4e0d\u540c\u3002</li> </ul> <p> \u8868 16-1 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u91cd\u8981\u540d\u8bcd </p> English \u7b80\u4f53\u4e2d\u6587 \u7e41\u4f53\u4e2d\u6587 algorithm \u7b97\u6cd5 \u6f14\u7b97\u6cd5 data structure \u6570\u636e\u7ed3\u6784 \u8cc7\u6599\u7d50\u69cb code \u4ee3\u7801 \u7a0b\u5f0f\u78bc file \u6587\u4ef6 \u6a94\u6848 function \u51fd\u6570 \u51fd\u5f0f method \u65b9\u6cd5 \u65b9\u6cd5 variable \u53d8\u91cf \u8b8a\u6578 asymptotic complexity analysis \u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 \u6f38\u8fd1\u8907\u96dc\u5ea6\u5206\u6790 time complexity \u65f6\u95f4\u590d\u6742\u5ea6 \u6642\u9593\u8907\u96dc\u5ea6 space complexity \u7a7a\u95f4\u590d\u6742\u5ea6 \u7a7a\u9593\u8907\u96dc\u5ea6 loop \u5faa\u73af \u8ff4\u5708 iteration \u8fed\u4ee3 \u8fed\u4ee3 recursion \u9012\u5f52 \u905e\u8ff4 tail recursion \u5c3e\u9012\u5f52 \u5c3e\u905e\u8ff4 recursion tree \u9012\u5f52\u6811 \u905e\u8ff4\u6a39 big-\\(O\\) notation \u5927 \\(O\\) \u8bb0\u53f7 \u5927 \\(O\\) \u8a18\u865f asymptotic upper bound \u6e10\u8fd1\u4e0a\u754c \u6f38\u8fd1\u4e0a\u754c sign-magnitude \u539f\u7801 \u539f\u78bc 1\u2019s complement \u53cd\u7801 \u53cd\u78bc 2\u2019s complement \u8865\u7801 \u88dc\u78bc array \u6570\u7ec4 \u9663\u5217 index \u7d22\u5f15 \u7d22\u5f15 linked list \u94fe\u8868 \u93c8\u7d50\u4e32\u5217 linked list node, list node \u94fe\u8868\u8282\u70b9 \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede head node \u5934\u8282\u70b9 \u982d\u7bc0\u9ede tail node \u5c3e\u8282\u70b9 \u5c3e\u7bc0\u9ede list \u5217\u8868 \u5217\u8868 dynamic array \u52a8\u6001\u6570\u7ec4 \u52d5\u614b\u9663\u5217 hard disk \u786c\u76d8 \u786c\u789f random-access memory (RAM) \u5185\u5b58 \u5167\u5b58 cache memory \u7f13\u5b58 \u5feb\u53d6 cache miss \u7f13\u5b58\u672a\u547d\u4e2d \u5feb\u53d6\u672a\u547d\u4e2d cache hit rate \u7f13\u5b58\u547d\u4e2d\u7387 \u5feb\u53d6\u547d\u4e2d\u7387 stack \u6808 \u5806\u758a top of the stack \u6808\u9876 \u5806\u758a\u9802 bottom of the stack \u6808\u5e95 \u5806\u758a\u5e95 queue \u961f\u5217 \u4f47\u5217 double-ended queue \u53cc\u5411\u961f\u5217 \u96d9\u5411\u4f47\u5217 front of the queue \u961f\u9996 \u4f47\u5217\u9996 rear of the queue \u961f\u5c3e \u4f47\u5217\u5c3e hash table \u54c8\u5e0c\u8868 \u96dc\u6e4a\u8868 bucket \u6876 \u6876 hash function \u54c8\u5e0c\u51fd\u6570 \u96dc\u6e4a\u51fd\u5f0f hash collision \u54c8\u5e0c\u51b2\u7a81 \u96dc\u6e4a\u885d\u7a81 load factor \u8d1f\u8f7d\u56e0\u5b50 \u8ca0\u8f09\u56e0\u5b50 separate chaining \u94fe\u5f0f\u5730\u5740 \u93c8\u7d50\u4f4d\u5740 open addressing \u5f00\u653e\u5bfb\u5740 \u958b\u653e\u5b9a\u5740 linear probing \u7ebf\u6027\u63a2\u6d4b \u7dda\u6027\u63a2\u67e5 lazy deletion \u61d2\u5220\u9664 \u61f6\u522a\u9664 binary tree \u4e8c\u53c9\u6811 \u4e8c\u5143\u6a39 tree node \u6811\u8282\u70b9 \u6a39\u7bc0\u9ede left-child node \u5de6\u5b50\u8282\u70b9 \u5de6\u5b50\u7bc0\u9ede right-child node \u53f3\u5b50\u8282\u70b9 \u53f3\u5b50\u7bc0\u9ede parent node \u7236\u8282\u70b9 \u7236\u7bc0\u9ede left subtree \u5de6\u5b50\u6811 \u5de6\u5b50\u6a39 right subtree \u53f3\u5b50\u6811 \u53f3\u5b50\u6a39 root node \u6839\u8282\u70b9 \u6839\u7bc0\u9ede leaf node \u53f6\u8282\u70b9 \u8449\u7bc0\u9ede edge \u8fb9 \u908a level \u5c42 \u5c64 degree \u5ea6 \u5ea6 height \u9ad8\u5ea6 \u9ad8\u5ea6 depth \u6df1\u5ea6 \u6df1\u5ea6 perfect binary tree \u5b8c\u7f8e\u4e8c\u53c9\u6811 \u5b8c\u7f8e\u4e8c\u5143\u6a39 complete binary tree \u5b8c\u5168\u4e8c\u53c9\u6811 \u5b8c\u5168\u4e8c\u5143\u6a39 full binary tree \u5b8c\u6ee1\u4e8c\u53c9\u6811 \u5b8c\u6eff\u4e8c\u5143\u6a39 balanced binary tree \u5e73\u8861\u4e8c\u53c9\u6811 \u5e73\u8861\u4e8c\u5143\u6a39 binary search tree \u4e8c\u53c9\u641c\u7d22\u6811 \u4e8c\u5143\u641c\u5c0b\u6a39 AVL tree AVL \u6811 AVL \u6a39 red-black tree \u7ea2\u9ed1\u6811 \u7d05\u9ed1\u6a39 level-order traversal \u5c42\u5e8f\u904d\u5386 \u5c64\u5e8f\u8d70\u8a2a breadth-first traversal \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a depth-first traversal \u6df1\u5ea6\u4f18\u5148\u904d\u5386 \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a binary search tree \u4e8c\u53c9\u641c\u7d22\u6811 \u4e8c\u5143\u641c\u5c0b\u6a39 balanced binary search tree \u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811 \u5e73\u8861\u4e8c\u5143\u641c\u5c0b\u6a39 balance factor \u5e73\u8861\u56e0\u5b50 \u5e73\u8861\u56e0\u5b50 heap \u5806 \u5806\u7a4d max heap \u5927\u9876\u5806 \u5927\u9802\u5806\u7a4d min heap \u5c0f\u9876\u5806 \u5c0f\u9802\u5806\u7a4d priority queue \u4f18\u5148\u961f\u5217 \u512a\u5148\u4f47\u5217 heapify \u5806\u5316 \u5806\u7a4d\u5316 top-\\(k\\) problem Top-\\(k\\) \u95ee\u9898 Top-\\(k\\) \u554f\u984c graph \u56fe \u5716 vertex \u9876\u70b9 \u9802\u9ede undirected graph \u65e0\u5411\u56fe \u7121\u5411\u5716 directed graph \u6709\u5411\u56fe \u6709\u5411\u5716 connected graph \u8fde\u901a\u56fe \u9023\u901a\u5716 disconnected graph \u975e\u8fde\u901a\u56fe \u975e\u9023\u901a\u5716 weighted graph \u6709\u6743\u56fe \u6709\u6b0a\u5716 adjacency \u90bb\u63a5 \u9130\u63a5 path \u8def\u5f84 \u8def\u5f91 in-degree \u5165\u5ea6 \u5165\u5ea6 out-degree \u51fa\u5ea6 \u51fa\u5ea6 adjacency matrix \u90bb\u63a5\u77e9\u9635 \u9130\u63a5\u77e9\u9663 adjacency list \u90bb\u63a5\u8868 \u9130\u63a5\u8868 breadth-first search \u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 \u5ee3\u5ea6\u512a\u5148\u641c\u5c0b depth-first search \u6df1\u5ea6\u4f18\u5148\u641c\u7d22 \u6df1\u5ea6\u512a\u5148\u641c\u5c0b binary search \u4e8c\u5206\u67e5\u627e \u4e8c\u5206\u641c\u5c0b searching algorithm \u641c\u7d22\u7b97\u6cd5 \u641c\u5c0b\u6f14\u7b97\u6cd5 sorting algorithm \u6392\u5e8f\u7b97\u6cd5 \u6392\u5e8f\u6f14\u7b97\u6cd5 selection sort \u9009\u62e9\u6392\u5e8f \u9078\u64c7\u6392\u5e8f bubble sort \u5192\u6ce1\u6392\u5e8f \u6ce1\u6cab\u6392\u5e8f insertion sort \u63d2\u5165\u6392\u5e8f \u63d2\u5165\u6392\u5e8f quick sort \u5feb\u901f\u6392\u5e8f \u5feb\u901f\u6392\u5e8f merge sort \u5f52\u5e76\u6392\u5e8f \u5408\u4f75\u6392\u5e8f heap sort \u5806\u6392\u5e8f \u5806\u7a4d\u6392\u5e8f bucket sort \u6876\u6392\u5e8f \u6876\u6392\u5e8f counting sort \u8ba1\u6570\u6392\u5e8f \u8a08\u6578\u6392\u5e8f radix sort \u57fa\u6570\u6392\u5e8f \u57fa\u6578\u6392\u5e8f divide and conquer \u5206\u6cbb \u5206\u6cbb hanota problem \u6c49\u8bfa\u5854\u95ee\u9898 \u6cb3\u5167\u5854\u554f\u984c backtracking algorithm \u56de\u6eaf\u7b97\u6cd5 \u56de\u6eaf\u6f14\u7b97\u6cd5 constraint \u7ea6\u675f \u7d04\u675f solution \u89e3 \u89e3 state \u72b6\u6001 \u72c0\u614b pruning \u526a\u679d \u526a\u679d permutations problem \u5168\u6392\u5217\u95ee\u9898 \u5168\u6392\u5217\u554f\u984c subset-sum problem \u5b50\u96c6\u548c\u95ee\u9898 \u5b50\u96c6\u5408\u554f\u984c \\(n\\)-queens problem \\(n\\) \u7687\u540e\u95ee\u9898 \\(n\\) \u7687\u540e\u554f\u984c dynamic programming \u52a8\u6001\u89c4\u5212 \u52d5\u614b\u898f\u5283 initial state \u521d\u59cb\u72b6\u6001 \u521d\u59cb\u72c0\u614b state-transition equation \u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b \u72c0\u614b\u8f49\u79fb\u65b9\u7a0b knapsack problem \u80cc\u5305\u95ee\u9898 \u80cc\u5305\u554f\u984c edit distance problem \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898 \u7de8\u8f2f\u8ddd\u96e2\u554f\u984c greedy algorithm \u8d2a\u5fc3\u7b97\u6cd5 \u8caa\u5a6a\u6f14\u7b97\u6cd5"},{"location":"chapter_array_and_linkedlist/","title":"\u7b2c 4 \u7ae0 \u00a0 \u6570\u7ec4\u4e0e\u94fe\u8868","text":"<p>Abstract</p> <p>\u6570\u636e\u7ed3\u6784\u7684\u4e16\u754c\u5982\u540c\u4e00\u5835\u539a\u5b9e\u7684\u7816\u5899\u3002</p> <p>\u6570\u7ec4\u7684\u7816\u5757\u6574\u9f50\u6392\u5217\uff0c\u9010\u4e2a\u7d27\u8d34\u3002\u94fe\u8868\u7684\u7816\u5757\u5206\u6563\u5404\u5904\uff0c\u8fde\u63a5\u7684\u85e4\u8513\u81ea\u7531\u5730\u7a7f\u68ad\u4e8e\u7816\u7f1d\u4e4b\u95f4\u3002</p>"},{"location":"chapter_array_and_linkedlist/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>4.1 \u00a0 \u6570\u7ec4</li> <li>4.2 \u00a0 \u94fe\u8868</li> <li>4.3 \u00a0 \u5217\u8868</li> <li>4.4 \u00a0 \u5185\u5b58\u4e0e\u7f13\u5b58 *</li> <li>4.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/","title":"4.1 \u00a0 \u6570\u7ec4","text":"<p>\u300c\u6570\u7ec4 array\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5c06\u76f8\u540c\u7c7b\u578b\u7684\u5143\u7d20\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\u3002\u6211\u4eec\u5c06\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u4f4d\u7f6e\u79f0\u4e3a\u8be5\u5143\u7d20\u7684\u300c\u7d22\u5f15 index\u300d\u3002\u56fe 4-1 \u5c55\u793a\u4e86\u6570\u7ec4\u7684\u4e3b\u8981\u6982\u5ff5\u548c\u5b58\u50a8\u65b9\u5f0f\u3002</p> <p></p> <p> \u56fe 4-1 \u00a0 \u6570\u7ec4\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f </p>"},{"location":"chapter_array_and_linkedlist/array/#411","title":"4.1.1 \u00a0 \u6570\u7ec4\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/array/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u6570\u7ec4","text":"<p>\u6211\u4eec\u53ef\u4ee5\u6839\u636e\u9700\u6c42\u9009\u7528\u6570\u7ec4\u7684\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u5f0f\uff1a\u65e0\u521d\u59cb\u503c\u3001\u7ed9\u5b9a\u521d\u59cb\u503c\u3002\u5728\u672a\u6307\u5b9a\u521d\u59cb\u503c\u7684\u60c5\u51b5\u4e0b\uff0c\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4f1a\u5c06\u6570\u7ec4\u5143\u7d20\u521d\u59cb\u5316\u4e3a \\(0\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code># \u521d\u59cb\u5316\u6570\u7ec4\narr: list[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: list[int] = [1, 3, 2, 5, 4]  \n</code></pre> array.cpp<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\n// \u5b58\u50a8\u5728\u6808\u4e0a\nint arr[5];\nint nums[5] = { 1, 3, 2, 5, 4 };\n// \u5b58\u50a8\u5728\u5806\u4e0a\uff08\u9700\u8981\u624b\u52a8\u91ca\u653e\u7a7a\u95f4\uff09\nint* arr1 = new int[5];\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n</code></pre> array.java<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n</code></pre> array.cs<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // [ 0, 0, 0, 0, 0 ]\nint[] nums = [1, 3, 2, 5, 4];\n</code></pre> array.go<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr [5]int\n// \u5728 Go \u4e2d\uff0c\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[5]int\uff09\u4e3a\u6570\u7ec4\uff0c\u4e0d\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[]int\uff09\u4e3a\u5207\u7247\n// \u7531\u4e8e Go \u7684\u6570\u7ec4\u88ab\u8bbe\u8ba1\u4e3a\u5728\u7f16\u8bd1\u671f\u786e\u5b9a\u957f\u5ea6\uff0c\u56e0\u6b64\u53ea\u80fd\u4f7f\u7528\u5e38\u91cf\u6765\u6307\u5b9a\u957f\u5ea6\n// \u4e3a\u4e86\u65b9\u4fbf\u5b9e\u73b0\u6269\u5bb9 extend() \u65b9\u6cd5\uff0c\u4ee5\u4e0b\u5c06\u5207\u7247\uff08Slice\uff09\u770b\u4f5c\u6570\u7ec4\uff08Array\uff09\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> array.swift<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n</code></pre> array.js<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n</code></pre> array.ts<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> array.dart<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nList&lt;int&gt; arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> array.rs<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: Vec&lt;i32&gt; = vec![0; 5]; // [0, 0, 0, 0, 0]\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> array.c<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n</code></pre> array.kt<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr = IntArray(5) // { 0, 0, 0, 0, 0 }\nvar nums = intArrayOf(1, 3, 2, 5, 4)\n</code></pre> array.rb<pre><code># \u521d\u59cb\u5316\u6570\u7ec4\narr = Array.new(5, 0)\nnums = [1, 3, 2, 5, 4]\n</code></pre> array.zig<pre><code>// \u521d\u59cb\u5316\u6570\u7ec4\nvar arr = [_]i32{0} ** 5; // { 0, 0, 0, 0, 0 }\nvar nums = [_]i32{ 1, 3, 2, 5, 4 };\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#2","title":"2. \u00a0 \u8bbf\u95ee\u5143\u7d20","text":"<p>\u6570\u7ec4\u5143\u7d20\u88ab\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u8fd9\u610f\u5473\u7740\u8ba1\u7b97\u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u975e\u5e38\u5bb9\u6613\u3002\u7ed9\u5b9a\u6570\u7ec4\u5185\u5b58\u5730\u5740\uff08\u9996\u5143\u7d20\u5185\u5b58\u5730\u5740\uff09\u548c\u67d0\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u56fe 4-2 \u6240\u793a\u7684\u516c\u5f0f\u8ba1\u7b97\u5f97\u5230\u8be5\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u76f4\u63a5\u8bbf\u95ee\u8be5\u5143\u7d20\u3002</p> <p></p> <p> \u56fe 4-2 \u00a0 \u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u8ba1\u7b97 </p> <p>\u89c2\u5bdf\u56fe 4-2 \uff0c\u6211\u4eec\u53d1\u73b0\u6570\u7ec4\u9996\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u4e3a \\(0\\) \uff0c\u8fd9\u4f3c\u4e4e\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u4ece \\(1\\) \u5f00\u59cb\u8ba1\u6570\u4f1a\u66f4\u81ea\u7136\u3002\u4f46\u4ece\u5730\u5740\u8ba1\u7b97\u516c\u5f0f\u7684\u89d2\u5ea6\u770b\uff0c\u7d22\u5f15\u672c\u8d28\u4e0a\u662f\u5185\u5b58\u5730\u5740\u7684\u504f\u79fb\u91cf\u3002\u9996\u4e2a\u5143\u7d20\u7684\u5730\u5740\u504f\u79fb\u91cf\u662f \\(0\\) \uff0c\u56e0\u6b64\u5b83\u7684\u7d22\u5f15\u4e3a \\(0\\) \u662f\u5408\u7406\u7684\u3002</p> <p>\u5728\u6570\u7ec4\u4e2d\u8bbf\u95ee\u5143\u7d20\u975e\u5e38\u9ad8\u6548\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u968f\u673a\u8bbf\u95ee\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\u5143\u7d20\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def random_access(nums: list[int]) -&gt; int:\n    \"\"\"\u968f\u673a\u8bbf\u95ee\u5143\u7d20\"\"\"\n    # \u5728\u533a\u95f4 [0, len(nums)-1] \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    random_index = random.randint(0, len(nums) - 1)\n    # \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    random_num = nums[random_index]\n    return random_num\n</code></pre> array.cpp<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n    // \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = rand() % size;\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.java<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int[] nums) {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.cs<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint RandomAccess(int[] nums) {\n    Random random = new();\n    // \u5728\u533a\u95f4 [0, nums.Length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = random.Next(nums.Length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.go<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums []int) (randomNum int) {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    randomIndex := rand.Intn(len(nums))\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    randomNum = nums[randomIndex]\n    return\n}\n</code></pre> array.swift<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums: [Int]) -&gt; Int {\n    // \u5728\u533a\u95f4 [0, nums.count) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    let randomIndex = nums.indices.randomElement()!\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    let randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.js<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums) {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    const random_index = Math.floor(Math.random() * nums.length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.ts<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums: number[]): number {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    const random_index = Math.floor(Math.random() * nums.length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.dart<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(List&lt;int&gt; nums) {\n  // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n  int randomIndex = Random().nextInt(nums.length);\n  // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n  int randomNum = nums[randomIndex];\n  return randomNum;\n}\n</code></pre> array.rs<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfn random_access(nums: &amp;[i32]) -&gt; i32 {\n    // \u5728\u533a\u95f4 [0, nums.len()) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    let random_index = rand::thread_rng().gen_range(0..nums.len());\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    let random_num = nums[random_index];\n    random_num\n}\n</code></pre> array.c<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n    // \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = rand() % size;\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.kt<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfun randomAccess(nums: IntArray): Int {\n    // \u5728\u533a\u95f4 [0, nums.size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    val randomIndex = ThreadLocalRandom.current().nextInt(0, nums.size)\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    val randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.rb<pre><code>### \u968f\u673a\u8bbf\u95ee\u5143\u7d20 ###\ndef random_access(nums)\n  # \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n  random_index = Random.rand 0...(nums.length - 1)\n\n  # \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n  nums[random_index]\nend\n</code></pre> array.zig<pre><code>// \u968f\u673a\u8bbf\u95ee\u5143\u7d20\nfn randomAccess(nums: []i32) i32 {\n    // \u5728\u533a\u95f4 [0, nums.len) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6574\u6570\n    var randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    var randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#3","title":"3. \u00a0 \u63d2\u5165\u5143\u7d20","text":"<p>\u6570\u7ec4\u5143\u7d20\u5728\u5185\u5b58\u4e2d\u662f\u201c\u7d27\u6328\u7740\u7684\u201d\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u7a7a\u95f4\u518d\u5b58\u653e\u4efb\u4f55\u6570\u636e\u3002\u5982\u56fe 4-3 \u6240\u793a\uff0c\u5982\u679c\u60f3\u5728\u6570\u7ec4\u4e2d\u95f4\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\uff0c\u5219\u9700\u8981\u5c06\u8be5\u5143\u7d20\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u4e4b\u540e\u518d\u628a\u5143\u7d20\u8d4b\u503c\u7ed9\u8be5\u7d22\u5f15\u3002</p> <p></p> <p> \u56fe 4-3 \u00a0 \u6570\u7ec4\u63d2\u5165\u5143\u7d20\u793a\u4f8b </p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u662f\u56fa\u5b9a\u7684\uff0c\u56e0\u6b64\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u4f1a\u5bfc\u81f4\u6570\u7ec4\u5c3e\u90e8\u5143\u7d20\u201c\u4e22\u5931\u201d\u3002\u6211\u4eec\u5c06\u8fd9\u4e2a\u95ee\u9898\u7684\u89e3\u51b3\u65b9\u6848\u7559\u5728\u201c\u5217\u8868\u201d\u7ae0\u8282\u4e2d\u8ba8\u8bba\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def insert(nums: list[int], num: int, index: int):\n    \"\"\"\u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\"\"\"\n    # \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i in range(len(nums) - 1, index, -1):\n        nums[i] = nums[i - 1]\n    # \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n</code></pre> array.cpp<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.java<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid Insert(int[] nums, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = nums.Length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.go<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums []int, num int, index int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i := len(nums) - 1; i &gt; index; i-- {\n        nums[i] = nums[i-1]\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.swift<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i in nums.indices.dropFirst(index).reversed() {\n        nums[i] = nums[i - 1]\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.js<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums, num, index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums: number[], num: number, index: number): void {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 _num */\nvoid insert(List&lt;int&gt; nums, int _num, int index) {\n  // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n  for (var i = nums.length - 1; i &gt; index; i--) {\n    nums[i] = nums[i - 1];\n  }\n  // \u5c06 _num \u8d4b\u7ed9 index \u5904\u5143\u7d20\n  nums[index] = _num;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfn insert(nums: &amp;mut Vec&lt;i32&gt;, num: i32, index: usize) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i in (index + 1..nums.len()).rev() {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.c<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.kt<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfun insert(nums: IntArray, num: Int, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (i in nums.size - 1 downTo index + 1) {\n        nums[i] = nums[i - 1]\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.rb<pre><code>### \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num ###\ndef insert(nums, num, index)\n  # \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n  for i in (nums.length - 1).downto(index + 1)\n    nums[i] = nums[i - 1]\n  end\n\n  # \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n  nums[index] = num\nend\n</code></pre> array.zig<pre><code>// \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\nfn insert(nums: []i32, num: i32, index: usize) void {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    var i = nums.len - 1;\n    while (i &gt; index) : (i -= 1) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#4","title":"4. \u00a0 \u5220\u9664\u5143\u7d20","text":"<p>\u540c\u7406\uff0c\u5982\u56fe 4-4 \u6240\u793a\uff0c\u82e5\u60f3\u5220\u9664\u7d22\u5f15 \\(i\\) \u5904\u7684\u5143\u7d20\uff0c\u5219\u9700\u8981\u628a\u7d22\u5f15 \\(i\\) \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\u3002</p> <p></p> <p> \u56fe 4-4 \u00a0 \u6570\u7ec4\u5220\u9664\u5143\u7d20\u793a\u4f8b </p> <p>\u8bf7\u6ce8\u610f\uff0c\u5220\u9664\u5143\u7d20\u5b8c\u6210\u540e\uff0c\u539f\u5148\u672b\u5c3e\u7684\u5143\u7d20\u53d8\u5f97\u201c\u65e0\u610f\u4e49\u201d\u4e86\uff0c\u6240\u4ee5\u6211\u4eec\u65e0\u987b\u7279\u610f\u53bb\u4fee\u6539\u5b83\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def remove(nums: list[int], index: int):\n    \"\"\"\u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\"\"\"\n    # \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i in range(index, len(nums) - 1):\n        nums[i] = nums[i + 1]\n</code></pre> array.cpp<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int *nums, int size, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.java<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.cs<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid Remove(int[] nums, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; nums.Length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.go<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums []int, index int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i := index; i &lt; len(nums)-1; i++ {\n        nums[i] = nums[i+1]\n    }\n}\n</code></pre> array.swift<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums: inout [Int], index: Int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i in nums.indices.dropFirst(index).dropLast() {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.js<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums, index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.ts<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums: number[], index: number): void {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.dart<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(List&lt;int&gt; nums, int index) {\n  // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n  for (var i = index; i &lt; nums.length - 1; i++) {\n    nums[i] = nums[i + 1];\n  }\n}\n</code></pre> array.rs<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfn remove(nums: &amp;mut Vec&lt;i32&gt;, index: usize) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i in index..nums.len() - 1 {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.c<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(int *nums, int size, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.kt<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfun remove(nums: IntArray, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (i in index..&lt;nums.size - 1) {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.rb<pre><code>### \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 ###\ndef remove(nums, index)\n  # \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n  for i in index...nums.length\n    nums[i] = nums[i + 1] || 0\n  end\nend\n</code></pre> array.zig<pre><code>// \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\nfn remove(nums: []i32, index: usize) void {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    var i = index;\n    while (i &lt; nums.len - 1) : (i += 1) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u603b\u7684\u6765\u770b\uff0c\u6570\u7ec4\u7684\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u6709\u4ee5\u4e0b\u7f3a\u70b9\u3002</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u9ad8\uff1a\u6570\u7ec4\u7684\u63d2\u5165\u548c\u5220\u9664\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u957f\u5ea6\u3002</li> <li>\u4e22\u5931\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u5728\u63d2\u5165\u5143\u7d20\u540e\uff0c\u8d85\u51fa\u6570\u7ec4\u957f\u5ea6\u8303\u56f4\u7684\u5143\u7d20\u4f1a\u4e22\u5931\u3002</li> <li>\u5185\u5b58\u6d6a\u8d39\uff1a\u6211\u4eec\u53ef\u4ee5\u521d\u59cb\u5316\u4e00\u4e2a\u6bd4\u8f83\u957f\u7684\u6570\u7ec4\uff0c\u53ea\u7528\u524d\u9762\u4e00\u90e8\u5206\uff0c\u8fd9\u6837\u5728\u63d2\u5165\u6570\u636e\u65f6\uff0c\u4e22\u5931\u7684\u672b\u5c3e\u5143\u7d20\u90fd\u662f\u201c\u65e0\u610f\u4e49\u201d\u7684\uff0c\u4f46\u8fd9\u6837\u505a\u4f1a\u9020\u6210\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#5","title":"5. \u00a0 \u904d\u5386\u6570\u7ec4","text":"<p>\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u65e2\u53ef\u4ee5\u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u904d\u5386\u83b7\u53d6\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def traverse(nums: list[int]):\n    \"\"\"\u904d\u5386\u6570\u7ec4\"\"\"\n    count = 0\n    # \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i in range(len(nums)):\n        count += nums[i]\n    # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for num in nums:\n        count += num\n    # \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n    for i, num in enumerate(nums):\n        count += nums[i]\n        count += num\n</code></pre> array.cpp<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.java<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (int num : nums) {\n        count += num;\n    }\n}\n</code></pre> array.cs<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid Traverse(int[] nums) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; nums.Length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    foreach (int num in nums) {\n        count += num;\n    }\n}\n</code></pre> array.go<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums []int) {\n    count := 0\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i := 0; i &lt; len(nums); i++ {\n        count += nums[i]\n    }\n    count = 0\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for _, num := range nums {\n        count += num\n    }\n    // \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n    for i, num := range nums {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.swift<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums: [Int]) {\n    var count = 0\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i in nums.indices {\n        count += nums[i]\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for num in nums {\n        count += num\n    }\n    // \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n    for (i, num) in nums.enumerated() {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.js<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums) {\n    let count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.ts<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums: number[]): void {\n    let count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.dart<pre><code>/* \u904d\u5386\u6570\u7ec4\u5143\u7d20 */\nvoid traverse(List&lt;int&gt; nums) {\n  int count = 0;\n  // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n  for (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n  }\n  // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n  for (int _num in nums) {\n    count += _num;\n  }\n  // \u901a\u8fc7 forEach \u65b9\u6cd5\u904d\u5386\u6570\u7ec4\n  nums.forEach((_num) {\n    count += _num;\n  });\n}\n</code></pre> array.rs<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfn traverse(nums: &amp;[i32]) {\n    let mut _count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i in 0..nums.len() {\n        _count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for num in nums {\n        _count += num;\n    }\n}\n</code></pre> array.c<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.kt<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfun traverse(nums: IntArray) {\n    var count = 0\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (i in nums.indices) {\n        count += nums[i]\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (j: Int in nums) {\n        count += j\n    }\n}\n</code></pre> array.rb<pre><code>### \u904d\u5386\u6570\u7ec4 ###\ndef traverse(nums)\n  count = 0\n\n  # \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n  for i in 0...nums.length\n    count += nums[i]\n  end\n\n  # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n  for num in nums\n    count += num\n  end\nend\n</code></pre> array.zig<pre><code>// \u904d\u5386\u6570\u7ec4\nfn traverse(nums: []i32) void {\n    var count: i32 = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    var i: i32 = 0;\n    while (i &lt; nums.len) : (i += 1) {\n        count += nums[i];\n    }\n    count = 0;\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (nums) |num| {\n        count += num;\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#6","title":"6. \u00a0 \u67e5\u627e\u5143\u7d20","text":"<p>\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\u9700\u8981\u904d\u5386\u6570\u7ec4\uff0c\u6bcf\u8f6e\u5224\u65ad\u5143\u7d20\u503c\u662f\u5426\u5339\u914d\uff0c\u82e5\u5339\u914d\u5219\u8f93\u51fa\u5bf9\u5e94\u7d22\u5f15\u3002</p> <p>\u56e0\u4e3a\u6570\u7ec4\u662f\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u6240\u4ee5\u4e0a\u8ff0\u67e5\u627e\u64cd\u4f5c\u88ab\u79f0\u4e3a\u201c\u7ebf\u6027\u67e5\u627e\u201d\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def find(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\"\"\"\n    for i in range(len(nums)):\n        if nums[i] == target:\n            return i\n    return -1\n</code></pre> array.cpp<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.java<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint Find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.go<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums []int, target int) (index int) {\n    index = -1\n    for i := 0; i &lt; len(nums); i++ {\n        if nums[i] == target {\n            index = i\n            break\n        }\n    }\n    return\n}\n</code></pre> array.swift<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums: [Int], target: Int) -&gt; Int {\n    for i in nums.indices {\n        if nums[i] == target {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> array.js<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums, target) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) return i;\n    }\n    return -1;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums: number[], target: number): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(List&lt;int&gt; nums, int target) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    if (nums[i] == target) return i;\n  }\n  return -1;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfn find(nums: &amp;[i32], target: i32) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        if nums[i] == target {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> array.c<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.kt<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfun find(nums: IntArray, target: Int): Int {\n    for (i in nums.indices) {\n        if (nums[i] == target) return i\n    }\n    return -1\n}\n</code></pre> array.rb<pre><code>### \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 ###\ndef find(nums, target)\n  for i in 0...nums.length\n    return i if nums[i] == target\n  end\n\n  -1\nend\n</code></pre> array.zig<pre><code>// \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\nfn find(nums: []i32, target: i32) i32 {\n    for (nums, 0..) |num, i| {\n        if (num == target) return @intCast(i);\n    }\n    return -1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#7","title":"7. \u00a0 \u6269\u5bb9\u6570\u7ec4","text":"<p>\u5728\u590d\u6742\u7684\u7cfb\u7edf\u73af\u5883\u4e2d\uff0c\u7a0b\u5e8f\u96be\u4ee5\u4fdd\u8bc1\u6570\u7ec4\u4e4b\u540e\u7684\u5185\u5b58\u7a7a\u95f4\u662f\u53ef\u7528\u7684\uff0c\u4ece\u800c\u65e0\u6cd5\u5b89\u5168\u5730\u6269\u5c55\u6570\u7ec4\u5bb9\u91cf\u3002\u56e0\u6b64\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6570\u7ec4\u7684\u957f\u5ea6\u662f\u4e0d\u53ef\u53d8\u7684\u3002</p> <p>\u5982\u679c\u6211\u4eec\u5e0c\u671b\u6269\u5bb9\u6570\u7ec4\uff0c\u5219\u9700\u91cd\u65b0\u5efa\u7acb\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u7136\u540e\u628a\u539f\u6570\u7ec4\u5143\u7d20\u4f9d\u6b21\u590d\u5236\u5230\u65b0\u6570\u7ec4\u3002\u8fd9\u662f\u4e00\u4e2a \\(O(n)\\) \u7684\u64cd\u4f5c\uff0c\u5728\u6570\u7ec4\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\u975e\u5e38\u8017\u65f6\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def extend(nums: list[int], enlarge: int) -&gt; list[int]:\n    \"\"\"\u6269\u5c55\u6570\u7ec4\u957f\u5ea6\"\"\"\n    # \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    res = [0] * (len(nums) + enlarge)\n    # \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for i in range(len(nums)):\n        res[i] = nums[i]\n    # \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n</code></pre> array.cpp<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int *res = new int[size + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // \u91ca\u653e\u5185\u5b58\n    delete[] nums;\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.java<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int[] res = new int[nums.length + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.cs<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] Extend(int[] nums, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int[] res = new int[nums.Length + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; nums.Length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.go<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums []int, enlarge int) []int {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    res := make([]int, len(nums)+enlarge)\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for i, num := range nums {\n        res[i] = num\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n}\n</code></pre> array.swift<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums: [Int], enlarge: Int) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    var res = Array(repeating: 0, count: nums.count + enlarge)\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for i in nums.indices {\n        res[i] = nums[i]\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n}\n</code></pre> array.js<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cJavaScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums, enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    const res = new Array(nums.length + enlarge).fill(0);\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.ts<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cTypeScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums: number[], enlarge: number): number[] {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    const res = new Array(nums.length + enlarge).fill(0);\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.dart<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nList&lt;int&gt; extend(List&lt;int&gt; nums, int enlarge) {\n  // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n  List&lt;int&gt; res = List.filled(nums.length + enlarge, 0);\n  // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n  for (var i = 0; i &lt; nums.length; i++) {\n    res[i] = nums[i];\n  }\n  // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n  return res;\n}\n</code></pre> array.rs<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfn extend(nums: Vec&lt;i32&gt;, enlarge: usize) -&gt; Vec&lt;i32&gt; {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    let mut res: Vec&lt;i32&gt; = vec![0; nums.len() + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\n    for i in 0..nums.len() {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    res\n}\n</code></pre> array.c<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int *res = (int *)malloc(sizeof(int) * (size + enlarge));\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // \u521d\u59cb\u5316\u6269\u5c55\u540e\u7684\u7a7a\u95f4\n    for (int i = size; i &lt; size + enlarge; i++) {\n        res[i] = 0;\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.kt<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfun extend(nums: IntArray, enlarge: Int): IntArray {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    val res = IntArray(nums.size + enlarge)\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (i in nums.indices) {\n        res[i] = nums[i]\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n}\n</code></pre> array.rb<pre><code>### \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 ###\n# \u8bf7\u6ce8\u610f\uff0cRuby \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n# \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\ndef extend(nums, enlarge)\n  # \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n  res = Array.new(nums.length + enlarge, 0)\n\n  # \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n  for i in 0...nums.length\n    res[i] = nums[i]\n  end\n\n  # \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n  res\nend\n</code></pre> array.zig<pre><code>// \u6269\u5c55\u6570\u7ec4\u957f\u5ea6\nfn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    var res = try mem_allocator.alloc(i32, nums.len + enlarge);\n    @memset(res, 0);\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    std.mem.copy(i32, res, nums);\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#412","title":"4.1.2 \u00a0 \u6570\u7ec4\u7684\u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u6570\u7ec4\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u5185\uff0c\u4e14\u5143\u7d20\u7c7b\u578b\u76f8\u540c\u3002\u8fd9\u79cd\u505a\u6cd5\u5305\u542b\u4e30\u5bcc\u7684\u5148\u9a8c\u4fe1\u606f\uff0c\u7cfb\u7edf\u53ef\u4ee5\u5229\u7528\u8fd9\u4e9b\u4fe1\u606f\u6765\u4f18\u5316\u6570\u636e\u7ed3\u6784\u7684\u64cd\u4f5c\u6548\u7387\u3002</p> <ul> <li>\u7a7a\u95f4\u6548\u7387\u9ad8\uff1a\u6570\u7ec4\u4e3a\u6570\u636e\u5206\u914d\u4e86\u8fde\u7eed\u7684\u5185\u5b58\u5757\uff0c\u65e0\u987b\u989d\u5916\u7684\u7ed3\u6784\u5f00\u9500\u3002</li> <li>\u652f\u6301\u968f\u673a\u8bbf\u95ee\uff1a\u6570\u7ec4\u5141\u8bb8\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u4efb\u4f55\u5143\u7d20\u3002</li> <li>\u7f13\u5b58\u5c40\u90e8\u6027\uff1a\u5f53\u8bbf\u95ee\u6570\u7ec4\u5143\u7d20\u65f6\uff0c\u8ba1\u7b97\u673a\u4e0d\u4ec5\u4f1a\u52a0\u8f7d\u5b83\uff0c\u8fd8\u4f1a\u7f13\u5b58\u5176\u5468\u56f4\u7684\u5176\u4ed6\u6570\u636e\uff0c\u4ece\u800c\u501f\u52a9\u9ad8\u901f\u7f13\u5b58\u6765\u63d0\u5347\u540e\u7eed\u64cd\u4f5c\u7684\u6267\u884c\u901f\u5ea6\u3002</li> </ul> <p>\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u662f\u4e00\u628a\u53cc\u5203\u5251\uff0c\u5176\u5b58\u5728\u4ee5\u4e0b\u5c40\u9650\u6027\u3002</p> <ul> <li>\u63d2\u5165\u4e0e\u5220\u9664\u6548\u7387\u4f4e\uff1a\u5f53\u6570\u7ec4\u4e2d\u5143\u7d20\u8f83\u591a\u65f6\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u9700\u8981\u79fb\u52a8\u5927\u91cf\u7684\u5143\u7d20\u3002</li> <li>\u957f\u5ea6\u4e0d\u53ef\u53d8\uff1a\u6570\u7ec4\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u5c31\u56fa\u5b9a\u4e86\uff0c\u6269\u5bb9\u6570\u7ec4\u9700\u8981\u5c06\u6240\u6709\u6570\u636e\u590d\u5236\u5230\u65b0\u6570\u7ec4\uff0c\u5f00\u9500\u5f88\u5927\u3002</li> <li>\u7a7a\u95f4\u6d6a\u8d39\uff1a\u5982\u679c\u6570\u7ec4\u5206\u914d\u7684\u5927\u5c0f\u8d85\u8fc7\u5b9e\u9645\u6240\u9700\uff0c\u90a3\u4e48\u591a\u4f59\u7684\u7a7a\u95f4\u5c31\u88ab\u6d6a\u8d39\u4e86\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#413","title":"4.1.3 \u00a0 \u6570\u7ec4\u5178\u578b\u5e94\u7528","text":"<p>\u6570\u7ec4\u662f\u4e00\u79cd\u57fa\u7840\u4e14\u5e38\u89c1\u7684\u6570\u636e\u7ed3\u6784\uff0c\u65e2\u9891\u7e41\u5e94\u7528\u5728\u5404\u7c7b\u7b97\u6cd5\u4e4b\u4e2d\uff0c\u4e5f\u53ef\u7528\u4e8e\u5b9e\u73b0\u5404\u79cd\u590d\u6742\u6570\u636e\u7ed3\u6784\u3002</p> <ul> <li>\u968f\u673a\u8bbf\u95ee\uff1a\u5982\u679c\u6211\u4eec\u60f3\u968f\u673a\u62bd\u53d6\u4e00\u4e9b\u6837\u672c\uff0c\u90a3\u4e48\u53ef\u4ee5\u7528\u6570\u7ec4\u5b58\u50a8\uff0c\u5e76\u751f\u6210\u4e00\u4e2a\u968f\u673a\u5e8f\u5217\uff0c\u6839\u636e\u7d22\u5f15\u5b9e\u73b0\u968f\u673a\u62bd\u6837\u3002</li> <li>\u6392\u5e8f\u548c\u641c\u7d22\uff1a\u6570\u7ec4\u662f\u6392\u5e8f\u548c\u641c\u7d22\u7b97\u6cd5\u6700\u5e38\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u4e8c\u5206\u67e5\u627e\u7b49\u90fd\u4e3b\u8981\u5728\u6570\u7ec4\u4e0a\u8fdb\u884c\u3002</li> <li>\u67e5\u627e\u8868\uff1a\u5f53\u9700\u8981\u5feb\u901f\u67e5\u627e\u4e00\u4e2a\u5143\u7d20\u6216\u5176\u5bf9\u5e94\u5173\u7cfb\u65f6\uff0c\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u4f5c\u4e3a\u67e5\u627e\u8868\u3002\u5047\u5982\u6211\u4eec\u60f3\u5b9e\u73b0\u5b57\u7b26\u5230 ASCII \u7801\u7684\u6620\u5c04\uff0c\u5219\u53ef\u4ee5\u5c06\u5b57\u7b26\u7684 ASCII \u7801\u503c\u4f5c\u4e3a\u7d22\u5f15\uff0c\u5bf9\u5e94\u7684\u5143\u7d20\u5b58\u653e\u5728\u6570\u7ec4\u4e2d\u7684\u5bf9\u5e94\u4f4d\u7f6e\u3002</li> <li>\u673a\u5668\u5b66\u4e60\uff1a\u795e\u7ecf\u7f51\u7edc\u4e2d\u5927\u91cf\u4f7f\u7528\u4e86\u5411\u91cf\u3001\u77e9\u9635\u3001\u5f20\u91cf\u4e4b\u95f4\u7684\u7ebf\u6027\u4ee3\u6570\u8fd0\u7b97\uff0c\u8fd9\u4e9b\u6570\u636e\u90fd\u662f\u4ee5\u6570\u7ec4\u7684\u5f62\u5f0f\u6784\u5efa\u7684\u3002\u6570\u7ec4\u662f\u795e\u7ecf\u7f51\u7edc\u7f16\u7a0b\u4e2d\u6700\u5e38\u4f7f\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff1a\u6570\u7ec4\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u3002\u4f8b\u5982\uff0c\u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a\u5b9e\u9645\u4e0a\u662f\u4e00\u4e2a\u4e8c\u7ef4\u6570\u7ec4\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/linked_list/","title":"4.2 \u00a0 \u94fe\u8868","text":"<p>\u5185\u5b58\u7a7a\u95f4\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u516c\u5171\u8d44\u6e90\uff0c\u5728\u4e00\u4e2a\u590d\u6742\u7684\u7cfb\u7edf\u8fd0\u884c\u73af\u5883\u4e0b\uff0c\u7a7a\u95f2\u7684\u5185\u5b58\u7a7a\u95f4\u53ef\u80fd\u6563\u843d\u5728\u5185\u5b58\u5404\u5904\u3002\u6211\u4eec\u77e5\u9053\uff0c\u5b58\u50a8\u6570\u7ec4\u7684\u5185\u5b58\u7a7a\u95f4\u5fc5\u987b\u662f\u8fde\u7eed\u7684\uff0c\u800c\u5f53\u6570\u7ec4\u975e\u5e38\u5927\u65f6\uff0c\u5185\u5b58\u53ef\u80fd\u65e0\u6cd5\u63d0\u4f9b\u5982\u6b64\u5927\u7684\u8fde\u7eed\u7a7a\u95f4\u3002\u6b64\u65f6\u94fe\u8868\u7684\u7075\u6d3b\u6027\u4f18\u52bf\u5c31\u4f53\u73b0\u51fa\u6765\u4e86\u3002</p> <p>\u300c\u94fe\u8868 linked list\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u662f\u4e00\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u5404\u4e2a\u8282\u70b9\u901a\u8fc7\u201c\u5f15\u7528\u201d\u76f8\u8fde\u63a5\u3002\u5f15\u7528\u8bb0\u5f55\u4e86\u4e0b\u4e00\u4e2a\u8282\u70b9\u7684\u5185\u5b58\u5730\u5740\uff0c\u901a\u8fc7\u5b83\u53ef\u4ee5\u4ece\u5f53\u524d\u8282\u70b9\u8bbf\u95ee\u5230\u4e0b\u4e00\u4e2a\u8282\u70b9\u3002</p> <p>\u94fe\u8868\u7684\u8bbe\u8ba1\u4f7f\u5f97\u5404\u4e2a\u8282\u70b9\u53ef\u4ee5\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\uff0c\u5b83\u4eec\u7684\u5185\u5b58\u5730\u5740\u65e0\u987b\u8fde\u7eed\u3002</p> <p></p> <p> \u56fe 4-5 \u00a0 \u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f </p> <p>\u89c2\u5bdf\u56fe 4-5 \uff0c\u94fe\u8868\u7684\u7ec4\u6210\u5355\u4f4d\u662f\u300c\u8282\u70b9 node\u300d\u5bf9\u8c61\u3002\u6bcf\u4e2a\u8282\u70b9\u90fd\u5305\u542b\u4e24\u9879\u6570\u636e\uff1a\u8282\u70b9\u7684\u201c\u503c\u201d\u548c\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u201c\u5f15\u7528\u201d\u3002</p> <ul> <li>\u94fe\u8868\u7684\u9996\u4e2a\u8282\u70b9\u88ab\u79f0\u4e3a\u201c\u5934\u8282\u70b9\u201d\uff0c\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u88ab\u79f0\u4e3a\u201c\u5c3e\u8282\u70b9\u201d\u3002</li> <li>\u5c3e\u8282\u70b9\u6307\u5411\u7684\u662f\u201c\u7a7a\u201d\uff0c\u5b83\u5728 Java\u3001C++ \u548c Python \u4e2d\u5206\u522b\u88ab\u8bb0\u4e3a <code>null</code>\u3001<code>nullptr</code> \u548c <code>None</code> \u3002</li> <li>\u5728 C\u3001C++\u3001Go \u548c Rust \u7b49\u652f\u6301\u6307\u9488\u7684\u8bed\u8a00\u4e2d\uff0c\u4e0a\u8ff0\u201c\u5f15\u7528\u201d\u5e94\u88ab\u66ff\u6362\u4e3a\u201c\u6307\u9488\u201d\u3002</li> </ul> <p>\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u94fe\u8868\u8282\u70b9 <code>ListNode</code> \u9664\u4e86\u5305\u542b\u503c\uff0c\u8fd8\u9700\u989d\u5916\u4fdd\u5b58\u4e00\u4e2a\u5f15\u7528\uff08\u6307\u9488\uff09\u3002\u56e0\u6b64\u5728\u76f8\u540c\u6570\u636e\u91cf\u4e0b\uff0c\u94fe\u8868\u6bd4\u6570\u7ec4\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class ListNode:\n    \"\"\"\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val               # \u8282\u70b9\u503c\n        self.next: ListNode | None = None # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\n    int val;         // \u8282\u70b9\u503c\n    ListNode *next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n    ListNode(int x) : val(x), next(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    int val;        // \u8282\u70b9\u503c\n    ListNode next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n    ListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode(int x) {  //\u6784\u9020\u51fd\u6570\n    int val = x;         // \u8282\u70b9\u503c\n    ListNode? next;      // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype ListNode struct {\n    Val  int       // \u8282\u70b9\u503c\n    Next *ListNode // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n}\n\n// NewListNode \u6784\u9020\u51fd\u6570\uff0c\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u94fe\u8868\nfunc NewListNode(val int) *ListNode {\n    return &amp;ListNode{\n        Val:  val,\n        Next: nil,\n    }\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    var val: Int // \u8282\u70b9\u503c\n    var next: ListNode? // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n\n    init(x: Int) { // \u6784\u9020\u51fd\u6570\n        val = x\n    }\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    constructor(val, next) {\n        this.val = (val === undefined ? 0 : val);       // \u8282\u70b9\u503c\n        this.next = (next === undefined ? null : next); // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    constructor(val?: number, next?: ListNode | null) {\n        this.val = val === undefined ? 0 : val;        // \u8282\u70b9\u503c\n        this.next = next === undefined ? null : next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n  int val; // \u8282\u70b9\u503c\n  ListNode? next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n  ListNode(this.val, [this.next]); // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // \u8282\u70b9\u503c\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct ListNode {\n    int val;               // \u8282\u70b9\u503c\n    struct ListNode *next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n} ListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\n// \u6784\u9020\u65b9\u6cd5\nclass ListNode(x: Int) {\n    val _val: Int = x          // \u8282\u70b9\u503c\n    val next: ListNode? = null // \u6307\u5411\u4e0b\u4e00\u4e2a\u8282\u70b9\u7684\u5f15\u7528\n}\n</code></pre> <pre><code># \u94fe\u8868\u8282\u70b9\u7c7b\nclass ListNode\n  attr_accessor :val  # \u8282\u70b9\u503c\n  attr_accessor :next # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n\n  def initialize(val=0, next_node=nil)\n    @val = val\n    @next = next_node\n  end\nend\n</code></pre> <pre><code>// \u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = 0, // \u8282\u70b9\u503c\n        next: ?*Self = null, // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n\n        // \u6784\u9020\u51fd\u6570\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#421","title":"4.2.1 \u00a0 \u94fe\u8868\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/linked_list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u94fe\u8868","text":"<p>\u5efa\u7acb\u94fe\u8868\u5206\u4e3a\u4e24\u6b65\uff0c\u7b2c\u4e00\u6b65\u662f\u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u7b2c\u4e8c\u6b65\u662f\u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\u5173\u7cfb\u3002\u521d\u59cb\u5316\u5b8c\u6210\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u4ece\u94fe\u8868\u7684\u5934\u8282\u70b9\u51fa\u53d1\uff0c\u901a\u8fc7\u5f15\u7528\u6307\u5411 <code>next</code> \u4f9d\u6b21\u8bbf\u95ee\u6240\u6709\u8282\u70b9\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code># \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.cpp<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.java<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.cs<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = new(1);\nListNode n1 = new(3);\nListNode n2 = new(2);\nListNode n3 = new(5);\nListNode n4 = new(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.go<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n</code></pre> linked_list.swift<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.js<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.ts<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.dart<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\\\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rs<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nlet n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None }));\nlet n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None }));\nlet n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None }));\nlet n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None }));\nlet n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None }));\n\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.borrow_mut().next = Some(n1.clone());\nn1.borrow_mut().next = Some(n2.clone());\nn2.borrow_mut().next = Some(n3.clone());\nn3.borrow_mut().next = Some(n4.clone());\n</code></pre> linked_list.c<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.kt<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nval n0 = ListNode(1)\nval n1 = ListNode(3)\nval n2 = ListNode(2)\nval n3 = ListNode(5)\nval n4 = ListNode(4)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rb<pre><code># \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 = ListNode.new 1\nn1 = ListNode.new 3\nn2 = ListNode.new 2\nn3 = ListNode.new 5\nn4 = ListNode.new 4\n# \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.zig<pre><code>// \u521d\u59cb\u5316\u94fe\u8868\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nvar n0 = inc.ListNode(i32){.val = 1};\nvar n1 = inc.ListNode(i32){.val = 3};\nvar n2 = inc.ListNode(i32){.val = 2};\nvar n3 = inc.ListNode(i32){.val = 5};\nvar n4 = inc.ListNode(i32){.val = 4};\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = &amp;n1;\nn1.next = &amp;n2;\nn2.next = &amp;n3;\nn3.next = &amp;n4;\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u6570\u7ec4\u6574\u4f53\u662f\u4e00\u4e2a\u53d8\u91cf\uff0c\u6bd4\u5982\u6570\u7ec4 <code>nums</code> \u5305\u542b\u5143\u7d20 <code>nums[0]</code> \u548c <code>nums[1]</code> \u7b49\uff0c\u800c\u94fe\u8868\u662f\u7531\u591a\u4e2a\u72ec\u7acb\u7684\u8282\u70b9\u5bf9\u8c61\u7ec4\u6210\u7684\u3002\u6211\u4eec\u901a\u5e38\u5c06\u5934\u8282\u70b9\u5f53\u4f5c\u94fe\u8868\u7684\u4ee3\u79f0\uff0c\u6bd4\u5982\u4ee5\u4e0a\u4ee3\u7801\u4e2d\u7684\u94fe\u8868\u53ef\u8bb0\u4f5c\u94fe\u8868 <code>n0</code> \u3002</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#2","title":"2. \u00a0 \u63d2\u5165\u8282\u70b9","text":"<p>\u5728\u94fe\u8868\u4e2d\u63d2\u5165\u8282\u70b9\u975e\u5e38\u5bb9\u6613\u3002\u5982\u56fe 4-6 \u6240\u793a\uff0c\u5047\u8bbe\u6211\u4eec\u60f3\u5728\u76f8\u90bb\u7684\u4e24\u4e2a\u8282\u70b9 <code>n0</code> \u548c <code>n1</code> \u4e4b\u95f4\u63d2\u5165\u4e00\u4e2a\u65b0\u8282\u70b9 <code>P</code> \uff0c\u5219\u53ea\u9700\u6539\u53d8\u4e24\u4e2a\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</p> <p>\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5728\u6570\u7ec4\u4e2d\u63d2\u5165\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u7684\u6548\u7387\u8f83\u4f4e\u3002</p> <p></p> <p> \u56fe 4-6 \u00a0 \u94fe\u8868\u63d2\u5165\u8282\u70b9\u793a\u4f8b </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def insert(n0: ListNode, P: ListNode):\n    \"\"\"\u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\"\"\"\n    n1 = n0.next\n    P.next = n1\n    n0.next = P\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\n    ListNode n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid Insert(ListNode n0, ListNode P) {\n    ListNode? n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insertNode(n0 *ListNode, P *ListNode) {\n    n1 := n0.Next\n    P.Next = n1\n    n0.Next = P\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insert(n0: ListNode, P: ListNode) {\n    let n1 = n0.next\n    P.next = n1\n    n0.next = P\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0, P) {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0: ListNode, P: ListNode): void {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\n  ListNode? n1 = n0.next;\n  P.next = n1;\n  n0.next = P;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\n#[allow(non_snake_case)]\npub fn insert&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, P: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    let n1 = n0.borrow_mut().next.take();\n    P.borrow_mut().next = n1;\n    n0.borrow_mut().next = Some(P);\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.kt<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9p */\nfun insert(n0: ListNode?, p: ListNode?) {\n    val n1 = n0?.next\n    p?.next = n1\n    n0?.next = p\n}\n</code></pre> linked_list.rb<pre><code>### \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 _p ###\n# Ruby \u7684 `p` \u662f\u4e00\u4e2a\u5185\u7f6e\u51fd\u6570\uff0c `P` \u662f\u4e00\u4e2a\u5e38\u91cf\uff0c\u6240\u4ee5\u53ef\u4ee5\u4f7f\u7528 `_p` \u4ee3\u66ff\ndef insert(n0, _p)\n  n1 = n0.next\n  _p.next = n1\n  n0.next = _p\nend\n</code></pre> linked_list.zig<pre><code>// \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\nfn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {\n    var n1 = n0.?.next;\n    P.?.next = n1;\n    n0.?.next = P;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#3","title":"3. \u00a0 \u5220\u9664\u8282\u70b9","text":"<p>\u5982\u56fe 4-7 \u6240\u793a\uff0c\u5728\u94fe\u8868\u4e2d\u5220\u9664\u8282\u70b9\u4e5f\u975e\u5e38\u65b9\u4fbf\uff0c\u53ea\u9700\u6539\u53d8\u4e00\u4e2a\u8282\u70b9\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\u3002</p> <p>\u8bf7\u6ce8\u610f\uff0c\u5c3d\u7ba1\u5728\u5220\u9664\u64cd\u4f5c\u5b8c\u6210\u540e\u8282\u70b9 <code>P</code> \u4ecd\u7136\u6307\u5411 <code>n1</code> \uff0c\u4f46\u5b9e\u9645\u4e0a\u904d\u5386\u6b64\u94fe\u8868\u5df2\u7ecf\u65e0\u6cd5\u8bbf\u95ee\u5230 <code>P</code> \uff0c\u8fd9\u610f\u5473\u7740 <code>P</code> \u5df2\u7ecf\u4e0d\u518d\u5c5e\u4e8e\u8be5\u94fe\u8868\u4e86\u3002</p> <p></p> <p> \u56fe 4-7 \u00a0 \u94fe\u8868\u5220\u9664\u8282\u70b9 </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def remove(n0: ListNode):\n    \"\"\"\u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\"\"\"\n    if not n0.next:\n        return\n    # n0 -&gt; P -&gt; n1\n    P = n0.next\n    n1 = P.next\n    n0.next = n1\n</code></pre> linked_list.cpp<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode *n0) {\n    if (n0-&gt;next == nullptr)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // \u91ca\u653e\u5185\u5b58\n    delete P;\n}\n</code></pre> linked_list.java<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid Remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode? n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.go<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc removeItem(n0 *ListNode) {\n    if n0.Next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    P := n0.Next\n    n1 := P.Next\n    n0.Next = n1\n}\n</code></pre> linked_list.swift<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc remove(n0: ListNode) {\n    if n0.next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    let P = n0.next\n    let n1 = P?.next\n    n0.next = n1\n}\n</code></pre> linked_list.js<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0) {\n    if (!n0.next) return;\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0: ListNode): void {\n    if (!n0.next) {\n        return;\n    }\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\n  if (n0.next == null) return;\n  // n0 -&gt; P -&gt; n1\n  ListNode P = n0.next!;\n  ListNode? n1 = P.next;\n  n0.next = n1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n#[allow(non_snake_case)]\npub fn remove&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    if n0.borrow().next.is_none() {\n        return;\n    };\n    // n0 -&gt; P -&gt; n1\n    let P = n0.borrow_mut().next.take();\n    if let Some(node) = P {\n        let n1 = node.borrow_mut().next.take();\n        n0.borrow_mut().next = n1;\n    }\n}\n</code></pre> linked_list.c<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(ListNode *n0) {\n    if (!n0-&gt;next)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // \u91ca\u653e\u5185\u5b58\n    free(P);\n}\n</code></pre> linked_list.kt<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfun remove(n0: ListNode?) {\n    val p = n0?.next\n    val n1 = p?.next\n    n0?.next = n1\n}\n</code></pre> linked_list.rb<pre><code>### \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 ###\ndef remove(n0)\n  return if n0.next.nil?\n\n  # n0 -&gt; remove_node -&gt; n1\n  remove_node = n0.next\n  n1 = remove_node.next\n  n0.next = n1\nend\n</code></pre> linked_list.zig<pre><code>// \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\nfn remove(n0: ?*inc.ListNode(i32)) void {\n    if (n0.?.next == null) return;\n    // n0 -&gt; P -&gt; n1\n    var P = n0.?.next;\n    var n1 = P.?.next;\n    n0.?.next = n1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#4","title":"4. \u00a0 \u8bbf\u95ee\u8282\u70b9","text":"<p>\u5728\u94fe\u8868\u4e2d\u8bbf\u95ee\u8282\u70b9\u7684\u6548\u7387\u8f83\u4f4e\u3002\u5982\u4e0a\u4e00\u8282\u6240\u8ff0\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u4e0b\u8bbf\u95ee\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u5143\u7d20\u3002\u94fe\u8868\u5219\u4e0d\u7136\uff0c\u7a0b\u5e8f\u9700\u8981\u4ece\u5934\u8282\u70b9\u51fa\u53d1\uff0c\u9010\u4e2a\u5411\u540e\u904d\u5386\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6807\u8282\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u8bbf\u95ee\u94fe\u8868\u7684\u7b2c \\(i\\) \u4e2a\u8282\u70b9\u9700\u8981\u5faa\u73af \\(i - 1\\) \u8f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def access(head: ListNode, index: int) -&gt; ListNode | None:\n    \"\"\"\u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\"\"\"\n    for _ in range(index):\n        if not head:\n            return None\n        head = head.next\n    return head\n</code></pre> linked_list.cpp<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == nullptr)\n            return nullptr;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.java<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode access(ListNode head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.cs<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? Access(ListNode? head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.go<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head *ListNode, index int) *ListNode {\n    for i := 0; i &lt; index; i++ {\n        if head == nil {\n            return nil\n        }\n        head = head.Next\n    }\n    return head\n}\n</code></pre> linked_list.swift<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head: ListNode, index: Int) -&gt; ListNode? {\n    var head: ListNode? = head\n    for _ in 0 ..&lt; index {\n        if head == nil {\n            return nil\n        }\n        head = head?.next\n    }\n    return head\n}\n</code></pre> linked_list.js<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head, index) {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.ts<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head: ListNode | null, index: number): ListNode | null {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.dart<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? access(ListNode? head, int index) {\n  for (var i = 0; i &lt; index; i++) {\n    if (head == null) return null;\n    head = head.next;\n  }\n  return head;\n}\n</code></pre> linked_list.rs<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\npub fn access&lt;T&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, index: i32) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n    if index &lt;= 0 {\n        return head;\n    };\n    if let Some(node) = &amp;head.borrow().next {\n        return access(node.clone(), index - 1);\n    }\n\n    return head;\n}\n</code></pre> linked_list.c<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == NULL)\n            return NULL;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.kt<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfun access(head: ListNode?, index: Int): ListNode? {\n    var h = head\n    for (i in 0..&lt;index) {\n        h = h?.next\n    }\n    return h\n}\n</code></pre> linked_list.rb<pre><code>### \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 ###\ndef access(head, index)\n  for i in 0...index\n    return nil if head.nil?\n    head = head.next\n  end\n\n  head\nend\n</code></pre> linked_list.zig<pre><code>// \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\nfn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {\n    var head = node;\n    var i: i32 = 0;\n    while (i &lt; index) : (i += 1) {\n        head = head.?.next;\n        if (head == null) return null;\n    }\n    return head;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#5","title":"5. \u00a0 \u67e5\u627e\u8282\u70b9","text":"<p>\u904d\u5386\u94fe\u8868\uff0c\u67e5\u627e\u5176\u4e2d\u503c\u4e3a <code>target</code> \u7684\u8282\u70b9\uff0c\u8f93\u51fa\u8be5\u8282\u70b9\u5728\u94fe\u8868\u4e2d\u7684\u7d22\u5f15\u3002\u6b64\u8fc7\u7a0b\u4e5f\u5c5e\u4e8e\u7ebf\u6027\u67e5\u627e\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def find(head: ListNode, target: int) -&gt; int:\n    \"\"\"\u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\"\"\"\n    index = 0\n    while head:\n        if head.val == target:\n            return index\n        head = head.next\n        index += 1\n    return -1\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head != nullptr) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint Find(ListNode? head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc findNode(head *ListNode, target int) int {\n    index := 0\n    for head != nil {\n        if head.Val == target {\n            return index\n        }\n        head = head.Next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc find(head: ListNode, target: Int) -&gt; Int {\n    var head: ListNode? = head\n    var index = 0\n    while head != nil {\n        if head?.val == target {\n            return index\n        }\n        head = head?.next\n        index += 1\n    }\n    return -1\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head, target) {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head: ListNode | null, target: number): number {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode? head, int target) {\n  int index = 0;\n  while (head != null) {\n    if (head.val == target) {\n      return index;\n    }\n    head = head.next;\n    index++;\n  }\n  return -1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\npub fn find&lt;T: PartialEq&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, target: T, index: i32) -&gt; i32 {\n    if head.borrow().val == target {\n        return index;\n    };\n    if let Some(node) = &amp;head.borrow_mut().next {\n        return find(node.clone(), target, index + 1);\n    }\n    return -1;\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.kt<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfun find(head: ListNode?, target: Int): Int {\n    var index = 0\n    var h = head\n    while (h != null) {\n        if (h.value == target) return index\n        h = h.next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.rb<pre><code>### \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 ###\ndef find(head, target)\n  index = 0\n  while head\n    return index if head.val == target\n    head = head.next\n    index += 1\n  end\n\n  -1\nend\n</code></pre> linked_list.zig<pre><code>// \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\nfn find(node: ?*inc.ListNode(i32), target: i32) i32 {\n    var head = node;\n    var index: i32 = 0;\n    while (head != null) {\n        if (head.?.val == target) return index;\n        head = head.?.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#422-vs","title":"4.2.2 \u00a0 \u6570\u7ec4 vs. \u94fe\u8868","text":"<p>\u8868 4-1 \u603b\u7ed3\u4e86\u6570\u7ec4\u548c\u94fe\u8868\u7684\u5404\u9879\u7279\u70b9\u5e76\u5bf9\u6bd4\u4e86\u64cd\u4f5c\u6548\u7387\u3002\u7531\u4e8e\u5b83\u4eec\u91c7\u7528\u4e24\u79cd\u76f8\u53cd\u7684\u5b58\u50a8\u7b56\u7565\uff0c\u56e0\u6b64\u5404\u79cd\u6027\u8d28\u548c\u64cd\u4f5c\u6548\u7387\u4e5f\u5448\u73b0\u5bf9\u7acb\u7684\u7279\u70b9\u3002</p> <p> \u8868 4-1 \u00a0 \u6570\u7ec4\u4e0e\u94fe\u8868\u7684\u6548\u7387\u5bf9\u6bd4 </p> \u6570\u7ec4 \u94fe\u8868 \u5b58\u50a8\u65b9\u5f0f \u8fde\u7eed\u5185\u5b58\u7a7a\u95f4 \u5206\u6563\u5185\u5b58\u7a7a\u95f4 \u5bb9\u91cf\u6269\u5c55 \u957f\u5ea6\u4e0d\u53ef\u53d8 \u53ef\u7075\u6d3b\u6269\u5c55 \u5185\u5b58\u6548\u7387 \u5143\u7d20\u5360\u7528\u5185\u5b58\u5c11\u3001\u4f46\u53ef\u80fd\u6d6a\u8d39\u7a7a\u95f4 \u5143\u7d20\u5360\u7528\u5185\u5b58\u591a \u8bbf\u95ee\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(1)\\)"},{"location":"chapter_array_and_linkedlist/linked_list/#423","title":"4.2.3 \u00a0 \u5e38\u89c1\u94fe\u8868\u7c7b\u578b","text":"<p>\u5982\u56fe 4-8 \u6240\u793a\uff0c\u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u5305\u62ec\u4e09\u79cd\u3002</p> <ul> <li>\u5355\u5411\u94fe\u8868\uff1a\u5373\u524d\u9762\u4ecb\u7ecd\u7684\u666e\u901a\u94fe\u8868\u3002\u5355\u5411\u94fe\u8868\u7684\u8282\u70b9\u5305\u542b\u503c\u548c\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\u4e24\u9879\u6570\u636e\u3002\u6211\u4eec\u5c06\u9996\u4e2a\u8282\u70b9\u79f0\u4e3a\u5934\u8282\u70b9\uff0c\u5c06\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u79f0\u4e3a\u5c3e\u8282\u70b9\uff0c\u5c3e\u8282\u70b9\u6307\u5411\u7a7a <code>None</code> \u3002</li> <li>\u73af\u5f62\u94fe\u8868\uff1a\u5982\u679c\u6211\u4eec\u4ee4\u5355\u5411\u94fe\u8868\u7684\u5c3e\u8282\u70b9\u6307\u5411\u5934\u8282\u70b9\uff08\u9996\u5c3e\u76f8\u63a5\uff09\uff0c\u5219\u5f97\u5230\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\u3002\u5728\u73af\u5f62\u94fe\u8868\u4e2d\uff0c\u4efb\u610f\u8282\u70b9\u90fd\u53ef\u4ee5\u89c6\u4f5c\u5934\u8282\u70b9\u3002</li> <li>\u53cc\u5411\u94fe\u8868\uff1a\u4e0e\u5355\u5411\u94fe\u8868\u76f8\u6bd4\uff0c\u53cc\u5411\u94fe\u8868\u8bb0\u5f55\u4e86\u4e24\u4e2a\u65b9\u5411\u7684\u5f15\u7528\u3002\u53cc\u5411\u94fe\u8868\u7684\u8282\u70b9\u5b9a\u4e49\u540c\u65f6\u5305\u542b\u6307\u5411\u540e\u7ee7\u8282\u70b9\uff08\u4e0b\u4e00\u4e2a\u8282\u70b9\uff09\u548c\u524d\u9a71\u8282\u70b9\uff08\u4e0a\u4e00\u4e2a\u8282\u70b9\uff09\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\u3002\u76f8\u8f83\u4e8e\u5355\u5411\u94fe\u8868\uff0c\u53cc\u5411\u94fe\u8868\u66f4\u5177\u7075\u6d3b\u6027\uff0c\u53ef\u4ee5\u671d\u4e24\u4e2a\u65b9\u5411\u904d\u5386\u94fe\u8868\uff0c\u4f46\u76f8\u5e94\u5730\u4e5f\u9700\u8981\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class ListNode:\n    \"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # \u8282\u70b9\u503c\n        self.next: ListNode | None = None  # \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n        self.prev: ListNode | None = None  # \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\n    int val;         // \u8282\u70b9\u503c\n    ListNode *next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n    ListNode *prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n    ListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    int val;        // \u8282\u70b9\u503c\n    ListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n    ListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n    ListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode(int x) {  // \u6784\u9020\u51fd\u6570\n    int val = x;    // \u8282\u70b9\u503c\n    ListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n    ListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype DoublyListNode struct {\n    Val  int             // \u8282\u70b9\u503c\n    Next *DoublyListNode // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n    Prev *DoublyListNode // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n}\n\n// NewDoublyListNode \u521d\u59cb\u5316\nfunc NewDoublyListNode(val int) *DoublyListNode {\n    return &amp;DoublyListNode{\n        Val:  val,\n        Next: nil,\n        Prev: nil,\n    }\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    var val: Int // \u8282\u70b9\u503c\n    var next: ListNode? // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n    var prev: ListNode? // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n\n    init(x: Int) { // \u6784\u9020\u51fd\u6570\n        val = x\n    }\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    constructor(val, next, prev) {\n        this.val = val  ===  undefined ? 0 : val;        // \u8282\u70b9\u503c\n        this.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n        this.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    prev: ListNode | null;\n    constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\n        this.val = val  ===  undefined ? 0 : val;        // \u8282\u70b9\u503c\n        this.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n        this.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    int val;        // \u8282\u70b9\u503c\n    ListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n    ListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n    ListNode(this.val, [this.next, this.prev]);  // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\u578b */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // \u8282\u70b9\u503c\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n    prev: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nimpl ListNode {\n    fn new(val: i32) -&gt; Self {\n        ListNode {\n            val,\n            next: None,\n            prev: None,\n        }\n    }\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct ListNode {\n    int val;               // \u8282\u70b9\u503c\n    struct ListNode *next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n    struct ListNode *prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n} ListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    node-&gt;prev = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\n// \u6784\u9020\u65b9\u6cd5\nclass ListNode(x: Int) {\n    val _val: Int = x           // \u8282\u70b9\u503c\n    val next: ListNode? = null  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n    val prev: ListNode? = null  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n}\n</code></pre> <pre><code># \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\nclass ListNode\n  attr_accessor :val    # \u8282\u70b9\u503c\n  attr_accessor :next   # \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n  attr_accessor :prev   # \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n\n  def initialize(val=0, next_node=nil, prev_node=nil)\n    @val = val\n    @next = next_node\n    @prev = prev_node\n  end\nend\n</code></pre> <pre><code>// \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = 0, // \u8282\u70b9\u503c\n        next: ?*Self = null, // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n        prev: ?*Self = null, // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n\n        // \u6784\u9020\u51fd\u6570\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n            self.prev = null;\n        }\n    };\n}\n</code></pre> <p></p> <p> \u56fe 4-8 \u00a0 \u5e38\u89c1\u94fe\u8868\u79cd\u7c7b </p>"},{"location":"chapter_array_and_linkedlist/linked_list/#424","title":"4.2.4 \u00a0 \u94fe\u8868\u5178\u578b\u5e94\u7528","text":"<p>\u5355\u5411\u94fe\u8868\u901a\u5e38\u7528\u4e8e\u5b9e\u73b0\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u548c\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u3002</p> <ul> <li>\u6808\u4e0e\u961f\u5217\uff1a\u5f53\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u90fd\u5728\u94fe\u8868\u7684\u4e00\u7aef\u8fdb\u884c\u65f6\uff0c\u5b83\u8868\u73b0\u51fa\u5148\u8fdb\u540e\u51fa\u7684\u7279\u6027\uff0c\u5bf9\u5e94\u6808\uff1b\u5f53\u63d2\u5165\u64cd\u4f5c\u5728\u94fe\u8868\u7684\u4e00\u7aef\u8fdb\u884c\uff0c\u5220\u9664\u64cd\u4f5c\u5728\u94fe\u8868\u7684\u53e6\u4e00\u7aef\u8fdb\u884c\uff0c\u5b83\u8868\u73b0\u51fa\u5148\u8fdb\u5148\u51fa\u7684\u7279\u6027\uff0c\u5bf9\u5e94\u961f\u5217\u3002</li> <li>\u54c8\u5e0c\u8868\uff1a\u94fe\u5f0f\u5730\u5740\u662f\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\u7684\u4e3b\u6d41\u65b9\u6848\u4e4b\u4e00\uff0c\u5728\u8be5\u65b9\u6848\u4e2d\uff0c\u6240\u6709\u51b2\u7a81\u7684\u5143\u7d20\u90fd\u4f1a\u88ab\u653e\u5230\u4e00\u4e2a\u94fe\u8868\u4e2d\u3002</li> <li>\u56fe\uff1a\u90bb\u63a5\u8868\u662f\u8868\u793a\u56fe\u7684\u4e00\u79cd\u5e38\u7528\u65b9\u5f0f\uff0c\u5176\u4e2d\u56fe\u7684\u6bcf\u4e2a\u9876\u70b9\u90fd\u4e0e\u4e00\u4e2a\u94fe\u8868\u76f8\u5173\u8054\uff0c\u94fe\u8868\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u4ee3\u8868\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u5176\u4ed6\u9876\u70b9\u3002</li> </ul> <p>\u53cc\u5411\u94fe\u8868\u5e38\u7528\u4e8e\u9700\u8981\u5feb\u901f\u67e5\u627e\u524d\u4e00\u4e2a\u548c\u540e\u4e00\u4e2a\u5143\u7d20\u7684\u573a\u666f\u3002</p> <ul> <li>\u9ad8\u7ea7\u6570\u636e\u7ed3\u6784\uff1a\u6bd4\u5982\u5728\u7ea2\u9ed1\u6811\u3001B \u6811\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u8bbf\u95ee\u8282\u70b9\u7684\u7236\u8282\u70b9\uff0c\u8fd9\u53ef\u4ee5\u901a\u8fc7\u5728\u8282\u70b9\u4e2d\u4fdd\u5b58\u4e00\u4e2a\u6307\u5411\u7236\u8282\u70b9\u7684\u5f15\u7528\u6765\u5b9e\u73b0\uff0c\u7c7b\u4f3c\u4e8e\u53cc\u5411\u94fe\u8868\u3002</li> <li>\u6d4f\u89c8\u5668\u5386\u53f2\uff1a\u5728\u7f51\u9875\u6d4f\u89c8\u5668\u4e2d\uff0c\u5f53\u7528\u6237\u70b9\u51fb\u524d\u8fdb\u6216\u540e\u9000\u6309\u94ae\u65f6\uff0c\u6d4f\u89c8\u5668\u9700\u8981\u77e5\u9053\u7528\u6237\u8bbf\u95ee\u8fc7\u7684\u524d\u4e00\u4e2a\u548c\u540e\u4e00\u4e2a\u7f51\u9875\u3002\u53cc\u5411\u94fe\u8868\u7684\u7279\u6027\u4f7f\u5f97\u8fd9\u79cd\u64cd\u4f5c\u53d8\u5f97\u7b80\u5355\u3002</li> <li>LRU \u7b97\u6cd5\uff1a\u5728\u7f13\u5b58\u6dd8\u6c70\uff08LRU\uff09\u7b97\u6cd5\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u5feb\u901f\u627e\u5230\u6700\u8fd1\u6700\u5c11\u4f7f\u7528\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u652f\u6301\u5feb\u901f\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u3002\u8fd9\u65f6\u5019\u4f7f\u7528\u53cc\u5411\u94fe\u8868\u5c31\u975e\u5e38\u5408\u9002\u3002</li> </ul> <p>\u73af\u5f62\u94fe\u8868\u5e38\u7528\u4e8e\u9700\u8981\u5468\u671f\u6027\u64cd\u4f5c\u7684\u573a\u666f\uff0c\u6bd4\u5982\u64cd\u4f5c\u7cfb\u7edf\u7684\u8d44\u6e90\u8c03\u5ea6\u3002</p> <ul> <li>\u65f6\u95f4\u7247\u8f6e\u8f6c\u8c03\u5ea6\u7b97\u6cd5\uff1a\u5728\u64cd\u4f5c\u7cfb\u7edf\u4e2d\uff0c\u65f6\u95f4\u7247\u8f6e\u8f6c\u8c03\u5ea6\u7b97\u6cd5\u662f\u4e00\u79cd\u5e38\u89c1\u7684 CPU \u8c03\u5ea6\u7b97\u6cd5\uff0c\u5b83\u9700\u8981\u5bf9\u4e00\u7ec4\u8fdb\u7a0b\u8fdb\u884c\u5faa\u73af\u3002\u6bcf\u4e2a\u8fdb\u7a0b\u88ab\u8d4b\u4e88\u4e00\u4e2a\u65f6\u95f4\u7247\uff0c\u5f53\u65f6\u95f4\u7247\u7528\u5b8c\u65f6\uff0cCPU \u5c06\u5207\u6362\u5230\u4e0b\u4e00\u4e2a\u8fdb\u7a0b\u3002\u8fd9\u79cd\u5faa\u73af\u64cd\u4f5c\u53ef\u4ee5\u901a\u8fc7\u73af\u5f62\u94fe\u8868\u6765\u5b9e\u73b0\u3002</li> <li>\u6570\u636e\u7f13\u51b2\u533a\uff1a\u5728\u67d0\u4e9b\u6570\u636e\u7f13\u51b2\u533a\u7684\u5b9e\u73b0\u4e2d\uff0c\u4e5f\u53ef\u80fd\u4f1a\u4f7f\u7528\u73af\u5f62\u94fe\u8868\u3002\u6bd4\u5982\u5728\u97f3\u9891\u3001\u89c6\u9891\u64ad\u653e\u5668\u4e2d\uff0c\u6570\u636e\u6d41\u53ef\u80fd\u4f1a\u88ab\u5206\u6210\u591a\u4e2a\u7f13\u51b2\u5757\u5e76\u653e\u5165\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\uff0c\u4ee5\u4fbf\u5b9e\u73b0\u65e0\u7f1d\u64ad\u653e\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/list/","title":"4.3 \u00a0 \u5217\u8868","text":"<p>\u300c\u5217\u8868 list\u300d\u662f\u4e00\u4e2a\u62bd\u8c61\u7684\u6570\u636e\u7ed3\u6784\u6982\u5ff5\uff0c\u5b83\u8868\u793a\u5143\u7d20\u7684\u6709\u5e8f\u96c6\u5408\uff0c\u652f\u6301\u5143\u7d20\u8bbf\u95ee\u3001\u4fee\u6539\u3001\u6dfb\u52a0\u3001\u5220\u9664\u548c\u904d\u5386\u7b49\u64cd\u4f5c\uff0c\u65e0\u987b\u4f7f\u7528\u8005\u8003\u8651\u5bb9\u91cf\u9650\u5236\u7684\u95ee\u9898\u3002\u5217\u8868\u53ef\u4ee5\u57fa\u4e8e\u94fe\u8868\u6216\u6570\u7ec4\u5b9e\u73b0\u3002</p> <ul> <li>\u94fe\u8868\u5929\u7136\u53ef\u4ee5\u770b\u4f5c\u4e00\u4e2a\u5217\u8868\uff0c\u5176\u652f\u6301\u5143\u7d20\u589e\u5220\u67e5\u6539\u64cd\u4f5c\uff0c\u5e76\u4e14\u53ef\u4ee5\u7075\u6d3b\u52a8\u6001\u6269\u5bb9\u3002</li> <li>\u6570\u7ec4\u4e5f\u652f\u6301\u5143\u7d20\u589e\u5220\u67e5\u6539\uff0c\u4f46\u7531\u4e8e\u5176\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u53ea\u80fd\u770b\u4f5c\u4e00\u4e2a\u5177\u6709\u957f\u5ea6\u9650\u5236\u7684\u5217\u8868\u3002</li> </ul> <p>\u5f53\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u5217\u8868\u65f6\uff0c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6027\u8d28\u4f1a\u5bfc\u81f4\u5217\u8868\u7684\u5b9e\u7528\u6027\u964d\u4f4e\u3002\u8fd9\u662f\u56e0\u4e3a\u6211\u4eec\u901a\u5e38\u65e0\u6cd5\u4e8b\u5148\u786e\u5b9a\u9700\u8981\u5b58\u50a8\u591a\u5c11\u6570\u636e\uff0c\u4ece\u800c\u96be\u4ee5\u9009\u62e9\u5408\u9002\u7684\u5217\u8868\u957f\u5ea6\u3002\u82e5\u957f\u5ea6\u8fc7\u5c0f\uff0c\u5219\u5f88\u53ef\u80fd\u65e0\u6cd5\u6ee1\u8db3\u4f7f\u7528\u9700\u6c42\uff1b\u82e5\u957f\u5ea6\u8fc7\u5927\uff0c\u5219\u4f1a\u9020\u6210\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002</p> <p>\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u300c\u52a8\u6001\u6570\u7ec4 dynamic array\u300d\u6765\u5b9e\u73b0\u5217\u8868\u3002\u5b83\u7ee7\u627f\u4e86\u6570\u7ec4\u7684\u5404\u9879\u4f18\u70b9\uff0c\u5e76\u4e14\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u8fdb\u884c\u52a8\u6001\u6269\u5bb9\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u6807\u51c6\u5e93\u63d0\u4f9b\u7684\u5217\u8868\u662f\u57fa\u4e8e\u52a8\u6001\u6570\u7ec4\u5b9e\u73b0\u7684\uff0c\u4f8b\u5982 Python \u4e2d\u7684 <code>list</code> \u3001Java \u4e2d\u7684 <code>ArrayList</code> \u3001C++ \u4e2d\u7684 <code>vector</code> \u548c C# \u4e2d\u7684 <code>List</code> \u7b49\u3002\u5728\u63a5\u4e0b\u6765\u7684\u8ba8\u8bba\u4e2d\uff0c\u6211\u4eec\u5c06\u628a\u201c\u5217\u8868\u201d\u548c\u201c\u52a8\u6001\u6570\u7ec4\u201d\u89c6\u4e3a\u7b49\u540c\u7684\u6982\u5ff5\u3002</p>"},{"location":"chapter_array_and_linkedlist/list/#431","title":"4.3.1 \u00a0 \u5217\u8868\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u5217\u8868","text":"<p>\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u201c\u65e0\u521d\u59cb\u503c\u201d\u548c\u201c\u6709\u521d\u59cb\u503c\u201d\u8fd9\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u521d\u59cb\u5316\u5217\u8868\n# \u65e0\u521d\u59cb\u503c\nnums1: list[int] = []\n# \u6709\u521d\u59cb\u503c\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> list.cpp<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u9700\u6ce8\u610f\uff0cC++ \u4e2d vector \u5373\u662f\u672c\u6587\u63cf\u8ff0\u7684 nums\n// \u65e0\u521d\u59cb\u503c\nvector&lt;int&gt; nums1;\n// \u6709\u521d\u59cb\u503c\nvector&lt;int&gt; nums = { 1, 3, 2, 5, 4 };\n</code></pre> list.java<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;();\n// \u6709\u521d\u59cb\u503c\uff08\u6ce8\u610f\u6570\u7ec4\u7684\u5143\u7d20\u7c7b\u578b\u9700\u4e3a int[] \u7684\u5305\u88c5\u7c7b Integer[]\uff09\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList&lt;Integer&gt; nums = new ArrayList&lt;&gt;(Arrays.asList(numbers));\n</code></pre> list.cs<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList&lt;int&gt; nums1 = [];\n// \u6709\u521d\u59cb\u503c\nint[] numbers = [1, 3, 2, 5, 4];\nList&lt;int&gt; nums = [.. numbers];\n</code></pre> list_test.go<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nnums1 := []int{}\n// \u6709\u521d\u59cb\u503c\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> list.swift<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet nums1: [Int] = []\n// \u6709\u521d\u59cb\u503c\nvar nums = [1, 3, 2, 5, 4]\n</code></pre> list.js<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst nums1 = [];\n// \u6709\u521d\u59cb\u503c\nconst nums = [1, 3, 2, 5, 4];\n</code></pre> list.ts<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst nums1: number[] = [];\n// \u6709\u521d\u59cb\u503c\nconst nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> list.dart<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList&lt;int&gt; nums1 = [];\n// \u6709\u521d\u59cb\u503c\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> list.rs<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet nums1: Vec&lt;i32&gt; = Vec::new();\n// \u6709\u521d\u59cb\u503c\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.kt<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nvar nums1 = listOf&lt;Int&gt;()\n// \u6709\u521d\u59cb\u503c\nvar numbers = arrayOf(1, 3, 2, 5, 4)\nvar nums = numbers.toMutableList()\n</code></pre> list.rb<pre><code># \u521d\u59cb\u5316\u5217\u8868\n# \u65e0\u521d\u59cb\u503c\nnums1 = []\n# \u6709\u521d\u59cb\u503c\nnums = [1, 3, 2, 5, 4]\n</code></pre> list.zig<pre><code>// \u521d\u59cb\u5316\u5217\u8868\nvar nums = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums.deinit();\ntry nums.appendSlice(&amp;[_]i32{ 1, 3, 2, 5, 4 });\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#2","title":"2. \u00a0 \u8bbf\u95ee\u5143\u7d20","text":"<p>\u5217\u8868\u672c\u8d28\u4e0a\u662f\u6570\u7ec4\uff0c\u56e0\u6b64\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u548c\u66f4\u65b0\u5143\u7d20\uff0c\u6548\u7387\u5f88\u9ad8\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u8bbf\u95ee\u5143\u7d20\nnum: int = nums[1]  # \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n# \u66f4\u65b0\u5143\u7d20\nnums[1] = 0    # \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.cpp<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.java<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums.get(1);  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums.set(1, 0);  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.cs<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list_test.go<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nnum := nums[1]  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0     // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.swift<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nlet num = nums[1] // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0 // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.js<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nconst num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.ts<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nconst num: number = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.dart<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.rs<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nlet num: i32 = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;             // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.kt<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nval num = nums[1]       // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0             // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.rb<pre><code># \u8bbf\u95ee\u5143\u7d20\nnum = nums[1] # \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n# \u66f4\u65b0\u5143\u7d20\nnums[1] = 0 # \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.zig<pre><code>// \u8bbf\u95ee\u5143\u7d20\nvar num = nums.items[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n// \u66f4\u65b0\u5143\u7d20\nnums.items[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#3","title":"3. \u00a0 \u63d2\u5165\u4e0e\u5220\u9664\u5143\u7d20","text":"<p>\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5730\u6dfb\u52a0\u4e0e\u5220\u9664\u5143\u7d20\u3002\u5728\u5217\u8868\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u7684\u6548\u7387\u4ecd\u4e0e\u6570\u7ec4\u76f8\u540c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u6e05\u7a7a\u5217\u8868\nnums.clear()\n\n# \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n# \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\nnums.insert(3, 6)  # \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n# \u5220\u9664\u5143\u7d20\nnums.pop(3)        # \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.cpp<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push_back(1);\nnums.push_back(3);\nnums.push_back(2);\nnums.push_back(5);\nnums.push_back(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(nums.begin() + 3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.erase(nums.begin() + 3);      // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.java<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.add(3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(3);  // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.cs<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.Clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.Add(1);\nnums.Add(3);\nnums.Add(2);\nnums.Add(5);\nnums.Add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.Insert(3, 6);\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.RemoveAt(3);\n</code></pre> list_test.go<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums = nil\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums = append(nums, 1)\nnums = append(nums, 3)\nnums = append(nums, 2)\nnums = append(nums, 5)\nnums = append(nums, 4)\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums = append(nums[:3], append([]int{6}, nums[3:]...)...) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums = append(nums[:3], nums[4:]...) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.swift<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.removeAll()\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(6, at: 3) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(at: 3) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.js<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.length = 0;\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6);\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.splice(3, 1);\n</code></pre> list.ts<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.length = 0;\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6);\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.splice(3, 1);\n</code></pre> list.dart<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.removeAt(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.rs<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(3);    // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.kt<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.add(3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(3);  // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.rb<pre><code># \u6e05\u7a7a\u5217\u8868\nnums.clear\n\n# \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\nnums &lt;&lt; 1\nnums &lt;&lt; 3\nnums &lt;&lt; 2\nnums &lt;&lt; 5\nnums &lt;&lt; 4\n\n# \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\nnums.insert 3, 6 # \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n# \u5220\u9664\u5143\u7d20\nnums.delete_at 3 # \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.zig<pre><code>// \u6e05\u7a7a\u5217\u8868\nnums.clearRetainingCapacity();\n\n// \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\ntry nums.append(1);\ntry nums.append(3);\ntry nums.append(2);\ntry nums.append(5);\ntry nums.append(4);\n\n// \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\ntry nums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n// \u5220\u9664\u5143\u7d20\n_ = nums.orderedRemove(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#4","title":"4. \u00a0 \u904d\u5386\u5217\u8868","text":"<p>\u4e0e\u6570\u7ec4\u4e00\u6837\uff0c\u5217\u8868\u53ef\u4ee5\u6839\u636e\u7d22\u5f15\u904d\u5386\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u904d\u5386\u5404\u5143\u7d20\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\ncount = 0\nfor i in range(len(nums)):\n    count += nums[i]\n\n# \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\nfor num in nums:\n    count += num\n</code></pre> list.cpp<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.java<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums.get(i);\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.cs<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i &lt; nums.Count; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nforeach (int num in nums) {\n    count += num;\n}\n</code></pre> list_test.go<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\ncount := 0\nfor i := 0; i &lt; len(nums); i++ {\n    count += nums[i]\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor _, num := range nums {\n    count += num\n}\n</code></pre> list.swift<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nvar count = 0\nfor i in nums.indices {\n    count += nums[i]\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor num in nums {\n    count += num\n}\n</code></pre> list.js<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.ts<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.dart<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (var num in nums) {\n    count += num;\n}\n</code></pre> list.rs<pre><code>// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nlet mut _count = 0;\nfor i in 0..nums.len() {\n    _count += nums[i];\n}\n\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\n_count = 0;\nfor num in &amp;nums {\n    _count += num;\n}\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.kt<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nvar count = 0\nfor (i in nums.indices) {\n    count += nums[i]\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\nfor (num in nums) {\n    count += num\n}\n</code></pre> list.rb<pre><code># \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\ncount = 0\nfor i in 0...nums.length\n    count += nums[i]\nend\n\n# \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\ncount = 0\nfor num in nums\n    count += num\nend\n</code></pre> list.zig<pre><code>// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i &lt; nums.items.len) : (i += 1) {\n    count += nums[i];\n}\n\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\ncount = 0;\nfor (nums.items) |num| {\n    count += num;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#5","title":"5. \u00a0 \u62fc\u63a5\u5217\u8868","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u65b0\u5217\u8868 <code>nums1</code> \uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u62fc\u63a5\u5230\u539f\u5217\u8868\u7684\u5c3e\u90e8\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nnums1: list[int] = [6, 8, 7, 10, 9]\nnums += nums1  # \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.cpp<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nvector&lt;int&gt; nums1 = { 6, 8, 7, 10, 9 };\n// \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\nnums.insert(nums.end(), nums1.begin(), nums1.end());\n</code></pre> list.java<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nnums.addAll(nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.cs<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.AddRange(nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list_test.go<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nnums1 := []int{6, 8, 7, 10, 9}\nnums = append(nums, nums1...)  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.swift<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet nums1 = [6, 8, 7, 10, 9]\nnums.append(contentsOf: nums1) // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.js<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst nums1 = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.ts<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst nums1: number[] = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.dart<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.addAll(nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.rs<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet nums1: Vec&lt;i32&gt; = vec![6, 8, 7, 10, 9];\nnums.extend(nums1);\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.kt<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nval nums1 = intArrayOf(6, 8, 7, 10, 9).toMutableList()\nnums.addAll(nums1)  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.rb<pre><code># \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nnums1 = [6, 8, 7, 10, 9]\nnums += nums1\n</code></pre> list.zig<pre><code>// \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nvar nums1 = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums1.deinit();\ntry nums1.appendSlice(&amp;[_]i32{ 6, 8, 7, 10, 9 });\ntry nums.insertSlice(nums.items.len, nums1.items); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#6","title":"6. \u00a0 \u6392\u5e8f\u5217\u8868","text":"<p>\u5b8c\u6210\u5217\u8868\u6392\u5e8f\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u4ee5\u4f7f\u7528\u5728\u6570\u7ec4\u7c7b\u7b97\u6cd5\u9898\u4e2d\u7ecf\u5e38\u8003\u67e5\u7684\u201c\u4e8c\u5206\u67e5\u627e\u201d\u548c\u201c\u53cc\u6307\u9488\u201d\u7b97\u6cd5\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u6392\u5e8f\u5217\u8868\nnums.sort()  # \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.cpp<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nsort(nums.begin(), nums.end());  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.java<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nCollections.sort(nums);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.cs<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.Sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list_test.go<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nsort.Ints(nums)  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.swift<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort() // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.js<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort((a, b) =&gt; a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.ts<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort((a, b) =&gt; a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.dart<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.rs<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.kt<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort() // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.rb<pre><code># \u6392\u5e8f\u5217\u8868\nnums = nums.sort { |a, b| a &lt;=&gt; b } # \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.zig<pre><code>// \u6392\u5e8f\u5217\u8868\nstd.sort.sort(i32, nums.items, {}, comptime std.sort.asc(i32));\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#432","title":"4.3.2 \u00a0 \u5217\u8868\u5b9e\u73b0","text":"<p>\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u4e86\u5217\u8868\uff0c\u4f8b\u5982 Java\u3001C++\u3001Python \u7b49\u3002\u5b83\u4eec\u7684\u5b9e\u73b0\u6bd4\u8f83\u590d\u6742\uff0c\u5404\u4e2a\u53c2\u6570\u7684\u8bbe\u5b9a\u4e5f\u975e\u5e38\u8003\u7a76\uff0c\u4f8b\u5982\u521d\u59cb\u5bb9\u91cf\u3001\u6269\u5bb9\u500d\u6570\u7b49\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u6e90\u7801\u8fdb\u884c\u5b66\u4e60\u3002</p> <p>\u4e3a\u4e86\u52a0\u6df1\u5bf9\u5217\u8868\u5de5\u4f5c\u539f\u7406\u7684\u7406\u89e3\uff0c\u6211\u4eec\u5c1d\u8bd5\u5b9e\u73b0\u4e00\u4e2a\u7b80\u6613\u7248\u5217\u8868\uff0c\u5305\u62ec\u4ee5\u4e0b\u4e09\u4e2a\u91cd\u70b9\u8bbe\u8ba1\u3002</p> <ul> <li>\u521d\u59cb\u5bb9\u91cf\uff1a\u9009\u53d6\u4e00\u4e2a\u5408\u7406\u7684\u6570\u7ec4\u521d\u59cb\u5bb9\u91cf\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u9009\u62e9 10 \u4f5c\u4e3a\u521d\u59cb\u5bb9\u91cf\u3002</li> <li>\u6570\u91cf\u8bb0\u5f55\uff1a\u58f0\u660e\u4e00\u4e2a\u53d8\u91cf <code>size</code> \uff0c\u7528\u4e8e\u8bb0\u5f55\u5217\u8868\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff0c\u5e76\u968f\u7740\u5143\u7d20\u63d2\u5165\u548c\u5220\u9664\u5b9e\u65f6\u66f4\u65b0\u3002\u6839\u636e\u6b64\u53d8\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4f4d\u5217\u8868\u5c3e\u90e8\uff0c\u4ee5\u53ca\u5224\u65ad\u662f\u5426\u9700\u8981\u6269\u5bb9\u3002</li> <li>\u6269\u5bb9\u673a\u5236\uff1a\u82e5\u63d2\u5165\u5143\u7d20\u65f6\u5217\u8868\u5bb9\u91cf\u5df2\u6ee1\uff0c\u5219\u9700\u8981\u8fdb\u884c\u6269\u5bb9\u3002\u5148\u6839\u636e\u6269\u5bb9\u500d\u6570\u521b\u5efa\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u518d\u5c06\u5f53\u524d\u6570\u7ec4\u7684\u6240\u6709\u5143\u7d20\u4f9d\u6b21\u79fb\u52a8\u81f3\u65b0\u6570\u7ec4\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u89c4\u5b9a\u6bcf\u6b21\u5c06\u6570\u7ec4\u6269\u5bb9\u81f3\u4e4b\u524d\u7684 2 \u500d\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_list.py<pre><code>class MyList:\n    \"\"\"\u5217\u8868\u7c7b\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._capacity: int = 10  # \u5217\u8868\u5bb9\u91cf\n        self._arr: list[int] = [0] * self._capacity  # \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n        self._size: int = 0  # \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        self._extend_ratio: int = 2  # \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\"\"\"\n        return self._size\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u5217\u8868\u5bb9\u91cf\"\"\"\n        return self._capacity\n\n    def get(self, index: int) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u5143\u7d20\"\"\"\n        # \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        return self._arr[index]\n\n    def set(self, num: int, index: int):\n        \"\"\"\u66f4\u65b0\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        self._arr[index] = num\n\n    def add(self, num: int):\n        \"\"\"\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\"\"\"\n        # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self.size() == self.capacity():\n            self.extend_capacity()\n        self._arr[self._size] = num\n        self._size += 1\n\n    def insert(self, num: int, index: int):\n        \"\"\"\u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self._size == self.capacity():\n            self.extend_capacity()\n        # \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for j in range(self._size - 1, index - 1, -1):\n            self._arr[j + 1] = self._arr[j]\n        self._arr[index] = num\n        # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self._size += 1\n\n    def remove(self, index: int) -&gt; int:\n        \"\"\"\u5220\u9664\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        num = self._arr[index]\n        # \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for j in range(index, self._size - 1):\n            self._arr[j] = self._arr[j + 1]\n        # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self._size -= 1\n        # \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num\n\n    def extend_capacity(self):\n        \"\"\"\u5217\u8868\u6269\u5bb9\"\"\"\n        # \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 _extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        self._arr = self._arr + [0] * self.capacity() * (self._extend_ratio - 1)\n        # \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        self._capacity = len(self._arr)\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868\"\"\"\n        return self._arr[: self._size]\n</code></pre> my_list.cpp<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n  private:\n    int *arr;             // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    int arrCapacity = 10; // \u5217\u8868\u5bb9\u91cf\n    int arrSize = 0;      // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    int extendRatio = 2;   // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~MyList() {\n        delete[] arr;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    int size() {\n        return arrSize;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    int capacity() {\n        return arrCapacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    int get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    void add(int num) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size() == capacity())\n            extendCapacity();\n        arr[size()] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size() == capacity())\n            extendCapacity();\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (int j = size() - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    int remove(int index) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (int j = index; j &lt; size() - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    void extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\n        int newCapacity = capacity() * extendRatio;\n        int *tmp = arr;\n        arr = new int[newCapacity];\n        // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        for (int i = 0; i &lt; size(); i++) {\n            arr[i] = tmp[i];\n        }\n        // \u91ca\u653e\u5185\u5b58\n        delete[] tmp;\n        arrCapacity = newCapacity;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Vector \u7528\u4e8e\u6253\u5370 */\n    vector&lt;int&gt; toVector() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        vector&lt;int&gt; vec(size());\n        for (int i = 0; i &lt; size(); i++) {\n            vec[i] = arr[i];\n        }\n        return vec;\n    }\n};\n</code></pre> my_list.java<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private int[] arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private int capacity = 10; // \u5217\u8868\u5bb9\u91cf\n    private int size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private int extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public MyList() {\n        arr = new int[capacity];\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\n    public int size() {\n        return size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    public int capacity() {\n        return capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    public int get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    public void add(int num) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity();\n        arr[size] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    public void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity();\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (int j = size - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    public int remove(int index) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (int j = index; j &lt; size - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    public void extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        arr = Arrays.copyOf(arr, capacity() * extendRatio);\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        capacity = arr.length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    public int[] toArray() {\n        int size = size();\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] arr = new int[size];\n        for (int i = 0; i &lt; size; i++) {\n            arr[i] = get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.cs<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private int[] arr;           // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private int arrCapacity = 10;    // \u5217\u8868\u5bb9\u91cf\n    private int arrSize = 0;         // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private readonly int extendRatio = 2;  // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    public int Size() {\n        return arrSize;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    public int Capacity() {\n        return arrCapacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    public int Get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public void Set(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    public void Add(int num) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        arr[arrSize] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    public void Insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (int j = arrSize - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    public int Remove(int index) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (int j = index; j &lt; arrSize - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    public void ExtendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a arrCapacity * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        Array.Resize(ref arr, arrCapacity * extendRatio);\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        arrCapacity = arr.Length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    public int[] ToArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] arr = new int[arrSize];\n        for (int i = 0; i &lt; arrSize; i++) {\n            arr[i] = Get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.go<pre><code>/* \u5217\u8868\u7c7b */\ntype myList struct {\n    arrCapacity int\n    arr         []int\n    arrSize     int\n    extendRatio int\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newMyList() *myList {\n    return &amp;myList{\n        arrCapacity: 10,              // \u5217\u8868\u5bb9\u91cf\n        arr:         make([]int, 10), // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n        arrSize:     0,               // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        extendRatio: 2,               // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n    }\n}\n\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\nfunc (l *myList) size() int {\n    return l.arrSize\n}\n\n/*  \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc (l *myList) capacity() int {\n    return l.arrCapacity\n}\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc (l *myList) get(index int) int {\n    // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    return l.arr[index]\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc (l *myList) set(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    l.arr[index] = num\n}\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc (l *myList) add(num int) {\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    l.arr[l.arrSize] = num\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    l.arrSize++\n}\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc (l *myList) insert(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for j := l.arrSize - 1; j &gt;= index; j-- {\n        l.arr[j+1] = l.arr[j]\n    }\n    l.arr[index] = num\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    l.arrSize++\n}\n\n/* \u5220\u9664\u5143\u7d20 */\nfunc (l *myList) remove(index int) int {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    num := l.arr[index]\n    // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for j := index; j &lt; l.arrSize-1; j++ {\n        l.arr[j] = l.arr[j+1]\n    }\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    l.arrSize--\n    // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n    return num\n}\n\n/* \u5217\u8868\u6269\u5bb9 */\nfunc (l *myList) extendCapacity() {\n    // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    l.arr = append(l.arr, make([]int, l.arrCapacity*(l.extendRatio-1))...)\n    // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n    l.arrCapacity = len(l.arr)\n}\n\n/* \u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868 */\nfunc (l *myList) toArray() []int {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    return l.arr[:l.arrSize]\n}\n</code></pre> my_list.swift<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private var arr: [Int] // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private var _capacity: Int // \u5217\u8868\u5bb9\u91cf\n    private var _size: Int // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private let extendRatio: Int // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init() {\n        _capacity = 10\n        _size = 0\n        extendRatio = 2\n        arr = Array(repeating: 0, count: _capacity)\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        _capacity\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    func get(index: Int) -&gt; Int {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u9519\u8bef\uff0c\u4e0b\u540c\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        return arr[index]\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    func set(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        arr[index] = num\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    func add(num: Int) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if size() == capacity() {\n            extendCapacity()\n        }\n        arr[size()] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        _size += 1\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    func insert(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if size() == capacity() {\n            extendCapacity()\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for j in (index ..&lt; size()).reversed() {\n            arr[j + 1] = arr[j]\n        }\n        arr[index] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        _size += 1\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    @discardableResult\n    func remove(index: Int) -&gt; Int {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        let num = arr[index]\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for j in index ..&lt; (size() - 1) {\n            arr[j] = arr[j + 1]\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        _size -= 1\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    func extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        arr = arr + Array(repeating: 0, count: capacity() * (extendRatio - 1))\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        _capacity = arr.count\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    func toArray() -&gt; [Int] {\n        Array(arr.prefix(size()))\n    }\n}\n</code></pre> my_list.js<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    #arr = new Array(); // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    #capacity = 10; // \u5217\u8868\u5bb9\u91cf\n    #size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    #extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#arr = new Array(this.#capacity);\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    size() {\n        return this.#size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    capacity() {\n        return this.#capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    get(index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        return this.#arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    set(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        this.#arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    add(num) {\n        // \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\n        this.#arr[this.#size] = num;\n        this.#size++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    insert(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (let j = this.#size - 1; j &gt;= index; j--) {\n            this.#arr[j + 1] = this.#arr[j];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this.#arr[index] = num;\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    remove(index) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        let num = this.#arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (let j = index; j &lt; this.#size - 1; j++) {\n            this.#arr[j] = this.#arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this.#size--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        this.#arr = this.#arr.concat(\n            new Array(this.capacity() * (this.#extendRatio - 1))\n        );\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        this.#capacity = this.#arr.length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    toArray() {\n        let size = this.size();\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.ts<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private arr: Array&lt;number&gt;; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private _capacity: number = 10; // \u5217\u8868\u5bb9\u91cf\n    private _size: number = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private extendRatio: number = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.arr = new Array(this._capacity);\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    public size(): number {\n        return this._size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    public capacity(): number {\n        return this._capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    public get(index: number): number {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        return this.arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public set(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        this.arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    public add(num: number): void {\n        // \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\n        if (this._size === this._capacity) this.extendCapacity();\n        // \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\n        this.arr[this._size] = num;\n        this._size++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    public insert(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (this._size === this._capacity) {\n            this.extendCapacity();\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (let j = this._size - 1; j &gt;= index; j--) {\n            this.arr[j + 1] = this.arr[j];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this.arr[index] = num;\n        this._size++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    public remove(index: number): number {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        let num = this.arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (let j = index; j &lt; this._size - 1; j++) {\n            this.arr[j] = this.arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this._size--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    public extendCapacity(): void {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        this.arr = this.arr.concat(\n            new Array(this.capacity() * (this.extendRatio - 1))\n        );\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        this._capacity = this.arr.length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    public toArray(): number[] {\n        let size = this.size();\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.dart<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n  late List&lt;int&gt; _arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n  int _capacity = 10; // \u5217\u8868\u5bb9\u91cf\n  int _size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n  int _extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  MyList() {\n    _arr = List.filled(_capacity, 0);\n  }\n\n  /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n  int size() =&gt; _size;\n\n  /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n  int capacity() =&gt; _capacity;\n\n  /* \u8bbf\u95ee\u5143\u7d20 */\n  int get(int index) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    return _arr[index];\n  }\n\n  /* \u66f4\u65b0\u5143\u7d20 */\n  void set(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    _arr[index] = _num;\n  }\n\n  /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n  void add(int _num) {\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if (_size == _capacity) extendCapacity();\n    _arr[_size] = _num;\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    _size++;\n  }\n\n  /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n  void insert(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if (_size == _capacity) extendCapacity();\n    // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (var j = _size - 1; j &gt;= index; j--) {\n      _arr[j + 1] = _arr[j];\n    }\n    _arr[index] = _num;\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    _size++;\n  }\n\n  /* \u5220\u9664\u5143\u7d20 */\n  int remove(int index) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    int _num = _arr[index];\n    // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (var j = index; j &lt; _size - 1; j++) {\n      _arr[j] = _arr[j + 1];\n    }\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    _size--;\n    // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n    return _num;\n  }\n\n  /* \u5217\u8868\u6269\u5bb9 */\n  void extendCapacity() {\n    // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 _extendRatio \u500d\u7684\u65b0\u6570\u7ec4\n    final _newNums = List.filled(_capacity * _extendRatio, 0);\n    // \u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    List.copyRange(_newNums, 0, _arr);\n    // \u66f4\u65b0 _arr \u7684\u5f15\u7528\n    _arr = _newNums;\n    // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n    _capacity = _arr.length;\n  }\n\n  /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n  List&lt;int&gt; toArray() {\n    List&lt;int&gt; arr = [];\n    for (var i = 0; i &lt; _size; i++) {\n      arr.add(get(i));\n    }\n    return arr;\n  }\n}\n</code></pre> my_list.rs<pre><code>/* \u5217\u8868\u7c7b */\n#[allow(dead_code)]\nstruct MyList {\n    arr: Vec&lt;i32&gt;,       // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    capacity: usize,     // \u5217\u8868\u5bb9\u91cf\n    size: usize,         // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    extend_ratio: usize, // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n}\n\n#[allow(unused, unused_comparisons)]\nimpl MyList {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(capacity: usize) -&gt; Self {\n        let mut vec = Vec::new();\n        vec.resize(capacity, 0);\n        Self {\n            arr: vec,\n            capacity,\n            size: 0,\n            extend_ratio: 2,\n        }\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    pub fn size(&amp;self) -&gt; usize {\n        return self.size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    pub fn capacity(&amp;self) -&gt; usize {\n        return self.capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    pub fn get(&amp;self, index: usize) -&gt; i32 {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if index &gt;= self.size {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        return self.arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    pub fn set(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        self.arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    pub fn add(&amp;mut self, num: i32) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        self.arr[self.size] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self.size += 1;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    pub fn insert(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size() {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for j in (index..self.size).rev() {\n            self.arr[j + 1] = self.arr[j];\n        }\n        self.arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self.size += 1;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    pub fn remove(&amp;mut self, index: usize) -&gt; i32 {\n        if index &gt;= self.size() {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        let num = self.arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for j in (index..self.size - 1) {\n            self.arr[j] = self.arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self.size -= 1;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    pub fn extend_capacity(&amp;mut self) {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        let new_capacity = self.capacity * self.extend_ratio;\n        self.arr.resize(new_capacity, 0);\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        self.capacity = new_capacity;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    pub fn to_array(&amp;mut self) -&gt; Vec&lt;i32&gt; {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        let mut arr = Vec::new();\n        for i in 0..self.size {\n            arr.push(self.get(i));\n        }\n        arr\n    }\n}\n</code></pre> my_list.c<pre><code>/* \u5217\u8868\u7c7b */\ntypedef struct {\n    int *arr;        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    int capacity;    // \u5217\u8868\u5bb9\u91cf\n    int size;        // \u5217\u8868\u5927\u5c0f\n    int extendRatio; // \u5217\u8868\u6bcf\u6b21\u6269\u5bb9\u7684\u500d\u6570\n} MyList;\n\n/* \u6784\u9020\u51fd\u6570 */\nMyList *newMyList() {\n    MyList *nums = malloc(sizeof(MyList));\n    nums-&gt;capacity = 10;\n    nums-&gt;arr = malloc(sizeof(int) * nums-&gt;capacity);\n    nums-&gt;size = 0;\n    nums-&gt;extendRatio = 2;\n    return nums;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delMyList(MyList *nums) {\n    free(nums-&gt;arr);\n    free(nums);\n}\n\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6 */\nint size(MyList *nums) {\n    return nums-&gt;size;\n}\n\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity(MyList *nums) {\n    return nums-&gt;capacity;\n}\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    return nums-&gt;arr[index];\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    nums-&gt;arr[index] = num;\n}\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(MyList *nums, int num) {\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // \u6269\u5bb9\n    }\n    nums-&gt;arr[size(nums)] = num;\n    nums-&gt;size++;\n}\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // \u6269\u5bb9\n    }\n    for (int i = size(nums); i &gt; index; --i) {\n        nums-&gt;arr[i] = nums-&gt;arr[i - 1];\n    }\n    nums-&gt;arr[index] = num;\n    nums-&gt;size++;\n}\n\n/* \u5220\u9664\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nint removeItem(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    int num = nums-&gt;arr[index];\n    for (int i = index; i &lt; size(nums) - 1; i++) {\n        nums-&gt;arr[i] = nums-&gt;arr[i + 1];\n    }\n    nums-&gt;size--;\n    return num;\n}\n\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity(MyList *nums) {\n    // \u5148\u5206\u914d\u7a7a\u95f4\n    int newCapacity = capacity(nums) * nums-&gt;extendRatio;\n    int *extend = (int *)malloc(sizeof(int) * newCapacity);\n    int *temp = nums-&gt;arr;\n\n    // \u62f7\u8d1d\u65e7\u6570\u636e\u5230\u65b0\u6570\u636e\n    for (int i = 0; i &lt; size(nums); i++)\n        extend[i] = nums-&gt;arr[i];\n\n    // \u91ca\u653e\u65e7\u6570\u636e\n    free(temp);\n\n    // \u66f4\u65b0\u65b0\u6570\u636e\n    nums-&gt;arr = extend;\n    nums-&gt;capacity = newCapacity;\n}\n\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Array \u7528\u4e8e\u6253\u5370 */\nint *toArray(MyList *nums) {\n    return nums-&gt;arr;\n}\n</code></pre> my_list.kt<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private var arr: IntArray = intArrayOf() // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private var capacity = 10 // \u5217\u8868\u5bb9\u91cf\n    private var size = 0 // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private var extendRatio = 2 // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u51fd\u6570 */\n    init {\n        arr = IntArray(capacity)\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\n    fun size(): Int {\n        return size\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    fun capacity(): Int {\n        return capacity\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    fun get(index: Int): Int {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException()\n        return arr[index]\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    fun set(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        arr[index] = num\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    fun add(num: Int) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity()\n        arr[size] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    fun insert(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity()\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (j in size - 1 downTo index)\n            arr[j + 1] = arr[j]\n        arr[index] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    fun remove(index: Int): Int {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        val num: Int = arr[index]\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (j in index..&lt;size - 1)\n            arr[j] = arr[j + 1]\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size--\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    fun extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        arr = arr.copyOf(capacity() * extendRatio)\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        capacity = arr.size\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    fun toArray(): IntArray {\n        val size = size()\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        val arr = IntArray(size)\n        for (i in 0..&lt;size) {\n            arr[i] = get(i)\n        }\n        return arr\n    }\n}\n</code></pre> my_list.rb<pre><code>### \u5217\u8868\u7c7b ###\nclass MyList\n  attr_reader :size       # \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n  attr_reader :capacity   # \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\n\n  ### \u6784\u9020\u65b9\u6cd5 ###\n  def initialize\n    @capacity = 10\n    @size = 0\n    @extend_ratio = 2\n    @arr = Array.new capacity\n  end\n\n  ### \u8bbf\u95ee\u5143\u7d20 ###\n  def get(index)\n    # \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    @arr[index]\n  end\n\n  ### \u8bbf\u95ee\u5143\u7d20 ###\n  def set(index, num)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    @arr[index] = num\n  end\n\n  ### \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 ###\n  def add(num)\n    # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    extend_capacity if size == capacity\n    @arr[size] = num\n\n    # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    @size += 1\n  end\n\n  ### \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 ###\n  def insert(index, num)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n\n    # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    extend_capacity if size == capacity\n\n    # \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for j in (size - 1).downto(index)\n      @arr[j + 1] = @arr[j] \n    end\n    @arr[index] = num\n\n    # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    @size += 1\n  end\n\n  ### \u5220\u9664\u5143\u7d20 ###\n  def remove(index)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    num = @arr[index]\n\n    # \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for j in index...size\n      @arr[j] = @arr[j + 1]\n    end\n\n    # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    @size -= 1\n\n    # \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n    num\n  end\n\n  ### \u5217\u8868\u6269\u5bb9 ###\n  def extend_capacity\n    # \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    arr = @arr.dup + Array.new(capacity * (@extend_ratio - 1))\n    # \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n    @capacity = arr.length\n  end\n\n  ### \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 ###\n  def to_array\n    sz = size\n    # \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    arr = Array.new sz\n    for i in 0...sz\n      arr[i] = get i\n    end\n    arr\n  end\nend\n</code></pre> my_list.zig<pre><code>// \u5217\u8868\u7c7b\nfn MyList(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        arr: []T = undefined,                        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n        arrCapacity: usize = 10,                     // \u5217\u8868\u5bb9\u91cf\n        numSize: usize = 0,                           // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        extendRatio: usize = 2,                       // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u5217\u8868\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.arr = try self.mem_allocator.alloc(T, self.arrCapacity);\n            @memset(self.arr, @as(T, 0));\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        pub fn size(self: *Self) usize {\n            return self.numSize;\n        }\n\n        // \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\n        pub fn capacity(self: *Self) usize {\n            return self.arrCapacity;\n        }\n\n        // \u8bbf\u95ee\u5143\u7d20\n        pub fn get(self: *Self, index: usize) T {\n            // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            return self.arr[index];\n        }  \n\n        // \u66f4\u65b0\u5143\u7d20\n        pub fn set(self: *Self, index: usize, num: T) void {\n            // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            self.arr[index] = num;\n        }  \n\n        // \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\n        pub fn add(self: *Self, num: T) !void {\n            // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n            if (self.size() == self.capacity()) try self.extendCapacity();\n            self.arr[self.size()] = num;\n            // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n            self.numSize += 1;\n        }  \n\n        // \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\n        pub fn insert(self: *Self, index: usize, num: T) !void {\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n            if (self.size() == self.capacity()) try self.extendCapacity();\n            // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n            var j = self.size() - 1;\n            while (j &gt;= index) : (j -= 1) {\n                self.arr[j + 1] = self.arr[j];\n            }\n            self.arr[index] = num;\n            // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n            self.numSize += 1;\n        }\n\n        // \u5220\u9664\u5143\u7d20\n        pub fn remove(self: *Self, index: usize) T {\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            var num = self.arr[index];\n            // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n            var j = index;\n            while (j &lt; self.size() - 1) : (j += 1) {\n                self.arr[j] = self.arr[j + 1];\n            }\n            // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n            self.numSize -= 1;\n            // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n            return num;\n        }\n\n        // \u5217\u8868\u6269\u5bb9\n        pub fn extendCapacity(self: *Self) !void {\n            // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n            var newCapacity = self.capacity() * self.extendRatio;\n            var extend = try self.mem_allocator.alloc(T, newCapacity);\n            @memset(extend, @as(T, 0));\n            // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n            std.mem.copy(T, extend, self.arr);\n            self.arr = extend;\n            // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n            self.arrCapacity = newCapacity;\n        }\n\n        // \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n            var arr = try self.mem_allocator.alloc(T, self.size());\n           @memset(arr, @as(T, 0));\n            for (arr, 0..) |*num, i| {\n                num.* = self.get(i);\n            }\n            return arr;\n        }\n    };\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/","title":"4.4 \u00a0 \u5185\u5b58\u4e0e\u7f13\u5b58 *","text":"<p>\u5728\u672c\u7ae0\u7684\u524d\u4e24\u8282\u4e2d\uff0c\u6211\u4eec\u63a2\u8ba8\u4e86\u6570\u7ec4\u548c\u94fe\u8868\u8fd9\u4e24\u79cd\u57fa\u7840\u4e14\u91cd\u8981\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5b83\u4eec\u5206\u522b\u4ee3\u8868\u4e86\u201c\u8fde\u7eed\u5b58\u50a8\u201d\u548c\u201c\u5206\u6563\u5b58\u50a8\u201d\u4e24\u79cd\u7269\u7406\u7ed3\u6784\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u7269\u7406\u7ed3\u6784\u5728\u5f88\u5927\u7a0b\u5ea6\u4e0a\u51b3\u5b9a\u4e86\u7a0b\u5e8f\u5bf9\u5185\u5b58\u548c\u7f13\u5b58\u7684\u4f7f\u7528\u6548\u7387\uff0c\u8fdb\u800c\u5f71\u54cd\u7b97\u6cd5\u7a0b\u5e8f\u7684\u6574\u4f53\u6027\u80fd\u3002</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#441","title":"4.4.1 \u00a0 \u8ba1\u7b97\u673a\u5b58\u50a8\u8bbe\u5907","text":"<p>\u8ba1\u7b97\u673a\u4e2d\u5305\u62ec\u4e09\u79cd\u7c7b\u578b\u7684\u5b58\u50a8\u8bbe\u5907\uff1a\u300c\u786c\u76d8 hard disk\u300d\u3001\u300c\u5185\u5b58 random-access memory, RAM\u300d\u3001\u300c\u7f13\u5b58 cache memory\u300d\u3002\u8868 4-2 \u5c55\u793a\u4e86\u5b83\u4eec\u5728\u8ba1\u7b97\u673a\u7cfb\u7edf\u4e2d\u7684\u4e0d\u540c\u89d2\u8272\u548c\u6027\u80fd\u7279\u70b9\u3002</p> <p> \u8868 4-2 \u00a0 \u8ba1\u7b97\u673a\u7684\u5b58\u50a8\u8bbe\u5907 </p> \u786c\u76d8 \u5185\u5b58 \u7f13\u5b58 \u7528\u9014 \u957f\u671f\u5b58\u50a8\u6570\u636e\uff0c\u5305\u62ec\u64cd\u4f5c\u7cfb\u7edf\u3001\u7a0b\u5e8f\u3001\u6587\u4ef6\u7b49 \u4e34\u65f6\u5b58\u50a8\u5f53\u524d\u8fd0\u884c\u7684\u7a0b\u5e8f\u548c\u6b63\u5728\u5904\u7406\u7684\u6570\u636e \u5b58\u50a8\u7ecf\u5e38\u8bbf\u95ee\u7684\u6570\u636e\u548c\u6307\u4ee4\uff0c\u51cf\u5c11 CPU \u8bbf\u95ee\u5185\u5b58\u7684\u6b21\u6570 \u6613\u5931\u6027 \u65ad\u7535\u540e\u6570\u636e\u4e0d\u4f1a\u4e22\u5931 \u65ad\u7535\u540e\u6570\u636e\u4f1a\u4e22\u5931 \u65ad\u7535\u540e\u6570\u636e\u4f1a\u4e22\u5931 \u5bb9\u91cf \u8f83\u5927\uff0cTB \u7ea7\u522b \u8f83\u5c0f\uff0cGB \u7ea7\u522b \u975e\u5e38\u5c0f\uff0cMB \u7ea7\u522b \u901f\u5ea6 \u8f83\u6162\uff0c\u51e0\u767e\u5230\u51e0\u5343 MB/s \u8f83\u5feb\uff0c\u51e0\u5341 GB/s \u975e\u5e38\u5feb\uff0c\u51e0\u5341\u5230\u51e0\u767e GB/s \u4ef7\u683c \u8f83\u4fbf\u5b9c\uff0c\u51e0\u6bdb\u5230\u51e0\u5143 / GB \u8f83\u8d35\uff0c\u51e0\u5341\u5230\u51e0\u767e\u5143 / GB \u975e\u5e38\u8d35\uff0c\u968f CPU \u6253\u5305\u8ba1\u4ef7 <p>\u6211\u4eec\u53ef\u4ee5\u5c06\u8ba1\u7b97\u673a\u5b58\u50a8\u7cfb\u7edf\u60f3\u8c61\u4e3a\u56fe 4-9 \u6240\u793a\u7684\u91d1\u5b57\u5854\u7ed3\u6784\u3002\u8d8a\u9760\u8fd1\u91d1\u5b57\u5854\u9876\u7aef\u7684\u5b58\u50a8\u8bbe\u5907\u7684\u901f\u5ea6\u8d8a\u5feb\u3001\u5bb9\u91cf\u8d8a\u5c0f\u3001\u6210\u672c\u8d8a\u9ad8\u3002\u8fd9\u79cd\u591a\u5c42\u7ea7\u7684\u8bbe\u8ba1\u5e76\u975e\u5076\u7136\uff0c\u800c\u662f\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6\u548c\u5de5\u7a0b\u5e08\u4eec\u7ecf\u8fc7\u6df1\u601d\u719f\u8651\u7684\u7ed3\u679c\u3002</p> <ul> <li>\u786c\u76d8\u96be\u4ee5\u88ab\u5185\u5b58\u53d6\u4ee3\u3002\u9996\u5148\uff0c\u5185\u5b58\u4e2d\u7684\u6570\u636e\u5728\u65ad\u7535\u540e\u4f1a\u4e22\u5931\uff0c\u56e0\u6b64\u5b83\u4e0d\u9002\u5408\u957f\u671f\u5b58\u50a8\u6570\u636e\uff1b\u5176\u6b21\uff0c\u5185\u5b58\u7684\u6210\u672c\u662f\u786c\u76d8\u7684\u51e0\u5341\u500d\uff0c\u8fd9\u4f7f\u5f97\u5b83\u96be\u4ee5\u5728\u6d88\u8d39\u8005\u5e02\u573a\u666e\u53ca\u3002</li> <li>\u7f13\u5b58\u7684\u5927\u5bb9\u91cf\u548c\u9ad8\u901f\u5ea6\u96be\u4ee5\u517c\u5f97\u3002\u968f\u7740 L1\u3001L2\u3001L3 \u7f13\u5b58\u7684\u5bb9\u91cf\u9010\u6b65\u589e\u5927\uff0c\u5176\u7269\u7406\u5c3a\u5bf8\u4f1a\u53d8\u5927\uff0c\u4e0e CPU \u6838\u5fc3\u4e4b\u95f4\u7684\u7269\u7406\u8ddd\u79bb\u4f1a\u53d8\u8fdc\uff0c\u4ece\u800c\u5bfc\u81f4\u6570\u636e\u4f20\u8f93\u65f6\u95f4\u589e\u52a0\uff0c\u5143\u7d20\u8bbf\u95ee\u5ef6\u8fdf\u53d8\u9ad8\u3002\u5728\u5f53\u524d\u6280\u672f\u4e0b\uff0c\u591a\u5c42\u7ea7\u7684\u7f13\u5b58\u7ed3\u6784\u662f\u5bb9\u91cf\u3001\u901f\u5ea6\u548c\u6210\u672c\u4e4b\u95f4\u7684\u6700\u4f73\u5e73\u8861\u70b9\u3002</li> </ul> <p></p> <p> \u56fe 4-9 \u00a0 \u8ba1\u7b97\u673a\u5b58\u50a8\u7cfb\u7edf </p> <p>Note</p> <p>\u8ba1\u7b97\u673a\u7684\u5b58\u50a8\u5c42\u6b21\u7ed3\u6784\u4f53\u73b0\u4e86\u901f\u5ea6\u3001\u5bb9\u91cf\u548c\u6210\u672c\u4e09\u8005\u4e4b\u95f4\u7684\u7cbe\u5999\u5e73\u8861\u3002\u5b9e\u9645\u4e0a\uff0c\u8fd9\u79cd\u6743\u8861\u666e\u904d\u5b58\u5728\u4e8e\u6240\u6709\u5de5\u4e1a\u9886\u57df\uff0c\u5b83\u8981\u6c42\u6211\u4eec\u5728\u4e0d\u540c\u7684\u4f18\u52bf\u548c\u9650\u5236\u4e4b\u95f4\u627e\u5230\u6700\u4f73\u5e73\u8861\u70b9\u3002</p> <p>\u603b\u7684\u6765\u8bf4\uff0c\u786c\u76d8\u7528\u4e8e\u957f\u671f\u5b58\u50a8\u5927\u91cf\u6570\u636e\uff0c\u5185\u5b58\u7528\u4e8e\u4e34\u65f6\u5b58\u50a8\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u6b63\u5728\u5904\u7406\u7684\u6570\u636e\uff0c\u800c\u7f13\u5b58\u5219\u7528\u4e8e\u5b58\u50a8\u7ecf\u5e38\u8bbf\u95ee\u7684\u6570\u636e\u548c\u6307\u4ee4\uff0c\u4ee5\u63d0\u9ad8\u7a0b\u5e8f\u8fd0\u884c\u6548\u7387\u3002\u4e09\u8005\u5171\u540c\u534f\u4f5c\uff0c\u786e\u4fdd\u8ba1\u7b97\u673a\u7cfb\u7edf\u9ad8\u6548\u8fd0\u884c\u3002</p> <p>\u5982\u56fe 4-10 \u6240\u793a\uff0c\u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u4f1a\u4ece\u786c\u76d8\u4e2d\u88ab\u8bfb\u53d6\u5230\u5185\u5b58\u4e2d\uff0c\u4f9b CPU \u8ba1\u7b97\u4f7f\u7528\u3002\u7f13\u5b58\u53ef\u4ee5\u770b\u4f5c CPU \u7684\u4e00\u90e8\u5206\uff0c\u5b83\u901a\u8fc7\u667a\u80fd\u5730\u4ece\u5185\u5b58\u52a0\u8f7d\u6570\u636e\uff0c\u7ed9 CPU \u63d0\u4f9b\u9ad8\u901f\u7684\u6570\u636e\u8bfb\u53d6\uff0c\u4ece\u800c\u663e\u8457\u63d0\u5347\u7a0b\u5e8f\u7684\u6267\u884c\u6548\u7387\uff0c\u51cf\u5c11\u5bf9\u8f83\u6162\u7684\u5185\u5b58\u7684\u4f9d\u8d56\u3002</p> <p></p> <p> \u56fe 4-10 \u00a0 \u786c\u76d8\u3001\u5185\u5b58\u548c\u7f13\u5b58\u4e4b\u95f4\u7684\u6570\u636e\u6d41\u901a </p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#442","title":"4.4.2 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u5185\u5b58\u6548\u7387","text":"<p>\u5728\u5185\u5b58\u7a7a\u95f4\u5229\u7528\u65b9\u9762\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u5404\u81ea\u5177\u6709\u4f18\u52bf\u548c\u5c40\u9650\u6027\u3002</p> <p>\u4e00\u65b9\u9762\uff0c\u5185\u5b58\u662f\u6709\u9650\u7684\uff0c\u4e14\u540c\u4e00\u5757\u5185\u5b58\u4e0d\u80fd\u88ab\u591a\u4e2a\u7a0b\u5e8f\u5171\u4eab\uff0c\u56e0\u6b64\u6211\u4eec\u5e0c\u671b\u6570\u636e\u7ed3\u6784\u80fd\u591f\u5c3d\u53ef\u80fd\u9ad8\u6548\u5730\u5229\u7528\u7a7a\u95f4\u3002\u6570\u7ec4\u7684\u5143\u7d20\u7d27\u5bc6\u6392\u5217\uff0c\u4e0d\u9700\u8981\u989d\u5916\u7684\u7a7a\u95f4\u6765\u5b58\u50a8\u94fe\u8868\u8282\u70b9\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u56e0\u6b64\u7a7a\u95f4\u6548\u7387\u66f4\u9ad8\u3002\u7136\u800c\uff0c\u6570\u7ec4\u9700\u8981\u4e00\u6b21\u6027\u5206\u914d\u8db3\u591f\u7684\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\uff0c\u8fd9\u53ef\u80fd\u5bfc\u81f4\u5185\u5b58\u6d6a\u8d39\uff0c\u6570\u7ec4\u6269\u5bb9\u4e5f\u9700\u8981\u989d\u5916\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u6210\u672c\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u94fe\u8868\u4ee5\u201c\u8282\u70b9\u201d\u4e3a\u5355\u4f4d\u8fdb\u884c\u52a8\u6001\u5185\u5b58\u5206\u914d\u548c\u56de\u6536\uff0c\u63d0\u4f9b\u4e86\u66f4\u5927\u7684\u7075\u6d3b\u6027\u3002</p> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u968f\u7740\u53cd\u590d\u7533\u8bf7\u4e0e\u91ca\u653e\u5185\u5b58\uff0c\u7a7a\u95f2\u5185\u5b58\u7684\u788e\u7247\u5316\u7a0b\u5ea6\u4f1a\u8d8a\u6765\u8d8a\u9ad8\uff0c\u4ece\u800c\u5bfc\u81f4\u5185\u5b58\u7684\u5229\u7528\u6548\u7387\u964d\u4f4e\u3002\u6570\u7ec4\u7531\u4e8e\u5176\u8fde\u7eed\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u76f8\u5bf9\u4e0d\u5bb9\u6613\u5bfc\u81f4\u5185\u5b58\u788e\u7247\u5316\u3002\u76f8\u53cd\uff0c\u94fe\u8868\u7684\u5143\u7d20\u662f\u5206\u6563\u5b58\u50a8\u7684\uff0c\u5728\u9891\u7e41\u7684\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u4e2d\uff0c\u66f4\u5bb9\u6613\u5bfc\u81f4\u5185\u5b58\u788e\u7247\u5316\u3002</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#443","title":"4.4.3 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u7f13\u5b58\u6548\u7387","text":"<p>\u7f13\u5b58\u867d\u7136\u5728\u7a7a\u95f4\u5bb9\u91cf\u4e0a\u8fdc\u5c0f\u4e8e\u5185\u5b58\uff0c\u4f46\u5b83\u6bd4\u5185\u5b58\u5feb\u5f97\u591a\uff0c\u5728\u7a0b\u5e8f\u6267\u884c\u901f\u5ea6\u4e0a\u8d77\u7740\u81f3\u5173\u91cd\u8981\u7684\u4f5c\u7528\u3002\u7531\u4e8e\u7f13\u5b58\u7684\u5bb9\u91cf\u6709\u9650\uff0c\u53ea\u80fd\u5b58\u50a8\u4e00\u5c0f\u90e8\u5206\u9891\u7e41\u8bbf\u95ee\u7684\u6570\u636e\uff0c\u56e0\u6b64\u5f53 CPU \u5c1d\u8bd5\u8bbf\u95ee\u7684\u6570\u636e\u4e0d\u5728\u7f13\u5b58\u4e2d\u65f6\uff0c\u5c31\u4f1a\u53d1\u751f\u300c\u7f13\u5b58\u672a\u547d\u4e2d cache miss\u300d\uff0c\u6b64\u65f6 CPU \u4e0d\u5f97\u4e0d\u4ece\u901f\u5ea6\u8f83\u6162\u7684\u5185\u5b58\u4e2d\u52a0\u8f7d\u6240\u9700\u6570\u636e\u3002</p> <p>\u663e\u7136\uff0c\u201c\u7f13\u5b58\u672a\u547d\u4e2d\u201d\u8d8a\u5c11\uff0cCPU \u8bfb\u5199\u6570\u636e\u7684\u6548\u7387\u5c31\u8d8a\u9ad8\uff0c\u7a0b\u5e8f\u6027\u80fd\u4e5f\u5c31\u8d8a\u597d\u3002\u6211\u4eec\u5c06 CPU \u4ece\u7f13\u5b58\u4e2d\u6210\u529f\u83b7\u53d6\u6570\u636e\u7684\u6bd4\u4f8b\u79f0\u4e3a\u300c\u7f13\u5b58\u547d\u4e2d\u7387 cache hit rate\u300d\uff0c\u8fd9\u4e2a\u6307\u6807\u901a\u5e38\u7528\u6765\u8861\u91cf\u7f13\u5b58\u6548\u7387\u3002</p> <p>\u4e3a\u4e86\u5c3d\u53ef\u80fd\u8fbe\u5230\u66f4\u9ad8\u7684\u6548\u7387\uff0c\u7f13\u5b58\u4f1a\u91c7\u53d6\u4ee5\u4e0b\u6570\u636e\u52a0\u8f7d\u673a\u5236\u3002</p> <ul> <li>\u7f13\u5b58\u884c\uff1a\u7f13\u5b58\u4e0d\u662f\u5355\u4e2a\u5b57\u8282\u5730\u5b58\u50a8\u4e0e\u52a0\u8f7d\u6570\u636e\uff0c\u800c\u662f\u4ee5\u7f13\u5b58\u884c\u4e3a\u5355\u4f4d\u3002\u76f8\u6bd4\u4e8e\u5355\u4e2a\u5b57\u8282\u7684\u4f20\u8f93\uff0c\u7f13\u5b58\u884c\u7684\u4f20\u8f93\u5f62\u5f0f\u66f4\u52a0\u9ad8\u6548\u3002</li> <li>\u9884\u53d6\u673a\u5236\uff1a\u5904\u7406\u5668\u4f1a\u5c1d\u8bd5\u9884\u6d4b\u6570\u636e\u8bbf\u95ee\u6a21\u5f0f\uff08\u4f8b\u5982\u987a\u5e8f\u8bbf\u95ee\u3001\u56fa\u5b9a\u6b65\u957f\u8df3\u8dc3\u8bbf\u95ee\u7b49\uff09\uff0c\u5e76\u6839\u636e\u7279\u5b9a\u6a21\u5f0f\u5c06\u6570\u636e\u52a0\u8f7d\u81f3\u7f13\u5b58\u4e4b\u4e2d\uff0c\u4ece\u800c\u63d0\u5347\u547d\u4e2d\u7387\u3002</li> <li>\u7a7a\u95f4\u5c40\u90e8\u6027\uff1a\u5982\u679c\u4e00\u4e2a\u6570\u636e\u88ab\u8bbf\u95ee\uff0c\u90a3\u4e48\u5b83\u9644\u8fd1\u7684\u6570\u636e\u53ef\u80fd\u8fd1\u671f\u4e5f\u4f1a\u88ab\u8bbf\u95ee\u3002\u56e0\u6b64\uff0c\u7f13\u5b58\u5728\u52a0\u8f7d\u67d0\u4e00\u6570\u636e\u65f6\uff0c\u4e5f\u4f1a\u52a0\u8f7d\u5176\u9644\u8fd1\u7684\u6570\u636e\uff0c\u4ee5\u63d0\u9ad8\u547d\u4e2d\u7387\u3002</li> <li>\u65f6\u95f4\u5c40\u90e8\u6027\uff1a\u5982\u679c\u4e00\u4e2a\u6570\u636e\u88ab\u8bbf\u95ee\uff0c\u90a3\u4e48\u5b83\u5728\u4e0d\u4e45\u7684\u5c06\u6765\u5f88\u53ef\u80fd\u518d\u6b21\u88ab\u8bbf\u95ee\u3002\u7f13\u5b58\u5229\u7528\u8fd9\u4e00\u539f\u7406\uff0c\u901a\u8fc7\u4fdd\u7559\u6700\u8fd1\u8bbf\u95ee\u8fc7\u7684\u6570\u636e\u6765\u63d0\u9ad8\u547d\u4e2d\u7387\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u5bf9\u7f13\u5b58\u7684\u5229\u7528\u6548\u7387\u662f\u4e0d\u540c\u7684\uff0c\u4e3b\u8981\u4f53\u73b0\u5728\u4ee5\u4e0b\u51e0\u4e2a\u65b9\u9762\u3002</p> <ul> <li>\u5360\u7528\u7a7a\u95f4\uff1a\u94fe\u8868\u5143\u7d20\u6bd4\u6570\u7ec4\u5143\u7d20\u5360\u7528\u7a7a\u95f4\u66f4\u591a\uff0c\u5bfc\u81f4\u7f13\u5b58\u4e2d\u5bb9\u7eb3\u7684\u6709\u6548\u6570\u636e\u91cf\u66f4\u5c11\u3002</li> <li>\u7f13\u5b58\u884c\uff1a\u94fe\u8868\u6570\u636e\u5206\u6563\u5728\u5185\u5b58\u5404\u5904\uff0c\u800c\u7f13\u5b58\u662f\u201c\u6309\u884c\u52a0\u8f7d\u201d\u7684\uff0c\u56e0\u6b64\u52a0\u8f7d\u5230\u65e0\u6548\u6570\u636e\u7684\u6bd4\u4f8b\u66f4\u9ad8\u3002</li> <li>\u9884\u53d6\u673a\u5236\uff1a\u6570\u7ec4\u6bd4\u94fe\u8868\u7684\u6570\u636e\u8bbf\u95ee\u6a21\u5f0f\u66f4\u5177\u201c\u53ef\u9884\u6d4b\u6027\u201d\uff0c\u5373\u7cfb\u7edf\u66f4\u5bb9\u6613\u731c\u51fa\u5373\u5c06\u88ab\u52a0\u8f7d\u7684\u6570\u636e\u3002</li> <li>\u7a7a\u95f4\u5c40\u90e8\u6027\uff1a\u6570\u7ec4\u88ab\u5b58\u50a8\u5728\u96c6\u4e2d\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u56e0\u6b64\u88ab\u52a0\u8f7d\u6570\u636e\u9644\u8fd1\u7684\u6570\u636e\u66f4\u6709\u53ef\u80fd\u5373\u5c06\u88ab\u8bbf\u95ee\u3002</li> </ul> <p>\u603b\u4f53\u800c\u8a00\uff0c\u6570\u7ec4\u5177\u6709\u66f4\u9ad8\u7684\u7f13\u5b58\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u5728\u64cd\u4f5c\u6548\u7387\u4e0a\u901a\u5e38\u4f18\u4e8e\u94fe\u8868\u3002\u8fd9\u4f7f\u5f97\u5728\u89e3\u51b3\u7b97\u6cd5\u95ee\u9898\u65f6\uff0c\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u5f80\u5f80\u66f4\u53d7\u6b22\u8fce\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u9ad8\u7f13\u5b58\u6548\u7387\u5e76\u4e0d\u610f\u5473\u7740\u6570\u7ec4\u5728\u6240\u6709\u60c5\u51b5\u4e0b\u90fd\u4f18\u4e8e\u94fe\u8868\u3002\u5b9e\u9645\u5e94\u7528\u4e2d\u9009\u62e9\u54ea\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u5e94\u6839\u636e\u5177\u4f53\u9700\u6c42\u6765\u51b3\u5b9a\u3002\u4f8b\u5982\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u90fd\u53ef\u4ee5\u5b9e\u73b0\u201c\u6808\u201d\u6570\u636e\u7ed3\u6784\uff08\u4e0b\u4e00\u7ae0\u4f1a\u8be6\u7ec6\u4ecb\u7ecd\uff09\uff0c\u4f46\u5b83\u4eec\u9002\u7528\u4e8e\u4e0d\u540c\u573a\u666f\u3002</p> <ul> <li>\u5728\u505a\u7b97\u6cd5\u9898\u65f6\uff0c\u6211\u4eec\u4f1a\u503e\u5411\u4e8e\u9009\u62e9\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\uff0c\u56e0\u4e3a\u5b83\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u64cd\u4f5c\u6548\u7387\u548c\u968f\u673a\u8bbf\u95ee\u7684\u80fd\u529b\uff0c\u4ee3\u4ef7\u4ec5\u662f\u9700\u8981\u9884\u5148\u4e3a\u6570\u7ec4\u5206\u914d\u4e00\u5b9a\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u5982\u679c\u6570\u636e\u91cf\u975e\u5e38\u5927\u3001\u52a8\u6001\u6027\u5f88\u9ad8\u3001\u6808\u7684\u9884\u671f\u5927\u5c0f\u96be\u4ee5\u4f30\u8ba1\uff0c\u90a3\u4e48\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u66f4\u52a0\u5408\u9002\u3002\u94fe\u8868\u80fd\u591f\u5c06\u5927\u91cf\u6570\u636e\u5206\u6563\u5b58\u50a8\u4e8e\u5185\u5b58\u7684\u4e0d\u540c\u90e8\u5206\uff0c\u5e76\u4e14\u907f\u514d\u4e86\u6570\u7ec4\u6269\u5bb9\u4ea7\u751f\u7684\u989d\u5916\u5f00\u9500\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/","title":"4.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_array_and_linkedlist/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u6570\u7ec4\u548c\u94fe\u8868\u662f\u4e24\u79cd\u57fa\u672c\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5206\u522b\u4ee3\u8868\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u4e24\u79cd\u5b58\u50a8\u65b9\u5f0f\uff1a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\u3002\u4e24\u8005\u7684\u7279\u70b9\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u6027\u3002</li> <li>\u6570\u7ec4\u652f\u6301\u968f\u673a\u8bbf\u95ee\u3001\u5360\u7528\u5185\u5b58\u8f83\u5c11\uff1b\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u6548\u7387\u4f4e\uff0c\u4e14\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002</li> <li>\u94fe\u8868\u901a\u8fc7\u66f4\u6539\u5f15\u7528\uff08\u6307\u9488\uff09\u5b9e\u73b0\u9ad8\u6548\u7684\u8282\u70b9\u63d2\u5165\u4e0e\u5220\u9664\uff0c\u4e14\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\uff1b\u4f46\u8282\u70b9\u8bbf\u95ee\u6548\u7387\u4f4e\u3001\u5360\u7528\u5185\u5b58\u8f83\u591a\u3002\u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u5305\u62ec\u5355\u5411\u94fe\u8868\u3001\u73af\u5f62\u94fe\u8868\u3001\u53cc\u5411\u94fe\u8868\u3002</li> <li>\u5217\u8868\u662f\u4e00\u79cd\u652f\u6301\u589e\u5220\u67e5\u6539\u7684\u5143\u7d20\u6709\u5e8f\u96c6\u5408\uff0c\u901a\u5e38\u57fa\u4e8e\u52a8\u6001\u6570\u7ec4\u5b9e\u73b0\u3002\u5b83\u4fdd\u7559\u4e86\u6570\u7ec4\u7684\u4f18\u52bf\uff0c\u540c\u65f6\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\u3002</li> <li>\u5217\u8868\u7684\u51fa\u73b0\u5927\u5e45\u63d0\u9ad8\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5bfc\u81f4\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002</li> <li>\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u4e3b\u8981\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u6570\u7ec4\u53ef\u63d0\u4f9b\u66f4\u9ad8\u7684\u5185\u5b58\u7a7a\u95f4\u6548\u7387\uff0c\u800c\u94fe\u8868\u5219\u5728\u5185\u5b58\u4f7f\u7528\u4e0a\u66f4\u52a0\u7075\u6d3b\u3002</li> <li>\u7f13\u5b58\u901a\u8fc7\u7f13\u5b58\u884c\u3001\u9884\u53d6\u673a\u5236\u4ee5\u53ca\u7a7a\u95f4\u5c40\u90e8\u6027\u548c\u65f6\u95f4\u5c40\u90e8\u6027\u7b49\u6570\u636e\u52a0\u8f7d\u673a\u5236\uff0c\u4e3a CPU \u63d0\u4f9b\u5feb\u901f\u6570\u636e\u8bbf\u95ee\uff0c\u663e\u8457\u63d0\u5347\u7a0b\u5e8f\u7684\u6267\u884c\u6548\u7387\u3002</li> <li>\u7531\u4e8e\u6570\u7ec4\u5177\u6709\u66f4\u9ad8\u7684\u7f13\u5b58\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u901a\u5e38\u6bd4\u94fe\u8868\u66f4\u9ad8\u6548\u3002\u5728\u9009\u62e9\u6570\u636e\u7ed3\u6784\u65f6\uff0c\u5e94\u6839\u636e\u5177\u4f53\u9700\u6c42\u548c\u573a\u666f\u505a\u51fa\u6070\u5f53\u9009\u62e9\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u6570\u7ec4\u5b58\u50a8\u5728\u6808\u4e0a\u548c\u5b58\u50a8\u5728\u5806\u4e0a\uff0c\u5bf9\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u5426\u6709\u5f71\u54cd\uff1f</p> <p>\u5b58\u50a8\u5728\u6808\u4e0a\u548c\u5806\u4e0a\u7684\u6570\u7ec4\u90fd\u88ab\u5b58\u50a8\u5728\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\u5185\uff0c\u6570\u636e\u64cd\u4f5c\u6548\u7387\u57fa\u672c\u4e00\u81f4\u3002\u7136\u800c\uff0c\u6808\u548c\u5806\u5177\u6709\u5404\u81ea\u7684\u7279\u70b9\uff0c\u4ece\u800c\u5bfc\u81f4\u4ee5\u4e0b\u4e0d\u540c\u70b9\u3002</p> <ol> <li>\u5206\u914d\u548c\u91ca\u653e\u6548\u7387\uff1a\u6808\u662f\u4e00\u5757\u8f83\u5c0f\u7684\u5185\u5b58\uff0c\u5206\u914d\u7531\u7f16\u8bd1\u5668\u81ea\u52a8\u5b8c\u6210\uff1b\u800c\u5806\u5185\u5b58\u76f8\u5bf9\u66f4\u5927\uff0c\u53ef\u4ee5\u5728\u4ee3\u7801\u4e2d\u52a8\u6001\u5206\u914d\uff0c\u66f4\u5bb9\u6613\u788e\u7247\u5316\u3002\u56e0\u6b64\uff0c\u5806\u4e0a\u7684\u5206\u914d\u548c\u91ca\u653e\u64cd\u4f5c\u901a\u5e38\u6bd4\u6808\u4e0a\u7684\u6162\u3002</li> <li>\u5927\u5c0f\u9650\u5236\uff1a\u6808\u5185\u5b58\u76f8\u5bf9\u8f83\u5c0f\uff0c\u5806\u7684\u5927\u5c0f\u4e00\u822c\u53d7\u9650\u4e8e\u53ef\u7528\u5185\u5b58\u3002\u56e0\u6b64\u5806\u66f4\u52a0\u9002\u5408\u5b58\u50a8\u5927\u578b\u6570\u7ec4\u3002</li> <li>\u7075\u6d3b\u6027\uff1a\u6808\u4e0a\u7684\u6570\u7ec4\u7684\u5927\u5c0f\u9700\u8981\u5728\u7f16\u8bd1\u65f6\u786e\u5b9a\uff0c\u800c\u5806\u4e0a\u7684\u6570\u7ec4\u7684\u5927\u5c0f\u53ef\u4ee5\u5728\u8fd0\u884c\u65f6\u52a8\u6001\u786e\u5b9a\u3002</li> </ol> <p>Q\uff1a\u4e3a\u4ec0\u4e48\u6570\u7ec4\u8981\u6c42\u76f8\u540c\u7c7b\u578b\u7684\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u5374\u6ca1\u6709\u5f3a\u8c03\u76f8\u540c\u7c7b\u578b\u5462\uff1f</p> <p>\u94fe\u8868\u7531\u8282\u70b9\u7ec4\u6210\uff0c\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u5f15\u7528\uff08\u6307\u9488\uff09\u8fde\u63a5\uff0c\u5404\u4e2a\u8282\u70b9\u53ef\u4ee5\u5b58\u50a8\u4e0d\u540c\u7c7b\u578b\u7684\u6570\u636e\uff0c\u4f8b\u5982 <code>int</code>\u3001<code>double</code>\u3001<code>string</code>\u3001<code>object</code> \u7b49\u3002</p> <p>\u76f8\u5bf9\u5730\uff0c\u6570\u7ec4\u5143\u7d20\u5219\u5fc5\u987b\u662f\u76f8\u540c\u7c7b\u578b\u7684\uff0c\u8fd9\u6837\u624d\u80fd\u901a\u8fc7\u8ba1\u7b97\u504f\u79fb\u91cf\u6765\u83b7\u53d6\u5bf9\u5e94\u5143\u7d20\u4f4d\u7f6e\u3002\u4f8b\u5982\uff0c\u6570\u7ec4\u540c\u65f6\u5305\u542b <code>int</code> \u548c <code>long</code> \u4e24\u79cd\u7c7b\u578b\uff0c\u5355\u4e2a\u5143\u7d20\u5206\u522b\u5360\u7528 4 \u5b57\u8282 \u548c 8 \u5b57\u8282 \uff0c\u6b64\u65f6\u5c31\u4e0d\u80fd\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u504f\u79fb\u91cf\u4e86\uff0c\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5305\u542b\u4e86\u4e24\u79cd\u201c\u5143\u7d20\u957f\u5ea6\u201d\u3002</p> <pre><code># \u5143\u7d20\u5185\u5b58\u5730\u5740 = \u6570\u7ec4\u5185\u5b58\u5730\u5740\uff08\u9996\u5143\u7d20\u5185\u5b58\u5730\u5740\uff09 + \u5143\u7d20\u957f\u5ea6 * \u5143\u7d20\u7d22\u5f15\n</code></pre> <p>Q\uff1a\u5220\u9664\u8282\u70b9 <code>P</code> \u540e\uff0c\u662f\u5426\u9700\u8981\u628a <code>P.next</code> \u8bbe\u4e3a <code>None</code> \u5462\uff1f</p> <p>\u4e0d\u4fee\u6539 <code>P.next</code> \u4e5f\u53ef\u4ee5\u3002\u4ece\u8be5\u94fe\u8868\u7684\u89d2\u5ea6\u770b\uff0c\u4ece\u5934\u8282\u70b9\u904d\u5386\u5230\u5c3e\u8282\u70b9\u5df2\u7ecf\u4e0d\u4f1a\u9047\u5230 <code>P</code> \u4e86\u3002\u8fd9\u610f\u5473\u7740\u8282\u70b9 <code>P</code> \u5df2\u7ecf\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u4e86\uff0c\u6b64\u65f6\u8282\u70b9 <code>P</code> \u6307\u5411\u54ea\u91cc\u90fd\u4e0d\u4f1a\u5bf9\u8be5\u94fe\u8868\u4ea7\u751f\u5f71\u54cd\u3002</p> <p>\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\uff08\u505a\u9898\uff09\u7684\u89d2\u5ea6\u770b\uff0c\u4e0d\u65ad\u5f00\u6ca1\u6709\u5173\u7cfb\uff0c\u53ea\u8981\u4fdd\u8bc1\u7a0b\u5e8f\u7684\u903b\u8f91\u662f\u6b63\u786e\u7684\u5c31\u884c\u3002\u4ece\u6807\u51c6\u5e93\u7684\u89d2\u5ea6\u770b\uff0c\u65ad\u5f00\u66f4\u52a0\u5b89\u5168\u3001\u903b\u8f91\u66f4\u52a0\u6e05\u6670\u3002\u5982\u679c\u4e0d\u65ad\u5f00\uff0c\u5047\u8bbe\u88ab\u5220\u9664\u8282\u70b9\u672a\u88ab\u6b63\u5e38\u56de\u6536\uff0c\u90a3\u4e48\u5b83\u4f1a\u5f71\u54cd\u540e\u7ee7\u8282\u70b9\u7684\u5185\u5b58\u56de\u6536\u3002</p> <p>Q\uff1a\u5728\u94fe\u8868\u4e2d\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u4f46\u662f\u589e\u5220\u4e4b\u524d\u90fd\u9700\u8981 \\(O(n)\\) \u7684\u65f6\u95f4\u67e5\u627e\u5143\u7d20\uff0c\u90a3\u4e3a\u4ec0\u4e48\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u662f \\(O(n)\\) \u5462\uff1f</p> <p>\u5982\u679c\u662f\u5148\u67e5\u627e\u5143\u7d20\u3001\u518d\u5220\u9664\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u786e\u5b9e\u662f \\(O(n)\\) \u3002\u7136\u800c\uff0c\u94fe\u8868\u7684 \\(O(1)\\) \u589e\u5220\u7684\u4f18\u52bf\u53ef\u4ee5\u5728\u5176\u4ed6\u5e94\u7528\u4e0a\u5f97\u5230\u4f53\u73b0\u3002\u4f8b\u5982\uff0c\u53cc\u5411\u961f\u5217\u9002\u5408\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff0c\u6211\u4eec\u7ef4\u62a4\u4e00\u4e2a\u6307\u9488\u53d8\u91cf\u59cb\u7ec8\u6307\u5411\u5934\u8282\u70b9\u3001\u5c3e\u8282\u70b9\uff0c\u6bcf\u6b21\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u90fd\u662f \\(O(1)\\) \u3002</p> <p>Q\uff1a\u56fe\u201c\u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f\u201d\u4e2d\uff0c\u6d45\u84dd\u8272\u7684\u5b58\u50a8\u8282\u70b9\u6307\u9488\u662f\u5360\u7528\u4e00\u5757\u5185\u5b58\u5730\u5740\u5417\uff1f\u8fd8\u662f\u548c\u8282\u70b9\u503c\u5404\u5360\u4e00\u534a\u5462\uff1f</p> <p>\u8be5\u793a\u610f\u56fe\u53ea\u662f\u5b9a\u6027\u8868\u793a\uff0c\u5b9a\u91cf\u8868\u793a\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u8fdb\u884c\u5206\u6790\u3002</p> <ul> <li>\u4e0d\u540c\u7c7b\u578b\u7684\u8282\u70b9\u503c\u5360\u7528\u7684\u7a7a\u95f4\u662f\u4e0d\u540c\u7684\uff0c\u6bd4\u5982 <code>int</code>\u3001<code>long</code>\u3001<code>double</code> \u548c\u5b9e\u4f8b\u5bf9\u8c61\u7b49\u3002</li> <li>\u6307\u9488\u53d8\u91cf\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u5927\u5c0f\u6839\u636e\u6240\u4f7f\u7528\u7684\u64cd\u4f5c\u7cfb\u7edf\u53ca\u7f16\u8bd1\u73af\u5883\u800c\u5b9a\uff0c\u5927\u591a\u4e3a 8 \u5b57\u8282\u6216 4 \u5b57\u8282\u3002</li> </ul> <p>Q\uff1a\u5728\u5217\u8868\u672b\u5c3e\u6dfb\u52a0\u5143\u7d20\u662f\u5426\u65f6\u65f6\u523b\u523b\u90fd\u4e3a \\(O(1)\\) \uff1f</p> <p>\u5982\u679c\u6dfb\u52a0\u5143\u7d20\u65f6\u8d85\u51fa\u5217\u8868\u957f\u5ea6\uff0c\u5219\u9700\u8981\u5148\u6269\u5bb9\u5217\u8868\u518d\u6dfb\u52a0\u3002\u7cfb\u7edf\u4f1a\u7533\u8bf7\u4e00\u5757\u65b0\u7684\u5185\u5b58\uff0c\u5e76\u5c06\u539f\u5217\u8868\u7684\u6240\u6709\u5143\u7d20\u642c\u8fd0\u8fc7\u53bb\uff0c\u8fd9\u65f6\u5019\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u4f1a\u662f \\(O(n)\\) \u3002</p> <p>Q\uff1a\u201c\u5217\u8868\u7684\u51fa\u73b0\u6781\u5927\u5730\u63d0\u9ad8\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5bfc\u81f4\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u201d\uff0c\u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u662f\u6307\u989d\u5916\u589e\u52a0\u7684\u53d8\u91cf\u5982\u5bb9\u91cf\u3001\u957f\u5ea6\u3001\u6269\u5bb9\u500d\u6570\u6240\u5360\u7684\u5185\u5b58\u5417\uff1f</p> <p>\u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u4e3b\u8981\u6709\u4e24\u65b9\u9762\u542b\u4e49\uff1a\u4e00\u65b9\u9762\uff0c\u5217\u8868\u90fd\u4f1a\u8bbe\u5b9a\u4e00\u4e2a\u521d\u59cb\u957f\u5ea6\uff0c\u6211\u4eec\u4e0d\u4e00\u5b9a\u9700\u8981\u7528\u8fd9\u4e48\u591a\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u4e3a\u4e86\u9632\u6b62\u9891\u7e41\u6269\u5bb9\uff0c\u6269\u5bb9\u4e00\u822c\u4f1a\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u6bd4\u5982 \\(\\times 1.5\\) \u3002\u8fd9\u6837\u4e00\u6765\uff0c\u4e5f\u4f1a\u51fa\u73b0\u5f88\u591a\u7a7a\u4f4d\uff0c\u6211\u4eec\u901a\u5e38\u4e0d\u80fd\u5b8c\u5168\u586b\u6ee1\u5b83\u4eec\u3002</p> <p>Q\uff1a\u5728 Python \u4e2d\u521d\u59cb\u5316 <code>n = [1, 2, 3]</code> \u540e\uff0c\u8fd9 3 \u4e2a\u5143\u7d20\u7684\u5730\u5740\u662f\u76f8\u8fde\u7684\uff0c\u4f46\u662f\u521d\u59cb\u5316 <code>m = [2, 1, 3]</code> \u4f1a\u53d1\u73b0\u5b83\u4eec\u6bcf\u4e2a\u5143\u7d20\u7684 id \u5e76\u4e0d\u662f\u8fde\u7eed\u7684\uff0c\u800c\u662f\u5206\u522b\u8ddf <code>n</code> \u4e2d\u7684\u76f8\u540c\u3002\u8fd9\u4e9b\u5143\u7d20\u7684\u5730\u5740\u4e0d\u8fde\u7eed\uff0c\u90a3\u4e48 <code>m</code> \u8fd8\u662f\u6570\u7ec4\u5417\uff1f</p> <p>\u5047\u5982\u628a\u5217\u8868\u5143\u7d20\u6362\u6210\u94fe\u8868\u8282\u70b9 <code>n = [n1, n2, n3, n4, n5]</code> \uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u8fd9 5 \u4e2a\u8282\u70b9\u5bf9\u8c61\u4e5f\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\u3002\u7136\u800c\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u5217\u8868\u7d22\u5f15\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u8282\u70b9\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u8bbf\u95ee\u5230\u5bf9\u5e94\u7684\u8282\u70b9\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5b58\u50a8\u7684\u662f\u8282\u70b9\u7684\u5f15\u7528\uff0c\u800c\u975e\u8282\u70b9\u672c\u8eab\u3002</p> <p>\u4e0e\u8bb8\u591a\u8bed\u8a00\u4e0d\u540c\uff0cPython \u4e2d\u7684\u6570\u5b57\u4e5f\u88ab\u5305\u88c5\u4e3a\u5bf9\u8c61\uff0c\u5217\u8868\u4e2d\u5b58\u50a8\u7684\u4e0d\u662f\u6570\u5b57\u672c\u8eab\uff0c\u800c\u662f\u5bf9\u6570\u5b57\u7684\u5f15\u7528\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u4f1a\u53d1\u73b0\u4e24\u4e2a\u6570\u7ec4\u4e2d\u7684\u76f8\u540c\u6570\u5b57\u62e5\u6709\u540c\u4e00\u4e2a id \uff0c\u5e76\u4e14\u8fd9\u4e9b\u6570\u5b57\u7684\u5185\u5b58\u5730\u5740\u65e0\u987b\u8fde\u7eed\u3002</p> <p>Q\uff1aC++ STL \u91cc\u9762\u7684 <code>std::list</code> \u5df2\u7ecf\u5b9e\u73b0\u4e86\u53cc\u5411\u94fe\u8868\uff0c\u4f46\u597d\u50cf\u4e00\u4e9b\u7b97\u6cd5\u4e66\u4e0a\u4e0d\u600e\u4e48\u76f4\u63a5\u4f7f\u7528\u5b83\uff0c\u662f\u4e0d\u662f\u56e0\u4e3a\u6709\u4ec0\u4e48\u5c40\u9650\u6027\u5462\uff1f</p> <p>\u4e00\u65b9\u9762\uff0c\u6211\u4eec\u5f80\u5f80\u66f4\u9752\u7750\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u7b97\u6cd5\uff0c\u800c\u53ea\u5728\u5fc5\u8981\u65f6\u624d\u4f7f\u7528\u94fe\u8868\uff0c\u4e3b\u8981\u6709\u4e24\u4e2a\u539f\u56e0\u3002</p> <ul> <li>\u7a7a\u95f4\u5f00\u9500\uff1a\u7531\u4e8e\u6bcf\u4e2a\u5143\u7d20\u9700\u8981\u4e24\u4e2a\u989d\u5916\u7684\u6307\u9488\uff08\u4e00\u4e2a\u7528\u4e8e\u524d\u4e00\u4e2a\u5143\u7d20\uff0c\u4e00\u4e2a\u7528\u4e8e\u540e\u4e00\u4e2a\u5143\u7d20\uff09\uff0c\u6240\u4ee5 <code>std::list</code> \u901a\u5e38\u6bd4 <code>std::vector</code> \u66f4\u5360\u7528\u7a7a\u95f4\u3002</li> <li>\u7f13\u5b58\u4e0d\u53cb\u597d\uff1a\u7531\u4e8e\u6570\u636e\u4e0d\u662f\u8fde\u7eed\u5b58\u653e\u7684\uff0c\u56e0\u6b64 <code>std::list</code> \u5bf9\u7f13\u5b58\u7684\u5229\u7528\u7387\u8f83\u4f4e\u3002\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c<code>std::vector</code> \u7684\u6027\u80fd\u4f1a\u66f4\u597d\u3002</li> </ul> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5fc5\u8981\u4f7f\u7528\u94fe\u8868\u7684\u60c5\u51b5\u4e3b\u8981\u662f\u4e8c\u53c9\u6811\u548c\u56fe\u3002\u6808\u548c\u961f\u5217\u5f80\u5f80\u4f1a\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684 <code>stack</code> \u548c <code>queue</code> \uff0c\u800c\u975e\u94fe\u8868\u3002</p> <p>Q\uff1a\u521d\u59cb\u5316\u5217\u8868 <code>res = [0] * self.size()</code> \u64cd\u4f5c\uff0c\u4f1a\u5bfc\u81f4 <code>res</code> \u7684\u6bcf\u4e2a\u5143\u7d20\u5f15\u7528\u76f8\u540c\u7684\u5730\u5740\u5417\uff1f</p> <p>\u4e0d\u4f1a\u3002\u4f46\u4e8c\u7ef4\u6570\u7ec4\u4f1a\u6709\u8fd9\u4e2a\u95ee\u9898\uff0c\u4f8b\u5982\u521d\u59cb\u5316\u4e8c\u7ef4\u5217\u8868 <code>res = [[0] * self.size()]</code> \uff0c\u5219\u591a\u6b21\u5f15\u7528\u4e86\u540c\u4e00\u4e2a\u5217\u8868 <code>[0]</code> \u3002</p>"},{"location":"chapter_backtracking/","title":"\u7b2c 13 \u7ae0 \u00a0 \u56de\u6eaf","text":"<p>Abstract</p> <p>\u6211\u4eec\u5982\u540c\u8ff7\u5bab\u4e2d\u7684\u63a2\u7d22\u8005\uff0c\u5728\u524d\u8fdb\u7684\u9053\u8def\u4e0a\u53ef\u80fd\u4f1a\u9047\u5230\u56f0\u96be\u3002</p> <p>\u56de\u6eaf\u7684\u529b\u91cf\u8ba9\u6211\u4eec\u80fd\u591f\u91cd\u65b0\u5f00\u59cb\uff0c\u4e0d\u65ad\u5c1d\u8bd5\uff0c\u6700\u7ec8\u627e\u5230\u901a\u5f80\u5149\u660e\u7684\u51fa\u53e3\u3002</p>"},{"location":"chapter_backtracking/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>13.1 \u00a0 \u56de\u6eaf\u7b97\u6cd5</li> <li>13.2 \u00a0 \u5168\u6392\u5217\u95ee\u9898</li> <li>13.3 \u00a0 \u5b50\u96c6\u548c\u95ee\u9898</li> <li>13.4 \u00a0 N \u7687\u540e\u95ee\u9898</li> <li>13.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_backtracking/backtracking_algorithm/","title":"13.1 \u00a0 \u56de\u6eaf\u7b97\u6cd5","text":"<p>\u300c\u56de\u6eaf\u7b97\u6cd5 backtracking algorithm\u300d\u662f\u4e00\u79cd\u901a\u8fc7\u7a77\u4e3e\u6765\u89e3\u51b3\u95ee\u9898\u7684\u65b9\u6cd5\uff0c\u5b83\u7684\u6838\u5fc3\u601d\u60f3\u662f\u4ece\u4e00\u4e2a\u521d\u59cb\u72b6\u6001\u51fa\u53d1\uff0c\u66b4\u529b\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u5f53\u9047\u5230\u6b63\u786e\u7684\u89e3\u5219\u5c06\u5176\u8bb0\u5f55\uff0c\u76f4\u5230\u627e\u5230\u89e3\u6216\u8005\u5c1d\u8bd5\u4e86\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u90fd\u65e0\u6cd5\u627e\u5230\u89e3\u4e3a\u6b62\u3002</p> <p>\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u201c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u6765\u904d\u5386\u89e3\u7a7a\u95f4\u3002\u5728\u201c\u4e8c\u53c9\u6811\u201d\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5229\u7528\u524d\u5e8f\u904d\u5386\u6784\u9020\u4e00\u4e2a\u56de\u6eaf\u95ee\u9898\uff0c\u9010\u6b65\u4e86\u89e3\u56de\u6eaf\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002</p> <p>\u4f8b\u9898\u4e00</p> <p>\u7ed9\u5b9a\u4e00\u68f5\u4e8c\u53c9\u6811\uff0c\u641c\u7d22\u5e76\u8bb0\u5f55\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u8282\u70b9\u5217\u8868\u3002</p> <p>\u5bf9\u4e8e\u6b64\u9898\uff0c\u6211\u4eec\u524d\u5e8f\u904d\u5386\u8fd9\u68f5\u6811\uff0c\u5e76\u5224\u65ad\u5f53\u524d\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \uff0c\u82e5\u662f\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u7684\u503c\u52a0\u5165\u7ed3\u679c\u5217\u8868 <code>res</code> \u4e4b\u4e2d\u3002\u76f8\u5173\u8fc7\u7a0b\u5b9e\u73b0\u5982\u56fe 13-1 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_i_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00\"\"\"\n    if root is None:\n        return\n    if root.val == 7:\n        # \u8bb0\u5f55\u89e3\n        res.append(root)\n    pre_order(root.left)\n    pre_order(root.right)\n</code></pre> preorder_traversal_i_compact.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push_back(root);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.java<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.add(root);\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.cs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.Add(root);\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.go<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrderI(root *TreeNode, res *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    if (root.Val).(int) == 7 {\n        // \u8bb0\u5f55\u89e3\n        *res = append(*res, root)\n    }\n    preOrderI(root.Left, res)\n    preOrderI(root.Right, res)\n}\n</code></pre> preorder_traversal_i_compact.swift<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    if root.val == 7 {\n        // \u8bb0\u5f55\u89e3\n        res.append(root)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n</code></pre> preorder_traversal_i_compact.js<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root, res) {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.ts<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root: TreeNode | null, res: TreeNode[]): void {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.dart<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode? root, List&lt;TreeNode&gt; res) {\n  if (root == null) {\n    return;\n  }\n  if (root.val == 7) {\n    // \u8bb0\u5f55\u89e3\n    res.add(root);\n  }\n  preOrder(root.left, res);\n  preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.rs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfn pre_order(res: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, root: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        if node.borrow().val == 7 {\n            // \u8bb0\u5f55\u89e3\n            res.push(node.clone());\n        }\n        pre_order(res, node.borrow().left.clone());\n        pre_order(res, node.borrow().right.clone());\n    }\n}\n</code></pre> preorder_traversal_i_compact.c<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res[resSize++] = root;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.kt<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    if (root.value == 7) {\n        // \u8bb0\u5f55\u89e3\n        res!!.add(root)\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n}\n</code></pre> preorder_traversal_i_compact.rb<pre><code>[class]{}-[func]{pre_order}\n</code></pre> preorder_traversal_i_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 13-1 \u00a0 \u5728\u524d\u5e8f\u904d\u5386\u4e2d\u641c\u7d22\u8282\u70b9 </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1311","title":"13.1.1 \u00a0 \u5c1d\u8bd5\u4e0e\u56de\u9000","text":"<p>\u4e4b\u6240\u4ee5\u79f0\u4e4b\u4e3a\u56de\u6eaf\u7b97\u6cd5\uff0c\u662f\u56e0\u4e3a\u8be5\u7b97\u6cd5\u5728\u641c\u7d22\u89e3\u7a7a\u95f4\u65f6\u4f1a\u91c7\u7528\u201c\u5c1d\u8bd5\u201d\u4e0e\u201c\u56de\u9000\u201d\u7684\u7b56\u7565\u3002\u5f53\u7b97\u6cd5\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u9047\u5230\u67d0\u4e2a\u72b6\u6001\u65e0\u6cd5\u7ee7\u7eed\u524d\u8fdb\u6216\u65e0\u6cd5\u5f97\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u65f6\uff0c\u5b83\u4f1a\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u5c1d\u8bd5\u5176\u4ed6\u53ef\u80fd\u7684\u9009\u62e9\u3002</p> <p>\u5bf9\u4e8e\u4f8b\u9898\u4e00\uff0c\u8bbf\u95ee\u6bcf\u4e2a\u8282\u70b9\u90fd\u4ee3\u8868\u4e00\u6b21\u201c\u5c1d\u8bd5\u201d\uff0c\u800c\u8d8a\u8fc7\u53f6\u8282\u70b9\u6216\u8fd4\u56de\u7236\u8282\u70b9\u7684 <code>return</code> \u5219\u8868\u793a\u201c\u56de\u9000\u201d\u3002</p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u56de\u9000\u5e76\u4e0d\u4ec5\u4ec5\u5305\u62ec\u51fd\u6570\u8fd4\u56de\u3002\u4e3a\u89e3\u91ca\u8fd9\u4e00\u70b9\uff0c\u6211\u4eec\u5bf9\u4f8b\u9898\u4e00\u7a0d\u4f5c\u62d3\u5c55\u3002</p> <p>\u4f8b\u9898\u4e8c</p> <p>\u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\u3002</p> <p>\u5728\u4f8b\u9898\u4e00\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u5217\u8868 <code>path</code> \u8bb0\u5f55\u8bbf\u95ee\u8fc7\u7684\u8282\u70b9\u8def\u5f84\u3002\u5f53\u8bbf\u95ee\u5230\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u590d\u5236 <code>path</code> \u5e76\u6dfb\u52a0\u8fdb\u7ed3\u679c\u5217\u8868 <code>res</code> \u3002\u904d\u5386\u5b8c\u6210\u540e\uff0c<code>res</code> \u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6240\u6709\u7684\u89e3\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_ii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c\"\"\"\n    if root is None:\n        return\n    # \u5c1d\u8bd5\n    path.append(root)\n    if root.val == 7:\n        # \u8bb0\u5f55\u89e3\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # \u56de\u9000\n    path.pop()\n</code></pre> preorder_traversal_ii_compact.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    path.pop_back();\n}\n</code></pre> preorder_traversal_ii_compact.java<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.add(root);\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // \u56de\u9000\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_ii_compact.cs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.Add(root);\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // \u56de\u9000\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_ii_compact.go<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrderII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    // \u5c1d\u8bd5\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // \u8bb0\u5f55\u89e3\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderII(root.Left, res, path)\n    preOrderII(root.Right, res, path)\n    // \u56de\u9000\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_ii_compact.swift<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u5c1d\u8bd5\n    path.append(root)\n    if root.val == 7 {\n        // \u8bb0\u5f55\u89e3\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // \u56de\u9000\n    path.removeLast()\n}\n</code></pre> preorder_traversal_ii_compact.js<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(root, path, res) {\n    if (root === null) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push(root);\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.ts<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    if (root === null) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push(root);\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.dart<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null) {\n    return;\n  }\n\n  // \u5c1d\u8bd5\n  path.add(root);\n  if (root.val == 7) {\n    // \u8bb0\u5f55\u89e3\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // \u56de\u9000\n  path.removeLast();\n}\n</code></pre> preorder_traversal_ii_compact.rs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        // \u5c1d\u8bd5\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // \u8bb0\u5f55\u89e3\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.clone());\n        pre_order(res, path, node.borrow().right.clone());\n        // \u56de\u9000\n        path.remove(path.len() - 1);\n    }\n}\n</code></pre> preorder_traversal_ii_compact.c<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        for (int i = 0; i &lt; pathSize; ++i) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    pathSize--;\n}\n</code></pre> preorder_traversal_ii_compact.kt<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    // \u5c1d\u8bd5\n    path!!.add(root)\n    if (root.value == 7) {\n        // \u8bb0\u5f55\u89e3\n        res!!.add(ArrayList(path!!))\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // \u56de\u9000\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_ii_compact.rb<pre><code>[class]{}-[func]{pre_order}\n</code></pre> preorder_traversal_ii_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5728\u6bcf\u6b21\u201c\u5c1d\u8bd5\u201d\u4e2d\uff0c\u6211\u4eec\u901a\u8fc7\u5c06\u5f53\u524d\u8282\u70b9\u6dfb\u52a0\u8fdb <code>path</code> \u6765\u8bb0\u5f55\u8def\u5f84\uff1b\u800c\u5728\u201c\u56de\u9000\u201d\u524d\uff0c\u6211\u4eec\u9700\u8981\u5c06\u8be5\u8282\u70b9\u4ece <code>path</code> \u4e2d\u5f39\u51fa\uff0c\u4ee5\u6062\u590d\u672c\u6b21\u5c1d\u8bd5\u4e4b\u524d\u7684\u72b6\u6001\u3002</p> <p>\u89c2\u5bdf\u56fe 13-2 \u6240\u793a\u7684\u8fc7\u7a0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5c1d\u8bd5\u548c\u56de\u9000\u7406\u89e3\u4e3a\u201c\u524d\u8fdb\u201d\u4e0e\u201c\u64a4\u9500\u201d\uff0c\u4e24\u4e2a\u64cd\u4f5c\u4e92\u4e3a\u9006\u5411\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 13-2 \u00a0 \u5c1d\u8bd5\u4e0e\u56de\u9000 </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1312","title":"13.1.2 \u00a0 \u526a\u679d","text":"<p>\u590d\u6742\u7684\u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u4e00\u4e2a\u6216\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u7ea6\u675f\u6761\u4ef6\u901a\u5e38\u53ef\u7528\u4e8e\u201c\u526a\u679d\u201d\u3002</p> <p>\u4f8b\u9898\u4e09</p> <p>\u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\uff0c\u5e76\u8981\u6c42\u8def\u5f84\u4e2d\u4e0d\u5305\u542b\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u3002</p> <p>\u4e3a\u4e86\u6ee1\u8db3\u4ee5\u4e0a\u7ea6\u675f\u6761\u4ef6\uff0c\u6211\u4eec\u9700\u8981\u6dfb\u52a0\u526a\u679d\u64cd\u4f5c\uff1a\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u82e5\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\uff0c\u5219\u63d0\u524d\u8fd4\u56de\uff0c\u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_iii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09\"\"\"\n    # \u526a\u679d\n    if root is None or root.val == 3:\n        return\n    # \u5c1d\u8bd5\n    path.append(root)\n    if root.val == 7:\n        # \u8bb0\u5f55\u89e3\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # \u56de\u9000\n    path.pop()\n</code></pre> preorder_traversal_iii_compact.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode *root) {\n    // \u526a\u679d\n    if (root == nullptr || root-&gt;val == 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    path.pop_back();\n}\n</code></pre> preorder_traversal_iii_compact.java<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode root) {\n    // \u526a\u679d\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.add(root);\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // \u56de\u9000\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_iii_compact.cs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid PreOrder(TreeNode? root) {\n    // \u526a\u679d\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.Add(root);\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // \u56de\u9000\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_iii_compact.go<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrderIII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    // \u526a\u679d\n    if root == nil || root.Val == 3 {\n        return\n    }\n    // \u5c1d\u8bd5\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // \u8bb0\u5f55\u89e3\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderIII(root.Left, res, path)\n    preOrderIII(root.Right, res, path)\n    // \u56de\u9000\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_iii_compact.swift<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrder(root: TreeNode?) {\n    // \u526a\u679d\n    guard let root = root, root.val != 3 else {\n        return\n    }\n    // \u5c1d\u8bd5\n    path.append(root)\n    if root.val == 7 {\n        // \u8bb0\u5f55\u89e3\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // \u56de\u9000\n    path.removeLast()\n}\n</code></pre> preorder_traversal_iii_compact.js<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(root, path, res) {\n    // \u526a\u679d\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push(root);\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.ts<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // \u526a\u679d\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push(root);\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.dart<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null || root.val == 3) {\n    return;\n  }\n\n  // \u5c1d\u8bd5\n  path.add(root);\n  if (root.val == 7) {\n    // \u8bb0\u5f55\u89e3\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // \u56de\u9000\n  path.removeLast();\n}\n</code></pre> preorder_traversal_iii_compact.rs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    // \u526a\u679d\n    if root.is_none() || root.as_ref().unwrap().borrow().val == 3 {\n        return;\n    }\n    if let Some(node) = root {\n        // \u5c1d\u8bd5\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // \u8bb0\u5f55\u89e3\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.clone());\n        pre_order(res, path, node.borrow().right.clone());\n        // \u56de\u9000\n        path.remove(path.len() - 1);\n    }\n}\n</code></pre> preorder_traversal_iii_compact.c<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode *root) {\n    // \u526a\u679d\n    if (root == NULL || root-&gt;val == 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        for (int i = 0; i &lt; pathSize; i++) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    pathSize--;\n}\n</code></pre> preorder_traversal_iii_compact.kt<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfun preOrder(root: TreeNode?) {\n    // \u526a\u679d\n    if (root == null || root.value == 3) {\n        return\n    }\n    // \u5c1d\u8bd5\n    path!!.add(root)\n    if (root.value == 7) {\n        // \u8bb0\u5f55\u89e3\n        res!!.add(ArrayList(path!!))\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // \u56de\u9000\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_iii_compact.rb<pre><code>[class]{}-[func]{pre_order}\n</code></pre> preorder_traversal_iii_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u201c\u526a\u679d\u201d\u662f\u4e00\u4e2a\u975e\u5e38\u5f62\u8c61\u7684\u540d\u8bcd\u3002\u5982\u56fe 13-3 \u6240\u793a\uff0c\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u201c\u526a\u6389\u201d\u4e86\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u641c\u7d22\u5206\u652f\uff0c\u907f\u514d\u8bb8\u591a\u65e0\u610f\u4e49\u7684\u5c1d\u8bd5\uff0c\u4ece\u800c\u63d0\u9ad8\u4e86\u641c\u7d22\u6548\u7387\u3002</p> <p></p> <p> \u56fe 13-3 \u00a0 \u6839\u636e\u7ea6\u675f\u6761\u4ef6\u526a\u679d </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1313","title":"13.1.3 \u00a0 \u6846\u67b6\u4ee3\u7801","text":"<p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c1d\u8bd5\u5c06\u56de\u6eaf\u7684\u201c\u5c1d\u8bd5\u3001\u56de\u9000\u3001\u526a\u679d\u201d\u7684\u4e3b\u4f53\u6846\u67b6\u63d0\u70bc\u51fa\u6765\uff0c\u63d0\u5347\u4ee3\u7801\u7684\u901a\u7528\u6027\u3002</p> <p>\u5728\u4ee5\u4e0b\u6846\u67b6\u4ee3\u7801\u4e2d\uff0c<code>state</code> \u8868\u793a\u95ee\u9898\u7684\u5f53\u524d\u72b6\u6001\uff0c<code>choices</code> \u8868\u793a\u5f53\u524d\u72b6\u6001\u4e0b\u53ef\u4ee5\u505a\u51fa\u7684\u9009\u62e9\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def backtrack(state: State, choices: list[choice], res: list[state]):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\"\"\"\n    # \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if is_solution(state):\n        # \u8bb0\u5f55\u89e3\n        record_solution(state, res)\n        # \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices:\n        # \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice):\n            # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            make_choice(state, choice)\n            backtrack(state, choices, res)\n            # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undo_choice(state, choice)\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, vector&lt;Choice *&gt; &amp;choices, vector&lt;State *&gt; &amp;res) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (Choice choice : choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (Choice choice : choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid Backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (IsSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        RecordSolution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    foreach (Choice choice in choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (IsValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            MakeChoice(state, choice);\n            Backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state *State, choices []Choice, res *[]State) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if isSolution(state) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res)\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for _, choice := range choices {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if isValid(state, choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state: inout State, choices: [Choice], res: inout [State]) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if isSolution(state: state) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state: state, res: &amp;res)\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if isValid(state: state, choice: choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state: &amp;state, choice: choice)\n            backtrack(state: &amp;state, choices: choices, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state, choices, res) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (let choice of choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state: State, choices: Choice[], res: State[]): void {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (let choice of choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List&lt;Choice&gt;, List&lt;State&gt; res) {\n  // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n  if (isSolution(state)) {\n    // \u8bb0\u5f55\u89e3\n    recordSolution(state, res);\n    // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  for (Choice choice in choices) {\n    // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n    if (isValid(state, choice)) {\n      // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n      makeChoice(state, choice);\n      backtrack(state, choices, res);\n      // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfn backtrack(state: &amp;mut State, choices: &amp;Vec&lt;Choice&gt;, res: &amp;mut Vec&lt;State&gt;) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if is_solution(state) {\n        // \u8bb0\u5f55\u89e3\n        record_solution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            make_choice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undo_choice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res, numRes);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; numChoices; i++) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, &amp;choices[i])) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, &amp;choices[i]);\n            backtrack(state, choices, numChoices, res, numRes);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, &amp;choices[i]);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfun backtrack(state: State?, choices: List&lt;Choice?&gt;, res: List&lt;State?&gt;?) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res)\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (choice in choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u57fa\u4e8e\u6846\u67b6\u4ee3\u7801\u6765\u89e3\u51b3\u4f8b\u9898\u4e09\u3002\u72b6\u6001 <code>state</code> \u4e3a\u8282\u70b9\u904d\u5386\u8def\u5f84\uff0c\u9009\u62e9 <code>choices</code> \u4e3a\u5f53\u524d\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\uff0c\u7ed3\u679c <code>res</code> \u662f\u8def\u5f84\u5217\u8868\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_iii_template.py<pre><code>def is_solution(state: list[TreeNode]) -&gt; bool:\n    \"\"\"\u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3\"\"\"\n    return state and state[-1].val == 7\n\ndef record_solution(state: list[TreeNode], res: list[list[TreeNode]]):\n    \"\"\"\u8bb0\u5f55\u89e3\"\"\"\n    res.append(list(state))\n\ndef is_valid(state: list[TreeNode], choice: TreeNode) -&gt; bool:\n    \"\"\"\u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5\"\"\"\n    return choice is not None and choice.val != 3\n\ndef make_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"\u66f4\u65b0\u72b6\u6001\"\"\"\n    state.append(choice)\n\ndef undo_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"\u6062\u590d\u72b6\u6001\"\"\"\n    state.pop()\n\ndef backtrack(\n    state: list[TreeNode], choices: list[TreeNode], res: list[list[TreeNode]]\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09\"\"\"\n    # \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if is_solution(state):\n        # \u8bb0\u5f55\u89e3\n        record_solution(state, res)\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices:\n        # \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice):\n            # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            make_choice(state, choice)\n            # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, [choice.left, choice.right], res)\n            # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undo_choice(state, choice)\n</code></pre> preorder_traversal_iii_template.cpp<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(vector&lt;TreeNode *&gt; &amp;state) {\n    return !state.empty() &amp;&amp; state.back()-&gt;val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(vector&lt;TreeNode *&gt; &amp;state, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    res.push_back(state);\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    return choice != nullptr &amp;&amp; choice-&gt;val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.push_back(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.pop_back();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(vector&lt;TreeNode *&gt; &amp;state, vector&lt;TreeNode *&gt; &amp;choices, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (TreeNode *choice : choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            vector&lt;TreeNode *&gt; nextChoices{choice-&gt;left, choice-&gt;right};\n            backtrack(state, nextChoices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.java<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nboolean isSolution(List&lt;TreeNode&gt; state) {\n    return !state.isEmpty() &amp;&amp; state.get(state.size() - 1).val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.add(new ArrayList&lt;&gt;(state));\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nboolean isValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.add(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.remove(state.size() - 1);\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (TreeNode choice : choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, Arrays.asList(choice.left, choice.right), res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.cs<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool IsSolution(List&lt;TreeNode&gt; state) {\n    return state.Count != 0 &amp;&amp; state[^1].val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid RecordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.Add(new List&lt;TreeNode&gt;(state));\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool IsValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid MakeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.Add(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid UndoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.RemoveAt(state.Count - 1);\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid Backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (IsSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        RecordSolution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    foreach (TreeNode choice in choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (IsValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            MakeChoice(state, choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            Backtrack(state, [choice.left!, choice.right!], res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.go<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state *[]*TreeNode) bool {\n    return len(*state) != 0 &amp;&amp; (*state)[len(*state)-1].Val == 7\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state *[]*TreeNode, res *[][]*TreeNode) {\n    *res = append(*res, append([]*TreeNode{}, *state...))\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state *[]*TreeNode, choice *TreeNode) bool {\n    return choice != nil &amp;&amp; choice.Val != 3\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = append(*state, choice)\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = (*state)[:len(*state)-1]\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrackIII(state *[]*TreeNode, choices *[]*TreeNode, res *[][]*TreeNode) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if isSolution(state) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res)\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for _, choice := range *choices {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if isValid(state, choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            temp := make([]*TreeNode, 0)\n            temp = append(temp, choice.Left, choice.Right)\n            backtrackIII(state, &amp;temp, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.swift<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state: [TreeNode]) -&gt; Bool {\n    !state.isEmpty &amp;&amp; state.last!.val == 7\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state: [TreeNode], res: inout [[TreeNode]]) {\n    res.append(state)\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state: [TreeNode], choice: TreeNode?) -&gt; Bool {\n    choice != nil &amp;&amp; choice!.val != 3\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.append(choice)\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.removeLast()\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrack(state: inout [TreeNode], choices: [TreeNode], res: inout [[TreeNode]]) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if isSolution(state: state) {\n        recordSolution(state: state, res: &amp;res)\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if isValid(state: state, choice: choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state: &amp;state, choice: choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state: &amp;state, choices: [choice.left, choice.right].compactMap { $0 }, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.js<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state) {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state, res) {\n    res.push([...state]);\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state, choice) {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state, choice) {\n    state.push(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state) {\n    state.pop();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(state, choices, res) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, [choice.left, choice.right], res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.ts<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state: TreeNode[]): boolean {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state: TreeNode[], res: TreeNode[][]): void {\n    res.push([...state]);\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state: TreeNode[], choice: TreeNode): boolean {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state: TreeNode[], choice: TreeNode): void {\n    state.push(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state: TreeNode[]): void {\n    state.pop();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(\n    state: TreeNode[],\n    choices: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, [choice.left, choice.right], res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.dart<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(List&lt;TreeNode&gt; state) {\n  return state.isNotEmpty &amp;&amp; state.last.val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n  res.add(List.from(state));\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.add(choice!);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.removeLast();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(\n  List&lt;TreeNode&gt; state,\n  List&lt;TreeNode?&gt; choices,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n  if (isSolution(state)) {\n    // \u8bb0\u5f55\u89e3\n    recordSolution(state, res);\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  for (TreeNode? choice in choices) {\n    // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n    if (isValid(state, choice)) {\n      // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n      makeChoice(state, choice);\n      // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n      backtrack(state, [choice!.left, choice.right], res);\n      // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> preorder_traversal_iii_template.rs<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfn is_solution(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\n    return !state.is_empty() &amp;&amp; state.get(state.len() - 1).unwrap().borrow().val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfn record_solution(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    res.push(state.clone());\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfn is_valid(_: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; bool {\n    return choice.borrow().val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfn make_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.push(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfn undo_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, _: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.remove(state.len() - 1);\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfn backtrack(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    choices: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if is_solution(state) {\n        // \u8bb0\u5f55\u89e3\n        record_solution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice.clone()) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            make_choice(state, choice.clone());\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(\n                state,\n                &amp;mut vec![\n                    choice.borrow().left.clone().unwrap(),\n                    choice.borrow().right.clone().unwrap(),\n                ],\n                res,\n            );\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undo_choice(state, choice.clone());\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.c<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(void) {\n    return pathSize &gt; 0 &amp;&amp; path[pathSize - 1]-&gt;val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(void) {\n    for (int i = 0; i &lt; pathSize; i++) {\n        res[resSize][i] = path[i];\n    }\n    resSize++;\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(TreeNode *choice) {\n    return choice != NULL &amp;&amp; choice-&gt;val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(TreeNode *choice) {\n    path[pathSize++] = choice;\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(void) {\n    pathSize--;\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(TreeNode *choices[2]) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution()) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution();\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; 2; i++) {\n        TreeNode *choice = choices[i];\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            TreeNode *nextChoices[2] = {choice-&gt;left, choice-&gt;right};\n            backtrack(nextChoices);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice();\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.kt<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfun isSolution(state: List&lt;TreeNode?&gt;): Boolean {\n    return state.isNotEmpty() &amp;&amp; state[state.size - 1]?.value == 7\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfun recordSolution(state: MutableList&lt;TreeNode?&gt;?, res: MutableList&lt;List&lt;TreeNode?&gt;?&gt;) {\n    res.add(state?.let { ArrayList(it) })\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfun isValid(state: List&lt;TreeNode?&gt;?, choice: TreeNode?): Boolean {\n    return choice != null &amp;&amp; choice.value != 3\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfun makeChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.add(choice)\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfun undoChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.removeLast()\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfun backtrack(\n    state: MutableList&lt;TreeNode?&gt;,\n    choices: List&lt;TreeNode?&gt;,\n    res: MutableList&lt;List&lt;TreeNode?&gt;?&gt;\n) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res)\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (choice in choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, listOf(choice!!.left, choice.right), res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.rb<pre><code>[class]{}-[func]{is_solution}\n\n[class]{}-[func]{record_solution}\n\n[class]{}-[func]{is_valid}\n\n[class]{}-[func]{make_choice}\n\n[class]{}-[func]{undo_choice}\n\n[class]{}-[func]{backtrack}\n</code></pre> preorder_traversal_iii_template.zig<pre><code>[class]{}-[func]{isSolution}\n\n[class]{}-[func]{recordSolution}\n\n[class]{}-[func]{isValid}\n\n[class]{}-[func]{makeChoice}\n\n[class]{}-[func]{undoChoice}\n\n[class]{}-[func]{backtrack}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u6839\u636e\u9898\u610f\uff0c\u6211\u4eec\u5728\u627e\u5230\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\u540e\u5e94\u8be5\u7ee7\u7eed\u641c\u7d22\uff0c\u56e0\u6b64\u9700\u8981\u5c06\u8bb0\u5f55\u89e3\u4e4b\u540e\u7684 <code>return</code> \u8bed\u53e5\u5220\u9664\u3002\u56fe 13-4 \u5bf9\u6bd4\u4e86\u4fdd\u7559\u6216\u5220\u9664 <code>return</code> \u8bed\u53e5\u7684\u641c\u7d22\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 13-4 \u00a0 \u4fdd\u7559\u4e0e\u5220\u9664 return \u7684\u641c\u7d22\u8fc7\u7a0b\u5bf9\u6bd4 </p> <p>\u76f8\u6bd4\u57fa\u4e8e\u524d\u5e8f\u904d\u5386\u7684\u4ee3\u7801\u5b9e\u73b0\uff0c\u57fa\u4e8e\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\u7684\u4ee3\u7801\u5b9e\u73b0\u867d\u7136\u663e\u5f97\u5570\u5506\uff0c\u4f46\u901a\u7528\u6027\u66f4\u597d\u3002\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u56de\u6eaf\u95ee\u9898\u53ef\u4ee5\u5728\u8be5\u6846\u67b6\u4e0b\u89e3\u51b3\u3002\u6211\u4eec\u53ea\u9700\u6839\u636e\u5177\u4f53\u95ee\u9898\u6765\u5b9a\u4e49 <code>state</code> \u548c <code>choices</code> \uff0c\u5e76\u5b9e\u73b0\u6846\u67b6\u4e2d\u7684\u5404\u4e2a\u65b9\u6cd5\u5373\u53ef\u3002</p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1314","title":"13.1.4 \u00a0 \u5e38\u7528\u672f\u8bed","text":"<p>\u4e3a\u4e86\u66f4\u6e05\u6670\u5730\u5206\u6790\u7b97\u6cd5\u95ee\u9898\uff0c\u6211\u4eec\u603b\u7ed3\u4e00\u4e0b\u56de\u6eaf\u7b97\u6cd5\u4e2d\u5e38\u7528\u672f\u8bed\u7684\u542b\u4e49\uff0c\u5e76\u5bf9\u7167\u4f8b\u9898\u4e09\u7ed9\u51fa\u5bf9\u5e94\u793a\u4f8b\uff0c\u5982\u8868 13-1 \u6240\u793a\u3002</p> <p> \u8868 13-1 \u00a0 \u5e38\u89c1\u7684\u56de\u6eaf\u7b97\u6cd5\u672f\u8bed </p> \u540d\u8bcd \u5b9a\u4e49 \u4f8b\u9898\u4e09 \u89e3\uff08solution\uff09 \u89e3\u662f\u6ee1\u8db3\u95ee\u9898\u7279\u5b9a\u6761\u4ef6\u7684\u7b54\u6848\uff0c\u53ef\u80fd\u6709\u4e00\u4e2a\u6216\u591a\u4e2a \u6839\u8282\u70b9\u5230\u8282\u70b9 \\(7\\) \u7684\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u6240\u6709\u8def\u5f84 \u7ea6\u675f\u6761\u4ef6\uff08constraint\uff09 \u7ea6\u675f\u6761\u4ef6\u662f\u95ee\u9898\u4e2d\u9650\u5236\u89e3\u7684\u53ef\u884c\u6027\u7684\u6761\u4ef6\uff0c\u901a\u5e38\u7528\u4e8e\u526a\u679d \u8def\u5f84\u4e2d\u4e0d\u5305\u542b\u8282\u70b9 \\(3\\) \u72b6\u6001\uff08state\uff09 \u72b6\u6001\u8868\u793a\u95ee\u9898\u5728\u67d0\u4e00\u65f6\u523b\u7684\u60c5\u51b5\uff0c\u5305\u62ec\u5df2\u7ecf\u505a\u51fa\u7684\u9009\u62e9 \u5f53\u524d\u5df2\u8bbf\u95ee\u7684\u8282\u70b9\u8def\u5f84\uff0c\u5373 <code>path</code> \u8282\u70b9\u5217\u8868 \u5c1d\u8bd5\uff08attempt\uff09 \u5c1d\u8bd5\u662f\u6839\u636e\u53ef\u7528\u9009\u62e9\u6765\u63a2\u7d22\u89e3\u7a7a\u95f4\u7684\u8fc7\u7a0b\uff0c\u5305\u62ec\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\uff0c\u68c0\u67e5\u662f\u5426\u4e3a\u89e3 \u9012\u5f52\u8bbf\u95ee\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\uff0c\u5c06\u8282\u70b9\u6dfb\u52a0\u8fdb <code>path</code> \uff0c\u5224\u65ad\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \u56de\u9000\uff08backtracking\uff09 \u56de\u9000\u6307\u9047\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u72b6\u6001\u65f6\uff0c\u64a4\u9500\u524d\u9762\u505a\u51fa\u7684\u9009\u62e9\uff0c\u56de\u5230\u4e0a\u4e00\u4e2a\u72b6\u6001 \u5f53\u8d8a\u8fc7\u53f6\u8282\u70b9\u3001\u7ed3\u675f\u8282\u70b9\u8bbf\u95ee\u3001\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\u7ec8\u6b62\u641c\u7d22\uff0c\u51fd\u6570\u8fd4\u56de \u526a\u679d\uff08pruning\uff09 \u526a\u679d\u662f\u6839\u636e\u95ee\u9898\u7279\u6027\u548c\u7ea6\u675f\u6761\u4ef6\u907f\u514d\u65e0\u610f\u4e49\u7684\u641c\u7d22\u8def\u5f84\u7684\u65b9\u6cd5\uff0c\u53ef\u63d0\u9ad8\u641c\u7d22\u6548\u7387 \u5f53\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u4e0d\u518d\u7ee7\u7eed\u641c\u7d22 <p>Tip</p> <p>\u95ee\u9898\u3001\u89e3\u3001\u72b6\u6001\u7b49\u6982\u5ff5\u662f\u901a\u7528\u7684\uff0c\u5728\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u8d2a\u5fc3\u7b49\u7b97\u6cd5\u4e2d\u90fd\u6709\u6d89\u53ca\u3002</p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1315","title":"13.1.5 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u4e0a\u662f\u4e00\u79cd\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u7b97\u6cd5\uff0c\u5b83\u5c1d\u8bd5\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\u76f4\u5230\u627e\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u3002\u8fd9\u79cd\u65b9\u6cd5\u7684\u4f18\u70b9\u5728\u4e8e\u80fd\u591f\u627e\u5230\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u800c\u4e14\u5728\u5408\u7406\u7684\u526a\u679d\u64cd\u4f5c\u4e0b\uff0c\u5177\u6709\u5f88\u9ad8\u7684\u6548\u7387\u3002</p> <p>\u7136\u800c\uff0c\u5728\u5904\u7406\u5927\u89c4\u6a21\u6216\u8005\u590d\u6742\u95ee\u9898\u65f6\uff0c\u56de\u6eaf\u7b97\u6cd5\u7684\u8fd0\u884c\u6548\u7387\u53ef\u80fd\u96be\u4ee5\u63a5\u53d7\u3002</p> <ul> <li>\u65f6\u95f4\uff1a\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u9700\u8981\u904d\u5386\u72b6\u6001\u7a7a\u95f4\u7684\u6240\u6709\u53ef\u80fd\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230\u6307\u6570\u9636\u6216\u9636\u4e58\u9636\u3002</li> <li>\u7a7a\u95f4\uff1a\u5728\u9012\u5f52\u8c03\u7528\u4e2d\u9700\u8981\u4fdd\u5b58\u5f53\u524d\u7684\u72b6\u6001\uff08\u4f8b\u5982\u8def\u5f84\u3001\u7528\u4e8e\u526a\u679d\u7684\u8f85\u52a9\u53d8\u91cf\u7b49\uff09\uff0c\u5f53\u6df1\u5ea6\u5f88\u5927\u65f6\uff0c\u7a7a\u95f4\u9700\u6c42\u53ef\u80fd\u4f1a\u53d8\u5f97\u5f88\u5927\u3002</li> </ul> <p>\u5373\u4fbf\u5982\u6b64\uff0c\u56de\u6eaf\u7b97\u6cd5\u4ecd\u7136\u662f\u67d0\u4e9b\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u7684\u6700\u4f73\u89e3\u51b3\u65b9\u6848\u3002\u5bf9\u4e8e\u8fd9\u4e9b\u95ee\u9898\uff0c\u7531\u4e8e\u65e0\u6cd5\u9884\u6d4b\u54ea\u4e9b\u9009\u62e9\u53ef\u751f\u6210\u6709\u6548\u7684\u89e3\uff0c\u56e0\u6b64\u6211\u4eec\u5fc5\u987b\u5bf9\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u8fdb\u884c\u904d\u5386\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5173\u952e\u662f\u5982\u4f55\u4f18\u5316\u6548\u7387\uff0c\u5e38\u89c1\u7684\u6548\u7387\u4f18\u5316\u65b9\u6cd5\u6709\u4e24\u79cd\u3002</p> <ul> <li>\u526a\u679d\uff1a\u907f\u514d\u641c\u7d22\u90a3\u4e9b\u80af\u5b9a\u4e0d\u4f1a\u4ea7\u751f\u89e3\u7684\u8def\u5f84\uff0c\u4ece\u800c\u8282\u7701\u65f6\u95f4\u548c\u7a7a\u95f4\u3002</li> <li>\u542f\u53d1\u5f0f\u641c\u7d22\uff1a\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u5f15\u5165\u4e00\u4e9b\u7b56\u7565\u6216\u8005\u4f30\u8ba1\u503c\uff0c\u4ece\u800c\u4f18\u5148\u641c\u7d22\u6700\u6709\u53ef\u80fd\u4ea7\u751f\u6709\u6548\u89e3\u7684\u8def\u5f84\u3002</li> </ul>"},{"location":"chapter_backtracking/backtracking_algorithm/#1316","title":"13.1.6 \u00a0 \u56de\u6eaf\u5178\u578b\u4f8b\u9898","text":"<p>\u56de\u6eaf\u7b97\u6cd5\u53ef\u7528\u4e8e\u89e3\u51b3\u8bb8\u591a\u641c\u7d22\u95ee\u9898\u3001\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u548c\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u3002</p> <p>\u641c\u7d22\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u89e3\u51b3\u65b9\u6848\u3002</p> <ul> <li>\u5168\u6392\u5217\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff0c\u6c42\u51fa\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u7ec4\u5408\u3002</li> <li>\u5b50\u96c6\u548c\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\u548c\u4e00\u4e2a\u76ee\u6807\u548c\uff0c\u627e\u5230\u96c6\u5408\u4e2d\u6240\u6709\u548c\u4e3a\u76ee\u6807\u548c\u7684\u5b50\u96c6\u3002</li> <li>\u6c49\u8bfa\u5854\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e09\u6839\u67f1\u5b50\u548c\u4e00\u7cfb\u5217\u5927\u5c0f\u4e0d\u540c\u7684\u5706\u76d8\uff0c\u8981\u6c42\u5c06\u6240\u6709\u5706\u76d8\u4ece\u4e00\u6839\u67f1\u5b50\u79fb\u52a8\u5230\u53e6\u4e00\u6839\u67f1\u5b50\uff0c\u6bcf\u6b21\u53ea\u80fd\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\uff0c\u4e14\u4e0d\u80fd\u5c06\u5927\u5706\u76d8\u653e\u5728\u5c0f\u5706\u76d8\u4e0a\u3002</li> </ul> <p>\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u6240\u6709\u7ea6\u675f\u6761\u4ef6\u7684\u89e3\u3002</p> <ul> <li>\\(n\\) \u7687\u540e\uff1a\u5728 \\(n \\times n\\) \u7684\u68cb\u76d8\u4e0a\u653e\u7f6e \\(n\\) \u4e2a\u7687\u540e\uff0c\u4f7f\u5f97\u5b83\u4eec\u4e92\u4e0d\u653b\u51fb\u3002</li> <li>\u6570\u72ec\uff1a\u5728 \\(9 \\times 9\\) \u7684\u7f51\u683c\u4e2d\u586b\u5165\u6570\u5b57 \\(1\\) ~ \\(9\\) \uff0c\u4f7f\u5f97\u6bcf\u884c\u3001\u6bcf\u5217\u548c\u6bcf\u4e2a \\(3 \\times 3\\) \u5b50\u7f51\u683c\u4e2d\u7684\u6570\u5b57\u4e0d\u91cd\u590d\u3002</li> <li>\u56fe\u7740\u8272\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u7528\u6700\u5c11\u7684\u989c\u8272\u7ed9\u56fe\u7684\u6bcf\u4e2a\u9876\u70b9\u7740\u8272\uff0c\u4f7f\u5f97\u76f8\u90bb\u9876\u70b9\u989c\u8272\u4e0d\u540c\u3002</li> </ul> <p>\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u4e00\u4e2a\u7ec4\u5408\u7a7a\u95f4\u4e2d\u627e\u5230\u6ee1\u8db3\u67d0\u4e9b\u6761\u4ef6\u7684\u6700\u4f18\u89e3\u3002</p> <ul> <li>0-1 \u80cc\u5305\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u7269\u54c1\u548c\u4e00\u4e2a\u80cc\u5305\uff0c\u6bcf\u4e2a\u7269\u54c1\u6709\u4e00\u5b9a\u7684\u4ef7\u503c\u548c\u91cd\u91cf\uff0c\u8981\u6c42\u5728\u80cc\u5305\u5bb9\u91cf\u9650\u5236\u5185\uff0c\u9009\u62e9\u7269\u54c1\u4f7f\u5f97\u603b\u4ef7\u503c\u6700\u5927\u3002</li> <li>\u65c5\u884c\u5546\u95ee\u9898\uff1a\u5728\u4e00\u4e2a\u56fe\u4e2d\uff0c\u4ece\u4e00\u4e2a\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u6240\u6709\u5176\u4ed6\u70b9\u6070\u597d\u4e00\u6b21\u540e\u8fd4\u56de\u8d77\u70b9\uff0c\u6c42\u6700\u77ed\u8def\u5f84\u3002</li> <li>\u6700\u5927\u56e2\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u627e\u5230\u6700\u5927\u7684\u5b8c\u5168\u5b50\u56fe\uff0c\u5373\u5b50\u56fe\u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u90fd\u6709\u8fb9\u76f8\u8fde\u3002</li> </ul> <p>\u8bf7\u6ce8\u610f\uff0c\u5bf9\u4e8e\u8bb8\u591a\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff0c\u56de\u6eaf\u4e0d\u662f\u6700\u4f18\u89e3\u51b3\u65b9\u6848\u3002</p> <ul> <li>0-1 \u80cc\u5305\u95ee\u9898\u901a\u5e38\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u89e3\u51b3\uff0c\u4ee5\u8fbe\u5230\u66f4\u9ad8\u7684\u65f6\u95f4\u6548\u7387\u3002</li> <li>\u65c5\u884c\u5546\u662f\u4e00\u4e2a\u8457\u540d\u7684 NP-Hard \u95ee\u9898\uff0c\u5e38\u7528\u89e3\u6cd5\u6709\u9057\u4f20\u7b97\u6cd5\u548c\u8681\u7fa4\u7b97\u6cd5\u7b49\u3002</li> <li>\u6700\u5927\u56e2\u95ee\u9898\u662f\u56fe\u8bba\u4e2d\u7684\u4e00\u4e2a\u7ecf\u5178\u95ee\u9898\uff0c\u53ef\u7528\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u542f\u53d1\u5f0f\u7b97\u6cd5\u6765\u89e3\u51b3\u3002</li> </ul>"},{"location":"chapter_backtracking/n_queens_problem/","title":"13.4 \u00a0 n \u7687\u540e\u95ee\u9898","text":"<p>Question</p> <p>\u6839\u636e\u56fd\u9645\u8c61\u68cb\u7684\u89c4\u5219\uff0c\u7687\u540e\u53ef\u4ee5\u653b\u51fb\u4e0e\u540c\u5904\u4e00\u884c\u3001\u4e00\u5217\u6216\u4e00\u6761\u659c\u7ebf\u4e0a\u7684\u68cb\u5b50\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u7687\u540e\u548c\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5bfb\u627e\u4f7f\u5f97\u6240\u6709\u7687\u540e\u4e4b\u95f4\u65e0\u6cd5\u76f8\u4e92\u653b\u51fb\u7684\u6446\u653e\u65b9\u6848\u3002</p> <p>\u5982\u56fe 13-15 \u6240\u793a\uff0c\u5f53 \\(n = 4\\) \u65f6\uff0c\u5171\u53ef\u4ee5\u627e\u5230\u4e24\u4e2a\u89e3\u3002\u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\u5171\u6709 \\(n^2\\) \u4e2a\u683c\u5b50\uff0c\u7ed9\u51fa\u4e86\u6240\u6709\u7684\u9009\u62e9 <code>choices</code> \u3002\u5728\u9010\u4e2a\u653e\u7f6e\u7687\u540e\u7684\u8fc7\u7a0b\u4e2d\uff0c\u68cb\u76d8\u72b6\u6001\u5728\u4e0d\u65ad\u5730\u53d8\u5316\uff0c\u6bcf\u4e2a\u65f6\u523b\u7684\u68cb\u76d8\u5c31\u662f\u72b6\u6001 <code>state</code> \u3002</p> <p></p> <p> \u56fe 13-15 \u00a0 4 \u7687\u540e\u95ee\u9898\u7684\u89e3 </p> <p>\u56fe 13-16 \u5c55\u793a\u4e86\u672c\u9898\u7684\u4e09\u4e2a\u7ea6\u675f\u6761\u4ef6\uff1a\u591a\u4e2a\u7687\u540e\u4e0d\u80fd\u5728\u540c\u4e00\u884c\u3001\u540c\u4e00\u5217\u3001\u540c\u4e00\u6761\u5bf9\u89d2\u7ebf\u4e0a\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5bf9\u89d2\u7ebf\u5206\u4e3a\u4e3b\u5bf9\u89d2\u7ebf <code>\\</code> \u548c\u6b21\u5bf9\u89d2\u7ebf <code>/</code> \u4e24\u79cd\u3002</p> <p></p> <p> \u56fe 13-16 \u00a0 n \u7687\u540e\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6 </p>"},{"location":"chapter_backtracking/n_queens_problem/#1","title":"1. \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565","text":"<p>\u7687\u540e\u7684\u6570\u91cf\u548c\u68cb\u76d8\u7684\u884c\u6570\u90fd\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u6211\u4eec\u5bb9\u6613\u5f97\u5230\u4e00\u4e2a\u63a8\u8bba\uff1a\u68cb\u76d8\u6bcf\u884c\u90fd\u5141\u8bb8\u4e14\u53ea\u5141\u8bb8\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002</p> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u53d6\u9010\u884c\u653e\u7f6e\u7b56\u7565\uff1a\u4ece\u7b2c\u4e00\u884c\u5f00\u59cb\uff0c\u5728\u6bcf\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\uff0c\u76f4\u81f3\u6700\u540e\u4e00\u884c\u7ed3\u675f\u3002</p> <p>\u56fe 13-17 \u6240\u793a\u4e3a \\(4\\) \u7687\u540e\u95ee\u9898\u7684\u9010\u884c\u653e\u7f6e\u8fc7\u7a0b\u3002\u53d7\u753b\u5e45\u9650\u5236\uff0c\u56fe 13-17 \u4ec5\u5c55\u5f00\u4e86\u7b2c\u4e00\u884c\u7684\u5176\u4e2d\u4e00\u4e2a\u641c\u7d22\u5206\u652f\uff0c\u5e76\u4e14\u5c06\u4e0d\u6ee1\u8db3\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u65b9\u6848\u90fd\u8fdb\u884c\u4e86\u526a\u679d\u3002</p> <p></p> <p> \u56fe 13-17 \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565 </p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9010\u884c\u653e\u7f6e\u7b56\u7565\u8d77\u5230\u4e86\u526a\u679d\u7684\u4f5c\u7528\uff0c\u5b83\u907f\u514d\u4e86\u540c\u4e00\u884c\u51fa\u73b0\u591a\u4e2a\u7687\u540e\u7684\u6240\u6709\u641c\u7d22\u5206\u652f\u3002</p>"},{"location":"chapter_backtracking/n_queens_problem/#2","title":"2. \u00a0 \u5217\u4e0e\u5bf9\u89d2\u7ebf\u526a\u679d","text":"<p>\u4e3a\u4e86\u6ee1\u8db3\u5217\u7ea6\u675f\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5e03\u5c14\u578b\u6570\u7ec4 <code>cols</code> \u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\u3002\u5728\u6bcf\u6b21\u51b3\u5b9a\u653e\u7f6e\u524d\uff0c\u6211\u4eec\u901a\u8fc7 <code>cols</code> \u5c06\u5df2\u6709\u7687\u540e\u7684\u5217\u8fdb\u884c\u526a\u679d\uff0c\u5e76\u5728\u56de\u6eaf\u4e2d\u52a8\u6001\u66f4\u65b0 <code>cols</code> \u7684\u72b6\u6001\u3002</p> <p>\u90a3\u4e48\uff0c\u5982\u4f55\u5904\u7406\u5bf9\u89d2\u7ebf\u7ea6\u675f\u5462\uff1f\u8bbe\u68cb\u76d8\u4e2d\u67d0\u4e2a\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u4e3a \\((row, col)\\) \uff0c\u9009\u5b9a\u77e9\u9635\u4e2d\u7684\u67d0\u6761\u4e3b\u5bf9\u89d2\u7ebf\uff0c\u6211\u4eec\u53d1\u73b0\u8be5\u5bf9\u89d2\u7ebf\u4e0a\u6240\u6709\u683c\u5b50\u7684\u884c\u7d22\u5f15\u51cf\u5217\u7d22\u5f15\u90fd\u76f8\u7b49\uff0c\u5373\u5bf9\u89d2\u7ebf\u4e0a\u6240\u6709\u683c\u5b50\u7684 \\(row - col\\) \u4e3a\u6052\u5b9a\u503c\u3002</p> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u4e24\u4e2a\u683c\u5b50\u6ee1\u8db3 \\(row_1 - col_1 = row_2 - col_2\\) \uff0c\u5219\u5b83\u4eec\u4e00\u5b9a\u5904\u5728\u540c\u4e00\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u3002\u5229\u7528\u8be5\u89c4\u5f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u501f\u52a9\u56fe 13-18 \u6240\u793a\u7684\u6570\u7ec4 <code>diags1</code> \u8bb0\u5f55\u6bcf\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\u3002</p> <p>\u540c\u7406\uff0c\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u7684\u6240\u6709\u683c\u5b50\u7684 \\(row + col\\) \u662f\u6052\u5b9a\u503c\u3002\u6211\u4eec\u540c\u6837\u4e5f\u53ef\u4ee5\u501f\u52a9\u6570\u7ec4 <code>diags2</code> \u6765\u5904\u7406\u6b21\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002</p> <p></p> <p> \u56fe 13-18 \u00a0 \u5904\u7406\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f </p>"},{"location":"chapter_backtracking/n_queens_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u8bf7\u6ce8\u610f\uff0c\\(n\\) \u7ef4\u65b9\u9635\u4e2d \\(row - col\\) \u7684\u8303\u56f4\u662f \\([-n + 1, n - 1]\\) \uff0c\\(row + col\\) \u7684\u8303\u56f4\u662f \\([0, 2n - 2]\\) \uff0c\u6240\u4ee5\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\u7684\u6570\u91cf\u90fd\u4e3a \\(2n - 1\\) \uff0c\u5373\u6570\u7ec4 <code>diags1</code> \u548c <code>diags2</code> \u7684\u957f\u5ea6\u90fd\u4e3a \\(2n - 1\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig n_queens.py<pre><code>def backtrack(\n    row: int,\n    n: int,\n    state: list[list[str]],\n    res: list[list[list[str]]],\n    cols: list[bool],\n    diags1: list[bool],\n    diags2: list[bool],\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e\"\"\"\n    # \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if row == n:\n        res.append([list(row) for row in state])\n        return\n    # \u904d\u5386\u6240\u6709\u5217\n    for col in range(n):\n        # \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        diag1 = row - col + n - 1\n        diag2 = row + col\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if not cols[col] and not diags1[diag1] and not diags2[diag2]:\n            # \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = \"Q\"\n            cols[col] = diags1[diag1] = diags2[diag2] = True\n            # \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            # \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = \"#\"\n            cols[col] = diags1[diag1] = diags2[diag2] = False\n\ndef n_queens(n: int) -&gt; list[list[list[str]]]:\n    \"\"\"\u6c42\u89e3 n \u7687\u540e\"\"\"\n    # \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    state = [[\"#\" for _ in range(n)] for _ in range(n)]\n    cols = [False] * n  # \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    diags1 = [False] * (2 * n - 1)  # \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    diags2 = [False] * (2 * n - 1)  # \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    res = []\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n</code></pre> n_queens.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(int row, int n, vector&lt;vector&lt;string&gt;&gt; &amp;state, vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; &amp;res, vector&lt;bool&gt; &amp;cols,\n               vector&lt;bool&gt; &amp;diags1, vector&lt;bool&gt; &amp;diags2) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row == n) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nvector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; nQueens(int n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    vector&lt;vector&lt;string&gt;&gt; state(n, vector&lt;string&gt;(n, \"#\"));\n    vector&lt;bool&gt; cols(n, false);           // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    vector&lt;bool&gt; diags1(2 * n - 1, false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    vector&lt;bool&gt; diags2(2 * n - 1, false); // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; res;\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(int row, int n, List&lt;List&lt;String&gt;&gt; state, List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n        boolean[] cols, boolean[] diags1, boolean[] diags2) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row == n) {\n        List&lt;List&lt;String&gt;&gt; copyState = new ArrayList&lt;&gt;();\n        for (List&lt;String&gt; sRow : state) {\n            copyState.add(new ArrayList&lt;&gt;(sRow));\n        }\n        res.add(copyState);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state.get(row).set(col, \"Q\");\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state.get(row).set(col, \"#\");\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    List&lt;List&lt;String&gt;&gt; state = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;String&gt; row = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            row.add(\"#\");\n        }\n        state.add(row);\n    }\n    boolean[] cols = new boolean[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    boolean[] diags1 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    boolean[] diags2 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = new ArrayList&lt;&gt;();\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid Backtrack(int row, int n, List&lt;List&lt;string&gt;&gt; state, List&lt;List&lt;List&lt;string&gt;&gt;&gt; res,\n        bool[] cols, bool[] diags1, bool[] diags2) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row == n) {\n        List&lt;List&lt;string&gt;&gt; copyState = [];\n        foreach (List&lt;string&gt; sRow in state) {\n            copyState.Add(new List&lt;string&gt;(sRow));\n        }\n        res.Add(copyState);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            Backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nList&lt;List&lt;List&lt;string&gt;&gt;&gt; NQueens(int n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    List&lt;List&lt;string&gt;&gt; state = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;string&gt; row = [];\n        for (int j = 0; j &lt; n; j++) {\n            row.Add(\"#\");\n        }\n        state.Add(row);\n    }\n    bool[] cols = new bool[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    bool[] diags1 = new bool[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    bool[] diags2 = new bool[2 * n - 1]; // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    List&lt;List&lt;List&lt;string&gt;&gt;&gt; res = [];\n\n    Backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if row == n {\n        newState := make([][]string, len(*state))\n        for i, _ := range newState {\n            newState[i] = make([]string, len((*state)[0]))\n            copy(newState[i], (*state)[i])\n\n        }\n        *res = append(*res, newState)\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for col := 0; col &lt; n; col++ {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        diag1 := row - col + n - 1\n        diag2 := row + col\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if !(*cols)[col] &amp;&amp; !(*diags1)[diag1] &amp;&amp; !(*diags2)[diag2] {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            (*state)[row][col] = \"Q\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row+1, n, state, res, cols, diags1, diags2)\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            (*state)[row][col] = \"#\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunc nQueens(n int) [][][]string {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    state := make([][]string, n)\n    for i := 0; i &lt; n; i++ {\n        row := make([]string, n)\n        for i := 0; i &lt; n; i++ {\n            row[i] = \"#\"\n        }\n        state[i] = row\n    }\n    // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    cols := make([]bool, n)\n    diags1 := make([]bool, 2*n-1)\n    diags2 := make([]bool, 2*n-1)\n    res := make([][][]string, 0)\n    backtrack(0, n, &amp;state, &amp;res, &amp;cols, &amp;diags1, &amp;diags2)\n    return res\n}\n</code></pre> n_queens.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nfunc backtrack(row: Int, n: Int, state: inout [[String]], res: inout [[[String]]], cols: inout [Bool], diags1: inout [Bool], diags2: inout [Bool]) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if row == n {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for col in 0 ..&lt; n {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        let diag1 = row - col + n - 1\n        let diag2 = row + col\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = \"Q\"\n            cols[col] = true\n            diags1[diag1] = true\n            diags2[diag2] = true\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row: row + 1, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = \"#\"\n            cols[col] = false\n            diags1[diag1] = false\n            diags2[diag2] = false\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunc nQueens(n: Int) -&gt; [[[String]]] {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    var state = Array(repeating: Array(repeating: \"#\", count: n), count: n)\n    var cols = Array(repeating: false, count: n) // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    var diags1 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    var diags2 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    var res: [[[String]]] = []\n\n    backtrack(row: 0, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n\n    return res\n}\n</code></pre> n_queens.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nfunction backtrack(row, n, state, res, cols, diags1, diags2) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (let col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunction nQueens(n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    const diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const res = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nfunction backtrack(\n    row: number,\n    n: number,\n    state: string[][],\n    res: string[][][],\n    cols: boolean[],\n    diags1: boolean[],\n    diags2: boolean[]\n): void {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (let col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunction nQueens(n: number): string[][][] {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    const diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const res: string[][][] = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(\n  int row,\n  int n,\n  List&lt;List&lt;String&gt;&gt; state,\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n  List&lt;bool&gt; cols,\n  List&lt;bool&gt; diags1,\n  List&lt;bool&gt; diags2,\n) {\n  // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (row == n) {\n    List&lt;List&lt;String&gt;&gt; copyState = [];\n    for (List&lt;String&gt; sRow in state) {\n      copyState.add(List.from(sRow));\n    }\n    res.add(copyState);\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u5217\n  for (int col = 0; col &lt; n; col++) {\n    // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n    int diag1 = row - col + n - 1;\n    int diag2 = row + col;\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n    if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n      // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n      state[row][col] = \"Q\";\n      cols[col] = true;\n      diags1[diag1] = true;\n      diags2[diag2] = true;\n      // \u653e\u7f6e\u4e0b\u4e00\u884c\n      backtrack(row + 1, n, state, res, cols, diags1, diags2);\n      // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n      state[row][col] = \"#\";\n      cols[col] = false;\n      diags1[diag1] = false;\n      diags2[diag2] = false;\n    }\n  }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n  // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n  List&lt;List&lt;String&gt;&gt; state = List.generate(n, (index) =&gt; List.filled(n, \"#\"));\n  List&lt;bool&gt; cols = List.filled(n, false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n  List&lt;bool&gt; diags1 = List.filled(2 * n - 1, false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n  List&lt;bool&gt; diags2 = List.filled(2 * n - 1, false); // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = [];\n\n  backtrack(0, n, state, res, cols, diags1, diags2);\n\n  return res;\n}\n</code></pre> n_queens.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nfn backtrack(\n    row: usize,\n    n: usize,\n    state: &amp;mut Vec&lt;Vec&lt;String&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt;,\n    cols: &amp;mut [bool],\n    diags1: &amp;mut [bool],\n    diags2: &amp;mut [bool],\n) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if row == n {\n        let mut copy_state: Vec&lt;Vec&lt;String&gt;&gt; = Vec::new();\n        for s_row in state.clone() {\n            copy_state.push(s_row);\n        }\n        res.push(copy_state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for col in 0..n {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        let diag1 = row + n - 1 - col;\n        let diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state.get_mut(row).unwrap()[col] = \"Q\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (true, true, true);\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state.get_mut(row).unwrap()[col] = \"#\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (false, false, false);\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfn n_queens(n: usize) -&gt; Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    let mut state: Vec&lt;Vec&lt;String&gt;&gt; = Vec::new();\n    for _ in 0..n {\n        let mut row: Vec&lt;String&gt; = Vec::new();\n        for _ in 0..n {\n            row.push(\"#\".into());\n        }\n        state.push(row);\n    }\n    let mut cols = vec![false; n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    let mut diags1 = vec![false; 2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    let mut diags2 = vec![false; 2 * n - 1]; // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    let mut res: Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; = Vec::new();\n\n    backtrack(\n        0,\n        n,\n        &amp;mut state,\n        &amp;mut res,\n        &amp;mut cols,\n        &amp;mut diags1,\n        &amp;mut diags2,\n    );\n\n    res\n}\n</code></pre> n_queens.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(int row, int n, char state[MAX_SIZE][MAX_SIZE], char ***res, int *resSize, bool cols[MAX_SIZE],\n               bool diags1[2 * MAX_SIZE - 1], bool diags2[2 * MAX_SIZE - 1]) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row == n) {\n        res[*resSize] = (char **)malloc(sizeof(char *) * n);\n        for (int i = 0; i &lt; n; ++i) {\n            res[*resSize][i] = (char *)malloc(sizeof(char) * (n + 1));\n            strcpy(res[*resSize][i], state[i]);\n        }\n        (*resSize)++;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, resSize, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nchar ***nQueens(int n, int *returnSize) {\n    char state[MAX_SIZE][MAX_SIZE];\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    for (int i = 0; i &lt; n; ++i) {\n        for (int j = 0; j &lt; n; ++j) {\n            state[i][j] = '#';\n        }\n        state[i][n] = '\\0';\n    }\n    bool cols[MAX_SIZE] = {false};           // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    bool diags1[2 * MAX_SIZE - 1] = {false}; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    bool diags2[2 * MAX_SIZE - 1] = {false}; // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n\n    char ***res = (char ***)malloc(sizeof(char **) * MAX_SIZE);\n    *returnSize = 0;\n    backtrack(0, n, state, res, returnSize, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.kt<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nfun backtrack(\n    row: Int,\n    n: Int,\n    state: List&lt;MutableList&lt;String&gt;&gt;,\n    res: MutableList&lt;List&lt;List&lt;String&gt;&gt;?&gt;,\n    cols: BooleanArray,\n    diags1: BooleanArray,\n    diags2: BooleanArray\n) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row == n) {\n        val copyState: MutableList&lt;List&lt;String&gt;&gt; = ArrayList()\n        for (sRow in state) {\n            copyState.add(ArrayList(sRow))\n        }\n        res.add(copyState)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (col in 0..&lt;n) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        val diag1 = row - col + n - 1\n        val diag2 = row + col\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = \"Q\"\n            diags2[diag2] = true\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = \"#\"\n            diags2[diag2] = false\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfun nQueens(n: Int): List&lt;List&lt;List&lt;String&gt;&gt;?&gt; {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    val state: MutableList&lt;MutableList&lt;String&gt;&gt; = ArrayList()\n    for (i in 0..&lt;n) {\n        val row: MutableList&lt;String&gt; = ArrayList()\n        for (j in 0..&lt;n) {\n            row.add(\"#\")\n        }\n        state.add(row)\n    }\n    val cols = BooleanArray(n) // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    val diags1 = BooleanArray(2 * n - 1) // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    val diags2 = BooleanArray(2 * n - 1) // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    val res: MutableList&lt;List&lt;List&lt;String&gt;&gt;?&gt; = ArrayList()\n\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n}\n</code></pre> n_queens.rb<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{n_queens}\n</code></pre> n_queens.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{nQueens}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u9010\u884c\u653e\u7f6e \\(n\\) \u6b21\uff0c\u8003\u8651\u5217\u7ea6\u675f\uff0c\u5219\u4ece\u7b2c\u4e00\u884c\u5230\u6700\u540e\u4e00\u884c\u5206\u522b\u6709 \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u4e2a\u9009\u62e9\uff0c\u4f7f\u7528 \\(O(n!)\\) \u65f6\u95f4\u3002\u5f53\u8bb0\u5f55\u89e3\u65f6\uff0c\u9700\u8981\u590d\u5236\u77e9\u9635 <code>state</code> \u5e76\u6dfb\u52a0\u8fdb <code>res</code> \uff0c\u590d\u5236\u64cd\u4f5c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002\u56e0\u6b64\uff0c\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n! \\cdot n^2)\\) \u3002\u5b9e\u9645\u4e0a\uff0c\u6839\u636e\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u526a\u679d\u4e5f\u80fd\u591f\u5927\u5e45\u7f29\u5c0f\u641c\u7d22\u7a7a\u95f4\uff0c\u56e0\u800c\u641c\u7d22\u6548\u7387\u5f80\u5f80\u4f18\u4e8e\u4ee5\u4e0a\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <p>\u6570\u7ec4 <code>state</code> \u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\uff0c\u6570\u7ec4 <code>cols</code>\u3001<code>diags1</code> \u548c <code>diags2</code> \u7686\u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/","title":"13.2 \u00a0 \u5168\u6392\u5217\u95ee\u9898","text":"<p>\u5168\u6392\u5217\u95ee\u9898\u662f\u56de\u6eaf\u7b97\u6cd5\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u7684\u5b9a\u4e49\u662f\u5728\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff08\u5982\u4e00\u4e2a\u6570\u7ec4\u6216\u5b57\u7b26\u4e32\uff09\u7684\u60c5\u51b5\u4e0b\uff0c\u627e\u51fa\u5176\u4e2d\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002</p> <p>\u8868 13-2 \u5217\u4e3e\u4e86\u51e0\u4e2a\u793a\u4f8b\u6570\u636e\uff0c\u5305\u62ec\u8f93\u5165\u6570\u7ec4\u548c\u5bf9\u5e94\u7684\u6240\u6709\u6392\u5217\u3002</p> <p> \u8868 13-2 \u00a0 \u5168\u6392\u5217\u793a\u4f8b </p> \u8f93\u5165\u6570\u7ec4 \u6240\u6709\u6392\u5217 \\([1]\\) \\([1]\\) \\([1, 2]\\) \\([1, 2], [2, 1]\\) \\([1, 2, 3]\\) \\([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\\)"},{"location":"chapter_backtracking/permutations_problem/#1321","title":"13.2.1 \u00a0 \u65e0\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u5176\u4e2d\u4e0d\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002</p> <p>\u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u751f\u6210\u6392\u5217\u7684\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\u3002\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 2, 3]\\) \uff0c\u5982\u679c\u6211\u4eec\u5148\u9009\u62e9 \\(1\\) \uff0c\u518d\u9009\u62e9 \\(3\\) \uff0c\u6700\u540e\u9009\u62e9 \\(2\\) \uff0c\u5219\u83b7\u5f97\u6392\u5217 \\([1, 3, 2]\\) \u3002\u56de\u9000\u8868\u793a\u64a4\u9500\u4e00\u4e2a\u9009\u62e9\uff0c\u4e4b\u540e\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002</p> <p>\u4ece\u56de\u6eaf\u4ee3\u7801\u7684\u89d2\u5ea6\u770b\uff0c\u5019\u9009\u96c6\u5408 <code>choices</code> \u662f\u8f93\u5165\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u72b6\u6001 <code>state</code> \u662f\u76f4\u81f3\u76ee\u524d\u5df2\u88ab\u9009\u62e9\u7684\u5143\u7d20\u3002\u8bf7\u6ce8\u610f\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u5141\u8bb8\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u56e0\u6b64 <code>state</code> \u4e2d\u7684\u6240\u6709\u5143\u7d20\u90fd\u5e94\u8be5\u662f\u552f\u4e00\u7684\u3002</p> <p>\u5982\u56fe 13-5 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u641c\u7d22\u8fc7\u7a0b\u5c55\u5f00\u6210\u4e00\u68f5\u9012\u5f52\u6811\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u5f53\u524d\u72b6\u6001 <code>state</code> \u3002\u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u7ecf\u8fc7\u4e09\u8f6e\u9009\u62e9\u540e\u5230\u8fbe\u53f6\u8282\u70b9\uff0c\u6bcf\u4e2a\u53f6\u8282\u70b9\u90fd\u5bf9\u5e94\u4e00\u4e2a\u6392\u5217\u3002</p> <p></p> <p> \u56fe 13-5 \u00a0 \u5168\u6392\u5217\u7684\u9012\u5f52\u6811 </p>"},{"location":"chapter_backtracking/permutations_problem/#1","title":"1. \u00a0 \u91cd\u590d\u9009\u62e9\u526a\u679d","text":"<p>\u4e3a\u4e86\u5b9e\u73b0\u6bcf\u4e2a\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u6211\u4eec\u8003\u8651\u5f15\u5165\u4e00\u4e2a\u5e03\u5c14\u578b\u6570\u7ec4 <code>selected</code> \uff0c\u5176\u4e2d <code>selected[i]</code> \u8868\u793a <code>choices[i]</code> \u662f\u5426\u5df2\u88ab\u9009\u62e9\uff0c\u5e76\u57fa\u4e8e\u5b83\u5b9e\u73b0\u4ee5\u4e0b\u526a\u679d\u64cd\u4f5c\u3002</p> <ul> <li>\u5728\u505a\u51fa\u9009\u62e9 <code>choice[i]</code> \u540e\uff0c\u6211\u4eec\u5c31\u5c06 <code>selected[i]</code> \u8d4b\u503c\u4e3a \\(\\text{True}\\) \uff0c\u4ee3\u8868\u5b83\u5df2\u88ab\u9009\u62e9\u3002</li> <li>\u904d\u5386\u9009\u62e9\u5217\u8868 <code>choices</code> \u65f6\uff0c\u8df3\u8fc7\u6240\u6709\u5df2\u88ab\u9009\u62e9\u7684\u8282\u70b9\uff0c\u5373\u526a\u679d\u3002</li> </ul> <p>\u5982\u56fe 13-6 \u6240\u793a\uff0c\u5047\u8bbe\u6211\u4eec\u7b2c\u4e00\u8f6e\u9009\u62e9 1 \uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9 3 \uff0c\u7b2c\u4e09\u8f6e\u9009\u62e9 2 \uff0c\u5219\u9700\u8981\u5728\u7b2c\u4e8c\u8f6e\u526a\u6389\u5143\u7d20 1 \u7684\u5206\u652f\uff0c\u5728\u7b2c\u4e09\u8f6e\u526a\u6389\u5143\u7d20 1 \u548c\u5143\u7d20 3 \u7684\u5206\u652f\u3002</p> <p></p> <p> \u56fe 13-6 \u00a0 \u5168\u6392\u5217\u526a\u679d\u793a\u4f8b </p> <p>\u89c2\u5bdf\u56fe 13-6 \u53d1\u73b0\uff0c\u8be5\u526a\u679d\u64cd\u4f5c\u5c06\u641c\u7d22\u7a7a\u95f4\u5927\u5c0f\u4ece \\(O(n^n)\\) \u51cf\u5c0f\u81f3 \\(O(n!)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u60f3\u6e05\u695a\u4ee5\u4e0a\u4fe1\u606f\u4e4b\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5728\u6846\u67b6\u4ee3\u7801\u4e2d\u505a\u201c\u5b8c\u5f62\u586b\u7a7a\u201d\u4e86\u3002\u4e3a\u4e86\u7f29\u77ed\u6574\u4f53\u4ee3\u7801\uff0c\u6211\u4eec\u4e0d\u5355\u72ec\u5b9e\u73b0\u6846\u67b6\u4ee3\u7801\u4e2d\u7684\u5404\u4e2a\u51fd\u6570\uff0c\u800c\u662f\u5c06\u5b83\u4eec\u5c55\u5f00\u5728 <code>backtrack()</code> \u51fd\u6570\u4e2d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig permutations_i.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I\"\"\"\n    # \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i, choice in enumerate(choices):\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if not selected[i]:\n            # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = True\n            state.append(choice)\n            # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res)\n            # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = False\n            state.pop()\n\ndef permutations_i(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"\u5168\u6392\u5217 I\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_i.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.push_back(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nvector&lt;vector&lt;int&gt;&gt; permutationsI(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_i.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.add(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;Integer&gt;&gt; permutationsI(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_i.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.Add(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            Backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;int&gt;&gt; PermutationsI(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_i.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrackI(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if !(*selected)[i] {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrackI(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackI(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_i.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (i, choice) in choices.enumerated() {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if !selected[i] {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true\n            state.append(choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_i.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(state, choices, selected, res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n    if (!selected[i]) {\n      // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n      selected[i] = true;\n      state.add(choice);\n      // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n      backtrack(state, choices, selected, res);\n      // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;int&gt;&gt; permutationsI(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_i.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if !selected[i] {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state.clone(), choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.remove(state.len() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfn permutations_i(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_i.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state[stateSize] = choice;\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nint **permutationsI(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_i.kt<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;List&lt;Int&gt;?&gt;\n) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size == choices.size) {\n        res.add(ArrayList(state))\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true\n            state.add(choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfun permutationsI(nums: IntArray): List&lt;List&lt;Int&gt;?&gt; {\n    val res: MutableList&lt;List&lt;Int&gt;?&gt; = ArrayList()\n    backtrack(ArrayList(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_i.rb<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutations_i}\n</code></pre> permutations_i.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutationsI}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_backtracking/permutations_problem/#1322","title":"13.2.2 \u00a0 \u8003\u8651\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u6570\u7ec4\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u4e0d\u91cd\u590d\u7684\u6392\u5217\u3002</p> <p>\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 1, 2]\\) \u3002\u4e3a\u4e86\u65b9\u4fbf\u533a\u5206\u4e24\u4e2a\u91cd\u590d\u5143\u7d20 \\(1\\) \uff0c\u6211\u4eec\u5c06\u7b2c\u4e8c\u4e2a \\(1\\) \u8bb0\u4e3a \\(\\hat{1}\\) \u3002</p> <p>\u5982\u56fe 13-7 \u6240\u793a\uff0c\u4e0a\u8ff0\u65b9\u6cd5\u751f\u6210\u7684\u6392\u5217\u6709\u4e00\u534a\u662f\u91cd\u590d\u7684\u3002</p> <p></p> <p> \u56fe 13-7 \u00a0 \u91cd\u590d\u6392\u5217 </p> <p>\u90a3\u4e48\u5982\u4f55\u53bb\u9664\u91cd\u590d\u7684\u6392\u5217\u5462\uff1f\u6700\u76f4\u63a5\u5730\uff0c\u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u76f4\u63a5\u5bf9\u6392\u5217\u7ed3\u679c\u8fdb\u884c\u53bb\u91cd\u3002\u7136\u800c\u8fd9\u6837\u505a\u4e0d\u591f\u4f18\u96c5\uff0c\u56e0\u4e3a\u751f\u6210\u91cd\u590d\u6392\u5217\u7684\u641c\u7d22\u5206\u652f\u6ca1\u6709\u5fc5\u8981\uff0c\u5e94\u5f53\u63d0\u524d\u8bc6\u522b\u5e76\u526a\u679d\uff0c\u8fd9\u6837\u53ef\u4ee5\u8fdb\u4e00\u6b65\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"<p>\u89c2\u5bdf\u56fe 13-8 \uff0c\u5728\u7b2c\u4e00\u8f6e\u4e2d\uff0c\u9009\u62e9 \\(1\\) \u6216\u9009\u62e9 \\(\\hat{1}\\) \u662f\u7b49\u4ef7\u7684\uff0c\u5728\u8fd9\u4e24\u4e2a\u9009\u62e9\u4e4b\u4e0b\u751f\u6210\u7684\u6240\u6709\u6392\u5217\u90fd\u662f\u91cd\u590d\u7684\u3002\u56e0\u6b64\u5e94\u8be5\u628a \\(\\hat{1}\\) \u526a\u679d\u3002</p> <p>\u540c\u7406\uff0c\u5728\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(2\\) \u4e4b\u540e\uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9\u4e2d\u7684 \\(1\\) \u548c \\(\\hat{1}\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5206\u652f\uff0c\u56e0\u6b64\u4e5f\u5e94\u5c06\u7b2c\u4e8c\u8f6e\u7684 \\(\\hat{1}\\) \u526a\u679d\u3002</p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u5728\u67d0\u4e00\u8f6e\u9009\u62e9\u4e2d\uff0c\u4fdd\u8bc1\u591a\u4e2a\u76f8\u7b49\u7684\u5143\u7d20\u4ec5\u88ab\u9009\u62e9\u4e00\u6b21\u3002</p> <p></p> <p> \u56fe 13-8 \u00a0 \u91cd\u590d\u6392\u5217\u526a\u679d </p>"},{"location":"chapter_backtracking/permutations_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5728\u4e0a\u4e00\u9898\u7684\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u8003\u8651\u5728\u6bcf\u4e00\u8f6e\u9009\u62e9\u4e2d\u5f00\u542f\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>duplicated</code> \uff0c\u7528\u4e8e\u8bb0\u5f55\u8be5\u8f6e\u4e2d\u5df2\u7ecf\u5c1d\u8bd5\u8fc7\u7684\u5143\u7d20\uff0c\u5e76\u5c06\u91cd\u590d\u5143\u7d20\u526a\u679d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig permutations_ii.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II\"\"\"\n    # \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    duplicated = set[int]()\n    for i, choice in enumerate(choices):\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if not selected[i] and choice not in duplicated:\n            # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.add(choice)  # \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = True\n            state.append(choice)\n            # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res)\n            # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = False\n            state.pop()\n\ndef permutations_ii(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"\u5168\u6392\u5217 II\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_ii.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    unordered_set&lt;int&gt; duplicated;\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; duplicated.find(choice) == duplicated.end()) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.emplace(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push_back(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nvector&lt;vector&lt;int&gt;&gt; permutationsII(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_ii.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    Set&lt;Integer&gt; duplicated = new HashSet&lt;Integer&gt;();\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.add(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;Integer&gt;&gt; permutationsII(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_ii.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    HashSet&lt;int&gt; duplicated = [];\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.Contains(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.Add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.Add(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            Backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;int&gt;&gt; PermutationsII(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_ii.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrackII(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    duplicated := make(map[int]struct{}, 0)\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if _, ok := duplicated[choice]; !ok &amp;&amp; !(*selected)[i] {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            duplicated[choice] = struct{}{}\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrackI(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackII(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_ii.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    var duplicated: Set&lt;Int&gt; = []\n    for (i, choice) in choices.enumerated() {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if !selected[i], !duplicated.contains(choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.insert(choice) // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true\n            state.append(choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_ii.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(state, choices, selected, res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  Set&lt;int&gt; duplicated = {};\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n    if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n      // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n      duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n      selected[i] = true;\n      state.add(choice);\n      // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n      backtrack(state, choices, selected, res);\n      // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;int&gt;&gt; permutationsII(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_ii.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    let mut duplicated = HashSet::&lt;i32&gt;::new();\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if !selected[i] &amp;&amp; !duplicated.contains(&amp;choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.insert(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state.clone(), choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.remove(state.len() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfn permutations_ii(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new();\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_ii.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    bool duplicated[MAX_SIZE] = {false};\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated[choice]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated[choice] = true; // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state[stateSize] = choice;\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nint **permutationsII(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_ii.kt<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size == choices.size) {\n        res.add(ArrayList(state))\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    val duplicated: MutableSet&lt;Int&gt; = HashSet()\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.add(choice) // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true\n            state.add(choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfun permutationsII(nums: IntArray): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val res: MutableList&lt;MutableList&lt;Int&gt;?&gt; = ArrayList()\n    backtrack(ArrayList(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_ii.rb<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutations_ii}\n</code></pre> permutations_ii.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutationsII}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5047\u8bbe\u5143\u7d20\u4e24\u4e24\u4e4b\u95f4\u4e92\u4e0d\u76f8\u540c\uff0c\u5219 \\(n\\) \u4e2a\u5143\u7d20\u5171\u6709 \\(n!\\)  \u79cd\u6392\u5217\uff08\u9636\u4e58\uff09\uff1b\u5728\u8bb0\u5f55\u7ed3\u679c\u65f6\uff0c\u9700\u8981\u590d\u5236\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5217\u8868\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n!n)\\) \u3002</p> <p>\u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002<code>selected</code> \u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u540c\u4e00\u65f6\u523b\u6700\u591a\u5171\u6709 \\(n\\) \u4e2a <code>duplicated</code> \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\u3002\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#3","title":"3. \u00a0 \u4e24\u79cd\u526a\u679d\u5bf9\u6bd4","text":"<p>\u8bf7\u6ce8\u610f\uff0c\u867d\u7136 <code>selected</code> \u548c <code>duplicated</code> \u90fd\u7528\u4e8e\u526a\u679d\uff0c\u4f46\u4e24\u8005\u7684\u76ee\u6807\u4e0d\u540c\u3002</p> <ul> <li>\u91cd\u590d\u9009\u62e9\u526a\u679d\uff1a\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\u4e2d\u53ea\u6709\u4e00\u4e2a <code>selected</code> \u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5f53\u524d\u72b6\u6001\u4e2d\u5305\u542b\u54ea\u4e9b\u5143\u7d20\uff0c\u5176\u4f5c\u7528\u662f\u907f\u514d\u67d0\u4e2a\u5143\u7d20\u5728 <code>state</code> \u4e2d\u91cd\u590d\u51fa\u73b0\u3002</li> <li>\u76f8\u7b49\u5143\u7d20\u526a\u679d\uff1a\u6bcf\u8f6e\u9009\u62e9\uff08\u6bcf\u4e2a\u8c03\u7528\u7684 <code>backtrack</code> \u51fd\u6570\uff09\u90fd\u5305\u542b\u4e00\u4e2a <code>duplicated</code> \u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5728\u672c\u8f6e\u904d\u5386\uff08<code>for</code> \u5faa\u73af\uff09\u4e2d\u54ea\u4e9b\u5143\u7d20\u5df2\u88ab\u9009\u62e9\u8fc7\uff0c\u5176\u4f5c\u7528\u662f\u4fdd\u8bc1\u76f8\u7b49\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002</li> </ul> <p>\u56fe 13-9 \u5c55\u793a\u4e86\u4e24\u4e2a\u526a\u679d\u6761\u4ef6\u7684\u751f\u6548\u8303\u56f4\u3002\u6ce8\u610f\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u9009\u62e9\uff0c\u4ece\u6839\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\u6784\u6210\u4e00\u4e2a\u6392\u5217\u3002</p> <p></p> <p> \u56fe 13-9 \u00a0 \u4e24\u79cd\u526a\u679d\u6761\u4ef6\u7684\u4f5c\u7528\u8303\u56f4 </p>"},{"location":"chapter_backtracking/subset_sum_problem/","title":"13.3 \u00a0 \u5b50\u96c6\u548c\u95ee\u9898","text":""},{"location":"chapter_backtracking/subset_sum_problem/#1331","title":"13.3.1 \u00a0 \u65e0\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570 <code>target</code> \uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8e <code>target</code> \u3002\u7ed9\u5b9a\u6570\u7ec4\u65e0\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ef\u4ee5\u88ab\u9009\u53d6\u591a\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002</p> <p>\u4f8b\u5982\uff0c\u8f93\u5165\u96c6\u5408 \\(\\{3, 4, 5\\}\\) \u548c\u76ee\u6807\u6574\u6570 \\(9\\) \uff0c\u89e3\u4e3a \\(\\{3, 3, 3\\}, \\{4, 5\\}\\) \u3002\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u4e24\u70b9\u3002</p> <ul> <li>\u8f93\u5165\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u65e0\u9650\u6b21\u91cd\u590d\u9009\u53d6\u3002</li> <li>\u5b50\u96c6\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\uff0c\u6bd4\u5982 \\(\\{4, 5\\}\\) \u548c \\(\\{5, 4\\}\\) \u662f\u540c\u4e00\u4e2a\u5b50\u96c6\u3002</li> </ul>"},{"location":"chapter_backtracking/subset_sum_problem/#1","title":"1. \u00a0 \u53c2\u8003\u5168\u6392\u5217\u89e3\u6cd5","text":"<p>\u7c7b\u4f3c\u4e8e\u5168\u6392\u5217\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b50\u96c6\u7684\u751f\u6210\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\uff0c\u5e76\u5728\u9009\u62e9\u8fc7\u7a0b\u4e2d\u5b9e\u65f6\u66f4\u65b0\u201c\u5143\u7d20\u548c\u201d\uff0c\u5f53\u5143\u7d20\u548c\u7b49\u4e8e <code>target</code> \u65f6\uff0c\u5c31\u5c06\u5b50\u96c6\u8bb0\u5f55\u81f3\u7ed3\u679c\u5217\u8868\u3002</p> <p>\u800c\u4e0e\u5168\u6392\u5217\u95ee\u9898\u4e0d\u540c\u7684\u662f\uff0c\u672c\u9898\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u65e0\u9650\u6b21\u9009\u53d6\uff0c\u56e0\u6b64\u65e0\u987b\u501f\u52a9 <code>selected</code> \u5e03\u5c14\u5217\u8868\u6765\u8bb0\u5f55\u5143\u7d20\u662f\u5426\u5df2\u88ab\u9009\u62e9\u3002\u6211\u4eec\u53ef\u4ee5\u5bf9\u5168\u6392\u5217\u4ee3\u7801\u8fdb\u884c\u5c0f\u5e45\u4fee\u6539\uff0c\u521d\u6b65\u5f97\u5230\u89e3\u9898\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig subset_sum_i_naive.py<pre><code>def backtrack(\n    state: list[int],\n    target: int,\n    total: int,\n    choices: list[int],\n    res: list[list[int]],\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n    # \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if total == target:\n        res.append(list(state))\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i in range(len(choices)):\n        # \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if total + choices[i] &gt; target:\n            continue\n        # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.append(choices[i])\n        # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res)\n        # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop()\n\ndef subset_sum_i_naive(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09\"\"\"\n    state = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    total = 0  # \u5b50\u96c6\u548c\n    res = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res)\n    return res\n</code></pre> subset_sum_i_naive.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, int total, vector&lt;int&gt; &amp;choices, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total == target) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (size_t i = 0; i &lt; choices.size(); i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push_back(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop_back();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nvector&lt;vector&lt;int&gt;&gt; subsetSumINaive(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;       // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    int total = 0;           // \u5b50\u96c6\u548c\n    vector&lt;vector&lt;int&gt;&gt; res; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int total, int[] choices, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total == target) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choices.length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.remove(state.size() - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList&lt;List&lt;Integer&gt;&gt; subsetSumINaive(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    int total = 0; // \u5b50\u96c6\u548c\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List&lt;int&gt; state, int target, int total, int[] choices, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total == target) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choices.Length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.Add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        Backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList&lt;List&lt;int&gt;&gt; SubsetSumINaive(int[] nums, int target) {\n    List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    int total = 0; // \u5b50\u96c6\u548c\n    List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    Backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumINaive(total, target int, state, choices *[]int, res *[][]int) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == total {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i := 0; i &lt; len(*choices); i++ {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if total+(*choices)[i] &gt; target {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        *state = append(*state, (*choices)[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrackSubsetSumINaive(total+(*choices)[i], target, state, choices, res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums []int, target int) [][]int {\n    state := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    total := 0              // \u5b50\u96c6\u548c\n    res := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrackSubsetSumINaive(total, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, total: Int, choices: [Int], res: inout [[Int]]) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if total == target {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i in choices.indices {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if total + choices[i] &gt; target {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.append(choices[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state: &amp;state, target: target, total: total + choices[i], choices: choices, res: &amp;res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.removeLast()\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    let total = 0 // \u5b50\u96c6\u548c\n    var res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state: &amp;state, target: target, total: total, choices: nums, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, total, choices, res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (let i = 0; i &lt; choices.length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums, target) {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    const total = 0; // \u5b50\u96c6\u548c\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    total: number,\n    choices: number[],\n    res: number[][]\n): void {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (let i = 0; i &lt; choices.length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums: number[], target: number): number[][] {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    const total = 0; // \u5b50\u96c6\u548c\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  int total,\n  List&lt;int&gt; choices,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (total == target) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  for (int i = 0; i &lt; choices.length; i++) {\n    // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n    if (total + choices[i] &gt; target) {\n      continue;\n    }\n    // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n    state.add(choices[i]);\n    // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n    backtrack(state, target, total + choices[i], choices, res);\n    // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n    state.removeLast();\n  }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList&lt;List&lt;int&gt;&gt; subsetSumINaive(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n  int total = 0; // \u5143\u7d20\u548c\n  List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n  backtrack(state, target, total, nums, res);\n  return res;\n}\n</code></pre> subset_sum_i_naive.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(\n    mut state: Vec&lt;i32&gt;,\n    target: i32,\n    total: i32,\n    choices: &amp;[i32],\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if total == target {\n        res.push(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i in 0..choices.len() {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if total + choices[i] &gt; target {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state.clone(), target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfn subset_sum_i_naive(nums: &amp;[i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    let total = 0; // \u5b50\u96c6\u548c\n    let mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i_naive.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int total, int *choices, int choicesSize) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total == target) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choicesSize; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state[stateSize++] = choices[i];\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(target, total + choices[i], choices, choicesSize);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        stateSize--;\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nvoid subsetSumINaive(int *nums, int numsSize, int target) {\n    resSize = 0; // \u521d\u59cb\u5316\u89e3\u7684\u6570\u91cf\u4e3a0\n    backtrack(target, 0, nums, numsSize);\n}\n</code></pre> subset_sum_i_naive.kt<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    total: Int,\n    choices: IntArray,\n    res: MutableList&lt;List&lt;Int&gt;?&gt;\n) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total == target) {\n        res.add(ArrayList(state))\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (i in choices.indices) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.add(choices[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfun subsetSumINaive(nums: IntArray, target: Int): List&lt;List&lt;Int&gt;?&gt; {\n    val state: MutableList&lt;Int&gt; = ArrayList() // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    val total = 0 // \u5b50\u96c6\u548c\n    val res: MutableList&lt;List&lt;Int&gt;?&gt; = ArrayList() // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res)\n    return res\n}\n</code></pre> subset_sum_i_naive.rb<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subset_sum_i_naive}\n</code></pre> subset_sum_i_naive.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumINaive}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5411\u4ee5\u4e0a\u4ee3\u7801\u8f93\u5165\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u8f93\u51fa\u7ed3\u679c\u4e3a \\([3, 3, 3], [4, 5], [5, 4]\\) \u3002\u867d\u7136\u6210\u529f\u627e\u51fa\u4e86\u6240\u6709\u548c\u4e3a \\(9\\) \u7684\u5b50\u96c6\uff0c\u4f46\u5176\u4e2d\u5b58\u5728\u91cd\u590d\u7684\u5b50\u96c6 \\([4, 5]\\) \u548c \\([5, 4]\\) \u3002</p> <p>\u8fd9\u662f\u56e0\u4e3a\u641c\u7d22\u8fc7\u7a0b\u662f\u533a\u5206\u9009\u62e9\u987a\u5e8f\u7684\uff0c\u7136\u800c\u5b50\u96c6\u4e0d\u533a\u5206\u9009\u62e9\u987a\u5e8f\u3002\u5982\u56fe 13-10 \u6240\u793a\uff0c\u5148\u9009 \\(4\\) \u540e\u9009 \\(5\\) \u4e0e\u5148\u9009 \\(5\\) \u540e\u9009 \\(4\\) \u662f\u4e0d\u540c\u7684\u5206\u652f\uff0c\u4f46\u5bf9\u5e94\u540c\u4e00\u4e2a\u5b50\u96c6\u3002</p> <p></p> <p> \u56fe 13-10 \u00a0 \u5b50\u96c6\u641c\u7d22\u4e0e\u8d8a\u754c\u526a\u679d </p> <p>\u4e3a\u4e86\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u601d\u8def\u662f\u5bf9\u7ed3\u679c\u5217\u8868\u8fdb\u884c\u53bb\u91cd\u3002\u4f46\u8fd9\u4e2a\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\uff0c\u6709\u4e24\u65b9\u9762\u539f\u56e0\u3002</p> <ul> <li>\u5f53\u6570\u7ec4\u5143\u7d20\u8f83\u591a\uff0c\u5c24\u5176\u662f\u5f53 <code>target</code> \u8f83\u5927\u65f6\uff0c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u4ea7\u751f\u5927\u91cf\u7684\u91cd\u590d\u5b50\u96c6\u3002</li> <li>\u6bd4\u8f83\u5b50\u96c6\uff08\u6570\u7ec4\uff09\u7684\u5f02\u540c\u975e\u5e38\u8017\u65f6\uff0c\u9700\u8981\u5148\u6392\u5e8f\u6570\u7ec4\uff0c\u518d\u6bd4\u8f83\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u5143\u7d20\u7684\u5f02\u540c\u3002</li> </ul>"},{"location":"chapter_backtracking/subset_sum_problem/#2","title":"2. \u00a0 \u91cd\u590d\u5b50\u96c6\u526a\u679d","text":"<p>\u6211\u4eec\u8003\u8651\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u901a\u8fc7\u526a\u679d\u8fdb\u884c\u53bb\u91cd\u3002\u89c2\u5bdf\u56fe 13-11 \uff0c\u91cd\u590d\u5b50\u96c6\u662f\u5728\u4ee5\u4e0d\u540c\u987a\u5e8f\u9009\u62e9\u6570\u7ec4\u5143\u7d20\u65f6\u4ea7\u751f\u7684\uff0c\u4f8b\u5982\u4ee5\u4e0b\u60c5\u51b5\u3002</p> <ol> <li>\u5f53\u7b2c\u4e00\u8f6e\u548c\u7b2c\u4e8c\u8f6e\u5206\u522b\u9009\u62e9 \\(3\\) \u548c \\(4\\) \u65f6\uff0c\u4f1a\u751f\u6210\u5305\u542b\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u6240\u6709\u5b50\u96c6\uff0c\u8bb0\u4e3a \\([3, 4, \\dots]\\) \u3002</li> <li>\u4e4b\u540e\uff0c\u5f53\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(4\\) \u65f6\uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \uff0c\u56e0\u4e3a\u8be5\u9009\u62e9\u4ea7\u751f\u7684\u5b50\u96c6 \\([4, 3, \\dots]\\) \u548c\u7b2c <code>1.</code> \u6b65\u4e2d\u751f\u6210\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u590d\u3002</li> </ol> <p>\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e00\u5c42\u7684\u9009\u62e9\u90fd\u662f\u4ece\u5de6\u5230\u53f3\u88ab\u9010\u4e2a\u5c1d\u8bd5\u7684\uff0c\u56e0\u6b64\u8d8a\u9760\u53f3\u7684\u5206\u652f\u88ab\u526a\u6389\u7684\u8d8a\u591a\u3002</p> <ol> <li>\u524d\u4e24\u8f6e\u9009\u62e9 \\(3\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([3, 5, \\dots]\\) \u3002</li> <li>\u524d\u4e24\u8f6e\u9009\u62e9 \\(4\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([4, 5, \\dots]\\) \u3002</li> <li>\u82e5\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(5\\) \uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \u548c \\(4\\) \uff0c\u56e0\u4e3a\u5b50\u96c6 \\([5, 3, \\dots]\\) \u548c \\([5, 4, \\dots]\\) \u4e0e\u7b2c <code>1.</code> \u6b65\u548c\u7b2c <code>2.</code> \u6b65\u4e2d\u63cf\u8ff0\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u590d\u3002</li> </ol> <p></p> <p> \u56fe 13-11 \u00a0 \u4e0d\u540c\u9009\u62e9\u987a\u5e8f\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6 </p> <p>\u603b\u7ed3\u6765\u770b\uff0c\u7ed9\u5b9a\u8f93\u5165\u6570\u7ec4 \\([x_1, x_2, \\dots, x_n]\\) \uff0c\u8bbe\u641c\u7d22\u8fc7\u7a0b\u4e2d\u7684\u9009\u62e9\u5e8f\u5217\u4e3a \\([x_{i_1}, x_{i_2}, \\dots, x_{i_m}]\\) \uff0c\u5219\u8be5\u9009\u62e9\u5e8f\u5217\u9700\u8981\u6ee1\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u4e0d\u6ee1\u8db3\u8be5\u6761\u4ef6\u7684\u9009\u62e9\u5e8f\u5217\u90fd\u4f1a\u9020\u6210\u91cd\u590d\uff0c\u5e94\u5f53\u526a\u679d\u3002</p>"},{"location":"chapter_backtracking/subset_sum_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u4e3a\u5b9e\u73b0\u8be5\u526a\u679d\uff0c\u6211\u4eec\u521d\u59cb\u5316\u53d8\u91cf <code>start</code> \uff0c\u7528\u4e8e\u6307\u793a\u904d\u5386\u8d77\u59cb\u70b9\u3002\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i\\) \u5f00\u59cb\u904d\u5386\u3002\u8fd9\u6837\u505a\u5c31\u53ef\u4ee5\u8ba9\u9009\u62e9\u5e8f\u5217\u6ee1\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u4ece\u800c\u4fdd\u8bc1\u5b50\u96c6\u552f\u4e00\u3002</p> <p>\u9664\u6b64\u4e4b\u5916\uff0c\u6211\u4eec\u8fd8\u5bf9\u4ee3\u7801\u8fdb\u884c\u4e86\u4ee5\u4e0b\u4e24\u9879\u4f18\u5316\u3002</p> <ul> <li>\u5728\u5f00\u542f\u641c\u7d22\u524d\uff0c\u5148\u5c06\u6570\u7ec4 <code>nums</code> \u6392\u5e8f\u3002\u5728\u904d\u5386\u6240\u6709\u9009\u62e9\u65f6\uff0c\u5f53\u5b50\u96c6\u548c\u8d85\u8fc7 <code>target</code> \u65f6\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\uff0c\u56e0\u4e3a\u540e\u8fb9\u7684\u5143\u7d20\u66f4\u5927\uff0c\u5176\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 <code>target</code> \u3002</li> <li>\u7701\u53bb\u5143\u7d20\u548c\u53d8\u91cf <code>total</code> \uff0c\u901a\u8fc7\u5728 <code>target</code> \u4e0a\u6267\u884c\u51cf\u6cd5\u6765\u7edf\u8ba1\u5143\u7d20\u548c\uff0c\u5f53 <code>target</code> \u7b49\u4e8e \\(0\\) \u65f6\u8bb0\u5f55\u89e3\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig subset_sum_i.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n    # \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0:\n        res.append(list(state))\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    # \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for i in range(start, len(choices)):\n        # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        # \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0:\n            break\n        # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res)\n        # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop()\n\ndef subset_sum_i(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\"\"\"\n    state = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort()  # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    start = 0  # \u904d\u5386\u8d77\u59cb\u70b9\n    res = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_i.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (int i = start; i &lt; choices.size(); i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push_back(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop_back();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvector&lt;vector&lt;int&gt;&gt; subsetSumI(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    sort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0;                  // \u904d\u5386\u8d77\u59cb\u70b9\n    vector&lt;vector&lt;int&gt;&gt; res;        // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (int i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.remove(state.size() - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;Integer&gt;&gt; subsetSumI(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Arrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (int i = start; i &lt; choices.Length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.Add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        Backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;int&gt;&gt; SubsetSumI(int[] nums, int target) {\n    List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Array.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumI(start, target int, state, choices *[]int, res *[][]int) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for i := start; i &lt; len(*choices); i++ {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        *state = append(*state, (*choices)[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrackSubsetSumI(i, target-(*choices)[i], state, choices, res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums []int, target int) [][]int {\n    state := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    sort.Ints(nums)         // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    start := 0              // \u904d\u5386\u8d77\u59cb\u70b9\n    res := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrackSubsetSumI(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for i in choices.indices.dropFirst(start) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i, res: &amp;res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.removeLast()\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    let nums = nums.sorted() // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    let start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\n    var res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, choices, start, res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums, target) {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums: number[], target: number): number[][] {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n  for (int i = start; i &lt; choices.length; i++) {\n    // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n    // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n    state.add(choices[i]);\n    // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n    backtrack(state, target - choices[i], choices, i, res);\n    // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n    state.removeLast();\n  }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;int&gt;&gt; subsetSumI(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n  nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n  int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n  List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_i.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(\n    mut state: Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        res.push(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for i in start..choices.len() {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state.clone(), target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfn subset_sum_i(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    let start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    let mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; ++i) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (int i = start; i &lt; choicesSize; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state[stateSize] = choices[i];\n        stateSize++;\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(target - choices[i], choices, choicesSize, i);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        stateSize--;\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvoid subsetSumI(int *nums, int numsSize, int target) {\n    qsort(nums, numsSize, sizeof(int), cmp); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0;                           // \u904d\u5386\u8d77\u59cb\u70b9\n    backtrack(target, nums, numsSize, start);\n}\n</code></pre> subset_sum_i.kt<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;List&lt;Int&gt;?&gt;\n) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.add(ArrayList(state))\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (i in start..&lt;choices.size) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfun subsetSumI(nums: IntArray, target: Int): List&lt;List&lt;Int&gt;?&gt; {\n    val state: MutableList&lt;Int&gt; = ArrayList() // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Arrays.sort(nums) // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    val start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\n    val res: MutableList&lt;List&lt;Int&gt;?&gt; = ArrayList() // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_i.rb<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subset_sum_i}\n</code></pre> subset_sum_i.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumI}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 13-12 \u6240\u793a\u4e3a\u5c06\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u8f93\u5165\u4ee5\u4e0a\u4ee3\u7801\u540e\u7684\u6574\u4f53\u56de\u6eaf\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 13-12 \u00a0 \u5b50\u96c6\u548c I \u56de\u6eaf\u8fc7\u7a0b </p>"},{"location":"chapter_backtracking/subset_sum_problem/#1332","title":"13.3.2 \u00a0 \u8003\u8651\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570 <code>target</code> \uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8e <code>target</code> \u3002\u7ed9\u5b9a\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u53ef\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002</p> <p>\u76f8\u6bd4\u4e8e\u4e0a\u9898\uff0c\u672c\u9898\u7684\u8f93\u5165\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd9\u5f15\u5165\u4e86\u65b0\u7684\u95ee\u9898\u3002\u4f8b\u5982\uff0c\u7ed9\u5b9a\u6570\u7ec4 \\([4, \\hat{4}, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u5219\u73b0\u6709\u4ee3\u7801\u7684\u8f93\u51fa\u7ed3\u679c\u4e3a \\([4, 5], [\\hat{4}, 5]\\) \uff0c\u51fa\u73b0\u4e86\u91cd\u590d\u5b50\u96c6\u3002</p> <p>\u9020\u6210\u8fd9\u79cd\u91cd\u590d\u7684\u539f\u56e0\u662f\u76f8\u7b49\u5143\u7d20\u5728\u67d0\u8f6e\u4e2d\u88ab\u591a\u6b21\u9009\u62e9\u3002\u5728\u56fe 13-13 \u4e2d\uff0c\u7b2c\u4e00\u8f6e\u5171\u6709\u4e09\u4e2a\u9009\u62e9\uff0c\u5176\u4e2d\u4e24\u4e2a\u90fd\u4e3a \\(4\\) \uff0c\u4f1a\u4ea7\u751f\u4e24\u4e2a\u91cd\u590d\u7684\u641c\u7d22\u5206\u652f\uff0c\u4ece\u800c\u8f93\u51fa\u91cd\u590d\u5b50\u96c6\uff1b\u540c\u7406\uff0c\u7b2c\u4e8c\u8f6e\u7684\u4e24\u4e2a \\(4\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002</p> <p></p> <p> \u56fe 13-13 \u00a0 \u76f8\u7b49\u5143\u7d20\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6 </p>"},{"location":"chapter_backtracking/subset_sum_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"<p>\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u9650\u5236\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u4e00\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5b9e\u73b0\u65b9\u5f0f\u6bd4\u8f83\u5de7\u5999\uff1a\u7531\u4e8e\u6570\u7ec4\u662f\u5df2\u6392\u5e8f\u7684\uff0c\u56e0\u6b64\u76f8\u7b49\u5143\u7d20\u90fd\u662f\u76f8\u90bb\u7684\u3002\u8fd9\u610f\u5473\u7740\u5728\u67d0\u8f6e\u9009\u62e9\u4e2d\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u4e0e\u5176\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u5219\u8bf4\u660e\u5b83\u5df2\u7ecf\u88ab\u9009\u62e9\u8fc7\uff0c\u56e0\u6b64\u76f4\u63a5\u8df3\u8fc7\u5f53\u524d\u5143\u7d20\u3002</p> <p>\u4e0e\u6b64\u540c\u65f6\uff0c\u672c\u9898\u89c4\u5b9a\u6bcf\u4e2a\u6570\u7ec4\u5143\u7d20\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5e78\u8fd0\u7684\u662f\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u5229\u7528\u53d8\u91cf <code>start</code> \u6765\u6ee1\u8db3\u8be5\u7ea6\u675f\uff1a\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i + 1\\) \u5f00\u59cb\u5411\u540e\u904d\u5386\u3002\u8fd9\u6837\u65e2\u80fd\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e5f\u80fd\u907f\u514d\u91cd\u590d\u9009\u62e9\u5143\u7d20\u3002</p>"},{"location":"chapter_backtracking/subset_sum_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig subset_sum_ii.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II\"\"\"\n    # \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0:\n        res.append(list(state))\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    # \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    # \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for i in range(start, len(choices)):\n        # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        # \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0:\n            break\n        # \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if i &gt; start and choices[i] == choices[i - 1]:\n            continue\n        # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop()\n\ndef subset_sum_ii(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c II\"\"\"\n    state = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort()  # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    start = 0  # \u904d\u5386\u8d77\u59cb\u70b9\n    res = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_ii.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choices.size(); i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push_back(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop_back();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvector&lt;vector&lt;int&gt;&gt; subsetSumII(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    sort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0;                  // \u904d\u5386\u8d77\u59cb\u70b9\n    vector&lt;vector&lt;int&gt;&gt; res;        // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.remove(state.size() - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;Integer&gt;&gt; subsetSumII(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Arrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choices.Length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.Add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        Backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;int&gt;&gt; SubsetSumII(int[] nums, int target) {\n    List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Array.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrackSubsetSumII(start, target int, state, choices *[]int, res *[][]int) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for i := start; i &lt; len(*choices); i++ {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if i &gt; start &amp;&amp; (*choices)[i] == (*choices)[i-1] {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        *state = append(*state, (*choices)[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrackSubsetSumII(i+1, target-(*choices)[i], state, choices, res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums []int, target int) [][]int {\n    state := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    sort.Ints(nums)         // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    start := 0              // \u904d\u5386\u8d77\u59cb\u70b9\n    res := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrackSubsetSumII(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for i in choices.indices.dropFirst(start) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if i &gt; start, choices[i] == choices[i - 1] {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i + 1, res: &amp;res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.removeLast()\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    let nums = nums.sorted() // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    let start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\n    var res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(state, target, choices, start, res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums, target) {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums: number[], target: number): number[][] {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n  // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n  for (int i = start; i &lt; choices.length; i++) {\n    // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n    // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n    if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n      continue;\n    }\n    // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n    state.add(choices[i]);\n    // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n    backtrack(state, target - choices[i], choices, i + 1, res);\n    // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n    state.removeLast();\n  }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;int&gt;&gt; subsetSumII(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n  nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n  int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n  List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_ii.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfn backtrack(\n    mut state: Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        res.push(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for i in start..choices.len() {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if i &gt; start &amp;&amp; choices[i] == choices[i - 1] {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state.clone(), target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfn subset_sum_ii(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    let start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    let mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_ii.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choicesSize; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\n        if (target - choices[i] &lt; 0) {\n            continue;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state[stateSize] = choices[i];\n        stateSize++;\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(target - choices[i], choices, choicesSize, i + 1);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        stateSize--;\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvoid subsetSumII(int *nums, int numsSize, int target) {\n    // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    qsort(nums, numsSize, sizeof(int), cmp);\n    // \u5f00\u59cb\u56de\u6eaf\n    backtrack(target, nums, numsSize, 0);\n}\n</code></pre> subset_sum_ii.kt<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;List&lt;Int&gt;?&gt;\n) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.add(ArrayList(state))\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (i in start..&lt;choices.size) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfun subsetSumII(nums: IntArray, target: Int): List&lt;List&lt;Int&gt;?&gt; {\n    val state: MutableList&lt;Int&gt; = ArrayList() // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Arrays.sort(nums) // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    val start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\n    val res: MutableList&lt;List&lt;Int&gt;?&gt; = ArrayList() // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_ii.rb<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subset_sum_ii}\n</code></pre> subset_sum_ii.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumII}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 13-14 \u5c55\u793a\u4e86\u6570\u7ec4 \\([4, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u7684\u56de\u6eaf\u8fc7\u7a0b\uff0c\u5171\u5305\u542b\u56db\u79cd\u526a\u679d\u64cd\u4f5c\u3002\u8bf7\u4f60\u5c06\u56fe\u793a\u4e0e\u4ee3\u7801\u6ce8\u91ca\u76f8\u7ed3\u5408\uff0c\u7406\u89e3\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ee5\u53ca\u6bcf\u79cd\u526a\u679d\u64cd\u4f5c\u662f\u5982\u4f55\u5de5\u4f5c\u7684\u3002</p> <p></p> <p> \u56fe 13-14 \u00a0 \u5b50\u96c6\u548c II \u56de\u6eaf\u8fc7\u7a0b </p>"},{"location":"chapter_backtracking/summary/","title":"13.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_backtracking/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u662f\u7a77\u4e3e\u6cd5\uff0c\u901a\u8fc7\u5bf9\u89e3\u7a7a\u95f4\u8fdb\u884c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u6765\u5bfb\u627e\u7b26\u5408\u6761\u4ef6\u7684\u89e3\u3002\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u9047\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u5219\u8bb0\u5f55\uff0c\u76f4\u81f3\u627e\u5230\u6240\u6709\u89e3\u6216\u904d\u5386\u5b8c\u6210\u540e\u7ed3\u675f\u3002</li> <li>\u56de\u6eaf\u7b97\u6cd5\u7684\u641c\u7d22\u8fc7\u7a0b\u5305\u62ec\u5c1d\u8bd5\u4e0e\u56de\u9000\u4e24\u4e2a\u90e8\u5206\u3002\u5b83\u901a\u8fc7\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u6765\u5c1d\u8bd5\u5404\u79cd\u9009\u62e9\uff0c\u5f53\u9047\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u60c5\u51b5\u65f6\uff0c\u5219\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002\u5c1d\u8bd5\u4e0e\u56de\u9000\u662f\u4e24\u4e2a\u65b9\u5411\u76f8\u53cd\u7684\u64cd\u4f5c\u3002</li> <li>\u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u5b83\u4eec\u53ef\u7528\u4e8e\u5b9e\u73b0\u526a\u679d\u64cd\u4f5c\u3002\u526a\u679d\u53ef\u4ee5\u63d0\u524d\u7ed3\u675f\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\uff0c\u5927\u5e45\u63d0\u5347\u641c\u7d22\u6548\u7387\u3002</li> <li>\u56de\u6eaf\u7b97\u6cd5\u4e3b\u8981\u53ef\u7528\u4e8e\u89e3\u51b3\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u3002\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u867d\u7136\u53ef\u4ee5\u7528\u56de\u6eaf\u7b97\u6cd5\u89e3\u51b3\uff0c\u4f46\u5f80\u5f80\u5b58\u5728\u6548\u7387\u66f4\u9ad8\u6216\u6548\u679c\u66f4\u597d\u7684\u89e3\u6cd5\u3002</li> <li>\u5168\u6392\u5217\u95ee\u9898\u65e8\u5728\u641c\u7d22\u7ed9\u5b9a\u96c6\u5408\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e2a\u5143\u7d20\u662f\u5426\u88ab\u9009\u62e9\uff0c\u526a\u6389\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\u7684\u641c\u7d22\u5206\u652f\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002</li> <li>\u5728\u5168\u6392\u5217\u95ee\u9898\u4e2d\uff0c\u5982\u679c\u96c6\u5408\u4e2d\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff0c\u5219\u6700\u7ec8\u7ed3\u679c\u4f1a\u51fa\u73b0\u91cd\u590d\u6392\u5217\u3002\u6211\u4eec\u9700\u8981\u7ea6\u675f\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u8fd9\u901a\u5e38\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002</li> <li>\u5b50\u96c6\u548c\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u7ed9\u5b9a\u96c6\u5408\u4e2d\u627e\u5230\u548c\u4e3a\u76ee\u6807\u503c\u7684\u6240\u6709\u5b50\u96c6\u3002\u96c6\u5408\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\uff0c\u800c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u8f93\u51fa\u6240\u6709\u987a\u5e8f\u7684\u7ed3\u679c\uff0c\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002\u6211\u4eec\u5728\u56de\u6eaf\u524d\u5c06\u6570\u636e\u8fdb\u884c\u6392\u5e8f\uff0c\u5e76\u8bbe\u7f6e\u4e00\u4e2a\u53d8\u91cf\u6765\u6307\u793a\u6bcf\u4e00\u8f6e\u7684\u904d\u5386\u8d77\u59cb\u70b9\uff0c\u4ece\u800c\u5c06\u751f\u6210\u91cd\u590d\u5b50\u96c6\u7684\u641c\u7d22\u5206\u652f\u8fdb\u884c\u526a\u679d\u3002</li> <li>\u5bf9\u4e8e\u5b50\u96c6\u548c\u95ee\u9898\uff0c\u6570\u7ec4\u4e2d\u7684\u76f8\u7b49\u5143\u7d20\u4f1a\u4ea7\u751f\u91cd\u590d\u96c6\u5408\u3002\u6211\u4eec\u5229\u7528\u6570\u7ec4\u5df2\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\uff0c\u901a\u8fc7\u5224\u65ad\u76f8\u90bb\u5143\u7d20\u662f\u5426\u76f8\u7b49\u5b9e\u73b0\u526a\u679d\uff0c\u4ece\u800c\u786e\u4fdd\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u4e2d\u4e00\u6b21\u3002</li> <li>\\(n\\) \u7687\u540e\u95ee\u9898\u65e8\u5728\u5bfb\u627e\u5c06 \\(n\\) \u4e2a\u7687\u540e\u653e\u7f6e\u5230 \\(n \\times n\\) \u5c3a\u5bf8\u68cb\u76d8\u4e0a\u7684\u65b9\u6848\uff0c\u8981\u6c42\u6240\u6709\u7687\u540e\u4e24\u4e24\u4e4b\u95f4\u65e0\u6cd5\u653b\u51fb\u5bf9\u65b9\u3002\u8be5\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6\u6709\u884c\u7ea6\u675f\u3001\u5217\u7ea6\u675f\u3001\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002\u4e3a\u6ee1\u8db3\u884c\u7ea6\u675f\uff0c\u6211\u4eec\u91c7\u7528\u6309\u884c\u653e\u7f6e\u7684\u7b56\u7565\uff0c\u4fdd\u8bc1\u6bcf\u4e00\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002</li> <li>\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u5904\u7406\u65b9\u5f0f\u7c7b\u4f3c\u3002\u5bf9\u4e8e\u5217\u7ea6\u675f\uff0c\u6211\u4eec\u5229\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\uff0c\u4ece\u800c\u6307\u793a\u9009\u4e2d\u7684\u683c\u5b50\u662f\u5426\u5408\u6cd5\u3002\u5bf9\u4e8e\u5bf9\u89d2\u7ebf\u7ea6\u675f\uff0c\u6211\u4eec\u501f\u52a9\u4e24\u4e2a\u6570\u7ec4\u6765\u5206\u522b\u8bb0\u5f55\u8be5\u4e3b\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u5b58\u5728\u7687\u540e\uff1b\u96be\u70b9\u5728\u4e8e\u627e\u5904\u5728\u5230\u540c\u4e00\u4e3b\uff08\u526f\uff09\u5bf9\u89d2\u7ebf\u4e0a\u683c\u5b50\u6ee1\u8db3\u7684\u884c\u5217\u7d22\u5f15\u89c4\u5f8b\u3002</li> </ul>"},{"location":"chapter_backtracking/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u600e\u4e48\u7406\u89e3\u56de\u6eaf\u548c\u9012\u5f52\u7684\u5173\u7cfb\uff1f</p> <p>\u603b\u7684\u6765\u770b\uff0c\u56de\u6eaf\u662f\u4e00\u79cd\u201c\u7b97\u6cd5\u7b56\u7565\u201d\uff0c\u800c\u9012\u5f52\u66f4\u50cf\u662f\u4e00\u4e2a\u201c\u5de5\u5177\u201d\u3002</p> <ul> <li>\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\u3002\u7136\u800c\uff0c\u56de\u6eaf\u662f\u9012\u5f52\u7684\u5e94\u7528\u573a\u666f\u4e4b\u4e00\uff0c\u662f\u9012\u5f52\u5728\u641c\u7d22\u95ee\u9898\u4e2d\u7684\u5e94\u7528\u3002</li> <li>\u9012\u5f52\u7684\u7ed3\u6784\u4f53\u73b0\u4e86\u201c\u5b50\u95ee\u9898\u5206\u89e3\u201d\u7684\u89e3\u9898\u8303\u5f0f\uff0c\u5e38\u7528\u4e8e\u89e3\u51b3\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\uff08\u8bb0\u5fc6\u5316\u9012\u5f52\uff09\u7b49\u95ee\u9898\u3002</li> </ul>"},{"location":"chapter_computational_complexity/","title":"\u7b2c 2 \u7ae0 \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>Abstract</p> <p>\u590d\u6742\u5ea6\u5206\u6790\u72b9\u5982\u6d69\u701a\u7684\u7b97\u6cd5\u5b87\u5b99\u4e2d\u7684\u65f6\u7a7a\u5411\u5bfc\u3002</p> <p>\u5b83\u5e26\u9886\u6211\u4eec\u5728\u65f6\u95f4\u4e0e\u7a7a\u95f4\u8fd9\u4e24\u4e2a\u7ef4\u5ea6\u4e0a\u6df1\u5165\u63a2\u7d22\uff0c\u5bfb\u627e\u66f4\u4f18\u96c5\u7684\u89e3\u51b3\u65b9\u6848\u3002</p>"},{"location":"chapter_computational_complexity/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>2.1 \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30</li> <li>2.2 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52</li> <li>2.3 \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6</li> <li>2.4 \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6</li> <li>2.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/","title":"2.2 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52","text":"<p>\u5728\u7b97\u6cd5\u4e2d\uff0c\u91cd\u590d\u6267\u884c\u67d0\u4e2a\u4efb\u52a1\u662f\u5f88\u5e38\u89c1\u7684\uff0c\u5b83\u4e0e\u590d\u6742\u5ea6\u5206\u6790\u606f\u606f\u76f8\u5173\u3002\u56e0\u6b64\uff0c\u5728\u4ecb\u7ecd\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e4b\u524d\uff0c\u6211\u4eec\u5148\u6765\u4e86\u89e3\u5982\u4f55\u5728\u7a0b\u5e8f\u4e2d\u5b9e\u73b0\u91cd\u590d\u6267\u884c\u4efb\u52a1\uff0c\u5373\u4e24\u79cd\u57fa\u672c\u7684\u7a0b\u5e8f\u63a7\u5236\u7ed3\u6784\uff1a\u8fed\u4ee3\u3001\u9012\u5f52\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#221","title":"2.2.1 \u00a0 \u8fed\u4ee3","text":"<p>\u300c\u8fed\u4ee3 iteration\u300d\u662f\u4e00\u79cd\u91cd\u590d\u6267\u884c\u67d0\u4e2a\u4efb\u52a1\u7684\u63a7\u5236\u7ed3\u6784\u3002\u5728\u8fed\u4ee3\u4e2d\uff0c\u7a0b\u5e8f\u4f1a\u5728\u6ee1\u8db3\u4e00\u5b9a\u7684\u6761\u4ef6\u4e0b\u91cd\u590d\u6267\u884c\u67d0\u6bb5\u4ee3\u7801\uff0c\u76f4\u5230\u8fd9\u4e2a\u6761\u4ef6\u4e0d\u518d\u6ee1\u8db3\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for","title":"1. \u00a0 for \u5faa\u73af","text":"<p><code>for</code> \u5faa\u73af\u662f\u6700\u5e38\u89c1\u7684\u8fed\u4ee3\u5f62\u5f0f\u4e4b\u4e00\uff0c\u9002\u5408\u5728\u9884\u5148\u77e5\u9053\u8fed\u4ee3\u6b21\u6570\u65f6\u4f7f\u7528\u3002</p> <p>\u4ee5\u4e0b\u51fd\u6570\u57fa\u4e8e <code>for</code> \u5faa\u73af\u5b9e\u73b0\u4e86\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff0c\u6c42\u548c\u7ed3\u679c\u4f7f\u7528\u53d8\u91cf <code>res</code> \u8bb0\u5f55\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0cPython \u4e2d <code>range(a, b)</code> \u5bf9\u5e94\u7684\u533a\u95f4\u662f\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u7684\uff0c\u5bf9\u5e94\u7684\u904d\u5386\u8303\u56f4\u4e3a \\(a, a + 1, \\dots, b-1\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def for_loop(n: int) -&gt; int:\n    \"\"\"for \u5faa\u73af\"\"\"\n    res = 0\n    # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        res += i\n    return res\n</code></pre> iteration.cpp<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* for \u5faa\u73af */\nint ForLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* for \u5faa\u73af */\nfunc forLoop(n int) int {\n    res := 0\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* for \u5faa\u73af */\nfunc forLoop(n: Int) -&gt; Int {\n    var res = 0\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* for \u5faa\u73af */\nfunction forLoop(n) {\n    let res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* for \u5faa\u73af */\nfunction forLoop(n: number): number {\n    let res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n  int res = 0;\n  // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    res += i;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* for \u5faa\u73af */\nfn for_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i in 1..=n {\n        res += i;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* for \u5faa\u73af */\nfun forLoop(n: Int): Int {\n    var res = 0\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (i in 1..n) {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{for_loop}\n</code></pre> iteration.zig<pre><code>// for \u5faa\u73af\nfn forLoop(n: usize) i32 {\n    var res: i32 = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (1..n+1) |i| {\n        res = res + @as(i32, @intCast(i));\n    }\n    return res;\n} \n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 2-1 \u662f\u8be5\u6c42\u548c\u51fd\u6570\u7684\u6d41\u7a0b\u6846\u56fe\u3002</p> <p></p> <p> \u56fe 2-1 \u00a0 \u6c42\u548c\u51fd\u6570\u7684\u6d41\u7a0b\u6846\u56fe </p> <p>\u6b64\u6c42\u548c\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8bf4\u6210\u201c\u7ebf\u6027\u5173\u7cfb\u201d\u3002\u5b9e\u9645\u4e0a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u63cf\u8ff0\u7684\u5c31\u662f\u8fd9\u4e2a\u201c\u7ebf\u6027\u5173\u7cfb\u201d\u3002\u76f8\u5173\u5185\u5bb9\u5c06\u4f1a\u5728\u4e0b\u4e00\u8282\u4e2d\u8be6\u7ec6\u4ecb\u7ecd\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while","title":"2. \u00a0 while \u5faa\u73af","text":"<p>\u4e0e <code>for</code> \u5faa\u73af\u7c7b\u4f3c\uff0c<code>while</code> \u5faa\u73af\u4e5f\u662f\u4e00\u79cd\u5b9e\u73b0\u8fed\u4ee3\u7684\u65b9\u6cd5\u3002\u5728 <code>while</code> \u5faa\u73af\u4e2d\uff0c\u7a0b\u5e8f\u6bcf\u8f6e\u90fd\u4f1a\u5148\u68c0\u67e5\u6761\u4ef6\uff0c\u5982\u679c\u6761\u4ef6\u4e3a\u771f\uff0c\u5219\u7ee7\u7eed\u6267\u884c\uff0c\u5426\u5219\u5c31\u7ed3\u675f\u5faa\u73af\u3002</p> <p>\u4e0b\u9762\u6211\u4eec\u7528 <code>while</code> \u5faa\u73af\u6765\u5b9e\u73b0\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def while_loop(n: int) -&gt; int:\n    \"\"\"while \u5faa\u73af\"\"\"\n    res = 0\n    i = 1  # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n:\n        res += i\n        i += 1  # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    return res\n</code></pre> iteration.cpp<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u5faa\u73af */\nint WhileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while \u5faa\u73af */\nfunc whileLoop(n int) int {\n    res := 0\n    // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    i := 1\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while \u5faa\u73af */\nfunc whileLoop(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i\n        i += 1 // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while \u5faa\u73af */\nfunction whileLoop(n) {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u5faa\u73af */\nfunction whileLoop(n: number): number {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n  int res = 0;\n  int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n  // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n  while (i &lt;= n) {\n    res += i;\n    i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u5faa\u73af */\nfn while_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i;\n        i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while \u5faa\u73af */\nfun whileLoop(n: Int): Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i\n        i++ // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{while_loop}\n</code></pre> iteration.zig<pre><code>// while \u5faa\u73af\nfn whileLoop(n: i32) i32 {\n    var res: i32 = 0;\n    var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += @intCast(i);\n        i += 1;\n    }\n    return res;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p><code>while</code> \u5faa\u73af\u6bd4 <code>for</code> \u5faa\u73af\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\u3002\u5728 <code>while</code> \u5faa\u73af\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u81ea\u7531\u5730\u8bbe\u8ba1\u6761\u4ef6\u53d8\u91cf\u7684\u521d\u59cb\u5316\u548c\u66f4\u65b0\u6b65\u9aa4\u3002</p> <p>\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u6761\u4ef6\u53d8\u91cf \\(i\\) \u6bcf\u8f6e\u8fdb\u884c\u4e24\u6b21\u66f4\u65b0\uff0c\u8fd9\u79cd\u60c5\u51b5\u5c31\u4e0d\u592a\u65b9\u4fbf\u7528 <code>for</code> \u5faa\u73af\u5b9e\u73b0\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def while_loop_ii(n: int) -&gt; int:\n    \"\"\"while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\"\"\"\n    res = 0\n    i = 1  # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    # \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n:\n        res += i\n        # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1\n        i *= 2\n    return res\n</code></pre> iteration.cpp<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint WhileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1; \n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n int) int {\n    res := 0\n    // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    i := 1\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    for i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n) {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n: number): number {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n  int res = 0;\n  int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n  // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n  while (i &lt;= n) {\n    res += i;\n    // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    i++;\n    i *= 2;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfn while_loop_ii(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1;\n        i *= 2;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfun whileLoopII(n: Int): Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{while_loop_ii}\n</code></pre> iteration.zig<pre><code>//  while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\nfn whileLoopII(n: i32) i32 {\n    var res: i32 = 0;\n    var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += @intCast(i);\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u603b\u7684\u6765\u8bf4\uff0c<code>for</code> \u5faa\u73af\u7684\u4ee3\u7801\u66f4\u52a0\u7d27\u51d1\uff0c<code>while</code> \u5faa\u73af\u66f4\u52a0\u7075\u6d3b\uff0c\u4e24\u8005\u90fd\u53ef\u4ee5\u5b9e\u73b0\u8fed\u4ee3\u7ed3\u6784\u3002\u9009\u62e9\u4f7f\u7528\u54ea\u4e00\u4e2a\u5e94\u8be5\u6839\u636e\u7279\u5b9a\u95ee\u9898\u7684\u9700\u6c42\u6765\u51b3\u5b9a\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3","title":"3. \u00a0 \u5d4c\u5957\u5faa\u73af","text":"<p>\u6211\u4eec\u53ef\u4ee5\u5728\u4e00\u4e2a\u5faa\u73af\u7ed3\u6784\u5185\u5d4c\u5957\u53e6\u4e00\u4e2a\u5faa\u73af\u7ed3\u6784\uff0c\u4e0b\u9762\u4ee5 <code>for</code> \u5faa\u73af\u4e3a\u4f8b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def nested_for_loop(n: int) -&gt; str:\n    \"\"\"\u53cc\u5c42 for \u5faa\u73af\"\"\"\n    res = \"\"\n    # \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        # \u5faa\u73af j = 1, 2, ..., n-1, n\n        for j in range(1, n + 1):\n            res += f\"({i}, {j}), \"\n    return res\n</code></pre> iteration.cpp<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nstring nestedForLoop(int n) {\n    ostringstream res;\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; ++j) {\n            res &lt;&lt; \"(\" &lt;&lt; i &lt;&lt; \", \" &lt;&lt; j &lt;&lt; \"), \";\n        }\n    }\n    return res.str();\n}\n</code></pre> iteration.java<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n    StringBuilder res = new StringBuilder();\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.append(\"(\" + i + \", \" + j + \"), \");\n        }\n    }\n    return res.toString();\n}\n</code></pre> iteration.cs<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nstring NestedForLoop(int n) {\n    StringBuilder res = new();\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.Append($\"({i}, {j}), \");\n        }\n    }\n    return res.ToString();\n}\n</code></pre> iteration.go<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n int) string {\n    res := \"\"\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= n; j++ {\n            // \u5faa\u73af j = 1, 2, ..., n-1, n\n            res += fmt.Sprintf(\"(%d, %d), \", i, j)\n        }\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n: Int) -&gt; String {\n    var res = \"\"\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for j in 1 ... n {\n            res.append(\"(\\(i), \\(j)), \")\n        }\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n) {\n    let res = '';\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n: number): string {\n    let res = '';\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n  String res = \"\";\n  // \u5faa\u73af i = 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    // \u5faa\u73af j = 1, 2, ..., n-1, n\n    for (int j = 1; j &lt;= n; j++) {\n      res += \"($i, $j), \";\n    }\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfn nested_for_loop(n: i32) -&gt; String {\n    let mut res = vec![];\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i in 1..=n {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for j in 1..=n {\n            res.push(format!(\"({}, {}), \", i, j));\n        }\n    }\n    res.join(\"\")\n}\n</code></pre> iteration.c<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nchar *nestedForLoop(int n) {\n    // n * n \u4e3a\u5bf9\u5e94\u70b9\u6570\u91cf\uff0c\"(i, j), \" \u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u6700\u5927\u4e3a 6+10*2\uff0c\u52a0\u4e0a\u6700\u540e\u4e00\u4e2a\u7a7a\u5b57\u7b26 \\0 \u7684\u989d\u5916\u7a7a\u95f4\n    int size = n * n * 26 + 1;\n    char *res = malloc(size * sizeof(char));\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            char tmp[26];\n            snprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\n            strncat(res, tmp, size - strlen(res) - 1);\n        }\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfun nestedForLoop(n: Int): String {\n    val res = StringBuilder()\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (i in 1..n) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (j in 1..n) {\n            res.append(\" ($i, $j), \")\n        }\n    }\n    return res.toString()\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{nested_for_loop}\n</code></pre> iteration.zig<pre><code>// \u53cc\u5c42 for \u5faa\u73af\nfn nestedForLoop(allocator: Allocator, n: usize) ![]const u8 {\n    var res = std.ArrayList(u8).init(allocator);\n    defer res.deinit();\n    var buffer: [20]u8 = undefined;\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (1..n+1) |i| {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (1..n+1) |j| {\n            var _str = try std.fmt.bufPrint(&amp;buffer, \"({d}, {d}), \", .{i, j});\n            try res.appendSlice(_str);\n        }\n    }\n    return res.toOwnedSlice();\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 2-2 \u662f\u8be5\u5d4c\u5957\u5faa\u73af\u7684\u6d41\u7a0b\u6846\u56fe\u3002</p> <p></p> <p> \u56fe 2-2 \u00a0 \u5d4c\u5957\u5faa\u73af\u7684\u6d41\u7a0b\u6846\u56fe </p> <p>\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e \\(n^2\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8bf4\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u548c\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u6210\u201c\u5e73\u65b9\u5173\u7cfb\u201d\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u7ee7\u7eed\u6dfb\u52a0\u5d4c\u5957\u5faa\u73af\uff0c\u6bcf\u4e00\u6b21\u5d4c\u5957\u90fd\u662f\u4e00\u6b21\u201c\u5347\u7ef4\u201d\uff0c\u5c06\u4f1a\u4f7f\u65f6\u95f4\u590d\u6742\u5ea6\u63d0\u9ad8\u81f3\u201c\u7acb\u65b9\u5173\u7cfb\u201d\u201c\u56db\u6b21\u65b9\u5173\u7cfb\u201d\uff0c\u4ee5\u6b64\u7c7b\u63a8\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#222","title":"2.2.2 \u00a0 \u9012\u5f52","text":"<p>\u300c\u9012\u5f52 recursion\u300d\u662f\u4e00\u79cd\u7b97\u6cd5\u7b56\u7565\uff0c\u901a\u8fc7\u51fd\u6570\u8c03\u7528\u81ea\u8eab\u6765\u89e3\u51b3\u95ee\u9898\u3002\u5b83\u4e3b\u8981\u5305\u542b\u4e24\u4e2a\u9636\u6bb5\u3002</p> <ol> <li>\u9012\uff1a\u7a0b\u5e8f\u4e0d\u65ad\u6df1\u5165\u5730\u8c03\u7528\u81ea\u8eab\uff0c\u901a\u5e38\u4f20\u5165\u66f4\u5c0f\u6216\u66f4\u7b80\u5316\u7684\u53c2\u6570\uff0c\u76f4\u5230\u8fbe\u5230\u201c\u7ec8\u6b62\u6761\u4ef6\u201d\u3002</li> <li>\u5f52\uff1a\u89e6\u53d1\u201c\u7ec8\u6b62\u6761\u4ef6\u201d\u540e\uff0c\u7a0b\u5e8f\u4ece\u6700\u6df1\u5c42\u7684\u9012\u5f52\u51fd\u6570\u5f00\u59cb\u9010\u5c42\u8fd4\u56de\uff0c\u6c47\u805a\u6bcf\u4e00\u5c42\u7684\u7ed3\u679c\u3002</li> </ol> <p>\u800c\u4ece\u5b9e\u73b0\u7684\u89d2\u5ea6\u770b\uff0c\u9012\u5f52\u4ee3\u7801\u4e3b\u8981\u5305\u542b\u4e09\u4e2a\u8981\u7d20\u3002</p> <ol> <li>\u7ec8\u6b62\u6761\u4ef6\uff1a\u7528\u4e8e\u51b3\u5b9a\u4ec0\u4e48\u65f6\u5019\u7531\u201c\u9012\u201d\u8f6c\u201c\u5f52\u201d\u3002</li> <li>\u9012\u5f52\u8c03\u7528\uff1a\u5bf9\u5e94\u201c\u9012\u201d\uff0c\u51fd\u6570\u8c03\u7528\u81ea\u8eab\uff0c\u901a\u5e38\u8f93\u5165\u66f4\u5c0f\u6216\u66f4\u7b80\u5316\u7684\u53c2\u6570\u3002</li> <li>\u8fd4\u56de\u7ed3\u679c\uff1a\u5bf9\u5e94\u201c\u5f52\u201d\uff0c\u5c06\u5f53\u524d\u9012\u5f52\u5c42\u7ea7\u7684\u7ed3\u679c\u8fd4\u56de\u81f3\u4e0a\u4e00\u5c42\u3002</li> </ol> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6211\u4eec\u53ea\u9700\u8c03\u7528\u51fd\u6570 <code>recur(n)</code>  \uff0c\u5c31\u53ef\u4ee5\u5b8c\u6210 \\(1 + 2 + \\dots + n\\) \u7684\u8ba1\u7b97\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def recur(n: int) -&gt; int:\n    \"\"\"\u9012\u5f52\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6\n    if n == 1:\n        return 1\n    # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    res = recur(n - 1)\n    # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res\n</code></pre> recursion.cpp<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.java<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.cs<pre><code>/* \u9012\u5f52 */\nint Recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = Recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.go<pre><code>/* \u9012\u5f52 */\nfunc recur(n int) int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 1 {\n        return 1\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    res := recur(n - 1)\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res\n}\n</code></pre> recursion.swift<pre><code>/* \u9012\u5f52 */\nfunc recur(n: Int) -&gt; Int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 1 {\n        return 1\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    let res = recur(n: n - 1)\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res\n}\n</code></pre> recursion.js<pre><code>/* \u9012\u5f52 */\nfunction recur(n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 1) return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    const res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.ts<pre><code>/* \u9012\u5f52 */\nfunction recur(n: number): number {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 1) return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    const res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.dart<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n  // \u7ec8\u6b62\u6761\u4ef6\n  if (n == 1) return 1;\n  // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n  int res = recur(n - 1);\n  // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n  return n + res;\n}\n</code></pre> recursion.rs<pre><code>/* \u9012\u5f52 */\nfn recur(n: i32) -&gt; i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 1 {\n        return 1;\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    let res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    n + res\n}\n</code></pre> recursion.c<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.kt<pre><code>/* \u9012\u5f52 */\nfun recur(n: Int): Int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1\n    // \u9012: \u9012\u5f52\u8c03\u7528\n    val res = recur(n - 1)\n    // \u5f52: \u8fd4\u56de\u7ed3\u679c\n    return n + res\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{recur}\n</code></pre> recursion.zig<pre><code>// \u9012\u5f52\u51fd\u6570\nfn recur(n: i32) i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1) {\n        return 1;\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    var res: i32 = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 2-3 \u5c55\u793a\u4e86\u8be5\u51fd\u6570\u7684\u9012\u5f52\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 2-3 \u00a0 \u6c42\u548c\u51fd\u6570\u7684\u9012\u5f52\u8fc7\u7a0b </p> <p>\u867d\u7136\u4ece\u8ba1\u7b97\u89d2\u5ea6\u770b\uff0c\u8fed\u4ee3\u4e0e\u9012\u5f52\u53ef\u4ee5\u5f97\u5230\u76f8\u540c\u7684\u7ed3\u679c\uff0c\u4f46\u5b83\u4eec\u4ee3\u8868\u4e86\u4e24\u79cd\u5b8c\u5168\u4e0d\u540c\u7684\u601d\u8003\u548c\u89e3\u51b3\u95ee\u9898\u7684\u8303\u5f0f\u3002</p> <ul> <li>\u8fed\u4ee3\uff1a\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u5730\u89e3\u51b3\u95ee\u9898\u3002\u4ece\u6700\u57fa\u7840\u7684\u6b65\u9aa4\u5f00\u59cb\uff0c\u7136\u540e\u4e0d\u65ad\u91cd\u590d\u6216\u7d2f\u52a0\u8fd9\u4e9b\u6b65\u9aa4\uff0c\u76f4\u5230\u4efb\u52a1\u5b8c\u6210\u3002</li> <li>\u9012\u5f52\uff1a\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u5730\u89e3\u51b3\u95ee\u9898\u3002\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u8fd9\u4e9b\u5b50\u95ee\u9898\u548c\u539f\u95ee\u9898\u5177\u6709\u76f8\u540c\u7684\u5f62\u5f0f\u3002\u63a5\u4e0b\u6765\u5c06\u5b50\u95ee\u9898\u7ee7\u7eed\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u5230\u57fa\u672c\u60c5\u51b5\u65f6\u505c\u6b62\uff08\u57fa\u672c\u60c5\u51b5\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff09\u3002</li> </ul> <p>\u4ee5\u4e0a\u8ff0\u6c42\u548c\u51fd\u6570\u4e3a\u4f8b\uff0c\u8bbe\u95ee\u9898 \\(f(n) = 1 + 2 + \\dots + n\\) \u3002</p> <ul> <li>\u8fed\u4ee3\uff1a\u5728\u5faa\u73af\u4e2d\u6a21\u62df\u6c42\u548c\u8fc7\u7a0b\uff0c\u4ece \\(1\\) \u904d\u5386\u5230 \\(n\\) \uff0c\u6bcf\u8f6e\u6267\u884c\u6c42\u548c\u64cd\u4f5c\uff0c\u5373\u53ef\u6c42\u5f97 \\(f(n)\\) \u3002</li> <li>\u9012\u5f52\uff1a\u5c06\u95ee\u9898\u5206\u89e3\u4e3a\u5b50\u95ee\u9898 \\(f(n) = n + f(n-1)\\) \uff0c\u4e0d\u65ad\uff08\u9012\u5f52\u5730\uff09\u5206\u89e3\u4e0b\u53bb\uff0c\u76f4\u81f3\u57fa\u672c\u60c5\u51b5 \\(f(1) = 1\\) \u65f6\u7ec8\u6b62\u3002</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1","title":"1. \u00a0 \u8c03\u7528\u6808","text":"<p>\u9012\u5f52\u51fd\u6570\u6bcf\u6b21\u8c03\u7528\u81ea\u8eab\u65f6\uff0c\u7cfb\u7edf\u90fd\u4f1a\u4e3a\u65b0\u5f00\u542f\u7684\u51fd\u6570\u5206\u914d\u5185\u5b58\uff0c\u4ee5\u5b58\u50a8\u5c40\u90e8\u53d8\u91cf\u3001\u8c03\u7528\u5730\u5740\u548c\u5176\u4ed6\u4fe1\u606f\u7b49\u3002\u8fd9\u5c06\u5bfc\u81f4\u4e24\u65b9\u9762\u7684\u7ed3\u679c\u3002</p> <ul> <li>\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u90fd\u5b58\u50a8\u5728\u79f0\u4e3a\u201c\u6808\u5e27\u7a7a\u95f4\u201d\u7684\u5185\u5b58\u533a\u57df\u4e2d\uff0c\u76f4\u81f3\u51fd\u6570\u8fd4\u56de\u540e\u624d\u4f1a\u88ab\u91ca\u653e\u3002\u56e0\u6b64\uff0c\u9012\u5f52\u901a\u5e38\u6bd4\u8fed\u4ee3\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u9012\u5f52\u8c03\u7528\u51fd\u6570\u4f1a\u4ea7\u751f\u989d\u5916\u7684\u5f00\u9500\u3002\u56e0\u6b64\u9012\u5f52\u901a\u5e38\u6bd4\u5faa\u73af\u7684\u65f6\u95f4\u6548\u7387\u66f4\u4f4e\u3002</li> </ul> <p>\u5982\u56fe 2-4 \u6240\u793a\uff0c\u5728\u89e6\u53d1\u7ec8\u6b62\u6761\u4ef6\u524d\uff0c\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684\u9012\u5f52\u51fd\u6570\uff0c\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \u3002</p> <p></p> <p> \u56fe 2-4 \u00a0 \u9012\u5f52\u8c03\u7528\u6df1\u5ea6 </p> <p>\u5728\u5b9e\u9645\u4e2d\uff0c\u7f16\u7a0b\u8bed\u8a00\u5141\u8bb8\u7684\u9012\u5f52\u6df1\u5ea6\u901a\u5e38\u662f\u6709\u9650\u7684\uff0c\u8fc7\u6df1\u7684\u9012\u5f52\u53ef\u80fd\u5bfc\u81f4\u6808\u6ea2\u51fa\u9519\u8bef\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2","title":"2. \u00a0 \u5c3e\u9012\u5f52","text":"<p>\u6709\u8da3\u7684\u662f\uff0c\u5982\u679c\u51fd\u6570\u5728\u8fd4\u56de\u524d\u7684\u6700\u540e\u4e00\u6b65\u624d\u8fdb\u884c\u9012\u5f52\u8c03\u7528\uff0c\u5219\u8be5\u51fd\u6570\u53ef\u4ee5\u88ab\u7f16\u8bd1\u5668\u6216\u89e3\u91ca\u5668\u4f18\u5316\uff0c\u4f7f\u5176\u5728\u7a7a\u95f4\u6548\u7387\u4e0a\u4e0e\u8fed\u4ee3\u76f8\u5f53\u3002\u8fd9\u79cd\u60c5\u51b5\u88ab\u79f0\u4e3a\u300c\u5c3e\u9012\u5f52 tail recursion\u300d\u3002</p> <ul> <li>\u666e\u901a\u9012\u5f52\uff1a\u5f53\u51fd\u6570\u8fd4\u56de\u5230\u4e0a\u4e00\u5c42\u7ea7\u7684\u51fd\u6570\u540e\uff0c\u9700\u8981\u7ee7\u7eed\u6267\u884c\u4ee3\u7801\uff0c\u56e0\u6b64\u7cfb\u7edf\u9700\u8981\u4fdd\u5b58\u4e0a\u4e00\u5c42\u8c03\u7528\u7684\u4e0a\u4e0b\u6587\u3002</li> <li>\u5c3e\u9012\u5f52\uff1a\u9012\u5f52\u8c03\u7528\u662f\u51fd\u6570\u8fd4\u56de\u524d\u7684\u6700\u540e\u4e00\u4e2a\u64cd\u4f5c\uff0c\u8fd9\u610f\u5473\u7740\u51fd\u6570\u8fd4\u56de\u5230\u4e0a\u4e00\u5c42\u7ea7\u540e\uff0c\u65e0\u987b\u7ee7\u7eed\u6267\u884c\u5176\u4ed6\u64cd\u4f5c\uff0c\u56e0\u6b64\u7cfb\u7edf\u65e0\u987b\u4fdd\u5b58\u4e0a\u4e00\u5c42\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u3002</li> </ul> <p>\u4ee5\u8ba1\u7b97 \\(1 + 2 + \\dots + n\\) \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u7ed3\u679c\u53d8\u91cf <code>res</code> \u8bbe\u4e3a\u51fd\u6570\u53c2\u6570\uff0c\u4ece\u800c\u5b9e\u73b0\u5c3e\u9012\u5f52\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def tail_recur(n, res):\n    \"\"\"\u5c3e\u9012\u5f52\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6\n    if n == 0:\n        return res\n    # \u5c3e\u9012\u5f52\u8c03\u7528\n    return tail_recur(n - 1, res + n)\n</code></pre> recursion.cpp<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.java<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.cs<pre><code>/* \u5c3e\u9012\u5f52 */\nint TailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return TailRecur(n - 1, res + n);\n}\n</code></pre> recursion.go<pre><code>/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n int, res int) int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 0 {\n        return res\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n-1, res+n)\n}\n</code></pre> recursion.swift<pre><code>/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n: Int, res: Int) -&gt; Int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 0 {\n        return res\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n: n - 1, res: res + n)\n}\n</code></pre> recursion.js<pre><code>/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n, res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 0) return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.ts<pre><code>/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n: number, res: number): number {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 0) return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.dart<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n  // \u7ec8\u6b62\u6761\u4ef6\n  if (n == 0) return res;\n  // \u5c3e\u9012\u5f52\u8c03\u7528\n  return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.rs<pre><code>/* \u5c3e\u9012\u5f52 */\nfn tail_recur(n: i32, res: i32) -&gt; i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 0 {\n        return res;\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    tail_recur(n - 1, res + n)\n}\n</code></pre> recursion.c<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.kt<pre><code>/* \u5c3e\u9012\u5f52 */\ntailrec fun tailRecur(n: Int, res: Int): Int {\n    // \u6dfb\u52a0 tailrec \u5173\u952e\u8bcd\uff0c\u4ee5\u5f00\u542f\u5c3e\u9012\u5f52\u4f18\u5316\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n)\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{tail_recur}\n</code></pre> recursion.zig<pre><code>// \u5c3e\u9012\u5f52\u51fd\u6570\nfn tailRecur(n: i32, res: i32) i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0) {\n        return res;\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5c3e\u9012\u5f52\u7684\u6267\u884c\u8fc7\u7a0b\u5982\u56fe 2-5 \u6240\u793a\u3002\u5bf9\u6bd4\u666e\u901a\u9012\u5f52\u548c\u5c3e\u9012\u5f52\uff0c\u4e24\u8005\u7684\u6c42\u548c\u64cd\u4f5c\u7684\u6267\u884c\u70b9\u662f\u4e0d\u540c\u7684\u3002</p> <ul> <li>\u666e\u901a\u9012\u5f52\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u4e2d\u6267\u884c\u7684\uff0c\u6bcf\u5c42\u8fd4\u56de\u540e\u90fd\u8981\u518d\u6267\u884c\u4e00\u6b21\u6c42\u548c\u64cd\u4f5c\u3002</li> <li>\u5c3e\u9012\u5f52\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u9012\u201d\u7684\u8fc7\u7a0b\u4e2d\u6267\u884c\u7684\uff0c\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u53ea\u9700\u5c42\u5c42\u8fd4\u56de\u3002</li> </ul> <p></p> <p> \u56fe 2-5 \u00a0 \u5c3e\u9012\u5f52\u8fc7\u7a0b </p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u8bb8\u591a\u7f16\u8bd1\u5668\u6216\u89e3\u91ca\u5668\u5e76\u4e0d\u652f\u6301\u5c3e\u9012\u5f52\u4f18\u5316\u3002\u4f8b\u5982\uff0cPython \u9ed8\u8ba4\u4e0d\u652f\u6301\u5c3e\u9012\u5f52\u4f18\u5316\uff0c\u56e0\u6b64\u5373\u4f7f\u51fd\u6570\u662f\u5c3e\u9012\u5f52\u5f62\u5f0f\uff0c\u4ecd\u7136\u53ef\u80fd\u4f1a\u9047\u5230\u6808\u6ea2\u51fa\u95ee\u9898\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3_1","title":"3. \u00a0 \u9012\u5f52\u6811","text":"<p>\u5f53\u5904\u7406\u4e0e\u201c\u5206\u6cbb\u201d\u76f8\u5173\u7684\u7b97\u6cd5\u95ee\u9898\u65f6\uff0c\u9012\u5f52\u5f80\u5f80\u6bd4\u8fed\u4ee3\u7684\u601d\u8def\u66f4\u52a0\u76f4\u89c2\u3001\u4ee3\u7801\u66f4\u52a0\u6613\u8bfb\u3002\u4ee5\u201c\u6590\u6ce2\u90a3\u5951\u6570\u5217\u201d\u4e3a\u4f8b\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6590\u6ce2\u90a3\u5951\u6570\u5217 \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\) \uff0c\u6c42\u8be5\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\u3002</p> <p>\u8bbe\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\u4e3a \\(f(n)\\) \uff0c\u6613\u5f97\u4e24\u4e2a\u7ed3\u8bba\u3002</p> <ul> <li>\u6570\u5217\u7684\u524d\u4e24\u4e2a\u6570\u5b57\u4e3a \\(f(1) = 0\\) \u548c \\(f(2) = 1\\) \u3002</li> <li>\u6570\u5217\u4e2d\u7684\u6bcf\u4e2a\u6570\u5b57\u662f\u524d\u4e24\u4e2a\u6570\u5b57\u7684\u548c\uff0c\u5373 \\(f(n) = f(n - 1) + f(n - 2)\\) \u3002</li> </ul> <p>\u6309\u7167\u9012\u63a8\u5173\u7cfb\u8fdb\u884c\u9012\u5f52\u8c03\u7528\uff0c\u5c06\u524d\u4e24\u4e2a\u6570\u5b57\u4f5c\u4e3a\u7ec8\u6b62\u6761\u4ef6\uff0c\u4fbf\u53ef\u5199\u51fa\u9012\u5f52\u4ee3\u7801\u3002\u8c03\u7528 <code>fib(n)</code> \u5373\u53ef\u5f97\u5230\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def fib(n: int) -&gt; int:\n    \"\"\"\u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 or n == 2:\n        return n - 1\n    # \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    res = fib(n - 1) + fib(n - 2)\n    # \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n</code></pre> recursion.cpp<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint Fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = Fib(n - 1) + Fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n int) int {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    res := fib(n-1) + fib(n-2)\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n: Int) -&gt; Int {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    let res = fib(n: n - 1) + fib(n: n - 2)\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n}\n</code></pre> recursion.js<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n: number): number {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n  // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n  if (n == 1 || n == 2) return n - 1;\n  // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n  int res = fib(n - 1) + fib(n - 2);\n  // \u8fd4\u56de\u7ed3\u679c f(n)\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfn fib(n: i32) -&gt; i32 {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1;\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    let res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c\n    res\n}\n</code></pre> recursion.c<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfun fib(n: Int): Int {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    val res = fib(n - 1) + fib(n - 2)\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{fib}\n</code></pre> recursion.zig<pre><code>// \u6590\u6ce2\u90a3\u5951\u6570\u5217\nfn fib(n: i32) i32 {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 or n == 2) {\n        return n - 1;\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    var res: i32 = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u6211\u4eec\u5728\u51fd\u6570\u5185\u9012\u5f52\u8c03\u7528\u4e86\u4e24\u4e2a\u51fd\u6570\uff0c\u8fd9\u610f\u5473\u7740\u4ece\u4e00\u4e2a\u8c03\u7528\u4ea7\u751f\u4e86\u4e24\u4e2a\u8c03\u7528\u5206\u652f\u3002\u5982\u56fe 2-6 \u6240\u793a\uff0c\u8fd9\u6837\u4e0d\u65ad\u9012\u5f52\u8c03\u7528\u4e0b\u53bb\uff0c\u6700\u7ec8\u5c06\u4ea7\u751f\u4e00\u68f5\u5c42\u6570\u4e3a \\(n\\) \u7684\u300c\u9012\u5f52\u6811 recursion tree\u300d\u3002</p> <p></p> <p> \u56fe 2-6 \u00a0 \u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u9012\u5f52\u6811 </p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9012\u5f52\u4f53\u73b0\u4e86\u201c\u5c06\u95ee\u9898\u5206\u89e3\u4e3a\u66f4\u5c0f\u5b50\u95ee\u9898\u201d\u7684\u601d\u7ef4\u8303\u5f0f\uff0c\u8fd9\u79cd\u5206\u6cbb\u7b56\u7565\u81f3\u5173\u91cd\u8981\u3002</p> <ul> <li>\u4ece\u7b97\u6cd5\u89d2\u5ea6\u770b\uff0c\u641c\u7d22\u3001\u6392\u5e8f\u3001\u56de\u6eaf\u3001\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u7b49\u8bb8\u591a\u91cd\u8981\u7b97\u6cd5\u7b56\u7565\u76f4\u63a5\u6216\u95f4\u63a5\u5730\u5e94\u7528\u4e86\u8fd9\u79cd\u601d\u7ef4\u65b9\u5f0f\u3002</li> <li>\u4ece\u6570\u636e\u7ed3\u6784\u89d2\u5ea6\u770b\uff0c\u9012\u5f52\u5929\u7136\u9002\u5408\u5904\u7406\u94fe\u8868\u3001\u6811\u548c\u56fe\u7684\u76f8\u5173\u95ee\u9898\uff0c\u56e0\u4e3a\u5b83\u4eec\u975e\u5e38\u9002\u5408\u7528\u5206\u6cbb\u601d\u60f3\u8fdb\u884c\u5206\u6790\u3002</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#223","title":"2.2.3 \u00a0 \u4e24\u8005\u5bf9\u6bd4","text":"<p>\u603b\u7ed3\u4ee5\u4e0a\u5185\u5bb9\uff0c\u5982\u8868 2-1 \u6240\u793a\uff0c\u8fed\u4ee3\u548c\u9012\u5f52\u5728\u5b9e\u73b0\u3001\u6027\u80fd\u548c\u9002\u7528\u6027\u4e0a\u6709\u6240\u4e0d\u540c\u3002</p> <p> \u8868 2-1 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52\u7279\u70b9\u5bf9\u6bd4 </p> \u8fed\u4ee3 \u9012\u5f52 \u5b9e\u73b0\u65b9\u5f0f \u5faa\u73af\u7ed3\u6784 \u51fd\u6570\u8c03\u7528\u81ea\u8eab \u65f6\u95f4\u6548\u7387 \u6548\u7387\u901a\u5e38\u8f83\u9ad8\uff0c\u65e0\u51fd\u6570\u8c03\u7528\u5f00\u9500 \u6bcf\u6b21\u51fd\u6570\u8c03\u7528\u90fd\u4f1a\u4ea7\u751f\u5f00\u9500 \u5185\u5b58\u4f7f\u7528 \u901a\u5e38\u4f7f\u7528\u56fa\u5b9a\u5927\u5c0f\u7684\u5185\u5b58\u7a7a\u95f4 \u7d2f\u79ef\u51fd\u6570\u8c03\u7528\u53ef\u80fd\u4f7f\u7528\u5927\u91cf\u7684\u6808\u5e27\u7a7a\u95f4 \u9002\u7528\u95ee\u9898 \u9002\u7528\u4e8e\u7b80\u5355\u5faa\u73af\u4efb\u52a1\uff0c\u4ee3\u7801\u76f4\u89c2\u3001\u53ef\u8bfb\u6027\u597d \u9002\u7528\u4e8e\u5b50\u95ee\u9898\u5206\u89e3\uff0c\u5982\u6811\u3001\u56fe\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u7b49\uff0c\u4ee3\u7801\u7ed3\u6784\u7b80\u6d01\u3001\u6e05\u6670 <p>Tip</p> <p>\u5982\u679c\u611f\u89c9\u4ee5\u4e0b\u5185\u5bb9\u7406\u89e3\u56f0\u96be\uff0c\u53ef\u4ee5\u5728\u8bfb\u5b8c\u201c\u6808\u201d\u7ae0\u8282\u540e\u518d\u6765\u590d\u4e60\u3002</p> <p>\u90a3\u4e48\uff0c\u8fed\u4ee3\u548c\u9012\u5f52\u5177\u6709\u4ec0\u4e48\u5185\u5728\u8054\u7cfb\u5462\uff1f\u4ee5\u4e0a\u8ff0\u9012\u5f52\u51fd\u6570\u4e3a\u4f8b\uff0c\u6c42\u548c\u64cd\u4f5c\u5728\u9012\u5f52\u7684\u201c\u5f52\u201d\u9636\u6bb5\u8fdb\u884c\u3002\u8fd9\u610f\u5473\u7740\u6700\u521d\u88ab\u8c03\u7528\u7684\u51fd\u6570\u5b9e\u9645\u4e0a\u662f\u6700\u540e\u5b8c\u6210\u5176\u6c42\u548c\u64cd\u4f5c\u7684\uff0c\u8fd9\u79cd\u5de5\u4f5c\u673a\u5236\u4e0e\u6808\u7684\u201c\u5148\u5165\u540e\u51fa\u201d\u539f\u5219\u5f02\u66f2\u540c\u5de5\u3002</p> <p>\u4e8b\u5b9e\u4e0a\uff0c\u201c\u8c03\u7528\u6808\u201d\u548c\u201c\u6808\u5e27\u7a7a\u95f4\u201d\u8fd9\u7c7b\u9012\u5f52\u672f\u8bed\u5df2\u7ecf\u6697\u793a\u4e86\u9012\u5f52\u4e0e\u6808\u4e4b\u95f4\u7684\u5bc6\u5207\u5173\u7cfb\u3002</p> <ol> <li>\u9012\uff1a\u5f53\u51fd\u6570\u88ab\u8c03\u7528\u65f6\uff0c\u7cfb\u7edf\u4f1a\u5728\u201c\u8c03\u7528\u6808\u201d\u4e0a\u4e3a\u8be5\u51fd\u6570\u5206\u914d\u65b0\u7684\u6808\u5e27\uff0c\u7528\u4e8e\u5b58\u50a8\u51fd\u6570\u7684\u5c40\u90e8\u53d8\u91cf\u3001\u53c2\u6570\u3001\u8fd4\u56de\u5730\u5740\u7b49\u6570\u636e\u3002</li> <li>\u5f52\uff1a\u5f53\u51fd\u6570\u5b8c\u6210\u6267\u884c\u5e76\u8fd4\u56de\u65f6\uff0c\u5bf9\u5e94\u7684\u6808\u5e27\u4f1a\u88ab\u4ece\u201c\u8c03\u7528\u6808\u201d\u4e0a\u79fb\u9664\uff0c\u6062\u590d\u4e4b\u524d\u51fd\u6570\u7684\u6267\u884c\u73af\u5883\u3002</li> </ol> <p>\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u8c03\u7528\u6808\u7684\u884c\u4e3a\uff0c\u4ece\u800c\u5c06\u9012\u5f52\u8f6c\u5316\u4e3a\u8fed\u4ee3\u5f62\u5f0f\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def for_loop_recur(n: int) -&gt; int:\n    \"\"\"\u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\"\"\"\n    # \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    stack = []\n    res = 0\n    # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i in range(n, 0, -1):\n        # \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.append(i)\n    # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while stack:\n        # \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop()\n    # res = 1+2+3+...+n\n    return res\n</code></pre> recursion.cpp<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    stack&lt;int&gt; stack;\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (!stack.empty()) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.top();\n        stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    Stack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (!stack.isEmpty()) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint ForLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    Stack&lt;int&gt; stack = new();\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.Push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (stack.Count &gt; 0) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.Pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n int) int {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    stack := list.New()\n    res := 0\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i := n; i &gt; 0; i-- {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.PushBack(i)\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    for stack.Len() != 0 {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.Back().Value.(int)\n        stack.Remove(stack.Back())\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n: Int) -&gt; Int {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    var stack: [Int] = []\n    var res = 0\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i in (1 ... n).reversed() {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.append(i)\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while !stack.isEmpty {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.removeLast()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.js<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    const stack = [];\n    let res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (let i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (stack.length) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n: number): number {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808 \n    const stack: number[] = [];\n    let res: number = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (let i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (stack.length) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n  // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n  List&lt;int&gt; stack = [];\n  int res = 0;\n  // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n  for (int i = n; i &gt; 0; i--) {\n    // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n    stack.add(i);\n  }\n  // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n  while (!stack.isEmpty) {\n    // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n    res += stack.removeLast();\n  }\n  // res = 1+2+3+...+n\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfn for_loop_recur(n: i32) -&gt; i32 {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    let mut stack = Vec::new();\n    let mut res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i in (1..=n).rev() {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while !stack.is_empty() {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop().unwrap();\n    }\n    // res = 1+2+3+...+n\n    res\n}\n</code></pre> recursion.c<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n    int stack[1000]; // \u501f\u52a9\u4e00\u4e2a\u5927\u6570\u7ec4\u6765\u6a21\u62df\u6808\n    int top = -1;    // \u6808\u9876\u7d22\u5f15\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack[1 + top++] = i;\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (top &gt;= 0) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack[top--];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfun forLoopRecur(n: Int): Int {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    val stack = Stack&lt;Int&gt;()\n    var res = 0\n    // \u9012: \u9012\u5f52\u8c03\u7528\n    for (i in n downTo 0) {\n        stack.push(i)\n    }\n    // \u5f52: \u8fd4\u56de\u7ed3\u679c\n    while (stack.isNotEmpty()) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{for_loop_recur}\n</code></pre> recursion.zig<pre><code>// \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\nfn forLoopRecur(comptime n: i32) i32 {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    var stack: [n]i32 = undefined;\n    var res: i32 = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    var i: usize = n;\n    while (i &gt; 0) {\n        stack[i - 1] = @intCast(i);\n        i -= 1;\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    var index: usize = n;\n    while (index &gt; 0) {\n        index -= 1;\n        res += stack[index];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u5f53\u9012\u5f52\u8f6c\u5316\u4e3a\u8fed\u4ee3\u540e\uff0c\u4ee3\u7801\u53d8\u5f97\u66f4\u52a0\u590d\u6742\u4e86\u3002\u5c3d\u7ba1\u8fed\u4ee3\u548c\u9012\u5f52\u5728\u5f88\u591a\u60c5\u51b5\u4e0b\u53ef\u4ee5\u4e92\u76f8\u8f6c\u5316\uff0c\u4f46\u4e0d\u4e00\u5b9a\u503c\u5f97\u8fd9\u6837\u505a\uff0c\u6709\u4ee5\u4e0b\u4e24\u70b9\u539f\u56e0\u3002</p> <ul> <li>\u8f6c\u5316\u540e\u7684\u4ee3\u7801\u53ef\u80fd\u66f4\u52a0\u96be\u4ee5\u7406\u89e3\uff0c\u53ef\u8bfb\u6027\u66f4\u5dee\u3002</li> <li>\u5bf9\u4e8e\u67d0\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\u7684\u884c\u4e3a\u53ef\u80fd\u975e\u5e38\u56f0\u96be\u3002</li> </ul> <p>\u603b\u4e4b\uff0c\u9009\u62e9\u8fed\u4ee3\u8fd8\u662f\u9012\u5f52\u53d6\u51b3\u4e8e\u7279\u5b9a\u95ee\u9898\u7684\u6027\u8d28\u3002\u5728\u7f16\u7a0b\u5b9e\u8df5\u4e2d\uff0c\u6743\u8861\u4e24\u8005\u7684\u4f18\u52a3\u5e76\u6839\u636e\u60c5\u5883\u9009\u62e9\u5408\u9002\u7684\u65b9\u6cd5\u81f3\u5173\u91cd\u8981\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1 \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30","text":"<p>\u5728\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\uff0c\u6211\u4eec\u5148\u540e\u8ffd\u6c42\u4ee5\u4e0b\u4e24\u4e2a\u5c42\u9762\u7684\u76ee\u6807\u3002</p> <ol> <li>\u627e\u5230\u95ee\u9898\u89e3\u6cd5\uff1a\u7b97\u6cd5\u9700\u8981\u5728\u89c4\u5b9a\u7684\u8f93\u5165\u8303\u56f4\u5185\u53ef\u9760\u5730\u6c42\u5f97\u95ee\u9898\u7684\u6b63\u786e\u89e3\u3002</li> <li>\u5bfb\u6c42\u6700\u4f18\u89e3\u6cd5\uff1a\u540c\u4e00\u4e2a\u95ee\u9898\u53ef\u80fd\u5b58\u5728\u591a\u79cd\u89e3\u6cd5\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u5c3d\u53ef\u80fd\u9ad8\u6548\u7684\u7b97\u6cd5\u3002</li> </ol> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5728\u80fd\u591f\u89e3\u51b3\u95ee\u9898\u7684\u524d\u63d0\u4e0b\uff0c\u7b97\u6cd5\u6548\u7387\u5df2\u6210\u4e3a\u8861\u91cf\u7b97\u6cd5\u4f18\u52a3\u7684\u4e3b\u8981\u8bc4\u4ef7\u6307\u6807\uff0c\u5b83\u5305\u62ec\u4ee5\u4e0b\u4e24\u4e2a\u7ef4\u5ea6\u3002</p> <ul> <li>\u65f6\u95f4\u6548\u7387\uff1a\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u5feb\u6162\u3002</li> <li>\u7a7a\u95f4\u6548\u7387\uff1a\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u7684\u5927\u5c0f\u3002</li> </ul> <p>\u7b80\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u8bbe\u8ba1\u201c\u65e2\u5feb\u53c8\u7701\u201d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u3002\u800c\u6709\u6548\u5730\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u81f3\u5173\u91cd\u8981\uff0c\u56e0\u4e3a\u53ea\u6709\u8fd9\u6837\uff0c\u6211\u4eec\u624d\u80fd\u5c06\u5404\u79cd\u7b97\u6cd5\u8fdb\u884c\u5bf9\u6bd4\uff0c\u8fdb\u800c\u6307\u5bfc\u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u4f18\u5316\u8fc7\u7a0b\u3002</p> <p>\u6548\u7387\u8bc4\u4f30\u65b9\u6cd5\u4e3b\u8981\u5206\u4e3a\u4e24\u79cd\uff1a\u5b9e\u9645\u6d4b\u8bd5\u3001\u7406\u8bba\u4f30\u7b97\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#211","title":"2.1.1 \u00a0 \u5b9e\u9645\u6d4b\u8bd5","text":"<p>\u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u7b97\u6cd5 <code>A</code> \u548c\u7b97\u6cd5 <code>B</code> \uff0c\u5b83\u4eec\u90fd\u80fd\u89e3\u51b3\u540c\u4e00\u95ee\u9898\uff0c\u73b0\u5728\u9700\u8981\u5bf9\u6bd4\u8fd9\u4e24\u4e2a\u7b97\u6cd5\u7684\u6548\u7387\u3002\u6700\u76f4\u63a5\u7684\u65b9\u6cd5\u662f\u627e\u4e00\u53f0\u8ba1\u7b97\u673a\uff0c\u8fd0\u884c\u8fd9\u4e24\u4e2a\u7b97\u6cd5\uff0c\u5e76\u76d1\u63a7\u8bb0\u5f55\u5b83\u4eec\u7684\u8fd0\u884c\u65f6\u95f4\u548c\u5185\u5b58\u5360\u7528\u60c5\u51b5\u3002\u8fd9\u79cd\u8bc4\u4f30\u65b9\u5f0f\u80fd\u591f\u53cd\u6620\u771f\u5b9e\u60c5\u51b5\uff0c\u4f46\u4e5f\u5b58\u5728\u8f83\u5927\u7684\u5c40\u9650\u6027\u3002</p> <p>\u4e00\u65b9\u9762\uff0c\u96be\u4ee5\u6392\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5e72\u6270\u56e0\u7d20\u3002\u786c\u4ef6\u914d\u7f6e\u4f1a\u5f71\u54cd\u7b97\u6cd5\u7684\u6027\u80fd\u3002\u6bd4\u5982\u5728\u67d0\u53f0\u8ba1\u7b97\u673a\u4e2d\uff0c\u7b97\u6cd5 <code>A</code> \u7684\u8fd0\u884c\u65f6\u95f4\u6bd4\u7b97\u6cd5 <code>B</code> \u77ed\uff1b\u4f46\u5728\u53e6\u4e00\u53f0\u914d\u7f6e\u4e0d\u540c\u7684\u8ba1\u7b97\u673a\u4e2d\uff0c\u53ef\u80fd\u5f97\u5230\u76f8\u53cd\u7684\u6d4b\u8bd5\u7ed3\u679c\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u9700\u8981\u5728\u5404\u79cd\u673a\u5668\u4e0a\u8fdb\u884c\u6d4b\u8bd5\uff0c\u7edf\u8ba1\u5e73\u5747\u6548\u7387\uff0c\u800c\u8fd9\u662f\u4e0d\u73b0\u5b9e\u7684\u3002</p> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5c55\u5f00\u5b8c\u6574\u6d4b\u8bd5\u975e\u5e38\u8017\u8d39\u8d44\u6e90\u3002\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u7684\u53d8\u5316\uff0c\u7b97\u6cd5\u4f1a\u8868\u73b0\u51fa\u4e0d\u540c\u7684\u6548\u7387\u3002\u4f8b\u5982\uff0c\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 <code>A</code> \u7684\u8fd0\u884c\u65f6\u95f4\u6bd4\u7b97\u6cd5 <code>B</code> \u77ed\uff1b\u800c\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5927\u65f6\uff0c\u6d4b\u8bd5\u7ed3\u679c\u53ef\u80fd\u6070\u6070\u76f8\u53cd\u3002\u56e0\u6b64\uff0c\u4e3a\u4e86\u5f97\u5230\u6709\u8bf4\u670d\u529b\u7684\u7ed3\u8bba\uff0c\u6211\u4eec\u9700\u8981\u6d4b\u8bd5\u5404\u79cd\u89c4\u6a21\u7684\u8f93\u5165\u6570\u636e\uff0c\u800c\u8fd9\u9700\u8981\u8017\u8d39\u5927\u91cf\u7684\u8ba1\u7b97\u8d44\u6e90\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#212","title":"2.1.2 \u00a0 \u7406\u8bba\u4f30\u7b97","text":"<p>\u7531\u4e8e\u5b9e\u9645\u6d4b\u8bd5\u5177\u6709\u8f83\u5927\u7684\u5c40\u9650\u6027\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u4ec5\u901a\u8fc7\u4e00\u4e9b\u8ba1\u7b97\u6765\u8bc4\u4f30\u7b97\u6cd5\u7684\u6548\u7387\u3002\u8fd9\u79cd\u4f30\u7b97\u65b9\u6cd5\u88ab\u79f0\u4e3a\u300c\u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 asymptotic complexity analysis\u300d\uff0c\u7b80\u79f0\u300c\u590d\u6742\u5ea6\u5206\u6790\u300d\u3002</p> <p>\u590d\u6742\u5ea6\u5206\u6790\u80fd\u591f\u4f53\u73b0\u7b97\u6cd5\u8fd0\u884c\u6240\u9700\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u5b83\u63cf\u8ff0\u4e86\u968f\u7740\u8f93\u5165\u6570\u636e\u5927\u5c0f\u7684\u589e\u52a0\uff0c\u7b97\u6cd5\u6267\u884c\u6240\u9700\u65f6\u95f4\u548c\u7a7a\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u5b9a\u4e49\u6709\u4e9b\u62d7\u53e3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u5206\u4e3a\u4e09\u4e2a\u91cd\u70b9\u6765\u7406\u89e3\u3002</p> <ul> <li>\u201c\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\u201d\u5206\u522b\u5bf9\u5e94\u300c\u65f6\u95f4\u590d\u6742\u5ea6 time complexity\u300d\u548c\u300c\u7a7a\u95f4\u590d\u6742\u5ea6 space complexity\u300d\u3002</li> <li>\u201c\u968f\u7740\u8f93\u5165\u6570\u636e\u5927\u5c0f\u7684\u589e\u52a0\u201d\u610f\u5473\u7740\u590d\u6742\u5ea6\u53cd\u6620\u4e86\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u4e0e\u8f93\u5165\u6570\u636e\u4f53\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u3002</li> <li>\u201c\u65f6\u95f4\u548c\u7a7a\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u201d\u8868\u793a\u590d\u6742\u5ea6\u5206\u6790\u5173\u6ce8\u7684\u4e0d\u662f\u8fd0\u884c\u65f6\u95f4\u6216\u5360\u7528\u7a7a\u95f4\u7684\u5177\u4f53\u503c\uff0c\u800c\u662f\u65f6\u95f4\u6216\u7a7a\u95f4\u589e\u957f\u7684\u201c\u5feb\u6162\u201d\u3002</li> </ul> <p>\u590d\u6742\u5ea6\u5206\u6790\u514b\u670d\u4e86\u5b9e\u9645\u6d4b\u8bd5\u65b9\u6cd5\u7684\u5f0a\u7aef\uff0c\u4f53\u73b0\u5728\u4ee5\u4e0b\u4e24\u4e2a\u65b9\u9762\u3002</p> <ul> <li>\u5b83\u72ec\u7acb\u4e8e\u6d4b\u8bd5\u73af\u5883\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\u3002</li> <li>\u5b83\u53ef\u4ee5\u4f53\u73b0\u4e0d\u540c\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6548\u7387\uff0c\u5c24\u5176\u662f\u5728\u5927\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6027\u80fd\u3002</li> </ul> <p>Tip</p> <p>\u5982\u679c\u4f60\u4ecd\u5bf9\u590d\u6742\u5ea6\u7684\u6982\u5ff5\u611f\u5230\u56f0\u60d1\uff0c\u65e0\u987b\u62c5\u5fc3\uff0c\u6211\u4eec\u4f1a\u5728\u540e\u7eed\u7ae0\u8282\u4e2d\u8be6\u7ec6\u4ecb\u7ecd\u3002</p> <p>\u590d\u6742\u5ea6\u5206\u6790\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u4e00\u628a\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u7684\u201c\u6807\u5c3a\u201d\uff0c\u4f7f\u6211\u4eec\u53ef\u4ee5\u8861\u91cf\u6267\u884c\u67d0\u4e2a\u7b97\u6cd5\u6240\u9700\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\uff0c\u5bf9\u6bd4\u4e0d\u540c\u7b97\u6cd5\u4e4b\u95f4\u7684\u6548\u7387\u3002</p> <p>\u590d\u6742\u5ea6\u662f\u4e2a\u6570\u5b66\u6982\u5ff5\uff0c\u5bf9\u4e8e\u521d\u5b66\u8005\u53ef\u80fd\u6bd4\u8f83\u62bd\u8c61\uff0c\u5b66\u4e60\u96be\u5ea6\u76f8\u5bf9\u8f83\u9ad8\u3002\u4ece\u8fd9\u4e2a\u89d2\u5ea6\u770b\uff0c\u590d\u6742\u5ea6\u5206\u6790\u53ef\u80fd\u4e0d\u592a\u9002\u5408\u4f5c\u4e3a\u6700\u5148\u4ecb\u7ecd\u7684\u5185\u5bb9\u3002\u7136\u800c\uff0c\u5f53\u6211\u4eec\u8ba8\u8bba\u67d0\u4e2a\u6570\u636e\u7ed3\u6784\u6216\u7b97\u6cd5\u7684\u7279\u70b9\u65f6\uff0c\u96be\u4ee5\u907f\u514d\u8981\u5206\u6790\u5176\u8fd0\u884c\u901f\u5ea6\u548c\u7a7a\u95f4\u4f7f\u7528\u60c5\u51b5\u3002</p> <p>\u7efc\u4e0a\u6240\u8ff0\uff0c\u5efa\u8bae\u4f60\u5728\u6df1\u5165\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u524d\uff0c\u5148\u5bf9\u590d\u6742\u5ea6\u5206\u6790\u5efa\u7acb\u521d\u6b65\u7684\u4e86\u89e3\uff0c\u4ee5\u4fbf\u80fd\u591f\u5b8c\u6210\u7b80\u5355\u7b97\u6cd5\u7684\u590d\u6742\u5ea6\u5206\u6790\u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/","title":"2.4 \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6","text":"<p>\u300c\u7a7a\u95f4\u590d\u6742\u5ea6 space complexity\u300d\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u6982\u5ff5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u5e38\u7c7b\u4f3c\uff0c\u53ea\u9700\u5c06\u201c\u8fd0\u884c\u65f6\u95f4\u201d\u66ff\u6362\u4e3a\u201c\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u201d\u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/#241","title":"2.4.1 \u00a0 \u7b97\u6cd5\u76f8\u5173\u7a7a\u95f4","text":"<p>\u7b97\u6cd5\u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f7f\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u3002</p> <ul> <li>\u8f93\u5165\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u5165\u6570\u636e\u3002</li> <li>\u6682\u5b58\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u53d8\u91cf\u3001\u5bf9\u8c61\u3001\u51fd\u6570\u4e0a\u4e0b\u6587\u7b49\u6570\u636e\u3002</li> <li>\u8f93\u51fa\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u51fa\u6570\u636e\u3002</li> </ul> <p>\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u7edf\u8ba1\u8303\u56f4\u662f\u201c\u6682\u5b58\u7a7a\u95f4\u201d\u52a0\u4e0a\u201c\u8f93\u51fa\u7a7a\u95f4\u201d\u3002</p> <p>\u6682\u5b58\u7a7a\u95f4\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\u3002</p> <ul> <li>\u6682\u5b58\u6570\u636e\uff1a\u7528\u4e8e\u4fdd\u5b58\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u5404\u79cd\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u7b49\u3002</li> <li>\u6808\u5e27\u7a7a\u95f4\uff1a\u7528\u4e8e\u4fdd\u5b58\u8c03\u7528\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u3002\u7cfb\u7edf\u5728\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\u90fd\u4f1a\u5728\u6808\u9876\u90e8\u521b\u5efa\u4e00\u4e2a\u6808\u5e27\uff0c\u51fd\u6570\u8fd4\u56de\u540e\uff0c\u6808\u5e27\u7a7a\u95f4\u4f1a\u88ab\u91ca\u653e\u3002</li> <li>\u6307\u4ee4\u7a7a\u95f4\uff1a\u7528\u4e8e\u4fdd\u5b58\u7f16\u8bd1\u540e\u7684\u7a0b\u5e8f\u6307\u4ee4\uff0c\u5728\u5b9e\u9645\u7edf\u8ba1\u4e2d\u901a\u5e38\u5ffd\u7565\u4e0d\u8ba1\u3002</li> </ul> <p>\u5728\u5206\u6790\u4e00\u6bb5\u7a0b\u5e8f\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u7edf\u8ba1\u6682\u5b58\u6570\u636e\u3001\u6808\u5e27\u7a7a\u95f4\u548c\u8f93\u51fa\u6570\u636e\u4e09\u90e8\u5206\uff0c\u5982\u56fe 2-15 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 2-15 \u00a0 \u7b97\u6cd5\u4f7f\u7528\u7684\u76f8\u5173\u7a7a\u95f4 </p> <p>\u76f8\u5173\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class Node:\n    \"\"\"\u7c7b\"\"\"\n    def __init__(self, x: int):\n        self.val: int = x              # \u8282\u70b9\u503c\n        self.next: Node | None = None  # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n\ndef function() -&gt; int:\n    \"\"\"\u51fd\u6570\"\"\"\n    # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n\ndef algorithm(n) -&gt; int:  # \u8f93\u5165\u6570\u636e\n    A = 0                 # \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff0c\u4e00\u822c\u7528\u5927\u5199\u5b57\u6bcd\u8868\u793a\uff09\n    b = 0                 # \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    node = Node(0)        # \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    c = function()        # \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return A + b + c      # \u8f93\u51fa\u6570\u636e\n</code></pre> <pre><code>/* \u7ed3\u6784\u4f53 */\nstruct Node {\n    int val;\n    Node *next;\n    Node(int x) : val(x), next(nullptr) {}\n};\n\n/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\n    const int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    int b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    Node* node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    int c = func();           // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\n    int val;\n    Node next;\n    Node(int x) { val = x; }\n}\n\n/* \u51fd\u6570 */\nint function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\n    final int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    int b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    Node node = new Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    int c = function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node(int x) {\n    int val = x;\n    Node next;\n}\n\n/* \u51fd\u6570 */\nint Function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint Algorithm(int n) {        // \u8f93\u5165\u6570\u636e\n    const int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    int b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    Node node = new(0);       // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    int c = Function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7ed3\u6784\u4f53 */\ntype node struct {\n    val  int\n    next *node\n}\n\n/* \u521b\u5efa node \u7ed3\u6784\u4f53  */\nfunc newNode(val int) *node {\n    return &amp;node{val: val}\n}\n\n/* \u51fd\u6570 */\nfunc function() int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\nfunc algorithm(n int) int { // \u8f93\u5165\u6570\u636e\n    const a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    b := 0                  // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    newNode(0)              // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    c := function()         // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c        // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\n    var val: Int\n    var next: Node?\n\n    init(x: Int) {\n        val = x\n    }\n}\n\n/* \u51fd\u6570 */\nfunc function() -&gt; Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\nfunc algorithm(n: Int) -&gt; Int { // \u8f93\u5165\u6570\u636e\n    let a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    var b = 0             // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    let node = Node(x: 0) // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    let c = function()    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c      // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\n    val;\n    next;\n    constructor(val) {\n        this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n        this.next = null;                       // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n\n/* \u51fd\u6570 */\nfunction constFunc() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\nfunction algorithm(n) {       // \u8f93\u5165\u6570\u636e\n    const a = 0;              // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    let b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    const node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    const c = constFunc();    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\n    val: number;\n    next: Node | null;\n    constructor(val?: number) {\n        this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n        this.next = null;                       // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n\n/* \u51fd\u6570 */\nfunction constFunc(): number {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\nfunction algorithm(n: number): number { // \u8f93\u5165\u6570\u636e\n    const a = 0;                        // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    let b = 0;                          // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    const node = new Node(0);           // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    const c = constFunc();              // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;                   // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\n  int val;\n  Node next;\n  Node(this.val, [this.next]);\n}\n\n/* \u51fd\u6570 */\nint function() {\n  // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n  return 0;\n}\n\nint algorithm(int n) {  // \u8f93\u5165\u6570\u636e\n  const int a = 0;      // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n  int b = 0;            // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n  Node node = Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n  int c = function();   // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n  return a + b + c;     // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u7ed3\u6784\u4f53 */\nstruct Node {\n    val: i32,\n    next: Option&lt;Rc&lt;RefCell&lt;Node&gt;&gt;&gt;,\n}\n\n/* \u521b\u5efa Node \u7ed3\u6784\u4f53 */\nimpl Node {\n    fn new(val: i32) -&gt; Self {\n        Self { val: val, next: None }\n    }\n}\n\n/* \u51fd\u6570 */\nfn function() -&gt; i32 {      \n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nfn algorithm(n: i32) -&gt; i32 {       // \u8f93\u5165\u6570\u636e\n    const a: i32 = 0;               // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    let mut b = 0;                  // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    let node = Node::new(0);        // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    let c = function();             // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;               // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint algorithm(int n) { // \u8f93\u5165\u6570\u636e\n    const int a = 0;   // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    int b = 0;         // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    int c = func();    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;  // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node(var _val: Int) {\n    var next: Node? = null\n}\n\n/* \u51fd\u6570 */\nfun function(): Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\nfun algorithm(n: Int): Int { // \u8f93\u5165\u6570\u636e\n    val a = 0                // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    var b = 0                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    val node = Node(0)       // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    val c = function()       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre>"},{"location":"chapter_computational_complexity/space_complexity/#242","title":"2.4.2 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"<p>\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u5927\u81f4\u76f8\u540c\uff0c\u53ea\u9700\u5c06\u7edf\u8ba1\u5bf9\u8c61\u4ece\u201c\u64cd\u4f5c\u6570\u91cf\u201d\u8f6c\u4e3a\u201c\u4f7f\u7528\u7a7a\u95f4\u5927\u5c0f\u201d\u3002</p> <p>\u800c\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u7684\u662f\uff0c\u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u3002\u8fd9\u662f\u56e0\u4e3a\u5185\u5b58\u7a7a\u95f4\u662f\u4e00\u9879\u786c\u6027\u8981\u6c42\uff0c\u6211\u4eec\u5fc5\u987b\u786e\u4fdd\u5728\u6240\u6709\u8f93\u5165\u6570\u636e\u4e0b\u90fd\u6709\u8db3\u591f\u7684\u5185\u5b58\u7a7a\u95f4\u9884\u7559\u3002</p> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u201c\u6700\u5dee\u201d\u6709\u4e24\u5c42\u542b\u4e49\u3002</p> <ol> <li>\u4ee5\u6700\u5dee\u8f93\u5165\u6570\u636e\u4e3a\u51c6\uff1a\u5f53 \\(n &lt; 10\\) \u65f6\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff1b\u4f46\u5f53 \\(n &gt; 10\\) \u65f6\uff0c\u521d\u59cb\u5316\u7684\u6570\u7ec4 <code>nums</code> \u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</li> <li>\u4ee5\u7b97\u6cd5\u8fd0\u884c\u4e2d\u7684\u5cf0\u503c\u5185\u5b58\u4e3a\u51c6\uff1a\u4f8b\u5982\uff0c\u7a0b\u5e8f\u5728\u6267\u884c\u6700\u540e\u4e00\u884c\u4e4b\u524d\uff0c\u5360\u7528 \\(O(1)\\) \u7a7a\u95f4\uff1b\u5f53\u521d\u59cb\u5316\u6570\u7ec4 <code>nums</code> \u65f6\uff0c\u7a0b\u5e8f\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</li> </ol> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def algorithm(n: int):\n    a = 0               # O(1)\n    b = [0] * 10000     # O(1)\n    if n &gt; 10:\n        nums = [0] * n  # O(n)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    vector&lt;int&gt; b(10000);    // O(1)\n    if (n &gt; 10)\n        vector&lt;int&gt; nums(n); // O(n)\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10)\n        int[] nums = new int[n]; // O(n)\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10) {\n        int[] nums = new int[n]; // O(n)\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 0                      // O(1)\n    b := make([]int, 10000)     // O(1)\n    var nums []int\n    if n &gt; 10 {\n        nums := make([]int, n)  // O(n)\n    }\n    fmt.Println(a, b, nums)\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    let a = 0 // O(1)\n    let b = Array(repeating: 0, count: 10000) // O(1)\n    if n &gt; 10 {\n        let nums = Array(repeating: 0, count: n) // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 0;                            // O(1)\n  List&lt;int&gt; b = List.filled(10000, 0);  // O(1)\n  if (n &gt; 10) {\n    List&lt;int&gt; nums = List.filled(n, 0); // O(n)\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let a = 0;                              // O(1)\n    let b = [0; 10000];                     // O(1)\n    if n &gt; 10 {\n        let nums = vec![0; n as usize];     // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    int b[10000];            // O(1)\n    if (n &gt; 10)\n        int nums[n] = {0};   // O(n)\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    val a = 0                    // O(1)\n    val b = IntArray(10000)      // O(1)\n    if (n &gt; 10) {\n        val nums = IntArray(n)   // O(n)\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u7edf\u8ba1\u6808\u5e27\u7a7a\u95f4\u3002\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def function() -&gt; int:\n    # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n\ndef loop(n: int):\n    \"\"\"\u5faa\u73af\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(1)\"\"\"\n    for _ in range(n):\n        function()\n\ndef recur(n: int):\n    \"\"\"\u9012\u5f52\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\"\"\"\n    if n == 1:\n        return\n    return recur(n - 1)\n</code></pre> <pre><code>int func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int Function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid Loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n/* \u9012\u5f52 O(n) */\nint Recur(int n) {\n    if (n == 1) return 1;\n    return Recur(n - 1);\n}\n</code></pre> <pre><code>func function() int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5faa\u73af O(1) */\nfunc loop(n int) {\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n}\n\n/* \u9012\u5f52 O(n) */\nfunc recur(n int) {\n    if n == 1 {\n        return\n    }\n    recur(n - 1)\n}\n</code></pre> <pre><code>@discardableResult\nfunc function() -&gt; Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5faa\u73af O(1) */\nfunc loop(n: Int) {\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n\n/* \u9012\u5f52 O(n) */\nfunc recur(n: Int) {\n    if n == 1 {\n        return\n    }\n    recur(n: n - 1)\n}\n</code></pre> <pre><code>function constFunc() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n) {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n) {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>function constFunc(): number {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n: number): void {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n  // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n  return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n  if (n == 1) return;\n  return recur(n - 1);\n}\n</code></pre> <pre><code>fn function() -&gt; i32 {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nfn loop(n: i32) {\n    for i in 0..n {\n        function();\n    }\n}\n/* \u9012\u5f52 O(n) */\nfn recur(n: i32) {\n    if n == 1 {\n        return;\n    }\n    recur(n - 1);\n}\n</code></pre> <pre><code>int func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>fun function(): Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n/* \u5faa\u73af O(1) */\nfun loop(n: Int) {\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n/* \u9012\u5f52 O(n) */\nfun recur(n: Int) {\n    if (n == 1) return\n    return recur(n - 1)\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>\u51fd\u6570 <code>loop()</code> \u548c <code>recur()</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n)\\) \uff0c\u4f46\u7a7a\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u3002</p> <ul> <li>\u51fd\u6570 <code>loop()</code> \u5728\u5faa\u73af\u4e2d\u8c03\u7528\u4e86 \\(n\\) \u6b21 <code>function()</code> \uff0c\u6bcf\u8f6e\u4e2d\u7684 <code>function()</code> \u90fd\u8fd4\u56de\u5e76\u91ca\u653e\u4e86\u6808\u5e27\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002</li> <li>\u9012\u5f52\u51fd\u6570 <code>recur()</code> \u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f1a\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 <code>recur()</code> \uff0c\u4ece\u800c\u5360\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_computational_complexity/space_complexity/#243","title":"2.4.3 \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"<p>\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u56fe 2-16 \u5c55\u793a\u4e86\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\uff08\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\uff09\u3002</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \\newline \\text{\u5e38\u6570\u9636} &lt; \\text{\u5bf9\u6570\u9636} &lt; \\text{\u7ebf\u6027\u9636} &lt; \\text{\u5e73\u65b9\u9636} &lt; \\text{\u6307\u6570\u9636} \\end{aligned} \\] <p></p> <p> \u56fe 2-16 \u00a0 \u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b </p>"},{"location":"chapter_computational_complexity/space_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6570\u9636 \\(O(1)\\)","text":"<p>\u5e38\u6570\u9636\u5e38\u89c1\u4e8e\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u7684\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u5faa\u73af\u4e2d\u521d\u59cb\u5316\u53d8\u91cf\u6216\u8c03\u7528\u51fd\u6570\u800c\u5360\u7528\u7684\u5185\u5b58\uff0c\u5728\u8fdb\u5165\u4e0b\u4e00\u5faa\u73af\u540e\u5c31\u4f1a\u88ab\u91ca\u653e\uff0c\u56e0\u6b64\u4e0d\u4f1a\u7d2f\u79ef\u5360\u7528\u7a7a\u95f4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def function() -&gt; int:\n    \"\"\"\u51fd\u6570\"\"\"\n    # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n\ndef constant(n: int):\n    \"\"\"\u5e38\u6570\u9636\"\"\"\n    # \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    a = 0\n    nums = [0] * 10000\n    node = ListNode(0)\n    # \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in range(n):\n        c = 0\n    # \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in range(n):\n        function()\n</code></pre> space_complexity.cpp<pre><code>/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const int a = 0;\n    int b = 0;\n    vector&lt;int&gt; nums(10000);\n    ListNode node(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u51fd\u6570 */\nint function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    final int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new ListNode(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u51fd\u6570 */\nint Function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid Constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u51fd\u6570 */\nfunc function() int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc spaceConstant(n int) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a = 0\n    b := 0\n    nums := make([]int, 10000)\n    node := newNode(0)\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    var c int\n    for i := 0; i &lt; n; i++ {\n        c = 0\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n    b += 0\n    c += 0\n    nums[0] = 0\n    node.val = 0\n}\n</code></pre> space_complexity.swift<pre><code>/* \u51fd\u6570 */\n@discardableResult\nfunc function() -&gt; Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    let a = 0\n    var b = 0\n    let nums = Array(repeating: 0, count: 10000)\n    let node = ListNode(x: 0)\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in 0 ..&lt; n {\n        let c = 0\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n</code></pre> space_complexity.js<pre><code>/* \u51fd\u6570 */\nfunction constFunc() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u51fd\u6570 */\nfunction constFunc(): number {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n: number): void {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u51fd\u6570 */\nint function() {\n  // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n  return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n  // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n  final int a = 0;\n  int b = 0;\n  List&lt;int&gt; nums = List.filled(10000, 0);\n  ListNode node = ListNode(0);\n  // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n  for (var i = 0; i &lt; n; i++) {\n    int c = 0;\n  }\n  // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n  for (var i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u51fd\u6570 */\nfn function() -&gt; i32 {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\n#[allow(unused)]\nfn constant(n: i32) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const A: i32 = 0;\n    let b = 0;\n    let nums = vec![0; 10000];\n    let node = ListNode::new(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for i in 0..n {\n        let c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for i in 0..n {\n        function();\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const int a = 0;\n    int b = 0;\n    int nums[1000];\n    ListNode *node = newListNode(0);\n    free(node);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* \u51fd\u6570 */\nfun function(): Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfun constant(n: Int) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    val a = 0\n    var b = 0\n    val nums = Array(10000) { 0 }\n    val node = ListNode(0)\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (i in 0..&lt;n) {\n        val c = 0\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{function}\n\n[class]{}-[func]{constant}\n</code></pre> space_complexity.zig<pre><code>// \u51fd\u6570\nfn function() i32 {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n// \u5e38\u6570\u9636\nfn constant(n: i32) void {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a: i32 = 0;\n    var b: i32 = 0;\n    var nums = [_]i32{0}**10000;\n    var node = inc.ListNode(i32){.val = 0};\n    var i: i32 = 0;\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    while (i &lt; n) : (i += 1) {\n        var c: i32 = 0;\n        _ = c;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    i = 0;\n    while (i &lt; n) : (i += 1) {\n        _ = function();\n    }\n    _ = a;\n    _ = b;\n    _ = nums;\n    _ = node;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_computational_complexity/space_complexity/#2-on","title":"2. \u00a0 \u7ebf\u6027\u9636 \\(O(n)\\)","text":"<p>\u7ebf\u6027\u9636\u5e38\u89c1\u4e8e\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u6b63\u6bd4\u7684\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u7b49\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def linear(n: int):\n    \"\"\"\u7ebf\u6027\u9636\"\"\"\n    # \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    nums = [0] * n\n    # \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    hmap = dict[int, str]()\n    for i in range(n):\n        hmap[i] = str(i)\n</code></pre> space_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    vector&lt;int&gt; nums(n);\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    vector&lt;ListNode&gt; nodes;\n    for (int i = 0; i &lt; n; i++) {\n        nodes.push_back(ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    unordered_map&lt;int, string&gt; map;\n    for (int i = 0; i &lt; n; i++) {\n        map[i] = to_string(i);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    int[] nums = new int[n];\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    List&lt;ListNode&gt; nodes = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        nodes.add(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    Map&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        map.put(i, String.valueOf(i));\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7ebf\u6027\u9636 */\nvoid Linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    int[] nums = new int[n];\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    List&lt;ListNode&gt; nodes = [];\n    for (int i = 0; i &lt; n; i++) {\n        nodes.Add(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    Dictionary&lt;int, string&gt; map = [];\n    for (int i = 0; i &lt; n; i++) {\n        map.Add(i, i.ToString());\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u7ebf\u6027\u9636 */\nfunc spaceLinear(n int) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    _ = make([]int, n)\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    var nodes []*node\n    for i := 0; i &lt; n; i++ {\n        nodes = append(nodes, newNode(i))\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    m := make(map[int]string, n)\n    for i := 0; i &lt; n; i++ {\n        m[i] = strconv.Itoa(i)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    let nums = Array(repeating: 0, count: n)\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let nodes = (0 ..&lt; n).map { ListNode(x: $0) }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let map = Dictionary(uniqueKeysWithValues: (0 ..&lt; n).map { ($0, \"\\($0)\") })\n}\n</code></pre> space_complexity.js<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    const nums = new Array(n);\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const nodes = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): void {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    const nums = new Array(n);\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const nodes: ListNode[] = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n  // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n  List&lt;int&gt; nums = List.filled(n, 0);\n  // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n  List&lt;ListNode&gt; nodes = [];\n  for (var i = 0; i &lt; n; i++) {\n    nodes.add(ListNode(i));\n  }\n  // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n  Map&lt;int, String&gt; map = HashMap();\n  for (var i = 0; i &lt; n; i++) {\n    map.putIfAbsent(i, () =&gt; i.toString());\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7ebf\u6027\u9636 */\n#[allow(unused)]\nfn linear(n: i32) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    let mut nums = vec![0; n as usize];\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let mut nodes = Vec::new();\n    for i in 0..n {\n        nodes.push(ListNode::new(i))\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let mut map = HashMap::new();\n    for i in 0..n {\n        map.insert(i, i.to_string());\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u54c8\u5e0c\u8868 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n\n/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    int *nums = malloc(sizeof(int) * n);\n    free(nums);\n\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    ListNode **nodes = malloc(sizeof(ListNode *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        nodes[i] = newListNode(i);\n    }\n    // \u5185\u5b58\u91ca\u653e\n    for (int i = 0; i &lt; n; i++) {\n        free(nodes[i]);\n    }\n    free(nodes);\n\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    HashTable *h = NULL;\n    for (int i = 0; i &lt; n; i++) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = i;\n        tmp-&gt;val = i;\n        HASH_ADD_INT(h, key, tmp);\n    }\n\n    // \u5185\u5b58\u91ca\u653e\n    HashTable *curr, *tmp;\n    HASH_ITER(hh, h, curr, tmp) {\n        HASH_DEL(h, curr);\n        free(curr);\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* \u7ebf\u6027\u9636 */\nfun linear(n: Int) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    val nums = Array(n) { 0 }\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    val nodes = mutableListOf&lt;ListNode&gt;()\n    for (i in 0..&lt;n) {\n        nodes.add(ListNode(i))\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    val map = mutableMapOf&lt;Int, String&gt;()\n    for (i in 0..&lt;n) {\n        map[i] = i.toString()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{linear}\n</code></pre> space_complexity.zig<pre><code>// \u7ebf\u6027\u9636\nfn linear(comptime n: i32) !void {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    var nums = [_]i32{0}**n;\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    var nodes = std.ArrayList(i32).init(std.heap.page_allocator);\n    defer nodes.deinit();\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        try nodes.append(i);\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    var map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\n    defer map.deinit();\n    var j: i32 = 0;\n    while (j &lt; n) : (j += 1) {\n        const string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\n        defer std.heap.page_allocator.free(string);\n        try map.put(i, string);\n    }\n    _ = nums;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 2-17 \u6240\u793a\uff0c\u6b64\u51fd\u6570\u7684\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u5373\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 <code>linear_recur()</code> \u51fd\u6570\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def linear_recur(n: int):\n    \"\"\"\u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    print(\"\u9012\u5f52 n =\", n)\n    if n == 1:\n        return\n    linear_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n    cout &lt;&lt; \"\u9012\u5f52 n = \" &lt;&lt; n &lt;&lt; endl;\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n    System.out.println(\"\u9012\u5f52 n = \" + n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid LinearRecur(int n) {\n    Console.WriteLine(\"\u9012\u5f52 n = \" + n);\n    if (n == 1) return;\n    LinearRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceLinearRecur(n int) {\n    fmt.Println(\"\u9012\u5f52 n =\", n)\n    if n == 1 {\n        return\n    }\n    spaceLinearRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc linearRecur(n: Int) {\n    print(\"\u9012\u5f52 n = \\(n)\")\n    if n == 1 {\n        return\n    }\n    linearRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n) {\n    console.log(`\u9012\u5f52 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n: number): void {\n    console.log(`\u9012\u5f52 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n  print('\u9012\u5f52 n = $n');\n  if (n == 1) return;\n  linearRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn linear_recur(n: i32) {\n    println!(\"\u9012\u5f52 n = {}\", n);\n    if n == 1 {\n        return;\n    };\n    linear_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n    printf(\"\u9012\u5f52 n = %d\\r\\n\", n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.kt<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun linearRecur(n: Int) {\n    println(\"\u9012\u5f52 n = $n\")\n    if (n == 1)\n        return\n    linearRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{linear_recur}\n</code></pre> space_complexity.zig<pre><code>// \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn linearRecur(comptime n: i32) void {\n    std.debug.print(\"\u9012\u5f52 n = {}\\n\", .{n});\n    if (n == 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 2-17 \u00a0 \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u7ebf\u6027\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"<p>\u5e73\u65b9\u9636\u5e38\u89c1\u4e8e\u77e9\u9635\u548c\u56fe\uff0c\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u5e73\u65b9\u5173\u7cfb\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def quadratic(n: int):\n    \"\"\"\u5e73\u65b9\u9636\"\"\"\n    # \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    num_matrix = [[0] * n for _ in range(n)]\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    vector&lt;vector&lt;int&gt;&gt; numMatrix;\n    for (int i = 0; i &lt; n; i++) {\n        vector&lt;int&gt; tmp;\n        for (int j = 0; j &lt; n; j++) {\n            tmp.push_back(0);\n        }\n        numMatrix.push_back(tmp);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    int[][] numMatrix = new int[n][n];\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    List&lt;List&lt;Integer&gt;&gt; numList = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            tmp.add(0);\n        }\n        numList.add(tmp);\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u9636 */\nvoid Quadratic(int n) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    int[,] numMatrix = new int[n, n];\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    List&lt;List&lt;int&gt;&gt; numList = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;int&gt; tmp = [];\n        for (int j = 0; j &lt; n; j++) {\n            tmp.Add(0);\n        }\n        numList.Add(tmp);\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u9636 */\nfunc spaceQuadratic(n int) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    numMatrix := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        numMatrix[i] = make([]int, n)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    let numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): void {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n  // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n  List&lt;List&lt;int&gt;&gt; numMatrix = List.generate(n, (_) =&gt; List.filled(n, 0));\n  // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n  List&lt;List&lt;int&gt;&gt; numList = [];\n  for (var i = 0; i &lt; n; i++) {\n    List&lt;int&gt; tmp = [];\n    for (int j = 0; j &lt; n; j++) {\n      tmp.add(0);\n    }\n    numList.add(tmp);\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u9636 */\n#[allow(unused)]\nfn quadratic(n: i32) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    let num_matrix = vec![vec![0; n as usize]; n as usize];\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    let mut num_list = Vec::new();\n    for i in 0..n {\n        let mut tmp = Vec::new();\n        for j in 0..n {\n            tmp.push(0);\n        }\n        num_list.push(tmp);\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    int **numMatrix = malloc(sizeof(int *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        int *tmp = malloc(sizeof(int) * n);\n        for (int j = 0; j &lt; n; j++) {\n            tmp[j] = 0;\n        }\n        numMatrix[i] = tmp;\n    }\n\n    // \u5185\u5b58\u91ca\u653e\n    for (int i = 0; i &lt; n; i++) {\n        free(numMatrix[i]);\n    }\n    free(numMatrix);\n}\n</code></pre> space_complexity.kt<pre><code>/* \u5e73\u65b9\u9636 */\nfun quadratic(n: Int) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    val numMatrix: Array&lt;Array&lt;Int&gt;?&gt; = arrayOfNulls(n)\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    val numList: MutableList&lt;MutableList&lt;Int&gt;&gt; = arrayListOf()\n    for (i in 0..&lt;n) {\n        val tmp = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            tmp.add(0)\n        }\n        numList.add(tmp)\n    }\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{quadratic}\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u9636\nfn quadratic(n: i32) !void {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    var nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\n    defer nodes.deinit();\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        var tmp = std.ArrayList(i32).init(std.heap.page_allocator);\n        defer tmp.deinit();\n        var j: i32 = 0;\n        while (j &lt; n) : (j += 1) {\n            try tmp.append(0);\n        }\n        try nodes.append(tmp);\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 2-18 \u6240\u793a\uff0c\u8be5\u51fd\u6570\u7684\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u5728\u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u4e2d\u90fd\u521d\u59cb\u5316\u4e86\u4e00\u4e2a\u6570\u7ec4\uff0c\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u5e73\u5747\u957f\u5ea6\u4e3a \\(n / 2\\) \uff0c\u56e0\u6b64\u603b\u4f53\u5360\u7528 \\(O(n^2)\\) \u7a7a\u95f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def quadratic_recur(n: int) -&gt; int:\n    \"\"\"\u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n &lt;= 0:\n        return 0\n    # \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    nums = [0] * n\n    return quadratic_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    vector&lt;int&gt; nums(n);\n    cout &lt;&lt; \"\u9012\u5f52 n = \" &lt;&lt; n &lt;&lt; \" \u4e2d\u7684 nums \u957f\u5ea6 = \" &lt;&lt; nums.size() &lt;&lt; endl;\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    int[] nums = new int[n];\n    System.out.println(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.length);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint QuadraticRecur(int n) {\n    if (n &lt;= 0) return 0;\n    int[] nums = new int[n];\n    Console.WriteLine(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.Length);\n    return QuadraticRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceQuadraticRecur(n int) int {\n    if n &lt;= 0 {\n        return 0\n    }\n    nums := make([]int, n)\n    fmt.Printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d \\n\", n, len(nums))\n    return spaceQuadraticRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -&gt; Int {\n    if n &lt;= 0 {\n        return 0\n    }\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    let nums = Array(repeating: 0, count: n)\n    print(\"\u9012\u5f52 n = \\(n) \u4e2d\u7684 nums \u957f\u5ea6 = \\(nums.count)\")\n    return quadraticRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n) {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n: number): number {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n  if (n &lt;= 0) return 0;\n  List&lt;int&gt; nums = List.filled(n, 0);\n  print('\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}');\n  return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn quadratic_recur(n: i32) -&gt; i32 {\n    if n &lt;= 0 {\n        return 0;\n    };\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    let nums = vec![0; n as usize];\n    println!(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\", n, nums.len());\n    return quadratic_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    int *nums = malloc(sizeof(int) * n);\n    printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d\\r\\n\", n, n);\n    int res = quadraticRecur(n - 1);\n    free(nums);\n    return res;\n}\n</code></pre> space_complexity.kt<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\ntailrec fun quadraticRecur(n: Int): Int {\n    if (n &lt;= 0)\n        return 0\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    val nums = Array(n) { 0 }\n    println(\"\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.size}\")\n    return quadraticRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{quadratic_recur}\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn quadraticRecur(comptime n: i32) i32 {\n    if (n &lt;= 0) return 0;\n    var nums = [_]i32{0}**n;\n    std.debug.print(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\\n\", .{n, nums.len});\n    return quadraticRecur(n - 1);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 2-18 \u00a0 \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u5e73\u65b9\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6570\u9636 \\(O(2^n)\\)","text":"<p>\u6307\u6570\u9636\u5e38\u89c1\u4e8e\u4e8c\u53c9\u6811\u3002\u89c2\u5bdf\u56fe 2-19 \uff0c\u5c42\u6570\u4e3a \\(n\\) \u7684\u201c\u6ee1\u4e8c\u53c9\u6811\u201d\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(2^n - 1\\) \uff0c\u5360\u7528 \\(O(2^n)\\) \u7a7a\u95f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def build_tree(n: int) -&gt; TreeNode | None:\n    \"\"\"\u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\"\"\"\n    if n == 0:\n        return None\n    root = TreeNode(0)\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n    return root\n</code></pre> space_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return nullptr;\n    TreeNode *root = new TreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode buildTree(int n) {\n    if (n == 0)\n        return null;\n    TreeNode root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? BuildTree(int n) {\n    if (n == 0) return null;\n    TreeNode root = new(0) {\n        left = BuildTree(n - 1),\n        right = BuildTree(n - 1)\n    };\n    return root;\n}\n</code></pre> space_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n int) *TreeNode {\n    if n == 0 {\n        return nil\n    }\n    root := NewTreeNode(0)\n    root.Left = buildTree(n - 1)\n    root.Right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n: Int) -&gt; TreeNode? {\n    if n == 0 {\n        return nil\n    }\n    let root = TreeNode(x: 0)\n    root.left = buildTree(n: n - 1)\n    root.right = buildTree(n: n - 1)\n    return root\n}\n</code></pre> space_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n) {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n: number): TreeNode | null {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n) {\n  if (n == 0) return null;\n  TreeNode root = TreeNode(0);\n  root.left = buildTree(n - 1);\n  root.right = buildTree(n - 1);\n  return root;\n}\n</code></pre> space_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfn build_tree(n: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    if n == 0 {\n        return None;\n    };\n    let root = TreeNode::new(0);\n    root.borrow_mut().left = build_tree(n - 1);\n    root.borrow_mut().right = build_tree(n - 1);\n    return Some(root);\n}\n</code></pre> space_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return NULL;\n    TreeNode *root = newTreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.kt<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfun buildTree(n: Int): TreeNode? {\n    if (n == 0)\n        return null\n    val root = TreeNode(0)\n    root.left = buildTree(n - 1)\n    root.right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{build_tree}\n</code></pre> space_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\n    if (n == 0) return null;\n    const root = try mem_allocator.create(inc.TreeNode(i32));\n    root.init(0);\n    root.left = try buildTree(mem_allocator, n - 1);\n    root.right = try buildTree(mem_allocator, n - 1);\n    return root;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 2-19 \u00a0 \u6ee1\u4e8c\u53c9\u6811\u4ea7\u751f\u7684\u6307\u6570\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#5-olog-n","title":"5. \u00a0 \u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"<p>\u5bf9\u6570\u9636\u5e38\u89c1\u4e8e\u5206\u6cbb\u7b97\u6cd5\u3002\u4f8b\u5982\u5f52\u5e76\u6392\u5e8f\uff0c\u8f93\u5165\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u6bcf\u8f6e\u9012\u5f52\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5212\u5206\u4e3a\u4e24\u534a\uff0c\u5f62\u6210\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u6808\u5e27\u7a7a\u95f4\u3002</p> <p>\u518d\u4f8b\u5982\u5c06\u6570\u5b57\u8f6c\u5316\u4e3a\u5b57\u7b26\u4e32\uff0c\u8f93\u5165\u4e00\u4e2a\u6b63\u6574\u6570 \\(n\\) \uff0c\u5b83\u7684\u4f4d\u6570\u4e3a \\(\\lfloor \\log_{10} n \\rfloor + 1\\) \uff0c\u5373\u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u5ea6\u4e3a \\(\\lfloor \\log_{10} n \\rfloor + 1\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_{10} n + 1) = O(\\log n)\\) \u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/#244","title":"2.4.4 \u00a0 \u6743\u8861\u65f6\u95f4\u4e0e\u7a7a\u95f4","text":"<p>\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u80fd\u8fbe\u5230\u6700\u4f18\u3002\u7136\u800c\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u540c\u65f6\u4f18\u5316\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u975e\u5e38\u56f0\u96be\u3002</p> <p>\u964d\u4f4e\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u9700\u8981\u4ee5\u63d0\u5347\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a\u4ee3\u4ef7\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u6211\u4eec\u5c06\u727a\u7272\u5185\u5b58\u7a7a\u95f4\u6765\u63d0\u5347\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u601d\u8def\u79f0\u4e3a\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff1b\u53cd\u4e4b\uff0c\u5219\u79f0\u4e3a\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002</p> <p>\u9009\u62e9\u54ea\u79cd\u601d\u8def\u53d6\u51b3\u4e8e\u6211\u4eec\u66f4\u770b\u91cd\u54ea\u4e2a\u65b9\u9762\u3002\u5728\u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u65f6\u95f4\u6bd4\u7a7a\u95f4\u66f4\u5b9d\u8d35\uff0c\u56e0\u6b64\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u901a\u5e38\u662f\u66f4\u5e38\u7528\u7684\u7b56\u7565\u3002\u5f53\u7136\uff0c\u5728\u6570\u636e\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\uff0c\u63a7\u5236\u7a7a\u95f4\u590d\u6742\u5ea6\u4e5f\u975e\u5e38\u91cd\u8981\u3002</p>"},{"location":"chapter_computational_complexity/summary/","title":"2.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_computational_complexity/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<p>\u7b97\u6cd5\u6548\u7387\u8bc4\u4f30</p> <ul> <li>\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u8861\u91cf\u7b97\u6cd5\u4f18\u52a3\u7684\u4e24\u4e2a\u4e3b\u8981\u8bc4\u4ef7\u6307\u6807\u3002</li> <li>\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5b9e\u9645\u6d4b\u8bd5\u6765\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u96be\u4ee5\u6d88\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5f71\u54cd\uff0c\u4e14\u4f1a\u8017\u8d39\u5927\u91cf\u8ba1\u7b97\u8d44\u6e90\u3002</li> <li>\u590d\u6742\u5ea6\u5206\u6790\u53ef\u4ee5\u6d88\u9664\u5b9e\u9645\u6d4b\u8bd5\u7684\u5f0a\u7aef\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\uff0c\u5e76\u4e14\u80fd\u591f\u63ed\u793a\u7b97\u6cd5\u5728\u4e0d\u540c\u6570\u636e\u89c4\u6a21\u4e0b\u7684\u6548\u7387\u3002</li> </ul> <p>\u65f6\u95f4\u590d\u6742\u5ea6</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\uff0c\u53ef\u4ee5\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u53ef\u80fd\u5931\u6548\uff0c\u5982\u5728\u8f93\u5165\u7684\u6570\u636e\u91cf\u8f83\u5c0f\u6216\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\u65f6\uff0c\u65e0\u6cd5\u7cbe\u786e\u5bf9\u6bd4\u7b97\u6cd5\u6548\u7387\u7684\u4f18\u52a3\u3002</li> <li>\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f7f\u7528\u5927 \\(O\\) \u7b26\u53f7\u8868\u793a\uff0c\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u53cd\u6620\u5f53 \\(n\\) \u8d8b\u5411\u6b63\u65e0\u7a77\u65f6\uff0c\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u7684\u589e\u957f\u7ea7\u522b\u3002</li> <li>\u63a8\u7b97\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u4e24\u6b65\uff0c\u9996\u5148\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002</li> <li>\u5e38\u89c1\u65f6\u95f4\u590d\u6742\u5ea6\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n \\log n)\\)\u3001\\(O(n^2)\\)\u3001\\(O(2^n)\\) \u548c \\(O(n!)\\) \u7b49\u3002</li> <li>\u67d0\u4e9b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u56fa\u5b9a\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u51e0\u4e4e\u4e0d\u7528\uff0c\u56e0\u4e3a\u8f93\u5165\u6570\u636e\u4e00\u822c\u9700\u8981\u6ee1\u8db3\u4e25\u683c\u6761\u4ef6\u624d\u80fd\u8fbe\u5230\u6700\u4f73\u60c5\u51b5\u3002</li> <li>\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53cd\u6620\u7b97\u6cd5\u5728\u968f\u673a\u6570\u636e\u8f93\u5165\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u6700\u63a5\u8fd1\u5b9e\u9645\u5e94\u7528\u4e2d\u7684\u7b97\u6cd5\u6027\u80fd\u3002\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u9700\u8981\u7edf\u8ba1\u8f93\u5165\u6570\u636e\u5206\u5e03\u4ee5\u53ca\u7efc\u5408\u540e\u7684\u6570\u5b66\u671f\u671b\u3002</li> </ul> <p>\u7a7a\u95f4\u590d\u6742\u5ea6</p> <ul> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u4f5c\u7528\u7c7b\u4f3c\u4e8e\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\u3002</li> <li>\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u76f8\u5173\u5185\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u8f93\u5165\u7a7a\u95f4\u3001\u6682\u5b58\u7a7a\u95f4\u3001\u8f93\u51fa\u7a7a\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u8f93\u5165\u7a7a\u95f4\u4e0d\u7eb3\u5165\u7a7a\u95f4\u590d\u6742\u5ea6\u8ba1\u7b97\u3002\u6682\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u6682\u5b58\u6570\u636e\u3001\u6808\u5e27\u7a7a\u95f4\u548c\u6307\u4ee4\u7a7a\u95f4\uff0c\u5176\u4e2d\u6808\u5e27\u7a7a\u95f4\u901a\u5e38\u4ec5\u5728\u9012\u5f52\u51fd\u6570\u4e2d\u5f71\u54cd\u7a7a\u95f4\u590d\u6742\u5ea6\u3002</li> <li>\u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5373\u7edf\u8ba1\u7b97\u6cd5\u5728\u6700\u5dee\u8f93\u5165\u6570\u636e\u548c\u6700\u5dee\u8fd0\u884c\u65f6\u523b\u4e0b\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u3002</li> <li>\u5e38\u89c1\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n^2)\\) \u548c \\(O(2^n)\\) \u7b49\u3002</li> </ul>"},{"location":"chapter_computational_complexity/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u5c3e\u9012\u5f52\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u5417\uff1f</p> <p>\u7406\u8bba\u4e0a\uff0c\u5c3e\u9012\u5f52\u51fd\u6570\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \u3002\u4e0d\u8fc7\u7edd\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java\u3001Python\u3001C++\u3001Go\u3001C# \u7b49\uff09\u4e0d\u652f\u6301\u81ea\u52a8\u4f18\u5316\u5c3e\u9012\u5f52\uff0c\u56e0\u6b64\u901a\u5e38\u8ba4\u4e3a\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(n)\\) \u3002</p> <p>Q\uff1a\u51fd\u6570\u548c\u65b9\u6cd5\u8fd9\u4e24\u4e2a\u672f\u8bed\u7684\u533a\u522b\u662f\u4ec0\u4e48\uff1f</p> <p>\u300c\u51fd\u6570 function\u300d\u53ef\u4ee5\u88ab\u72ec\u7acb\u6267\u884c\uff0c\u6240\u6709\u53c2\u6570\u90fd\u4ee5\u663e\u5f0f\u4f20\u9012\u3002\u300c\u65b9\u6cd5 method\u300d\u4e0e\u4e00\u4e2a\u5bf9\u8c61\u5173\u8054\uff0c\u88ab\u9690\u5f0f\u4f20\u9012\u7ed9\u8c03\u7528\u5b83\u7684\u5bf9\u8c61\uff0c\u80fd\u591f\u5bf9\u7c7b\u7684\u5b9e\u4f8b\u4e2d\u5305\u542b\u7684\u6570\u636e\u8fdb\u884c\u64cd\u4f5c\u3002</p> <p>\u4e0b\u9762\u4ee5\u51e0\u79cd\u5e38\u89c1\u7684\u7f16\u7a0b\u8bed\u8a00\u4e3a\u4f8b\u6765\u8bf4\u660e\u3002</p> <ul> <li>C \u8bed\u8a00\u662f\u8fc7\u7a0b\u5f0f\u7f16\u7a0b\u8bed\u8a00\uff0c\u6ca1\u6709\u9762\u5411\u5bf9\u8c61\u7684\u6982\u5ff5\uff0c\u6240\u4ee5\u53ea\u6709\u51fd\u6570\u3002\u4f46\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u521b\u5efa\u7ed3\u6784\u4f53\uff08struct\uff09\u6765\u6a21\u62df\u9762\u5411\u5bf9\u8c61\u7f16\u7a0b\uff0c\u4e0e\u7ed3\u6784\u4f53\u76f8\u5173\u8054\u7684\u51fd\u6570\u5c31\u76f8\u5f53\u4e8e\u5176\u4ed6\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u65b9\u6cd5\u3002</li> <li>Java \u548c C# \u662f\u9762\u5411\u5bf9\u8c61\u7684\u7f16\u7a0b\u8bed\u8a00\uff0c\u4ee3\u7801\u5757\uff08\u65b9\u6cd5\uff09\u901a\u5e38\u4f5c\u4e3a\u67d0\u4e2a\u7c7b\u7684\u4e00\u90e8\u5206\u3002\u9759\u6001\u65b9\u6cd5\u7684\u884c\u4e3a\u7c7b\u4f3c\u4e8e\u51fd\u6570\uff0c\u56e0\u4e3a\u5b83\u88ab\u7ed1\u5b9a\u5728\u7c7b\u4e0a\uff0c\u4e0d\u80fd\u8bbf\u95ee\u7279\u5b9a\u7684\u5b9e\u4f8b\u53d8\u91cf\u3002</li> <li>C++ \u548c Python \u65e2\u652f\u6301\u8fc7\u7a0b\u5f0f\u7f16\u7a0b\uff08\u51fd\u6570\uff09\uff0c\u4e5f\u652f\u6301\u9762\u5411\u5bf9\u8c61\u7f16\u7a0b\uff08\u65b9\u6cd5\uff09\u3002</li> </ul> <p>Q\uff1a\u56fe\u89e3\u201c\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u201d\u53cd\u6620\u7684\u662f\u5426\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\uff1f</p> <p>\u4e0d\u662f\uff0c\u8be5\u56fe\u5c55\u793a\u7684\u662f\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5176\u53cd\u6620\u7684\u662f\u589e\u957f\u8d8b\u52bf\uff0c\u800c\u4e0d\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\u3002</p> <p>\u5047\u8bbe\u53d6 \\(n = 8\\) \uff0c\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\u6bcf\u6761\u66f2\u7ebf\u7684\u503c\u4e0e\u51fd\u6570\u5bf9\u5e94\u4e0d\u4e0a\u3002\u8fd9\u662f\u56e0\u4e3a\u6bcf\u6761\u66f2\u7ebf\u90fd\u5305\u542b\u4e00\u4e2a\u5e38\u6570\u9879\uff0c\u7528\u4e8e\u5c06\u53d6\u503c\u8303\u56f4\u538b\u7f29\u5230\u4e00\u4e2a\u89c6\u89c9\u8212\u9002\u7684\u8303\u56f4\u5185\u3002</p> <p>\u5728\u5b9e\u9645\u4e2d\uff0c\u56e0\u4e3a\u6211\u4eec\u901a\u5e38\u4e0d\u77e5\u9053\u6bcf\u4e2a\u65b9\u6cd5\u7684\u201c\u5e38\u6570\u9879\u201d\u590d\u6742\u5ea6\u662f\u591a\u5c11\uff0c\u6240\u4ee5\u4e00\u822c\u65e0\u6cd5\u4ec5\u51ed\u590d\u6742\u5ea6\u6765\u9009\u62e9 \\(n = 8\\) \u4e4b\u4e0b\u7684\u6700\u4f18\u89e3\u6cd5\u3002\u4f46\u5bf9\u4e8e \\(n = 8^5\\) \u5c31\u5f88\u597d\u9009\u4e86\uff0c\u8fd9\u65f6\u589e\u957f\u8d8b\u52bf\u5df2\u7ecf\u5360\u4e3b\u5bfc\u4e86\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/","title":"2.3 \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6","text":"<p>\u8fd0\u884c\u65f6\u95f4\u53ef\u4ee5\u76f4\u89c2\u4e14\u51c6\u786e\u5730\u53cd\u6620\u7b97\u6cd5\u7684\u6548\u7387\u3002\u5982\u679c\u6211\u4eec\u60f3\u51c6\u786e\u9884\u4f30\u4e00\u6bb5\u4ee3\u7801\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u5e94\u8be5\u5982\u4f55\u64cd\u4f5c\u5462\uff1f</p> <ol> <li>\u786e\u5b9a\u8fd0\u884c\u5e73\u53f0\uff0c\u5305\u62ec\u786c\u4ef6\u914d\u7f6e\u3001\u7f16\u7a0b\u8bed\u8a00\u3001\u7cfb\u7edf\u73af\u5883\u7b49\uff0c\u8fd9\u4e9b\u56e0\u7d20\u90fd\u4f1a\u5f71\u54cd\u4ee3\u7801\u7684\u8fd0\u884c\u6548\u7387\u3002</li> <li>\u8bc4\u4f30\u5404\u79cd\u8ba1\u7b97\u64cd\u4f5c\u6240\u9700\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u4f8b\u5982\u52a0\u6cd5\u64cd\u4f5c <code>+</code> \u9700\u8981 1 ns \uff0c\u4e58\u6cd5\u64cd\u4f5c <code>*</code> \u9700\u8981 10 ns \uff0c\u6253\u5370\u64cd\u4f5c <code>print()</code> \u9700\u8981 5 ns \u7b49\u3002</li> <li>\u7edf\u8ba1\u4ee3\u7801\u4e2d\u6240\u6709\u7684\u8ba1\u7b97\u64cd\u4f5c\uff0c\u5e76\u5c06\u6240\u6709\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u6c42\u548c\uff0c\u4ece\u800c\u5f97\u5230\u8fd0\u884c\u65f6\u95f4\u3002</li> </ol> <p>\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\ndef algorithm(n: int):\n    a = 2      # 1 ns\n    a = a + 1  # 1 ns\n    a = a * 2  # 10 ns\n    # \u5faa\u73af n \u6b21\n    for _ in range(n):  # 1 ns\n        print(0)        # 5 ns\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        cout &lt;&lt; 0 &lt;&lt; endl;         // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        System.out.println(0);     // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid Algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        Console.WriteLine(0);      // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n int) {\n    a := 2     // 1 ns\n    a = a + 1  // 1 ns\n    a = a * 2  // 10 ns\n    // \u5faa\u73af n \u6b21\n    for i := 0; i &lt; n; i++ {  // 1 ns\n        fmt.Println(a)        // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // \u5faa\u73af n \u6b21\n    for _ in 0 ..&lt; n { // 1 ns\n        print(0) // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n) {\n    var a = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // \u5faa\u73af n \u6b21\n    for(let i = 0; i &lt; n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n: number): void {\n    var a: number = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // \u5faa\u73af n \u6b21\n    for(let i = 0; i &lt; n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n  int a = 2; // 1 ns\n  a = a + 1; // 1 ns\n  a = a * 2; // 10 ns\n  // \u5faa\u73af n \u6b21\n  for (int i = 0; i &lt; n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n    print(0); // 5 ns\n  }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfn algorithm(n: i32) {\n    let mut a = 2;      // 1 ns\n    a = a + 1;          // 1 ns\n    a = a * 2;          // 10 ns\n    // \u5faa\u73af n \u6b21\n    for _ in 0..n {     // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        println!(\"{}\", 0);  // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {   // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        printf(\"%d\", 0);            // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfun algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (i in 0..&lt;n) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        println(0)      // 5 ns\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfn algorithm(n: usize) void {\n    var a: i32 = 2; // 1 ns\n    a += 1; // 1 ns\n    a *= 2; // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (0..n) |_| { // 1 ns\n        std.debug.print(\"{}\\n\", .{0}); // 5 ns\n    }\n}\n</code></pre> <p>\u6839\u636e\u4ee5\u4e0a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5f97\u5230\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u4e3a \\((6n + 12)\\) ns \uff1a</p> \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] <p>\u4f46\u5b9e\u9645\u4e0a\uff0c\u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u65e2\u4e0d\u5408\u7406\u4e5f\u4e0d\u73b0\u5b9e\u3002\u9996\u5148\uff0c\u6211\u4eec\u4e0d\u5e0c\u671b\u5c06\u9884\u4f30\u65f6\u95f4\u548c\u8fd0\u884c\u5e73\u53f0\u7ed1\u5b9a\uff0c\u56e0\u4e3a\u7b97\u6cd5\u9700\u8981\u5728\u5404\u79cd\u4e0d\u540c\u7684\u5e73\u53f0\u4e0a\u8fd0\u884c\u3002\u5176\u6b21\uff0c\u6211\u4eec\u5f88\u96be\u83b7\u77e5\u6bcf\u79cd\u64cd\u4f5c\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u8fd9\u7ed9\u9884\u4f30\u8fc7\u7a0b\u5e26\u6765\u4e86\u6781\u5927\u7684\u96be\u5ea6\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#231","title":"2.3.1 \u00a0 \u7edf\u8ba1\u65f6\u95f4\u589e\u957f\u8d8b\u52bf","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u7edf\u8ba1\u7684\u4e0d\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\uff0c\u800c\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002</p> <p>\u201c\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u201d\u8fd9\u4e2a\u6982\u5ff5\u6bd4\u8f83\u62bd\u8c61\uff0c\u6211\u4eec\u901a\u8fc7\u4e00\u4e2a\u4f8b\u5b50\u6765\u52a0\u4ee5\u7406\u89e3\u3002\u5047\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7ed9\u5b9a\u4e09\u4e2a\u7b97\u6cd5 <code>A</code>\u3001<code>B</code> \u548c <code>C</code> \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_A(n: int):\n    print(0)\n# \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\ndef algorithm_B(n: int):\n    for _ in range(n):\n        print(0)\n# \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_C(n: int):\n    for _ in range(1000000):\n        print(0)\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\n    cout &lt;&lt; 0 &lt;&lt; endl;\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\n    System.out.println(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        System.out.println(0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        System.out.println(0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid AlgorithmA(int n) {\n    Console.WriteLine(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid AlgorithmB(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Console.WriteLine(0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid AlgorithmC(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        Console.WriteLine(0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_A(n int) {\n    fmt.Println(0)\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithm_B(n int) {\n    for i := 0; i &lt; n; i++ {\n        fmt.Println(0)\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_C(n int) {\n    for i := 0; i &lt; 1000000; i++ {\n        fmt.Println(0)\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmA(n: Int) {\n    print(0)\n}\n\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithmB(n: Int) {\n    for _ in 0 ..&lt; n {\n        print(0)\n    }\n}\n\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmC(n: Int) {\n    for _ in 0 ..&lt; 1_000_000 {\n        print(0)\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n) {\n    console.log(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n) {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n) {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n: number): void {\n    console.log(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n: number): void {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmA(int n) {\n  print(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithmB(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    print(0);\n  }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmC(int n) {\n  for (int i = 0; i &lt; 1000000; i++) {\n    print(0);\n  }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_A(n: i32) {\n    println!(\"{}\", 0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfn algorithm_B(n: i32) {\n    for _ in 0..n {\n        println!(\"{}\", 0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_C(n: i32) {\n    for _ in 0..1000000 {\n        println!(\"{}\", 0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\n    printf(\"%d\", 0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        printf(\"%d\", 0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        printf(\"%d\", 0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfun algoritm_A(n: Int) {\n    println(0)\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfun algorithm_B(n: Int) {\n    for (i in 0..&lt;n){\n        println(0)\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfun algorithm_C(n: Int) {\n    for (i in 0..&lt;1000000) {\n        println(0)\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_A(n: usize) void {\n    _ = n;\n    std.debug.print(\"{}\\n\", .{0});\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfn algorithm_B(n: i32) void {\n    for (0..n) |_| {\n        std.debug.print(\"{}\\n\", .{0});\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_C(n: i32) void {\n    _ = n;\n    for (0..1000000) |_| { \n        std.debug.print(\"{}\\n\", .{0});\n    }\n}\n</code></pre> <p>\u56fe 2-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u4e09\u4e2a\u7b97\u6cd5\u51fd\u6570\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <ul> <li>\u7b97\u6cd5 <code>A</code> \u53ea\u6709 \\(1\\) \u4e2a\u6253\u5370\u64cd\u4f5c\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u4e0d\u968f\u7740 \\(n\\) \u589e\u5927\u800c\u589e\u957f\u3002\u6211\u4eec\u79f0\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u201c\u5e38\u6570\u9636\u201d\u3002</li> <li>\u7b97\u6cd5 <code>B</code> \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(n\\) \u6b21\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740 \\(n\\) \u589e\u5927\u5448\u7ebf\u6027\u589e\u957f\u3002\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u88ab\u79f0\u4e3a\u201c\u7ebf\u6027\u9636\u201d\u3002</li> <li>\u7b97\u6cd5 <code>C</code> \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(1000000\\) \u6b21\uff0c\u867d\u7136\u8fd0\u884c\u65f6\u95f4\u5f88\u957f\uff0c\u4f46\u5b83\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u3002\u56e0\u6b64 <code>C</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c <code>A</code> \u76f8\u540c\uff0c\u4ecd\u4e3a\u201c\u5e38\u6570\u9636\u201d\u3002</li> </ul> <p></p> <p> \u56fe 2-7 \u00a0 \u7b97\u6cd5 A\u3001B \u548c C \u7684\u65f6\u95f4\u589e\u957f\u8d8b\u52bf </p> <p>\u76f8\u8f83\u4e8e\u76f4\u63a5\u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u6709\u54ea\u4e9b\u7279\u70b9\u5462\uff1f</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u80fd\u591f\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u3002\u4f8b\u5982\uff0c\u7b97\u6cd5 <code>B</code> \u7684\u8fd0\u884c\u65f6\u95f4\u5448\u7ebf\u6027\u589e\u957f\uff0c\u5728 \\(n &gt; 1\\) \u65f6\u6bd4\u7b97\u6cd5 <code>A</code> \u66f4\u6162\uff0c\u5728 \\(n &gt; 1000000\\) \u65f6\u6bd4\u7b97\u6cd5 <code>C</code> \u66f4\u6162\u3002\u4e8b\u5b9e\u4e0a\uff0c\u53ea\u8981\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8db3\u591f\u5927\uff0c\u590d\u6742\u5ea6\u4e3a\u201c\u5e38\u6570\u9636\u201d\u7684\u7b97\u6cd5\u4e00\u5b9a\u4f18\u4e8e\u201c\u7ebf\u6027\u9636\u201d\u7684\u7b97\u6cd5\uff0c\u8fd9\u6b63\u662f\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u7684\u542b\u4e49\u3002</li> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u66f4\u7b80\u4fbf\u3002\u663e\u7136\uff0c\u8fd0\u884c\u5e73\u53f0\u548c\u8ba1\u7b97\u64cd\u4f5c\u7c7b\u578b\u90fd\u4e0e\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u65e0\u5173\u3002\u56e0\u6b64\u5728\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u5c06\u6240\u6709\u8ba1\u7b97\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u89c6\u4e3a\u76f8\u540c\u7684\u201c\u5355\u4f4d\u65f6\u95f4\u201d\uff0c\u4ece\u800c\u5c06\u201c\u8ba1\u7b97\u64cd\u4f5c\u8fd0\u884c\u65f6\u95f4\u7edf\u8ba1\u201d\u7b80\u5316\u4e3a\u201c\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u7edf\u8ba1\u201d\uff0c\u8fd9\u6837\u4e00\u6765\u4f30\u7b97\u96be\u5ea6\u5c31\u5927\u5927\u964d\u4f4e\u4e86\u3002</li> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u5b58\u5728\u4e00\u5b9a\u7684\u5c40\u9650\u6027\u3002\u4f8b\u5982\uff0c\u5c3d\u7ba1\u7b97\u6cd5 <code>A</code> \u548c <code>C</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\uff0c\u4f46\u5b9e\u9645\u8fd0\u884c\u65f6\u95f4\u5dee\u522b\u5f88\u5927\u3002\u540c\u6837\uff0c\u5c3d\u7ba1\u7b97\u6cd5 <code>B</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6bd4 <code>C</code> \u9ad8\uff0c\u4f46\u5728\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 <code>B</code> \u660e\u663e\u4f18\u4e8e\u7b97\u6cd5 <code>C</code> \u3002\u5728\u8fd9\u4e9b\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5f88\u96be\u4ec5\u51ed\u65f6\u95f4\u590d\u6742\u5ea6\u5224\u65ad\u7b97\u6cd5\u6548\u7387\u7684\u9ad8\u4f4e\u3002\u5f53\u7136\uff0c\u5c3d\u7ba1\u5b58\u5728\u4e0a\u8ff0\u95ee\u9898\uff0c\u590d\u6742\u5ea6\u5206\u6790\u4ecd\u7136\u662f\u8bc4\u5224\u7b97\u6cd5\u6548\u7387\u6700\u6709\u6548\u4e14\u5e38\u7528\u7684\u65b9\u6cd5\u3002</li> </ul>"},{"location":"chapter_computational_complexity/time_complexity/#232","title":"2.3.2 \u00a0 \u51fd\u6570\u6e10\u8fd1\u4e0a\u754c","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u8f93\u5165\u5927\u5c0f\u4e3a \\(n\\) \u7684\u51fd\u6570\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def algorithm(n: int):\n    a = 1      # +1\n    a = a + 1  # +1\n    a = a * 2  # +1\n    # \u5faa\u73af n \u6b21\n    for i in range(n):  # +1\n        print(0)        # +1\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        cout &lt;&lt; 0 &lt;&lt; endl;    // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        System.out.println(0);    // +1\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {   // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        Console.WriteLine(0);   // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1      // +1\n    a = a + 1   // +1\n    a = a * 2   // +1\n    // \u5faa\u73af n \u6b21\n    for i := 0; i &lt; n; i++ {   // +1\n        fmt.Println(a)         // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // \u5faa\u73af n \u6b21\n    for _ in 0 ..&lt; n { // +1\n        print(0) // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    var a = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // \u5faa\u73af n \u6b21\n    for(let i = 0; i &lt; n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void{\n    var a: number = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // \u5faa\u73af n \u6b21\n    for(let i = 0; i &lt; n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +1\n  a = a + 1; // +1\n  a = a * 2; // +1\n  // \u5faa\u73af n \u6b21\n  for (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n    print(0); // +1\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;   // +1\n    a = a + 1;      // +1\n    a = a * 2;      // +1\n\n    // \u5faa\u73af n \u6b21\n    for _ in 0..n { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        println!(\"{}\", 0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {   // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        printf(\"%d\", 0);            // +1\n    }\n}  \n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // \u5faa\u73af n \u6b21\n    for (i in 0..&lt;n) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        println(0) // +1\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>fn algorithm(n: usize) void {\n    var a: i32 = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // \u5faa\u73af n \u6b21\n    for (0..n) |_| { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        std.debug.print(\"{}\\n\", .{0}); // +1\n    }\n}\n</code></pre> <p>\u8bbe\u7b97\u6cd5\u7684\u64cd\u4f5c\u6570\u91cf\u662f\u4e00\u4e2a\u5173\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u7684\u51fd\u6570\uff0c\u8bb0\u4e3a \\(T(n)\\) \uff0c\u5219\u4ee5\u4e0a\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e3a\uff1a</p> \\[ T(n) = 3 + 2n \\] <p>\\(T(n)\\) \u662f\u4e00\u6b21\u51fd\u6570\uff0c\u8bf4\u660e\u5176\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u662f\u7ebf\u6027\u7684\uff0c\u56e0\u6b64\u5b83\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u7ebf\u6027\u9636\u3002</p> <p>\u6211\u4eec\u5c06\u7ebf\u6027\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u8bb0\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e2a\u6570\u5b66\u7b26\u53f7\u79f0\u4e3a\u300c\u5927 \\(O\\) \u8bb0\u53f7 big-\\(O\\) notation\u300d\uff0c\u8868\u793a\u51fd\u6570 \\(T(n)\\) \u7684\u300c\u6e10\u8fd1\u4e0a\u754c asymptotic upper bound\u300d\u3002</p> <p>\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u672c\u8d28\u4e0a\u662f\u8ba1\u7b97\u201c\u64cd\u4f5c\u6570\u91cf \\(T(n)\\)\u201d\u7684\u6e10\u8fd1\u4e0a\u754c\uff0c\u5b83\u5177\u6709\u660e\u786e\u7684\u6570\u5b66\u5b9a\u4e49\u3002</p> <p>\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c</p> <p>\u82e5\u5b58\u5728\u6b63\u5b9e\u6570 \\(c\\) \u548c\u5b9e\u6570 \\(n_0\\) \uff0c\u4f7f\u5f97\u5bf9\u4e8e\u6240\u6709\u7684 \\(n &gt; n_0\\) \uff0c\u5747\u6709 \\(T(n) \\leq c \\cdot f(n)\\) \uff0c\u5219\u53ef\u8ba4\u4e3a \\(f(n)\\) \u7ed9\u51fa\u4e86 \\(T(n)\\) \u7684\u4e00\u4e2a\u6e10\u8fd1\u4e0a\u754c\uff0c\u8bb0\u4e3a \\(T(n) = O(f(n))\\) \u3002</p> <p>\u5982\u56fe 2-8 \u6240\u793a\uff0c\u8ba1\u7b97\u6e10\u8fd1\u4e0a\u754c\u5c31\u662f\u5bfb\u627e\u4e00\u4e2a\u51fd\u6570 \\(f(n)\\) \uff0c\u4f7f\u5f97\u5f53 \\(n\\) \u8d8b\u5411\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\\(T(n)\\) \u548c \\(f(n)\\) \u5904\u4e8e\u76f8\u540c\u7684\u589e\u957f\u7ea7\u522b\uff0c\u4ec5\u76f8\u5dee\u4e00\u4e2a\u5e38\u6570\u9879 \\(c\\) \u7684\u500d\u6570\u3002</p> <p></p> <p> \u56fe 2-8 \u00a0 \u51fd\u6570\u7684\u6e10\u8fd1\u4e0a\u754c </p>"},{"location":"chapter_computational_complexity/time_complexity/#233","title":"2.3.3 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"<p>\u6e10\u8fd1\u4e0a\u754c\u7684\u6570\u5b66\u5473\u513f\u6709\u70b9\u91cd\uff0c\u5982\u679c\u4f60\u611f\u89c9\u6ca1\u6709\u5b8c\u5168\u7406\u89e3\uff0c\u4e5f\u65e0\u987b\u62c5\u5fc3\u3002\u6211\u4eec\u53ef\u4ee5\u5148\u638c\u63e1\u63a8\u7b97\u65b9\u6cd5\uff0c\u5728\u4e0d\u65ad\u7684\u5b9e\u8df5\u4e2d\uff0c\u5c31\u53ef\u4ee5\u9010\u6e10\u9886\u609f\u5176\u6570\u5b66\u610f\u4e49\u3002</p> <p>\u6839\u636e\u5b9a\u4e49\uff0c\u786e\u5b9a \\(f(n)\\) \u4e4b\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \u3002\u90a3\u4e48\u5982\u4f55\u786e\u5b9a\u6e10\u8fd1\u4e0a\u754c \\(f(n)\\) \u5462\uff1f\u603b\u4f53\u5206\u4e3a\u4e24\u6b65\uff1a\u9996\u5148\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#1","title":"1. \u00a0 \u7b2c\u4e00\u6b65\uff1a\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf","text":"<p>\u9488\u5bf9\u4ee3\u7801\uff0c\u9010\u884c\u4ece\u4e0a\u5230\u4e0b\u8ba1\u7b97\u5373\u53ef\u3002\u7136\u800c\uff0c\u7531\u4e8e\u4e0a\u8ff0 \\(c \\cdot f(n)\\) \u4e2d\u7684\u5e38\u6570\u9879 \\(c\\) \u53ef\u4ee5\u53d6\u4efb\u610f\u5927\u5c0f\uff0c\u56e0\u6b64\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u4e2d\u7684\u5404\u79cd\u7cfb\u6570\u3001\u5e38\u6570\u9879\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002\u6839\u636e\u6b64\u539f\u5219\uff0c\u53ef\u4ee5\u603b\u7ed3\u51fa\u4ee5\u4e0b\u8ba1\u6570\u7b80\u5316\u6280\u5de7\u3002</p> <ol> <li>\u5ffd\u7565 \\(T(n)\\) \u4e2d\u7684\u5e38\u6570\u9879\u3002\u56e0\u4e3a\u5b83\u4eec\u90fd\u4e0e \\(n\\) \u65e0\u5173\uff0c\u6240\u4ee5\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4ea7\u751f\u5f71\u54cd\u3002</li> <li>\u7701\u7565\u6240\u6709\u7cfb\u6570\u3002\u4f8b\u5982\uff0c\u5faa\u73af \\(2n\\) \u6b21\u3001\\(5n + 1\\) \u6b21\u7b49\uff0c\u90fd\u53ef\u4ee5\u7b80\u5316\u8bb0\u4e3a \\(n\\) \u6b21\uff0c\u56e0\u4e3a \\(n\\) \u524d\u9762\u7684\u7cfb\u6570\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u6ca1\u6709\u5f71\u54cd\u3002</li> <li>\u5faa\u73af\u5d4c\u5957\u65f6\u4f7f\u7528\u4e58\u6cd5\u3002\u603b\u64cd\u4f5c\u6570\u91cf\u7b49\u4e8e\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u64cd\u4f5c\u6570\u91cf\u4e4b\u79ef\uff0c\u6bcf\u4e00\u5c42\u5faa\u73af\u4f9d\u7136\u53ef\u4ee5\u5206\u522b\u5957\u7528\u7b2c <code>1.</code> \u70b9\u548c\u7b2c <code>2.</code> \u70b9\u7684\u6280\u5de7\u3002</li> </ol> <p>\u7ed9\u5b9a\u4e00\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u4e0a\u8ff0\u6280\u5de7\u6765\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def algorithm(n: int):\n    a = 1      # +0\uff08\u6280\u5de7 1\uff09\n    a = a + n  # +0\uff08\u6280\u5de7 1\uff09\n    # +n\uff08\u6280\u5de7 2\uff09\n    for i in range(5 * n + 1):\n        print(0)\n    # +n*n\uff08\u6280\u5de7 3\uff09\n    for i in range(2 * n):\n        for j in range(n + 1):\n            print(0)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            cout &lt;&lt; 0 &lt;&lt; endl;\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        System.out.println(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            System.out.println(0);\n        }\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        Console.WriteLine(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            Console.WriteLine(0);\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1     // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for i := 0; i &lt; 5 * n + 1; i++ {\n        fmt.Println(0)\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for i := 0; i &lt; 2 * n; i++ {\n        for j := 0; j &lt; n + 1; j++ {\n            fmt.Println(0)\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for _ in 0 ..&lt; (5 * n + 1) {\n        print(0)\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for _ in 0 ..&lt; (2 * n) {\n        for _ in 0 ..&lt; (n + 1) {\n            print(0)\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    let a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    let a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +0\uff08\u6280\u5de7 1\uff09\n  a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n  // +n\uff08\u6280\u5de7 2\uff09\n  for (int i = 0; i &lt; 5 * n + 1; i++) {\n    print(0);\n  }\n  // +n*n\uff08\u6280\u5de7 3\uff09\n  for (int i = 0; i &lt; 2 * n; i++) {\n    for (int j = 0; j &lt; n + 1; j++) {\n      print(0);\n    }\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;     // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;        // +0\uff08\u6280\u5de7 1\uff09\n\n    // +n\uff08\u6280\u5de7 2\uff09\n    for i in 0..(5 * n + 1) {\n        println!(\"{}\", 0);\n    }\n\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for i in 0..(2 * n) {\n        for j in 0..(n + 1) {\n            println!(\"{}\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        printf(\"%d\", 0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            printf(\"%d\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1   // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n   // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (i in 0..&lt;5 * n + 1) {\n        println(0)\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (i in 0..&lt;2 * n) {\n        for (j in 0..&lt;n + 1) {\n            println(0)\n        }\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>fn algorithm(n: usize) void {\n    var a: i32 = 1;     // +0\uff08\u6280\u5de7 1\uff09\n    a = a + @as(i32, @intCast(n));        // +0\uff08\u6280\u5de7 1\uff09\n\n    // +n\uff08\u6280\u5de7 2\uff09\n    for(0..(5 * n + 1)) |_| {\n        std.debug.print(\"{}\\n\", .{0}); \n    }\n\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for(0..(2 * n)) |_| {\n        for(0..(n + 1)) |_| {\n            std.debug.print(\"{}\\n\", .{0}); \n        }\n    }\n}\n</code></pre> <p>\u4ee5\u4e0b\u516c\u5f0f\u5c55\u793a\u4e86\u4f7f\u7528\u4e0a\u8ff0\u6280\u5de7\u524d\u540e\u7684\u7edf\u8ba1\u7ed3\u679c\uff0c\u4e24\u8005\u63a8\u7b97\u51fa\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \u3002</p> \\[ \\begin{aligned} T(n) &amp; = 2n(n + 1) + (5n + 1) + 2 &amp; \\text{\u5b8c\u6574\u7edf\u8ba1 (-.-|||)} \\newline &amp; = 2n^2 + 7n + 3 \\newline T(n) &amp; = n^2 + n &amp; \\text{\u5077\u61d2\u7edf\u8ba1 (o.O)} \\end{aligned} \\]"},{"location":"chapter_computational_complexity/time_complexity/#2","title":"2. \u00a0 \u7b2c\u4e8c\u6b65\uff1a\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\u7531 \\(T(n)\\) \u4e2d\u6700\u9ad8\u9636\u7684\u9879\u6765\u51b3\u5b9a\u3002\u8fd9\u662f\u56e0\u4e3a\u5728 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u6700\u9ad8\u9636\u7684\u9879\u5c06\u53d1\u6325\u4e3b\u5bfc\u4f5c\u7528\uff0c\u5176\u4ed6\u9879\u7684\u5f71\u54cd\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002</p> <p>\u8868 2-2 \u5c55\u793a\u4e86\u4e00\u4e9b\u4f8b\u5b50\uff0c\u5176\u4e2d\u4e00\u4e9b\u5938\u5f20\u7684\u503c\u662f\u4e3a\u4e86\u5f3a\u8c03\u201c\u7cfb\u6570\u65e0\u6cd5\u64bc\u52a8\u9636\u6570\u201d\u8fd9\u4e00\u7ed3\u8bba\u3002\u5f53 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u8fd9\u4e9b\u5e38\u6570\u53d8\u5f97\u65e0\u8db3\u8f7b\u91cd\u3002</p> <p> \u8868 2-2 \u00a0 \u4e0d\u540c\u64cd\u4f5c\u6570\u91cf\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> \u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#234","title":"2.3.4 \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"<p>\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u5982\u56fe 2-9 \u6240\u793a\uff08\u6309\u7167\u4ece\u4f4e\u5230\u9ad8\u7684\u987a\u5e8f\u6392\u5217\uff09\u3002</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n \\log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!) \\newline \\text{\u5e38\u6570\u9636} &lt; \\text{\u5bf9\u6570\u9636} &lt; \\text{\u7ebf\u6027\u9636} &lt; \\text{\u7ebf\u6027\u5bf9\u6570\u9636} &lt; \\text{\u5e73\u65b9\u9636} &lt; \\text{\u6307\u6570\u9636} &lt; \\text{\u9636\u4e58\u9636} \\end{aligned} \\] <p></p> <p> \u56fe 2-9 \u00a0 \u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b </p>"},{"location":"chapter_computational_complexity/time_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6570\u9636 \\(O(1)\\)","text":"<p>\u5e38\u6570\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u5373\u4e0d\u968f\u7740 \\(n\\) \u7684\u53d8\u5316\u800c\u53d8\u5316\u3002</p> <p>\u5728\u4ee5\u4e0b\u51fd\u6570\u4e2d\uff0c\u5c3d\u7ba1\u64cd\u4f5c\u6570\u91cf <code>size</code> \u53ef\u80fd\u5f88\u5927\uff0c\u4f46\u7531\u4e8e\u5176\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def constant(n: int) -&gt; int:\n    \"\"\"\u5e38\u6570\u9636\"\"\"\n    count = 0\n    size = 100000\n    for _ in range(size):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e38\u6570\u9636 */\nint Constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e38\u6570\u9636 */\nfunc constant(n int) int {\n    count := 0\n    size := 100000\n    for i := 0; i &lt; size; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) -&gt; Int {\n    var count = 0\n    let size = 100_000\n    for _ in 0 ..&lt; size {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e38\u6570\u9636 */\nfunction constant(n: number): number {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n  int count = 0;\n  int size = 100000;\n  for (var i = 0; i &lt; size; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e38\u6570\u9636 */\nfn constant(n: i32) -&gt; i32 {\n    _ = n;\n    let mut count = 0;\n    let size = 100_000;\n    for _ in 0..size {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    int i = 0;\n    for (int i = 0; i &lt; size; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e38\u6570\u9636 */\nfun constant(n: Int): Int {\n    var count = 0\n    val size = 10_0000\n    for (i in 0..&lt;size)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{constant}\n</code></pre> time_complexity.zig<pre><code>// \u5e38\u6570\u9636\nfn constant(n: i32) i32 {\n    _ = n;\n    var count: i32 = 0;\n    const size: i32 = 100_000;\n    var i: i32 = 0;\n    while(i&lt;size) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_computational_complexity/time_complexity/#2-on","title":"2. \u00a0 \u7ebf\u6027\u9636 \\(O(n)\\)","text":"<p>\u7ebf\u6027\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u4ee5\u7ebf\u6027\u7ea7\u522b\u589e\u957f\u3002\u7ebf\u6027\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5355\u5c42\u5faa\u73af\u4e2d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def linear(n: int) -&gt; int:\n    \"\"\"\u7ebf\u6027\u9636\"\"\"\n    count = 0\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u9636 */\nint Linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n int) int {\n    count := 0\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) -&gt; Int {\n    var count = 0\n    for _ in 0 ..&lt; n {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): number {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n  int count = 0;\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u9636 */\nfn linear(n: i32) -&gt; i32 {\n    let mut count = 0;\n    for _ in 0..n {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7ebf\u6027\u9636 */\nfun linear(n: Int): Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (i in 0..&lt;n)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{linear}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u9636\nfn linear(n: i32) i32 {\n    var count: i32 = 0;\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u904d\u5386\u6570\u7ec4\u548c\u904d\u5386\u94fe\u8868\u7b49\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u7684\u957f\u5ea6\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def array_traversal(nums: list[int]) -&gt; int:\n    \"\"\"\u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\"\"\"\n    count = 0\n    # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for num in nums:\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(vector&lt;int&gt; &amp;nums) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint ArrayTraversal(int[] nums) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    foreach (int num in nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums []int) int {\n    count := 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for range nums {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums: [Int]) -&gt; Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for _ in nums {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums) {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums: number[]): number {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(List&lt;int&gt; nums) {\n  int count = 0;\n  // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n  for (var _num in nums) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfn array_traversal(nums: &amp;[i32]) -&gt; i32 {\n    let mut count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for _ in nums {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int *nums, int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfun arrayTraversal(nums: IntArray): Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (num in nums) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{array_traversal}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\nfn arrayTraversal(nums: []i32) i32 {\n    var count: i32 = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (nums) |_| {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u9700\u6839\u636e\u8f93\u5165\u6570\u636e\u7684\u7c7b\u578b\u6765\u5177\u4f53\u786e\u5b9a\u3002\u6bd4\u5982\u5728\u7b2c\u4e00\u4e2a\u793a\u4f8b\u4e2d\uff0c\u53d8\u91cf \\(n\\) \u4e3a\u8f93\u5165\u6570\u636e\u5927\u5c0f\uff1b\u5728\u7b2c\u4e8c\u4e2a\u793a\u4f8b\u4e2d\uff0c\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u4e3a\u6570\u636e\u5927\u5c0f\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"<p>\u5e73\u65b9\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u4ee5\u5e73\u65b9\u7ea7\u522b\u589e\u957f\u3002\u5e73\u65b9\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n)\\) \uff0c\u56e0\u6b64\u603b\u4f53\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def quadratic(n: int) -&gt; int:\n    \"\"\"\u5e73\u65b9\u9636\"\"\"\n    count = 0\n    # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for i in range(n):\n        for j in range(n):\n            count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u9636 */\nint Quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n int) int {\n    count := 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; n; j++ {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) -&gt; Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; n {\n            count += 1\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): number {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n  int count = 0;\n  // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n  for (int i = 0; i &lt; n; i++) {\n    for (int j = 0; j &lt; n; j++) {\n      count++;\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u9636 */\nfn quadratic(n: i32) -&gt; i32 {\n    let mut count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for _ in 0..n {\n        for _ in 0..n {\n            count += 1;\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e73\u65b9\u9636 */\nfun quadratic(n: Int): Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;n) {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{quadratic}\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u9636\nfn quadratic(n: i32) i32 {\n    var count: i32 = 0;\n    var i: i32 = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    while (i &lt; n) : (i += 1) {\n        var j: i32 = 0;\n        while (j &lt; n) : (j += 1) {\n            count += 1;\n        }\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 2-10 \u5bf9\u6bd4\u4e86\u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u548c\u5e73\u65b9\u9636\u4e09\u79cd\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <p></p> <p> \u56fe 2-10 \u00a0 \u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u548c\u5e73\u65b9\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u4ee5\u5192\u6ce1\u6392\u5e8f\u4e3a\u4f8b\uff0c\u5916\u5c42\u5faa\u73af\u6267\u884c \\(n - 1\\) \u6b21\uff0c\u5185\u5c42\u5faa\u73af\u6267\u884c \\(n-1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u5e73\u5747\u4e3a \\(n / 2\\) \u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O((n - 1) n / 2) = O(n^2)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def bubble_sort(nums: list[int]) -&gt; int:\n    \"\"\"\u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\"\"\"\n    count = 0  # \u8ba1\u6570\u5668\n    # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in range(len(nums) - 1, 0, -1):\n        # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                tmp: int = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3  # \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(vector&lt;int&gt; &amp;nums) {\n    int count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\n    int count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint BubbleSort(int[] nums) {\n    int count = 0;  // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                count += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\n    count := 0 // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                tmp := nums[j]\n                nums[j] = nums[j+1]\n                nums[j+1] = tmp\n                count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -&gt; Int {\n    var count = 0 // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\n    let count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\n    let count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(List&lt;int&gt; nums) {\n  int count = 0; // \u8ba1\u6570\u5668\n  // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n  for (var i = nums.length - 1; i &gt; 0; i--) {\n    // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n    for (var j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n      }\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfn bubble_sort(nums: &amp;mut [i32]) -&gt; i32 {\n    let mut count = 0; // \u8ba1\u6570\u5668\n\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in (1..nums.len()).rev() {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int *nums, int n) {\n    int count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = n - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfun bubbleSort(nums: IntArray): Int {\n    var count = 0\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j] = nums[j + 1].also { nums[j + 1] = nums[j] }\n                count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{bubble_sort}\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\n    var count: i32 = 0;  // \u8ba1\u6570\u5668 \n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    var i: i32 = @as(i32, @intCast(nums.len)) - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var j: usize = 0;\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_computational_complexity/time_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6570\u9636 \\(O(2^n)\\)","text":"<p>\u751f\u7269\u5b66\u7684\u201c\u7ec6\u80de\u5206\u88c2\u201d\u662f\u6307\u6570\u9636\u589e\u957f\u7684\u5178\u578b\u4f8b\u5b50\uff1a\u521d\u59cb\u72b6\u6001\u4e3a \\(1\\) \u4e2a\u7ec6\u80de\uff0c\u5206\u88c2\u4e00\u8f6e\u540e\u53d8\u4e3a \\(2\\) \u4e2a\uff0c\u5206\u88c2\u4e24\u8f6e\u540e\u53d8\u4e3a \\(4\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5206\u88c2 \\(n\\) \u8f6e\u540e\u6709 \\(2^n\\) \u4e2a\u7ec6\u80de\u3002</p> <p>\u56fe 2-11 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6a21\u62df\u4e86\u7ec6\u80de\u5206\u88c2\u7684\u8fc7\u7a0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def exponential(n: int) -&gt; int:\n    \"\"\"\u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n    count = 0\n    base = 1\n    # \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in range(n):\n        for _ in range(base):\n            count += 1\n        base *= 2\n    # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Exponential(int n) {\n    int count = 0, bas = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc exponential(n int) int {\n    count, base := 0, 1\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; base; j++ {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc exponential(n: Int) -&gt; Int {\n    var count = 0\n    var base = 1\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; base {\n            count += 1\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n) {\n    let count = 0,\n        base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n: number): number {\n    let count = 0,\n        base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n  int count = 0, base = 1;\n  // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n  for (var i = 0; i &lt; n; i++) {\n    for (var j = 0; j &lt; base; j++) {\n      count++;\n    }\n    base *= 2;\n  }\n  // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn exponential(n: i32) -&gt; i32 {\n    let mut count = 0;\n    let mut base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0..n {\n        for _ in 0..base {\n            count += 1\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n    int count = 0;\n    int bas = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfun exponential(n: Int): Int {\n    var count = 0\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    var base = 1\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;base) {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{exponential}\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn exponential(n: i32) i32 {\n    var count: i32 = 0;\n    var bas: i32 = 1;\n    var i: i32 = 0;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    while (i &lt; n) : (i += 1) {\n        var j: i32 = 0;\n        while (j &lt; bas) : (j += 1) {\n            count += 1;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 2-11 \u00a0 \u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u5728\u5b9e\u9645\u7b97\u6cd5\u4e2d\uff0c\u6307\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u4e2d\u3002\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u5176\u9012\u5f52\u5730\u4e00\u5206\u4e3a\u4e8c\uff0c\u7ecf\u8fc7 \\(n\\) \u6b21\u5206\u88c2\u540e\u505c\u6b62\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def exp_recur(n: int) -&gt; int:\n    \"\"\"\u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n == 1:\n        return 1\n    return exp_recur(n - 1) + exp_recur(n - 1) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint ExpRecur(int n) {\n    if (n == 1) return 1;\n    return ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc expRecur(n int) int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n-1) + expRecur(n-1) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc expRecur(n: Int) -&gt; Int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n) {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n: number): number {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n  if (n == 1) return 1;\n  return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn exp_recur(n: i32) -&gt; i32 {\n    if n == 1 {\n        return 1;\n    }\n    exp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun expRecur(n: Int): Int {\n    if (n == 1) {\n        return 1\n    }\n    return expRecur(n - 1) + expRecur(n - 1) + 1\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{exp_recur}\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn expRecur(n: i32) i32 {\n    if (n == 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u6307\u6570\u9636\u589e\u957f\u975e\u5e38\u8fc5\u901f\uff0c\u5728\u7a77\u4e3e\u6cd5\uff08\u66b4\u529b\u641c\u7d22\u3001\u56de\u6eaf\u7b49\uff09\u4e2d\u6bd4\u8f83\u5e38\u89c1\u3002\u5bf9\u4e8e\u6570\u636e\u89c4\u6a21\u8f83\u5927\u7684\u95ee\u9898\uff0c\u6307\u6570\u9636\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\uff0c\u901a\u5e38\u9700\u8981\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u6216\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u6765\u89e3\u51b3\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#5-olog-n","title":"5. \u00a0 \u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"<p>\u4e0e\u6307\u6570\u9636\u76f8\u53cd\uff0c\u5bf9\u6570\u9636\u53cd\u6620\u4e86\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u201d\u7684\u60c5\u51b5\u3002\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7531\u4e8e\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\uff0c\u56e0\u6b64\u5faa\u73af\u6b21\u6570\u662f \\(\\log_2 n\\) \uff0c\u5373 \\(2^n\\) \u7684\u53cd\u51fd\u6570\u3002</p> <p>\u56fe 2-12 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6a21\u62df\u4e86\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u201d\u7684\u8fc7\u7a0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_2 n)\\) \uff0c\u7b80\u8bb0\u4e3a \\(O(\\log n)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def logarithmic(n: int) -&gt; int:\n    \"\"\"\u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n    count = 0\n    while n &gt; 1:\n        n = n / 2\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n /= 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc logarithmic(n int) int {\n    count := 0\n    for n &gt; 1 {\n        n = n / 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc logarithmic(n: Int) -&gt; Int {\n    var count = 0\n    var n = n\n    while n &gt; 1 {\n        n = n / 2\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n) {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n: number): number {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n  int count = 0;\n  while (n &gt; 1) {\n    n = n ~/ 2;\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn logarithmic(mut n: i32) -&gt; i32 {\n    let mut count = 0;\n    while n &gt; 1 {\n        n = n / 2;\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfun logarithmic(n: Int): Int {\n    var n1 = n\n    var count = 0\n    while (n1 &gt; 1) {\n        n1 /= 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{logarithmic}\n</code></pre> time_complexity.zig<pre><code>// \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn logarithmic(n: i32) i32 {\n    var count: i32 = 0;\n    var n_var = n;\n    while (n_var &gt; 1)\n    {\n        n_var = n_var / 2;\n        count +=1;\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 2-12 \u00a0 \u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u4e0e\u6307\u6570\u9636\u7c7b\u4f3c\uff0c\u5bf9\u6570\u9636\u4e5f\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u4e2d\u3002\u4ee5\u4e0b\u4ee3\u7801\u5f62\u6210\u4e86\u4e00\u68f5\u9ad8\u5ea6\u4e3a \\(\\log_2 n\\) \u7684\u9012\u5f52\u6811\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def log_recur(n: int) -&gt; int:\n    \"\"\"\u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n &lt;= 1:\n        return 0\n    return log_recur(n / 2) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint LogRecur(int n) {\n    if (n &lt;= 1) return 0;\n    return LogRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc logRecur(n int) int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n/2) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc logRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n: n / 2) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n) {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n: number): number {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n  if (n &lt;= 1) return 0;\n  return logRecur(n ~/ 2) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 0;\n    }\n    log_recur(n / 2) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun logRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 0\n    return logRecur(n / 2) + 1\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{log_recur}\n</code></pre> time_complexity.zig<pre><code>// \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn logRecur(n: i32) i32 {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u7b97\u6cd5\u4e2d\uff0c\u4f53\u73b0\u4e86\u201c\u4e00\u5206\u4e3a\u591a\u201d\u548c\u201c\u5316\u7e41\u4e3a\u7b80\u201d\u7684\u7b97\u6cd5\u601d\u60f3\u3002\u5b83\u589e\u957f\u7f13\u6162\uff0c\u662f\u4ec5\u6b21\u4e8e\u5e38\u6570\u9636\u7684\u7406\u60f3\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <p>\\(O(\\log n)\\) \u7684\u5e95\u6570\u662f\u591a\u5c11\uff1f</p> <p>\u51c6\u786e\u6765\u8bf4\uff0c\u201c\u4e00\u5206\u4e3a \\(m\\)\u201d\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(\\log_m n)\\) \u3002\u800c\u901a\u8fc7\u5bf9\u6570\u6362\u5e95\u516c\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5177\u6709\u4e0d\u540c\u5e95\u6570\u3001\u76f8\u7b49\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a</p> \\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\] <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5e95\u6570 \\(m\\) \u53ef\u4ee5\u5728\u4e0d\u5f71\u54cd\u590d\u6742\u5ea6\u7684\u524d\u63d0\u4e0b\u8f6c\u6362\u3002\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f1a\u7701\u7565\u5e95\u6570 \\(m\\) \uff0c\u5c06\u5bf9\u6570\u9636\u76f4\u63a5\u8bb0\u4e3a \\(O(\\log n)\\) \u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#6-on-log-n","title":"6. \u00a0 \u7ebf\u6027\u5bf9\u6570\u9636 \\(O(n \\log n)\\)","text":"<p>\u7ebf\u6027\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u4e24\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u522b\u4e3a \\(O(\\log n)\\) \u548c \\(O(n)\\) \u3002\u76f8\u5173\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def linear_log_recur(n: int) -&gt; int:\n    \"\"\"\u7ebf\u6027\u5bf9\u6570\u9636\"\"\"\n    if n &lt;= 1:\n        return 1\n    count: int = linear_log_recur(n // 2) + linear_log_recur(n // 2)\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint LinearLogRecur(int n) {\n    if (n &lt;= 1) return 1;\n    int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n int) int {\n    if n &lt;= 1 {\n        return 1\n    }\n    count := linearLogRecur(n/2) + linearLogRecur(n/2)\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 1\n    }\n    var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\n    for _ in stride(from: 0, to: n, by: 1) {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n) {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n: number): number {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n  if (n &lt;= 1) return 1;\n  int count = linearLogRecur(n ~/ 2) + linearLogRecur(n ~/ 2);\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfn linear_log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 1;\n    }\n    let mut count = linear_log_recur(n / 2) + linear_log_recur(n / 2);\n    for _ in 0..n as i32 {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfun linearLogRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 1\n    var count = linearLogRecur(n / 2) + linearLogRecur(n / 2)\n    for (i in 0..&lt;n.toInt()) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{linear_log_recur}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u5bf9\u6570\u9636\nfn linearLogRecur(n: i32) i32 {\n    if (n &lt;= 1) return 1;\n    var count: i32 = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 2-13 \u5c55\u793a\u4e86\u7ebf\u6027\u5bf9\u6570\u9636\u7684\u751f\u6210\u65b9\u5f0f\u3002\u4e8c\u53c9\u6811\u7684\u6bcf\u4e00\u5c42\u7684\u64cd\u4f5c\u603b\u6570\u90fd\u4e3a \\(n\\) \uff0c\u6811\u5171\u6709 \\(\\log_2 n + 1\\) \u5c42\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p> <p></p> <p> \u56fe 2-13 \u00a0 \u7ebf\u6027\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u7b49\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#7-on","title":"7. \u00a0 \u9636\u4e58\u9636 \\(O(n!)\\)","text":"<p>\u9636\u4e58\u9636\u5bf9\u5e94\u6570\u5b66\u4e0a\u7684\u201c\u5168\u6392\u5217\u201d\u95ee\u9898\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u4e92\u4e0d\u91cd\u590d\u7684\u5143\u7d20\uff0c\u6c42\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u65b9\u6848\uff0c\u65b9\u6848\u6570\u91cf\u4e3a\uff1a</p> \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\] <p>\u9636\u4e58\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u5b9e\u73b0\u3002\u5982\u56fe 2-14 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u7b2c\u4e00\u5c42\u5206\u88c2\u51fa \\(n\\) \u4e2a\uff0c\u7b2c\u4e8c\u5c42\u5206\u88c2\u51fa \\(n - 1\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u7b2c \\(n\\) \u5c42\u65f6\u505c\u6b62\u5206\u88c2\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def factorial_recur(n: int) -&gt; int:\n    \"\"\"\u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n == 0:\n        return 1\n    count = 0\n    # \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for _ in range(n):\n        count += factorial_recur(n - 1)\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint FactorialRecur(int n) {\n    if (n == 0) return 1;\n    int count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (int i = 0; i &lt; n; i++) {\n        count += FactorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n int) int {\n    if n == 0 {\n        return 1\n    }\n    count := 0\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for i := 0; i &lt; n; i++ {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n: Int) -&gt; Int {\n    if n == 0 {\n        return 1\n    }\n    var count = 0\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for _ in 0 ..&lt; n {\n        count += factorialRecur(n: n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n) {\n    if (n === 0) return 1;\n    let count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n: number): number {\n    if (n === 0) return 1;\n    let count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n  if (n == 0) return 1;\n  int count = 0;\n  // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n  for (var i = 0; i &lt; n; i++) {\n    count += factorialRecur(n - 1);\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn factorial_recur(n: i32) -&gt; i32 {\n    if n == 0 {\n        return 1;\n    }\n    let mut count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for _ in 0..n {\n        count += factorial_recur(n - 1);\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun factorialRecur(n: Int): Int {\n    if (n == 0)\n        return 1\n    var count = 0\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (i in 0..&lt;n) {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{factorial_recur}\n</code></pre> time_complexity.zig<pre><code>// \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn factorialRecur(n: i32) i32 {\n    if (n == 0) return 1;\n    var count: i32 = 0;\n    var i: i32 = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    while (i &lt; n) : (i += 1) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 2-14 \u00a0 \u9636\u4e58\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u8bf7\u6ce8\u610f\uff0c\u56e0\u4e3a\u5f53 \\(n \\geq 4\\) \u65f6\u6052\u6709 \\(n! &gt; 2^n\\) \uff0c\u6240\u4ee5\u9636\u4e58\u9636\u6bd4\u6307\u6570\u9636\u589e\u957f\u5f97\u66f4\u5feb\uff0c\u5728 \\(n\\) \u8f83\u5927\u65f6\u4e5f\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#235","title":"2.3.5 \u00a0 \u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6","text":"<p>\u7b97\u6cd5\u7684\u65f6\u95f4\u6548\u7387\u5f80\u5f80\u4e0d\u662f\u56fa\u5b9a\u7684\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u5047\u8bbe\u8f93\u5165\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d <code>nums</code> \u7531\u4ece \\(1\\) \u81f3 \\(n\\) \u7684\u6570\u5b57\u7ec4\u6210\uff0c\u6bcf\u4e2a\u6570\u5b57\u53ea\u51fa\u73b0\u4e00\u6b21\uff1b\u4f46\u5143\u7d20\u987a\u5e8f\u662f\u968f\u673a\u6253\u4e71\u7684\uff0c\u4efb\u52a1\u76ee\u6807\u662f\u8fd4\u56de\u5143\u7d20 \\(1\\) \u7684\u7d22\u5f15\u3002\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\u3002</p> <ul> <li>\u5f53 <code>nums = [?, ?, ..., 1]</code> \uff0c\u5373\u5f53\u672b\u5c3e\u5143\u7d20\u662f \\(1\\) \u65f6\uff0c\u9700\u8981\u5b8c\u6574\u904d\u5386\u6570\u7ec4\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002</li> <li>\u5f53 <code>nums = [1, ?, ?, ...]</code> \uff0c\u5373\u5f53\u9996\u4e2a\u5143\u7d20\u4e3a \\(1\\) \u65f6\uff0c\u65e0\u8bba\u6570\u7ec4\u591a\u957f\u90fd\u4e0d\u9700\u8981\u7ee7\u7eed\u904d\u5386\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(\\Omega(1)\\) \u3002</li> </ul> <p>\u201c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u201d\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u4f7f\u7528\u5927 \\(O\\) \u8bb0\u53f7\u8868\u793a\u3002\u76f8\u5e94\u5730\uff0c\u201c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u201d\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0b\u754c\uff0c\u7528 \\(\\Omega\\) \u8bb0\u53f7\u8868\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig worst_best_time_complexity.py<pre><code>def random_numbers(n: int) -&gt; list[int]:\n    \"\"\"\u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a: 1, 2, ..., n \uff0c\u987a\u5e8f\u88ab\u6253\u4e71\"\"\"\n    # \u751f\u6210\u6570\u7ec4 nums =: 1, 2, 3, ..., n\n    nums = [i for i in range(1, n + 1)]\n    # \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    random.shuffle(nums)\n    return nums\n\ndef find_one(nums: list[int]) -&gt; int:\n    \"\"\"\u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\"\"\"\n    for i in range(len(nums)):\n        # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1:\n            return i\n    return -1\n</code></pre> worst_best_time_complexity.cpp<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nvector&lt;int&gt; randomNumbers(int n) {\n    vector&lt;int&gt; nums(n);\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u4f7f\u7528\u7cfb\u7edf\u65f6\u95f4\u751f\u6210\u968f\u673a\u79cd\u5b50\n    unsigned seed = chrono::system_clock::now().time_since_epoch().count();\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    shuffle(nums.begin(), nums.end(), default_random_engine(seed));\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector&lt;int&gt; &amp;nums) {\n    for (int i = 0; i &lt; nums.size(); i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.java<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\n    Integer[] nums = new Integer[n];\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    Collections.shuffle(Arrays.asList(nums));\n    // Integer[] -&gt; int[]\n    int[] res = new int[n];\n    for (int i = 0; i &lt; n; i++) {\n        res[i] = nums[i];\n    }\n    return res;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.cs<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] RandomNumbers(int n) {\n    int[] nums = new int[n];\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (int i = 0; i &lt; nums.Length; i++) {\n        int index = new Random().Next(i, nums.Length);\n        (nums[i], nums[index]) = (nums[index], nums[i]);\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint FindOne(int[] nums) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.go<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n int) []int {\n    nums := make([]int, n)\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for i := 0; i &lt; n; i++ {\n        nums[i] = i + 1\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    rand.Shuffle(len(nums), func(i, j int) {\n        nums[i], nums[j] = nums[j], nums[i]\n    })\n    return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\n    for i := 0; i &lt; len(nums); i++ {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.swift<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n: Int) -&gt; [Int] {\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    var nums = Array(1 ... n)\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    nums.shuffle()\n    return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -&gt; Int {\n    for i in nums.indices {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.js<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n) {\n    const nums = Array(n);\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.ts<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n: number): number[] {\n    const nums = Array(n);\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.dart<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nList&lt;int&gt; randomNumbers(int n) {\n  final nums = List.filled(n, 0);\n  // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n  for (var i = 0; i &lt; n; i++) {\n    nums[i] = i + 1;\n  }\n  // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n  nums.shuffle();\n\n  return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(List&lt;int&gt; nums) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n    // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n    if (nums[i] == 1) return i;\n  }\n\n  return -1;\n}\n</code></pre> worst_best_time_complexity.rs<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfn random_numbers(n: i32) -&gt; Vec&lt;i32&gt; {\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    let mut nums = (1..=n).collect::&lt;Vec&lt;i32&gt;&gt;();\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    nums.shuffle(&amp;mut thread_rng());\n    nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfn find_one(nums: &amp;[i32]) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1 {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> worst_best_time_complexity.c<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint *randomNumbers(int n) {\n    // \u5206\u914d\u5806\u533a\u5185\u5b58\uff08\u521b\u5efa\u4e00\u7ef4\u53ef\u53d8\u957f\u6570\u7ec4\uff1a\u6570\u7ec4\u4e2d\u5143\u7d20\u6570\u91cf\u4e3a n \uff0c\u5143\u7d20\u7c7b\u578b\u4e3a int \uff09\n    int *nums = (int *)malloc(n * sizeof(int));\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (int i = n - 1; i &gt; 0; i--) {\n        int j = rand() % (i + 1);\n        int temp = nums[i];\n        nums[i] = nums[j];\n        nums[j] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int *nums, int n) {\n    for (int i = 0; i &lt; n; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.kt<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfun randomNumbers(n: Int): Array&lt;Int?&gt; {\n    val nums = IntArray(n)\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (i in 0..&lt;n) {\n        nums[i] = i + 1\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    val mutableList = nums.toMutableList()\n    mutableList.shuffle()\n    // Integer[] -&gt; int[]\n    val res = arrayOfNulls&lt;Int&gt;(n)\n    for (i in 0..&lt;n) {\n        res[i] = mutableList[i]\n    }\n    return res\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfun findOne(nums: Array&lt;Int?&gt;): Int {\n    for (i in nums.indices) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.rb<pre><code>[class]{}-[func]{random_numbers}\n\n[class]{}-[func]{find_one}\n</code></pre> worst_best_time_complexity.zig<pre><code>// \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71\nfn randomNumbers(comptime n: usize) [n]i32 {\n    var nums: [n]i32 = undefined;\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (&amp;nums, 0..) |*num, i| {\n        num.* = @as(i32, @intCast(i)) + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    const rand = std.crypto.random;\n    rand.shuffle(i32, &amp;nums);\n    return nums;\n}\n\n// \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\nfn findOne(nums: []i32) i32 {\n    for (nums, 0..) |num, i| {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (num == 1) return @intCast(i);\n    }\n    return -1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6211\u4eec\u5728\u5b9e\u9645\u4e2d\u5f88\u5c11\u4f7f\u7528\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u56e0\u4e3a\u901a\u5e38\u53ea\u6709\u5728\u5f88\u5c0f\u6982\u7387\u4e0b\u624d\u80fd\u8fbe\u5230\uff0c\u53ef\u80fd\u4f1a\u5e26\u6765\u4e00\u5b9a\u7684\u8bef\u5bfc\u6027\u3002\u800c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u66f4\u4e3a\u5b9e\u7528\uff0c\u56e0\u4e3a\u5b83\u7ed9\u51fa\u4e86\u4e00\u4e2a\u6548\u7387\u5b89\u5168\u503c\uff0c\u8ba9\u6211\u4eec\u53ef\u4ee5\u653e\u5fc3\u5730\u4f7f\u7528\u7b97\u6cd5\u3002</p> <p>\u4ece\u4e0a\u8ff0\u793a\u4f8b\u53ef\u4ee5\u770b\u51fa\uff0c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ea\u51fa\u73b0\u4e8e\u201c\u7279\u6b8a\u7684\u6570\u636e\u5206\u5e03\u201d\uff0c\u8fd9\u4e9b\u60c5\u51b5\u7684\u51fa\u73b0\u6982\u7387\u53ef\u80fd\u5f88\u5c0f\uff0c\u5e76\u4e0d\u80fd\u771f\u5b9e\u5730\u53cd\u6620\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f53\u73b0\u7b97\u6cd5\u5728\u968f\u673a\u8f93\u5165\u6570\u636e\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u7528 \\(\\Theta\\) \u8bb0\u53f7\u6765\u8868\u793a\u3002</p> <p>\u5bf9\u4e8e\u90e8\u5206\u7b97\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u63a8\u7b97\u51fa\u968f\u673a\u6570\u636e\u5206\u5e03\u4e0b\u7684\u5e73\u5747\u60c5\u51b5\u3002\u6bd4\u5982\u4e0a\u8ff0\u793a\u4f8b\uff0c\u7531\u4e8e\u8f93\u5165\u6570\u7ec4\u662f\u88ab\u6253\u4e71\u7684\uff0c\u56e0\u6b64\u5143\u7d20 \\(1\\) \u51fa\u73b0\u5728\u4efb\u610f\u7d22\u5f15\u7684\u6982\u7387\u90fd\u662f\u76f8\u7b49\u7684\uff0c\u90a3\u4e48\u7b97\u6cd5\u7684\u5e73\u5747\u5faa\u73af\u6b21\u6570\u5c31\u662f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a \\(n / 2\\) \uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(\\Theta(n / 2) = \\Theta(n)\\) \u3002</p> <p>\u4f46\u5bf9\u4e8e\u8f83\u4e3a\u590d\u6742\u7684\u7b97\u6cd5\uff0c\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5f80\u5f80\u6bd4\u8f83\u56f0\u96be\uff0c\u56e0\u4e3a\u5f88\u96be\u5206\u6790\u51fa\u5728\u6570\u636e\u5206\u5e03\u4e0b\u7684\u6574\u4f53\u6570\u5b66\u671f\u671b\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f5c\u4e3a\u7b97\u6cd5\u6548\u7387\u7684\u8bc4\u5224\u6807\u51c6\u3002</p> <p>\u4e3a\u4ec0\u4e48\u5f88\u5c11\u770b\u5230 \\(\\Theta\\) \u7b26\u53f7\uff1f</p> <p>\u53ef\u80fd\u7531\u4e8e \\(O\\) \u7b26\u53f7\u8fc7\u4e8e\u6717\u6717\u4e0a\u53e3\uff0c\u56e0\u6b64\u6211\u4eec\u5e38\u5e38\u4f7f\u7528\u5b83\u6765\u8868\u793a\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u4f46\u4ece\u4e25\u683c\u610f\u4e49\u4e0a\u8bb2\uff0c\u8fd9\u79cd\u505a\u6cd5\u5e76\u4e0d\u89c4\u8303\u3002\u5728\u672c\u4e66\u548c\u5176\u4ed6\u8d44\u6599\u4e2d\uff0c\u82e5\u9047\u5230\u7c7b\u4f3c\u201c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u201d\u7684\u8868\u8ff0\uff0c\u8bf7\u5c06\u5176\u76f4\u63a5\u7406\u89e3\u4e3a \\(\\Theta(n)\\) \u3002</p>"},{"location":"chapter_data_structure/","title":"\u7b2c 3 \u7ae0 \u00a0 \u6570\u636e\u7ed3\u6784","text":"<p>Abstract</p> <p>\u6570\u636e\u7ed3\u6784\u5982\u540c\u4e00\u526f\u7a33\u56fa\u800c\u591a\u6837\u7684\u6846\u67b6\u3002</p> <p>\u5b83\u4e3a\u6570\u636e\u7684\u6709\u5e8f\u7ec4\u7ec7\u63d0\u4f9b\u4e86\u84dd\u56fe\uff0c\u7b97\u6cd5\u5f97\u4ee5\u5728\u6b64\u57fa\u7840\u4e0a\u751f\u52a8\u8d77\u6765\u3002</p>"},{"location":"chapter_data_structure/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>3.1 \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b</li> <li>3.2 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b</li> <li>3.3 \u00a0 \u6570\u5b57\u7f16\u7801 *</li> <li>3.4 \u00a0 \u5b57\u7b26\u7f16\u7801 *</li> <li>3.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_data_structure/basic_data_types/","title":"3.2 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b","text":"<p>\u5f53\u8c08\u53ca\u8ba1\u7b97\u673a\u4e2d\u7684\u6570\u636e\u65f6\uff0c\u6211\u4eec\u4f1a\u60f3\u5230\u6587\u672c\u3001\u56fe\u7247\u3001\u89c6\u9891\u3001\u8bed\u97f3\u30013D \u6a21\u578b\u7b49\u5404\u79cd\u5f62\u5f0f\u3002\u5c3d\u7ba1\u8fd9\u4e9b\u6570\u636e\u7684\u7ec4\u7ec7\u5f62\u5f0f\u5404\u5f02\uff0c\u4f46\u5b83\u4eec\u90fd\u7531\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6784\u6210\u3002</p> <p>\u57fa\u672c\u6570\u636e\u7c7b\u578b\u662f CPU \u53ef\u4ee5\u76f4\u63a5\u8fdb\u884c\u8fd0\u7b97\u7684\u7c7b\u578b\uff0c\u5728\u7b97\u6cd5\u4e2d\u76f4\u63a5\u88ab\u4f7f\u7528\uff0c\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u3002</p> <ul> <li>\u6574\u6570\u7c7b\u578b <code>byte</code>\u3001<code>short</code>\u3001<code>int</code>\u3001<code>long</code> \u3002</li> <li>\u6d6e\u70b9\u6570\u7c7b\u578b <code>float</code>\u3001<code>double</code> \uff0c\u7528\u4e8e\u8868\u793a\u5c0f\u6570\u3002</li> <li>\u5b57\u7b26\u7c7b\u578b <code>char</code> \uff0c\u7528\u4e8e\u8868\u793a\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u6bcd\u3001\u6807\u70b9\u7b26\u53f7\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u7b49\u3002</li> <li>\u5e03\u5c14\u7c7b\u578b <code>bool</code> \uff0c\u7528\u4e8e\u8868\u793a\u201c\u662f\u201d\u4e0e\u201c\u5426\u201d\u5224\u65ad\u3002</li> </ul> <p>\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4ee5\u4e8c\u8fdb\u5236\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u3002\u4e00\u4e2a\u4e8c\u8fdb\u5236\u4f4d\u5373\u4e3a \\(1\\) \u6bd4\u7279\u3002\u5728\u7edd\u5927\u591a\u6570\u73b0\u4ee3\u64cd\u4f5c\u7cfb\u7edf\u4e2d\uff0c\\(1\\) \u5b57\u8282\uff08byte\uff09\u7531 \\(8\\) \u6bd4\u7279\uff08bit\uff09\u7ec4\u6210\u3002</p> <p>\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5176\u5360\u7528\u7684\u7a7a\u95f4\u5927\u5c0f\u3002\u4e0b\u9762\u4ee5 Java \u4e3a\u4f8b\u3002</p> <ul> <li>\u6574\u6570\u7c7b\u578b <code>byte</code> \u5360\u7528 \\(1\\) \u5b57\u8282 = \\(8\\) \u6bd4\u7279 \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{8}\\) \u4e2a\u6570\u5b57\u3002</li> <li>\u6574\u6570\u7c7b\u578b <code>int</code> \u5360\u7528 \\(4\\) \u5b57\u8282 = \\(32\\) \u6bd4\u7279 \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{32}\\) \u4e2a\u6570\u5b57\u3002</li> </ul> <p>\u8868 3-1 \u5217\u4e3e\u4e86 Java \u4e2d\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5360\u7528\u7a7a\u95f4\u3001\u53d6\u503c\u8303\u56f4\u548c\u9ed8\u8ba4\u503c\u3002\u6b64\u8868\u683c\u65e0\u987b\u6b7b\u8bb0\u786c\u80cc\uff0c\u5927\u81f4\u7406\u89e3\u5373\u53ef\uff0c\u9700\u8981\u65f6\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u6765\u56de\u5fc6\u3002</p> <p> \u8868 3-1 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5360\u7528\u7a7a\u95f4\u548c\u53d6\u503c\u8303\u56f4 </p> \u7c7b\u578b \u7b26\u53f7 \u5360\u7528\u7a7a\u95f4 \u6700\u5c0f\u503c \u6700\u5927\u503c \u9ed8\u8ba4\u503c \u6574\u6570 <code>byte</code> 1 \u5b57\u8282 \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) \\(0\\) <code>short</code> 2 \u5b57\u8282 \\(-2^{15}\\) \\(2^{15} - 1\\) \\(0\\) <code>int</code> 4 \u5b57\u8282 \\(-2^{31}\\) \\(2^{31} - 1\\) \\(0\\) <code>long</code> 8 \u5b57\u8282 \\(-2^{63}\\) \\(2^{63} - 1\\) \\(0\\) \u6d6e\u70b9\u6570 <code>float</code> 4 \u5b57\u8282 \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0\\text{f}\\) <code>double</code> 8 \u5b57\u8282 \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) \\(0.0\\) \u5b57\u7b26 <code>char</code> 2 \u5b57\u8282 \\(0\\) \\(2^{16} - 1\\) \\(0\\) \u5e03\u5c14 <code>bool</code> 1 \u5b57\u8282 \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\) <p>\u8bf7\u6ce8\u610f\uff0c\u8868 3-1 \u9488\u5bf9\u7684\u662f Java \u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u60c5\u51b5\u3002\u6bcf\u79cd\u7f16\u7a0b\u8bed\u8a00\u90fd\u6709\u5404\u81ea\u7684\u6570\u636e\u7c7b\u578b\u5b9a\u4e49\uff0c\u5b83\u4eec\u7684\u5360\u7528\u7a7a\u95f4\u3001\u53d6\u503c\u8303\u56f4\u548c\u9ed8\u8ba4\u503c\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002</p> <ul> <li>\u5728 Python \u4e2d\uff0c\u6574\u6570\u7c7b\u578b <code>int</code> \u53ef\u4ee5\u662f\u4efb\u610f\u5927\u5c0f\uff0c\u53ea\u53d7\u9650\u4e8e\u53ef\u7528\u5185\u5b58\uff1b\u6d6e\u70b9\u6570 <code>float</code> \u662f\u53cc\u7cbe\u5ea6 64 \u4f4d\uff1b\u6ca1\u6709 <code>char</code> \u7c7b\u578b\uff0c\u5355\u4e2a\u5b57\u7b26\u5b9e\u9645\u4e0a\u662f\u957f\u5ea6\u4e3a 1 \u7684\u5b57\u7b26\u4e32 <code>str</code> \u3002</li> <li>C \u548c C++ \u672a\u660e\u786e\u89c4\u5b9a\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5927\u5c0f\uff0c\u800c\u56e0\u5b9e\u73b0\u548c\u5e73\u53f0\u5404\u5f02\u3002\u8868 3-1 \u9075\u5faa LP64 \u6570\u636e\u6a21\u578b\uff0c\u5176\u7528\u4e8e\u5305\u62ec Linux \u548c macOS \u5728\u5185\u7684 Unix 64 \u4f4d\u64cd\u4f5c\u7cfb\u7edf\u3002</li> <li>\u5b57\u7b26 <code>char</code> \u7684\u5927\u5c0f\u5728 C \u548c C++ \u4e2d\u4e3a 1 \u5b57\u8282\uff0c\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\u53d6\u51b3\u4e8e\u7279\u5b9a\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\uff0c\u8be6\u89c1\u201c\u5b57\u7b26\u7f16\u7801\u201d\u7ae0\u8282\u3002</li> <li>\u5373\u4f7f\u8868\u793a\u5e03\u5c14\u91cf\u4ec5\u9700 1 \u4f4d\uff08\\(0\\) \u6216 \\(1\\)\uff09\uff0c\u5b83\u5728\u5185\u5b58\u4e2d\u901a\u5e38\u4e5f\u5b58\u50a8\u4e3a 1 \u5b57\u8282\u3002\u8fd9\u662f\u56e0\u4e3a\u73b0\u4ee3\u8ba1\u7b97\u673a CPU \u901a\u5e38\u5c06 1 \u5b57\u8282\u4f5c\u4e3a\u6700\u5c0f\u5bfb\u5740\u5185\u5b58\u5355\u5143\u3002</li> </ul> <p>\u90a3\u4e48\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4e0e\u6570\u636e\u7ed3\u6784\u4e4b\u95f4\u6709\u4ec0\u4e48\u8054\u7cfb\u5462\uff1f\u6211\u4eec\u77e5\u9053\uff0c\u6570\u636e\u7ed3\u6784\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u4e0e\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002\u8fd9\u53e5\u8bdd\u7684\u4e3b\u8bed\u662f\u201c\u7ed3\u6784\u201d\u800c\u975e\u201c\u6570\u636e\u201d\u3002</p> <p>\u5982\u679c\u60f3\u8868\u793a\u201c\u4e00\u6392\u6570\u5b57\u201d\uff0c\u6211\u4eec\u81ea\u7136\u4f1a\u60f3\u5230\u4f7f\u7528\u6570\u7ec4\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u7684\u7ebf\u6027\u7ed3\u6784\u53ef\u4ee5\u8868\u793a\u6570\u5b57\u7684\u76f8\u90bb\u5173\u7cfb\u548c\u987a\u5e8f\u5173\u7cfb\uff0c\u4f46\u81f3\u4e8e\u5b58\u50a8\u7684\u5185\u5bb9\u662f\u6574\u6570 <code>int</code>\u3001\u5c0f\u6570 <code>float</code> \u8fd8\u662f\u5b57\u7b26 <code>char</code> \uff0c\u5219\u4e0e\u201c\u6570\u636e\u7ed3\u6784\u201d\u65e0\u5173\u3002</p> <p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u5185\u5bb9\u7c7b\u578b\u201d\uff0c\u800c\u6570\u636e\u7ed3\u6784\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u7ec4\u7ec7\u65b9\u5f0f\u201d\u3002\u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6211\u4eec\u7528\u76f8\u540c\u7684\u6570\u636e\u7ed3\u6784\uff08\u6570\u7ec4\uff09\u6765\u5b58\u50a8\u4e0e\u8868\u793a\u4e0d\u540c\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u5305\u62ec <code>int</code>\u3001<code>float</code>\u3001<code>char</code>\u3001<code>bool</code> \u7b49\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nnumbers: list[int] = [0] * 5\ndecimals: list[float] = [0.0] * 5\n# Python \u7684\u5b57\u7b26\u5b9e\u9645\u4e0a\u662f\u957f\u5ea6\u4e3a 1 \u7684\u5b57\u7b26\u4e32\ncharacters: list[str] = ['0'] * 5\nbools: list[bool] = [False] * 5\n# Python \u7684\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61\u5f15\u7528\ndata = [0, 0.0, 'a', False, ListNode(0)]\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool bools[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] bools = new boolean[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] bools = new bool[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar bools = [5]bool{}\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nlet numbers = Array(repeating: 0, count: 5)\nlet decimals = Array(repeating: 0.0, count: 5)\nlet characters: [Character] = Array(repeating: \"a\", count: 5)\nlet bools = Array(repeating: false, count: 5)\n</code></pre> <pre><code>// JavaScript \u7684\u6570\u7ec4\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61\nconst array = [0, 0.0, 'a', false];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst bools: boolean[] = [];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nList&lt;int&gt; numbers = List.filled(5, 0);\nList&lt;double&gt; decimals = List.filled(5, 0.0);\nList&lt;String&gt; characters = List.filled(5, 'a');\nList&lt;bool&gt; bools = List.filled(5, false);\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nlet numbers: Vec&lt;i32&gt; = vec![0; 5];\nlet decimals: Vec&lt;f32&gt; = vec![0.0; 5];\nlet characters: Vec&lt;char&gt; = vec!['0'; 5];\nlet bools: Vec&lt;bool&gt; = vec![false; 5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool bools[10];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nval numbers = IntArray(5)\nval decinals = FloatArray(5)\nval characters = CharArray(5)\nval bools = BooleanArray(5)\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_data_structure/character_encoding/","title":"3.4 \u00a0 \u5b57\u7b26\u7f16\u7801 *","text":"<p>\u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u6240\u6709\u6570\u636e\u90fd\u662f\u4ee5\u4e8c\u8fdb\u5236\u6570\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\uff0c\u5b57\u7b26 <code>char</code> \u4e5f\u4e0d\u4f8b\u5916\u3002\u4e3a\u4e86\u8868\u793a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u5efa\u7acb\u4e00\u5957\u201c\u5b57\u7b26\u96c6\u201d\uff0c\u89c4\u5b9a\u6bcf\u4e2a\u5b57\u7b26\u548c\u4e8c\u8fdb\u5236\u6570\u4e4b\u95f4\u7684\u4e00\u4e00\u5bf9\u5e94\u5173\u7cfb\u3002\u6709\u4e86\u5b57\u7b26\u96c6\u4e4b\u540e\uff0c\u8ba1\u7b97\u673a\u5c31\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u5b8c\u6210\u4e8c\u8fdb\u5236\u6570\u5230\u5b57\u7b26\u7684\u8f6c\u6362\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#341-ascii","title":"3.4.1 \u00a0 ASCII \u5b57\u7b26\u96c6","text":"<p>\u300cASCII \u7801\u300d\u662f\u6700\u65e9\u51fa\u73b0\u7684\u5b57\u7b26\u96c6\uff0c\u5176\u5168\u79f0\u4e3a American Standard Code for Information Interchange\uff08\u7f8e\u56fd\u6807\u51c6\u4fe1\u606f\u4ea4\u6362\u4ee3\u7801\uff09\u3002\u5b83\u4f7f\u7528 7 \u4f4d\u4e8c\u8fdb\u5236\u6570\uff08\u4e00\u4e2a\u5b57\u8282\u7684\u4f4e 7 \u4f4d\uff09\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6700\u591a\u80fd\u591f\u8868\u793a 128 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002\u5982\u56fe 3-6 \u6240\u793a\uff0cASCII \u7801\u5305\u62ec\u82f1\u6587\u5b57\u6bcd\u7684\u5927\u5c0f\u5199\u3001\u6570\u5b57 0 ~ 9\u3001\u4e00\u4e9b\u6807\u70b9\u7b26\u53f7\uff0c\u4ee5\u53ca\u4e00\u4e9b\u63a7\u5236\u5b57\u7b26\uff08\u5982\u6362\u884c\u7b26\u548c\u5236\u8868\u7b26\uff09\u3002</p> <p></p> <p> \u56fe 3-6 \u00a0 ASCII \u7801 </p> <p>\u7136\u800c\uff0cASCII \u7801\u4ec5\u80fd\u591f\u8868\u793a\u82f1\u6587\u3002\u968f\u7740\u8ba1\u7b97\u673a\u7684\u5168\u7403\u5316\uff0c\u8bde\u751f\u4e86\u4e00\u79cd\u80fd\u591f\u8868\u793a\u66f4\u591a\u8bed\u8a00\u7684\u300cEASCII\u300d\u5b57\u7b26\u96c6\u3002\u5b83\u5728 ASCII \u7684 7 \u4f4d\u57fa\u7840\u4e0a\u6269\u5c55\u5230 8 \u4f4d\uff0c\u80fd\u591f\u8868\u793a 256 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002</p> <p>\u5728\u4e16\u754c\u8303\u56f4\u5185\uff0c\u9646\u7eed\u51fa\u73b0\u4e86\u4e00\u6279\u9002\u7528\u4e8e\u4e0d\u540c\u5730\u533a\u7684 EASCII \u5b57\u7b26\u96c6\u3002\u8fd9\u4e9b\u5b57\u7b26\u96c6\u7684\u524d 128 \u4e2a\u5b57\u7b26\u7edf\u4e00\u4e3a ASCII \u7801\uff0c\u540e 128 \u4e2a\u5b57\u7b26\u5b9a\u4e49\u4e0d\u540c\uff0c\u4ee5\u9002\u5e94\u4e0d\u540c\u8bed\u8a00\u7684\u9700\u6c42\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#342-gbk","title":"3.4.2 \u00a0 GBK \u5b57\u7b26\u96c6","text":"<p>\u540e\u6765\u4eba\u4eec\u53d1\u73b0\uff0cEASCII \u7801\u4ecd\u7136\u65e0\u6cd5\u6ee1\u8db3\u8bb8\u591a\u8bed\u8a00\u7684\u5b57\u7b26\u6570\u91cf\u8981\u6c42\u3002\u6bd4\u5982\u6c49\u5b57\u6709\u8fd1\u5341\u4e07\u4e2a\uff0c\u5149\u65e5\u5e38\u4f7f\u7528\u7684\u5c31\u6709\u51e0\u5343\u4e2a\u3002\u4e2d\u56fd\u56fd\u5bb6\u6807\u51c6\u603b\u5c40\u4e8e 1980 \u5e74\u53d1\u5e03\u4e86\u300cGB2312\u300d\u5b57\u7b26\u96c6\uff0c\u5176\u6536\u5f55\u4e86 6763 \u4e2a\u6c49\u5b57\uff0c\u57fa\u672c\u6ee1\u8db3\u4e86\u6c49\u5b57\u7684\u8ba1\u7b97\u673a\u5904\u7406\u9700\u8981\u3002</p> <p>\u7136\u800c\uff0cGB2312 \u65e0\u6cd5\u5904\u7406\u90e8\u5206\u7f55\u89c1\u5b57\u548c\u7e41\u4f53\u5b57\u3002\u300cGBK\u300d\u5b57\u7b26\u96c6\u662f\u5728 GB2312 \u7684\u57fa\u7840\u4e0a\u6269\u5c55\u5f97\u5230\u7684\uff0c\u5b83\u5171\u6536\u5f55\u4e86 21886 \u4e2a\u6c49\u5b57\u3002\u5728 GBK \u7684\u7f16\u7801\u65b9\u6848\u4e2d\uff0cASCII \u5b57\u7b26\u4f7f\u7528\u4e00\u4e2a\u5b57\u8282\u8868\u793a\uff0c\u6c49\u5b57\u4f7f\u7528\u4e24\u4e2a\u5b57\u8282\u8868\u793a\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#343-unicode","title":"3.4.3 \u00a0 Unicode \u5b57\u7b26\u96c6","text":"<p>\u968f\u7740\u8ba1\u7b97\u673a\u6280\u672f\u7684\u84ec\u52c3\u53d1\u5c55\uff0c\u5b57\u7b26\u96c6\u4e0e\u7f16\u7801\u6807\u51c6\u767e\u82b1\u9f50\u653e\uff0c\u800c\u8fd9\u5e26\u6765\u4e86\u8bb8\u591a\u95ee\u9898\u3002\u4e00\u65b9\u9762\uff0c\u8fd9\u4e9b\u5b57\u7b26\u96c6\u4e00\u822c\u53ea\u5b9a\u4e49\u4e86\u7279\u5b9a\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u65e0\u6cd5\u5728\u591a\u8bed\u8a00\u73af\u5883\u4e0b\u6b63\u5e38\u5de5\u4f5c\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u540c\u4e00\u79cd\u8bed\u8a00\u5b58\u5728\u591a\u79cd\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u5982\u679c\u4e24\u53f0\u8ba1\u7b97\u673a\u4f7f\u7528\u7684\u662f\u4e0d\u540c\u7684\u7f16\u7801\u6807\u51c6\uff0c\u5219\u5728\u4fe1\u606f\u4f20\u9012\u65f6\u5c31\u4f1a\u51fa\u73b0\u4e71\u7801\u3002</p> <p>\u90a3\u4e2a\u65f6\u4ee3\u7684\u7814\u7a76\u4eba\u5458\u5c31\u5728\u60f3\uff1a\u5982\u679c\u63a8\u51fa\u4e00\u4e2a\u8db3\u591f\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\uff0c\u5c06\u4e16\u754c\u8303\u56f4\u5185\u7684\u6240\u6709\u8bed\u8a00\u548c\u7b26\u53f7\u90fd\u6536\u5f55\u5176\u4e2d\uff0c\u4e0d\u5c31\u53ef\u4ee5\u89e3\u51b3\u8de8\u8bed\u8a00\u73af\u5883\u548c\u4e71\u7801\u95ee\u9898\u4e86\u5417\uff1f\u5728\u8fd9\u79cd\u60f3\u6cd5\u7684\u9a71\u52a8\u4e0b\uff0c\u4e00\u4e2a\u5927\u800c\u5168\u7684\u5b57\u7b26\u96c6 Unicode \u5e94\u8fd0\u800c\u751f\u3002</p> <p>\u300cUnicode\u300d\u7684\u4e2d\u6587\u540d\u79f0\u4e3a\u201c\u7edf\u4e00\u7801\u201d\uff0c\u7406\u8bba\u4e0a\u80fd\u5bb9\u7eb3 100 \u591a\u4e07\u4e2a\u5b57\u7b26\u3002\u5b83\u81f4\u529b\u4e8e\u5c06\u5168\u7403\u8303\u56f4\u5185\u7684\u5b57\u7b26\u7eb3\u5165\u7edf\u4e00\u7684\u5b57\u7b26\u96c6\u4e4b\u4e2d\uff0c\u63d0\u4f9b\u4e00\u79cd\u901a\u7528\u7684\u5b57\u7b26\u96c6\u6765\u5904\u7406\u548c\u663e\u793a\u5404\u79cd\u8bed\u8a00\u6587\u5b57\uff0c\u51cf\u5c11\u56e0\u4e3a\u7f16\u7801\u6807\u51c6\u4e0d\u540c\u800c\u4ea7\u751f\u7684\u4e71\u7801\u95ee\u9898\u3002</p> <p>\u81ea 1991 \u5e74\u53d1\u5e03\u4ee5\u6765\uff0cUnicode \u4e0d\u65ad\u6269\u5145\u65b0\u7684\u8bed\u8a00\u4e0e\u5b57\u7b26\u3002\u622a\u81f3 2022 \u5e74 9 \u6708\uff0cUnicode \u5df2\u7ecf\u5305\u542b 149186 \u4e2a\u5b57\u7b26\uff0c\u5305\u62ec\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\u3001\u7b26\u53f7\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u7b49\u3002\u5728\u5e9e\u5927\u7684 Unicode \u5b57\u7b26\u96c6\u4e2d\uff0c\u5e38\u7528\u7684\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\uff0c\u6709\u4e9b\u751f\u50fb\u7684\u5b57\u7b26\u5360\u7528 3 \u5b57\u8282\u751a\u81f3 4 \u5b57\u8282\u3002</p> <p>Unicode \u662f\u4e00\u79cd\u901a\u7528\u5b57\u7b26\u96c6\uff0c\u672c\u8d28\u4e0a\u662f\u7ed9\u6bcf\u4e2a\u5b57\u7b26\u5206\u914d\u4e00\u4e2a\u7f16\u53f7\uff08\u79f0\u4e3a\u201c\u7801\u70b9\u201d\uff09\uff0c\u4f46\u5b83\u5e76\u6ca1\u6709\u89c4\u5b9a\u5728\u8ba1\u7b97\u673a\u4e2d\u5982\u4f55\u5b58\u50a8\u8fd9\u4e9b\u5b57\u7b26\u7801\u70b9\u3002\u6211\u4eec\u4e0d\u7981\u4f1a\u95ee\uff1a\u5f53\u591a\u79cd\u957f\u5ea6\u7684 Unicode \u7801\u70b9\u540c\u65f6\u51fa\u73b0\u5728\u4e00\u4e2a\u6587\u672c\u4e2d\u65f6\uff0c\u7cfb\u7edf\u5982\u4f55\u89e3\u6790\u5b57\u7b26\uff1f\u4f8b\u5982\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a 2 \u5b57\u8282\u7684\u7f16\u7801\uff0c\u7cfb\u7edf\u5982\u4f55\u786e\u8ba4\u5b83\u662f\u4e00\u4e2a 2 \u5b57\u8282\u7684\u5b57\u7b26\u8fd8\u662f\u4e24\u4e2a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff1f</p> <p>\u5bf9\u4e8e\u4ee5\u4e0a\u95ee\u9898\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u89e3\u51b3\u65b9\u6848\u662f\u5c06\u6240\u6709\u5b57\u7b26\u5b58\u50a8\u4e3a\u7b49\u957f\u7684\u7f16\u7801\u3002\u5982\u56fe 3-7 \u6240\u793a\uff0c\u201cHello\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u5b57\u8282\uff0c\u201c\u7b97\u6cd5\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9ad8\u4f4d\u586b 0 \u5c06\u201cHello \u7b97\u6cd5\u201d\u4e2d\u7684\u6240\u6709\u5b57\u7b26\u90fd\u7f16\u7801\u4e3a 2 \u5b57\u8282\u957f\u5ea6\u3002\u8fd9\u6837\u7cfb\u7edf\u5c31\u53ef\u4ee5\u6bcf\u9694 2 \u5b57\u8282\u89e3\u6790\u4e00\u4e2a\u5b57\u7b26\uff0c\u6062\u590d\u8fd9\u4e2a\u77ed\u8bed\u7684\u5185\u5bb9\u4e86\u3002</p> <p></p> <p> \u56fe 3-7 \u00a0 Unicode \u7f16\u7801\u793a\u4f8b </p> <p>\u7136\u800c ASCII \u7801\u5df2\u7ecf\u5411\u6211\u4eec\u8bc1\u660e\uff0c\u7f16\u7801\u82f1\u6587\u53ea\u9700 1 \u5b57\u8282\u3002\u82e5\u91c7\u7528\u4e0a\u8ff0\u65b9\u6848\uff0c\u82f1\u6587\u6587\u672c\u5360\u7528\u7a7a\u95f4\u7684\u5927\u5c0f\u5c06\u4f1a\u662f ASCII \u7f16\u7801\u4e0b\u7684\u4e24\u500d\uff0c\u975e\u5e38\u6d6a\u8d39\u5185\u5b58\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u66f4\u52a0\u9ad8\u6548\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#344-utf-8","title":"3.4.4 \u00a0 UTF-8 \u7f16\u7801","text":"<p>\u76ee\u524d\uff0cUTF-8 \u5df2\u6210\u4e3a\u56fd\u9645\u4e0a\u4f7f\u7528\u6700\u5e7f\u6cdb\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002\u5b83\u662f\u4e00\u79cd\u53ef\u53d8\u957f\u5ea6\u7684\u7f16\u7801\uff0c\u4f7f\u7528 1 \u5230 4 \u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6839\u636e\u5b57\u7b26\u7684\u590d\u6742\u6027\u800c\u53d8\u3002ASCII \u5b57\u7b26\u53ea\u9700 1 \u5b57\u8282\uff0c\u62c9\u4e01\u5b57\u6bcd\u548c\u5e0c\u814a\u5b57\u6bcd\u9700\u8981 2 \u5b57\u8282\uff0c\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u9700\u8981 3 \u5b57\u8282\uff0c\u5176\u4ed6\u7684\u4e00\u4e9b\u751f\u50fb\u5b57\u7b26\u9700\u8981 4 \u5b57\u8282\u3002</p> <p>UTF-8 \u7684\u7f16\u7801\u89c4\u5219\u5e76\u4e0d\u590d\u6742\uff0c\u5206\u4e3a\u4ee5\u4e0b\u4e24\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u5bf9\u4e8e\u957f\u5ea6\u4e3a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff0c\u5c06\u6700\u9ad8\u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \uff0c\u5176\u4f59 7 \u4f4d\u8bbe\u7f6e\u4e3a Unicode \u7801\u70b9\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0cASCII \u5b57\u7b26\u5728 Unicode \u5b57\u7b26\u96c6\u4e2d\u5360\u636e\u4e86\u524d 128 \u4e2a\u7801\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0cUTF-8 \u7f16\u7801\u53ef\u4ee5\u5411\u4e0b\u517c\u5bb9 ASCII \u7801\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528 UTF-8 \u6765\u89e3\u6790\u5e74\u4ee3\u4e45\u8fdc\u7684 ASCII \u7801\u6587\u672c\u3002</li> <li>\u5bf9\u4e8e\u957f\u5ea6\u4e3a \\(n\\) \u5b57\u8282\u7684\u5b57\u7b26\uff08\u5176\u4e2d \\(n &gt; 1\\)\uff09\uff0c\u5c06\u9996\u4e2a\u5b57\u8282\u7684\u9ad8 \\(n\\) \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(1\\) \uff0c\u7b2c \\(n + 1\\) \u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \uff1b\u4ece\u7b2c\u4e8c\u4e2a\u5b57\u8282\u5f00\u59cb\uff0c\u5c06\u6bcf\u4e2a\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \uff1b\u5176\u4f59\u6240\u6709\u4f4d\u7528\u4e8e\u586b\u5145\u5b57\u7b26\u7684 Unicode \u7801\u70b9\u3002</li> </ul> <p>\u56fe 3-8 \u5c55\u793a\u4e86\u201cHello\u7b97\u6cd5\u201d\u5bf9\u5e94\u7684 UTF-8 \u7f16\u7801\u3002\u89c2\u5bdf\u53d1\u73b0\uff0c\u7531\u4e8e\u6700\u9ad8 \\(n\\) \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(1\\) \uff0c\u56e0\u6b64\u7cfb\u7edf\u53ef\u4ee5\u901a\u8fc7\u8bfb\u53d6\u6700\u9ad8\u4f4d \\(1\\) \u7684\u4e2a\u6570\u6765\u89e3\u6790\u51fa\u5b57\u7b26\u7684\u957f\u5ea6\u4e3a \\(n\\) \u3002</p> <p>\u4f46\u4e3a\u4ec0\u4e48\u8981\u5c06\u5176\u4f59\u6240\u6709\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \u5462\uff1f\u5b9e\u9645\u4e0a\uff0c\u8fd9\u4e2a \\(10\\) \u80fd\u591f\u8d77\u5230\u6821\u9a8c\u7b26\u7684\u4f5c\u7528\u3002\u5047\u8bbe\u7cfb\u7edf\u4ece\u4e00\u4e2a\u9519\u8bef\u7684\u5b57\u8282\u5f00\u59cb\u89e3\u6790\u6587\u672c\uff0c\u5b57\u8282\u5934\u90e8\u7684 \\(10\\) \u80fd\u591f\u5e2e\u52a9\u7cfb\u7edf\u5feb\u901f\u5224\u65ad\u51fa\u5f02\u5e38\u3002</p> <p>\u4e4b\u6240\u4ee5\u5c06 \\(10\\) \u5f53\u4f5c\u6821\u9a8c\u7b26\uff0c\u662f\u56e0\u4e3a\u5728 UTF-8 \u7f16\u7801\u89c4\u5219\u4e0b\uff0c\u4e0d\u53ef\u80fd\u6709\u5b57\u7b26\u7684\u6700\u9ad8\u4e24\u4f4d\u662f \\(10\\) \u3002\u8fd9\u4e2a\u7ed3\u8bba\u53ef\u4ee5\u7528\u53cd\u8bc1\u6cd5\u6765\u8bc1\u660e\uff1a\u5047\u8bbe\u4e00\u4e2a\u5b57\u7b26\u7684\u6700\u9ad8\u4e24\u4f4d\u662f \\(10\\) \uff0c\u8bf4\u660e\u8be5\u5b57\u7b26\u7684\u957f\u5ea6\u4e3a \\(1\\) \uff0c\u5bf9\u5e94 ASCII \u7801\u3002\u800c ASCII \u7801\u7684\u6700\u9ad8\u4f4d\u5e94\u8be5\u662f \\(0\\) \uff0c\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</p> <p></p> <p> \u56fe 3-8 \u00a0 UTF-8 \u7f16\u7801\u793a\u4f8b </p> <p>\u9664\u4e86 UTF-8 \u4e4b\u5916\uff0c\u5e38\u89c1\u7684\u7f16\u7801\u65b9\u5f0f\u8fd8\u5305\u62ec\u4ee5\u4e0b\u4e24\u79cd\u3002</p> <ul> <li>UTF-16 \u7f16\u7801\uff1a\u4f7f\u7528 2 \u6216 4 \u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\u3002\u6240\u6709\u7684 ASCII \u5b57\u7b26\u548c\u5e38\u7528\u7684\u975e\u82f1\u6587\u5b57\u7b26\uff0c\u90fd\u7528 2 \u5b57\u8282\u8868\u793a\uff1b\u5c11\u6570\u5b57\u7b26\u9700\u8981\u7528\u5230 4 \u5b57\u8282\u8868\u793a\u3002\u5bf9\u4e8e 2 \u5b57\u8282\u7684\u5b57\u7b26\uff0cUTF-16 \u7f16\u7801\u4e0e Unicode \u7801\u70b9\u76f8\u7b49\u3002</li> <li>UTF-32 \u7f16\u7801\uff1a\u6bcf\u4e2a\u5b57\u7b26\u90fd\u4f7f\u7528 4 \u5b57\u8282\u3002\u8fd9\u610f\u5473\u7740 UTF-32 \u6bd4 UTF-8 \u548c UTF-16 \u66f4\u5360\u7528\u7a7a\u95f4\uff0c\u7279\u522b\u662f\u5bf9\u4e8e ASCII \u5b57\u7b26\u5360\u6bd4\u8f83\u9ad8\u7684\u6587\u672c\u3002</li> </ul> <p>\u4ece\u5b58\u50a8\u7a7a\u95f4\u5360\u7528\u7684\u89d2\u5ea6\u770b\uff0c\u4f7f\u7528 UTF-8 \u8868\u793a\u82f1\u6587\u5b57\u7b26\u975e\u5e38\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u4ec5\u9700 1 \u5b57\u8282\uff1b\u4f7f\u7528 UTF-16 \u7f16\u7801\u67d0\u4e9b\u975e\u82f1\u6587\u5b57\u7b26\uff08\u4f8b\u5982\u4e2d\u6587\uff09\u4f1a\u66f4\u52a0\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u4ec5\u9700 2 \u5b57\u8282\uff0c\u800c UTF-8 \u53ef\u80fd\u9700\u8981 3 \u5b57\u8282\u3002</p> <p>\u4ece\u517c\u5bb9\u6027\u7684\u89d2\u5ea6\u770b\uff0cUTF-8 \u7684\u901a\u7528\u6027\u6700\u4f73\uff0c\u8bb8\u591a\u5de5\u5177\u548c\u5e93\u4f18\u5148\u652f\u6301 UTF-8 \u3002</p>"},{"location":"chapter_data_structure/character_encoding/#345","title":"3.4.5 \u00a0 \u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801","text":"<p>\u5bf9\u4e8e\u4ee5\u5f80\u7684\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff0c\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u7684\u5b57\u7b26\u4e32\u90fd\u91c7\u7528 UTF-16 \u6216 UTF-32 \u8fd9\u7c7b\u7b49\u957f\u7f16\u7801\u3002\u5728\u7b49\u957f\u7f16\u7801\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b57\u7b26\u4e32\u770b\u4f5c\u6570\u7ec4\u6765\u5904\u7406\uff0c\u8fd9\u79cd\u505a\u6cd5\u5177\u6709\u4ee5\u4e0b\u4f18\u70b9\u3002</p> <ul> <li>\u968f\u673a\u8bbf\u95ee\uff1aUTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u8fdb\u884c\u968f\u673a\u8bbf\u95ee\u3002UTF-8 \u662f\u4e00\u79cd\u53d8\u957f\u7f16\u7801\uff0c\u8981\u60f3\u627e\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u4ece\u5b57\u7b26\u4e32\u7684\u5f00\u59cb\u5904\u904d\u5386\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u8fd9\u9700\u8981 \\(O(n)\\) \u7684\u65f6\u95f4\u3002</li> <li>\u5b57\u7b26\u8ba1\u6570\uff1a\u4e0e\u968f\u673a\u8bbf\u95ee\u7c7b\u4f3c\uff0c\u8ba1\u7b97 UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u4e5f\u662f \\(O(1)\\) \u7684\u64cd\u4f5c\u3002\u4f46\u662f\uff0c\u8ba1\u7b97 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u9700\u8981\u904d\u5386\u6574\u4e2a\u5b57\u7b26\u4e32\u3002</li> <li>\u5b57\u7b26\u4e32\u64cd\u4f5c\uff1a\u5728 UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e0a\uff0c\u5f88\u591a\u5b57\u7b26\u4e32\u64cd\u4f5c\uff08\u5982\u5206\u5272\u3001\u8fde\u63a5\u3001\u63d2\u5165\u3001\u5220\u9664\u7b49\uff09\u66f4\u5bb9\u6613\u8fdb\u884c\u3002\u5728 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e0a\uff0c\u8fdb\u884c\u8fd9\u4e9b\u64cd\u4f5c\u901a\u5e38\u9700\u8981\u989d\u5916\u7684\u8ba1\u7b97\uff0c\u4ee5\u786e\u4fdd\u4e0d\u4f1a\u4ea7\u751f\u65e0\u6548\u7684 UTF-8 \u7f16\u7801\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6848\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5f88\u6709\u8da3\u7684\u8bdd\u9898\uff0c\u6d89\u53ca\u8bb8\u591a\u56e0\u7d20\u3002</p> <ul> <li>Java \u7684 <code>String</code> \u7c7b\u578b\u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u8fd9\u662f\u56e0\u4e3a Java \u8bed\u8a00\u8bbe\u8ba1\u4e4b\u521d\uff0c\u4eba\u4eec\u8ba4\u4e3a 16 \u4f4d\u8db3\u4ee5\u8868\u793a\u6240\u6709\u53ef\u80fd\u7684\u5b57\u7b26\u3002\u7136\u800c\uff0c\u8fd9\u662f\u4e00\u4e2a\u4e0d\u6b63\u786e\u7684\u5224\u65ad\u3002\u540e\u6765 Unicode \u89c4\u8303\u6269\u5c55\u5230\u4e86\u8d85\u8fc7 16 \u4f4d\uff0c\u6240\u4ee5 Java \u4e2d\u7684\u5b57\u7b26\u73b0\u5728\u53ef\u80fd\u7531\u4e00\u5bf9 16 \u4f4d\u7684\u503c\uff08\u79f0\u4e3a\u201c\u4ee3\u7406\u5bf9\u201d\uff09\u8868\u793a\u3002</li> <li>JavaScript \u548c TypeScript \u7684\u5b57\u7b26\u4e32\u4f7f\u7528 UTF-16 \u7f16\u7801\u7684\u539f\u56e0\u4e0e Java \u7c7b\u4f3c\u3002\u5f53 1995 \u5e74 Netscape \u516c\u53f8\u9996\u6b21\u63a8\u51fa JavaScript \u8bed\u8a00\u65f6\uff0cUnicode \u8fd8\u5904\u4e8e\u53d1\u5c55\u65e9\u671f\uff0c\u90a3\u65f6\u5019\u4f7f\u7528 16 \u4f4d\u7684\u7f16\u7801\u5c31\u8db3\u4ee5\u8868\u793a\u6240\u6709\u7684 Unicode \u5b57\u7b26\u4e86\u3002</li> <li>C# \u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u4e3b\u8981\u662f\u56e0\u4e3a .NET \u5e73\u53f0\u662f\u7531 Microsoft \u8bbe\u8ba1\u7684\uff0c\u800c Microsoft \u7684\u5f88\u591a\u6280\u672f\uff08\u5305\u62ec Windows \u64cd\u4f5c\u7cfb\u7edf\uff09\u90fd\u5e7f\u6cdb\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002</li> </ul> <p>\u7531\u4e8e\u4ee5\u4e0a\u7f16\u7a0b\u8bed\u8a00\u5bf9\u5b57\u7b26\u6570\u91cf\u7684\u4f4e\u4f30\uff0c\u5b83\u4eec\u4e0d\u5f97\u4e0d\u91c7\u53d6\u201c\u4ee3\u7406\u5bf9\u201d\u7684\u65b9\u5f0f\u6765\u8868\u793a\u8d85\u8fc7 16 \u4f4d\u957f\u5ea6\u7684 Unicode \u5b57\u7b26\u3002\u8fd9\u662f\u4e00\u4e2a\u4e0d\u5f97\u5df2\u4e3a\u4e4b\u7684\u65e0\u5948\u4e4b\u4e3e\u3002\u4e00\u65b9\u9762\uff0c\u5305\u542b\u4ee3\u7406\u5bf9\u7684\u5b57\u7b26\u4e32\u4e2d\uff0c\u4e00\u4e2a\u5b57\u7b26\u53ef\u80fd\u5360\u7528 2 \u5b57\u8282\u6216 4 \u5b57\u8282\uff0c\u4ece\u800c\u4e27\u5931\u4e86\u7b49\u957f\u7f16\u7801\u7684\u4f18\u52bf\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5904\u7406\u4ee3\u7406\u5bf9\u9700\u8981\u989d\u5916\u589e\u52a0\u4ee3\u7801\uff0c\u8fd9\u63d0\u9ad8\u4e86\u7f16\u7a0b\u7684\u590d\u6742\u6027\u548c\u8c03\u8bd5\u96be\u5ea6\u3002</p> <p>\u51fa\u4e8e\u4ee5\u4e0a\u539f\u56e0\uff0c\u90e8\u5206\u7f16\u7a0b\u8bed\u8a00\u63d0\u51fa\u4e86\u4e00\u4e9b\u4e0d\u540c\u7684\u7f16\u7801\u65b9\u6848\u3002</p> <ul> <li>Python \u4e2d\u7684 <code>str</code> \u4f7f\u7528 Unicode \u7f16\u7801\uff0c\u5e76\u91c7\u7528\u4e00\u79cd\u7075\u6d3b\u7684\u5b57\u7b26\u4e32\u8868\u793a\uff0c\u5b58\u50a8\u7684\u5b57\u7b26\u957f\u5ea6\u53d6\u51b3\u4e8e\u5b57\u7b26\u4e32\u4e2d\u6700\u5927\u7684 Unicode \u7801\u70b9\u3002\u82e5\u5b57\u7b26\u4e32\u4e2d\u5168\u90e8\u662f ASCII \u5b57\u7b26\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u5b57\u8282\uff1b\u5982\u679c\u6709\u5b57\u7b26\u8d85\u51fa\u4e86 ASCII \u8303\u56f4\uff0c\u4f46\u5168\u90e8\u5728\u57fa\u672c\u591a\u8bed\u8a00\u5e73\u9762\uff08BMP\uff09\u5185\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\uff1b\u5982\u679c\u6709\u8d85\u51fa BMP \u7684\u5b57\u7b26\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 4 \u5b57\u8282\u3002</li> <li>Go \u8bed\u8a00\u7684 <code>string</code> \u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Go \u8bed\u8a00\u8fd8\u63d0\u4f9b\u4e86 <code>rune</code> \u7c7b\u578b\uff0c\u5b83\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002</li> <li>Rust \u8bed\u8a00\u7684 <code>str</code> \u548c <code>String</code> \u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Rust \u4e5f\u63d0\u4f9b\u4e86 <code>char</code> \u7c7b\u578b\uff0c\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002</li> </ul> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ee5\u4e0a\u8ba8\u8bba\u7684\u90fd\u662f\u5b57\u7b26\u4e32\u5728\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u8fd9\u548c\u5b57\u7b26\u4e32\u5982\u4f55\u5728\u6587\u4ef6\u4e2d\u5b58\u50a8\u6216\u5728\u7f51\u7edc\u4e2d\u4f20\u8f93\u662f\u4e0d\u540c\u7684\u95ee\u9898\u3002\u5728\u6587\u4ef6\u5b58\u50a8\u6216\u7f51\u7edc\u4f20\u8f93\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u5b57\u7b26\u4e32\u7f16\u7801\u4e3a UTF-8 \u683c\u5f0f\uff0c\u4ee5\u8fbe\u5230\u6700\u4f18\u7684\u517c\u5bb9\u6027\u548c\u7a7a\u95f4\u6548\u7387\u3002</p>"},{"location":"chapter_data_structure/classification_of_data_structure/","title":"3.1 \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b","text":"<p>\u5e38\u89c1\u7684\u6570\u636e\u7ed3\u6784\u5305\u62ec\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\uff0c\u5b83\u4eec\u53ef\u4ee5\u4ece\u201c\u903b\u8f91\u7ed3\u6784\u201d\u548c\u201c\u7269\u7406\u7ed3\u6784\u201d\u4e24\u4e2a\u7ef4\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002</p>"},{"location":"chapter_data_structure/classification_of_data_structure/#311","title":"3.1.1 \u00a0 \u903b\u8f91\u7ed3\u6784\uff1a\u7ebf\u6027\u4e0e\u975e\u7ebf\u6027","text":"<p>\u903b\u8f91\u7ed3\u6784\u63ed\u793a\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\u3002\u5728\u6570\u7ec4\u548c\u94fe\u8868\u4e2d\uff0c\u6570\u636e\u6309\u7167\u4e00\u5b9a\u987a\u5e8f\u6392\u5217\uff0c\u4f53\u73b0\u4e86\u6570\u636e\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\uff1b\u800c\u5728\u6811\u4e2d\uff0c\u6570\u636e\u4ece\u9876\u90e8\u5411\u4e0b\u6309\u5c42\u6b21\u6392\u5217\uff0c\u8868\u73b0\u51fa\u201c\u7956\u5148\u201d\u4e0e\u201c\u540e\u4ee3\u201d\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff1b\u56fe\u5219\u7531\u8282\u70b9\u548c\u8fb9\u6784\u6210\uff0c\u53cd\u6620\u4e86\u590d\u6742\u7684\u7f51\u7edc\u5173\u7cfb\u3002</p> <p>\u5982\u56fe 3-1 \u6240\u793a\uff0c\u903b\u8f91\u7ed3\u6784\u53ef\u5206\u4e3a\u201c\u7ebf\u6027\u201d\u548c\u201c\u975e\u7ebf\u6027\u201d\u4e24\u5927\u7c7b\u3002\u7ebf\u6027\u7ed3\u6784\u6bd4\u8f83\u76f4\u89c2\uff0c\u6307\u6570\u636e\u5728\u903b\u8f91\u5173\u7cfb\u4e0a\u5448\u7ebf\u6027\u6392\u5217\uff1b\u975e\u7ebf\u6027\u7ed3\u6784\u5219\u76f8\u53cd\uff0c\u5448\u975e\u7ebf\u6027\u6392\u5217\u3002</p> <ul> <li>\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u4e00\u5bf9\u4e00\u7684\u987a\u5e8f\u5173\u7cfb\u3002</li> <li>\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6811\u3001\u5806\u3001\u56fe\u3001\u54c8\u5e0c\u8868\u3002</li> </ul> <p>\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a\u6811\u5f62\u7ed3\u6784\u548c\u7f51\u72b6\u7ed3\u6784\u3002</p> <ul> <li>\u6811\u5f62\u7ed3\u6784\uff1a\u6811\u3001\u5806\u3001\u54c8\u5e0c\u8868\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u4e00\u5bf9\u591a\u7684\u5173\u7cfb\u3002</li> <li>\u7f51\u72b6\u7ed3\u6784\uff1a\u56fe\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u591a\u5bf9\u591a\u7684\u5173\u7cfb\u3002</li> </ul> <p></p> <p> \u56fe 3-1 \u00a0 \u7ebf\u6027\u6570\u636e\u7ed3\u6784\u4e0e\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784 </p>"},{"location":"chapter_data_structure/classification_of_data_structure/#312","title":"3.1.2 \u00a0 \u7269\u7406\u7ed3\u6784\uff1a\u8fde\u7eed\u4e0e\u5206\u6563","text":"<p>\u5f53\u7b97\u6cd5\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6b63\u5728\u5904\u7406\u7684\u6570\u636e\u4e3b\u8981\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u56fe 3-2 \u5c55\u793a\u4e86\u4e00\u4e2a\u8ba1\u7b97\u673a\u5185\u5b58\u6761\uff0c\u5176\u4e2d\u6bcf\u4e2a\u9ed1\u8272\u65b9\u5757\u90fd\u5305\u542b\u4e00\u5757\u5185\u5b58\u7a7a\u95f4\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u5185\u5b58\u60f3\u8c61\u6210\u4e00\u4e2a\u5de8\u5927\u7684 Excel \u8868\u683c\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5355\u5143\u683c\u90fd\u53ef\u4ee5\u5b58\u50a8\u4e00\u5b9a\u5927\u5c0f\u7684\u6570\u636e\u3002</p> <p>\u7cfb\u7edf\u901a\u8fc7\u5185\u5b58\u5730\u5740\u6765\u8bbf\u95ee\u76ee\u6807\u4f4d\u7f6e\u7684\u6570\u636e\u3002\u5982\u56fe 3-2 \u6240\u793a\uff0c\u8ba1\u7b97\u673a\u6839\u636e\u7279\u5b9a\u89c4\u5219\u4e3a\u8868\u683c\u4e2d\u7684\u6bcf\u4e2a\u5355\u5143\u683c\u5206\u914d\u7f16\u53f7\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u6709\u552f\u4e00\u7684\u5185\u5b58\u5730\u5740\u3002\u6709\u4e86\u8fd9\u4e9b\u5730\u5740\uff0c\u7a0b\u5e8f\u4fbf\u53ef\u4ee5\u8bbf\u95ee\u5185\u5b58\u4e2d\u7684\u6570\u636e\u3002</p> <p></p> <p> \u56fe 3-2 \u00a0 \u5185\u5b58\u6761\u3001\u5185\u5b58\u7a7a\u95f4\u3001\u5185\u5b58\u5730\u5740 </p> <p>Tip</p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5c06\u5185\u5b58\u6bd4\u4f5c Excel \u8868\u683c\u662f\u4e00\u4e2a\u7b80\u5316\u7684\u7c7b\u6bd4\uff0c\u5b9e\u9645\u5185\u5b58\u7684\u5de5\u4f5c\u673a\u5236\u6bd4\u8f83\u590d\u6742\uff0c\u6d89\u53ca\u5730\u5740\u7a7a\u95f4\u3001\u5185\u5b58\u7ba1\u7406\u3001\u7f13\u5b58\u673a\u5236\u3001\u865a\u62df\u5185\u5b58\u548c\u7269\u7406\u5185\u5b58\u7b49\u6982\u5ff5\u3002</p> <p>\u5185\u5b58\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u5171\u4eab\u8d44\u6e90\uff0c\u5f53\u67d0\u5757\u5185\u5b58\u88ab\u67d0\u4e2a\u7a0b\u5e8f\u5360\u7528\u65f6\uff0c\u5219\u65e0\u6cd5\u88ab\u5176\u4ed6\u7a0b\u5e8f\u540c\u65f6\u4f7f\u7528\u4e86\u3002\u56e0\u6b64\u5728\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8bbe\u8ba1\u4e2d\uff0c\u5185\u5b58\u8d44\u6e90\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u8003\u8651\u56e0\u7d20\u3002\u6bd4\u5982\uff0c\u7b97\u6cd5\u6240\u5360\u7528\u7684\u5185\u5b58\u5cf0\u503c\u4e0d\u5e94\u8d85\u8fc7\u7cfb\u7edf\u5269\u4f59\u7a7a\u95f2\u5185\u5b58\uff1b\u5982\u679c\u7f3a\u5c11\u8fde\u7eed\u5927\u5757\u7684\u5185\u5b58\u7a7a\u95f4\uff0c\u90a3\u4e48\u6240\u9009\u7528\u7684\u6570\u636e\u7ed3\u6784\u5fc5\u987b\u80fd\u591f\u5b58\u50a8\u5728\u5206\u6563\u7684\u5185\u5b58\u7a7a\u95f4\u5185\u3002</p> <p>\u5982\u56fe 3-3 \u6240\u793a\uff0c\u7269\u7406\u7ed3\u6784\u53cd\u6620\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u53ef\u5206\u4e3a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\u3002\u7269\u7406\u7ed3\u6784\u4ece\u5e95\u5c42\u51b3\u5b9a\u4e86\u6570\u636e\u7684\u8bbf\u95ee\u3001\u66f4\u65b0\u3001\u589e\u5220\u7b49\u64cd\u4f5c\u65b9\u6cd5\uff0c\u4e24\u79cd\u7269\u7406\u7ed3\u6784\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u65b9\u9762\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u70b9\u3002</p> <p></p> <p> \u56fe 3-3 \u00a0 \u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u4e0e\u5206\u6563\u7a7a\u95f4\u5b58\u50a8 </p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u57fa\u4e8e\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e8c\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002\u4f8b\u5982\uff0c\u6808\u548c\u961f\u5217\u65e2\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff1b\u800c\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u6570\u7ec4\u548c\u94fe\u8868\u3002</p> <ul> <li>\u57fa\u4e8e\u6570\u7ec4\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u3001\u77e9\u9635\u3001\u5f20\u91cf\uff08\u7ef4\u5ea6 \\(\\geq 3\\) \u7684\u6570\u7ec4\uff09\u7b49\u3002</li> <li>\u57fa\u4e8e\u94fe\u8868\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u3002</li> </ul> <p>\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u79f0\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d\uff0c\u8fd9\u610f\u5473\u7740\u6b64\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u76f8\u5bf9\u5e94\u5730\uff0c\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u79f0\u201c\u52a8\u6001\u6570\u636e\u7ed3\u6784\u201d\uff0c\u8fd9\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\uff0c\u4ecd\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u5bf9\u5176\u957f\u5ea6\u8fdb\u884c\u8c03\u6574\u3002</p> <p>Tip</p> <p>\u5982\u679c\u4f60\u611f\u89c9\u7269\u7406\u7ed3\u6784\u7406\u89e3\u8d77\u6765\u6709\u56f0\u96be\uff0c\u5efa\u8bae\u5148\u9605\u8bfb\u4e0b\u4e00\u7ae0\uff0c\u7136\u540e\u518d\u56de\u987e\u672c\u8282\u5185\u5bb9\u3002</p>"},{"location":"chapter_data_structure/number_encoding/","title":"3.3 \u00a0 \u6570\u5b57\u7f16\u7801 *","text":"<p>Note</p> <p>\u5728\u672c\u4e66\u4e2d\uff0c\u6807\u9898\u5e26\u6709 * \u7b26\u53f7\u7684\u662f\u9009\u8bfb\u7ae0\u8282\u3002\u5982\u679c\u4f60\u65f6\u95f4\u6709\u9650\u6216\u611f\u5230\u7406\u89e3\u56f0\u96be\uff0c\u53ef\u4ee5\u5148\u8df3\u8fc7\uff0c\u7b49\u5b66\u5b8c\u5fc5\u8bfb\u7ae0\u8282\u540e\u518d\u5355\u72ec\u653b\u514b\u3002</p>"},{"location":"chapter_data_structure/number_encoding/#331","title":"3.3.1 \u00a0 \u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801","text":"<p>\u5728\u4e0a\u4e00\u8282\u7684\u8868\u683c\u4e2d\u6211\u4eec\u53d1\u73b0\uff0c\u6240\u6709\u6574\u6570\u7c7b\u578b\u80fd\u591f\u8868\u793a\u7684\u8d1f\u6570\u90fd\u6bd4\u6b63\u6570\u591a\u4e00\u4e2a\uff0c\u4f8b\u5982 <code>byte</code> \u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \u3002\u8fd9\u4e2a\u73b0\u8c61\u6bd4\u8f83\u53cd\u76f4\u89c9\uff0c\u5b83\u7684\u5185\u5728\u539f\u56e0\u6d89\u53ca\u539f\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u7684\u76f8\u5173\u77e5\u8bc6\u3002</p> <p>\u9996\u5148\u9700\u8981\u6307\u51fa\uff0c\u6570\u5b57\u662f\u4ee5\u201c\u8865\u7801\u201d\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u7684\u3002\u5728\u5206\u6790\u8fd9\u6837\u505a\u7684\u539f\u56e0\u4e4b\u524d\uff0c\u9996\u5148\u7ed9\u51fa\u4e09\u8005\u7684\u5b9a\u4e49\u3002</p> <ul> <li>\u539f\u7801\uff1a\u6211\u4eec\u5c06\u6570\u5b57\u7684\u4e8c\u8fdb\u5236\u8868\u793a\u7684\u6700\u9ad8\u4f4d\u89c6\u4e3a\u7b26\u53f7\u4f4d\uff0c\u5176\u4e2d \\(0\\) \u8868\u793a\u6b63\u6570\uff0c\\(1\\) \u8868\u793a\u8d1f\u6570\uff0c\u5176\u4f59\u4f4d\u8868\u793a\u6570\u5b57\u7684\u503c\u3002</li> <li>\u53cd\u7801\uff1a\u6b63\u6570\u7684\u53cd\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u53cd\u7801\u662f\u5bf9\u5176\u539f\u7801\u9664\u7b26\u53f7\u4f4d\u5916\u7684\u6240\u6709\u4f4d\u53d6\u53cd\u3002</li> <li>\u8865\u7801\uff1a\u6b63\u6570\u7684\u8865\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u8865\u7801\u662f\u5728\u5176\u53cd\u7801\u7684\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u3002</li> </ul> <p>\u56fe 3-4 \u5c55\u793a\u4e86\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\u4e4b\u95f4\u7684\u8f6c\u6362\u65b9\u6cd5\u3002</p> <p></p> <p> \u56fe 3-4 \u00a0 \u539f\u7801\u3001\u53cd\u7801\u4e0e\u8865\u7801\u4e4b\u95f4\u7684\u76f8\u4e92\u8f6c\u6362 </p> <p>\u300c\u539f\u7801 sign-magnitude\u300d\u867d\u7136\u6700\u76f4\u89c2\uff0c\u4f46\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\u3002\u4e00\u65b9\u9762\uff0c\u8d1f\u6570\u7684\u539f\u7801\u4e0d\u80fd\u76f4\u63a5\u7528\u4e8e\u8fd0\u7b97\u3002\u4f8b\u5982\u5728\u539f\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u5f97\u5230\u7684\u7ed3\u679c\u662f \\(-3\\) \uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 + 1000 \\; 0010 \\newline &amp; = 1000 \\; 0011 \\newline &amp; \\rightarrow -3 \\end{aligned} \\] <p>\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u8ba1\u7b97\u673a\u5f15\u5165\u4e86\u300c\u53cd\u7801 1's complement\u300d\u3002\u5982\u679c\u6211\u4eec\u5148\u5c06\u539f\u7801\u8f6c\u6362\u4e3a\u53cd\u7801\uff0c\u5e76\u5728\u53cd\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u6700\u540e\u5c06\u7ed3\u679c\u4ece\u53cd\u7801\u8f6c\u6362\u56de\u539f\u7801\uff0c\u5219\u53ef\u5f97\u5230\u6b63\u786e\u7ed3\u679c \\(-1\\) \u3002</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 \\; \\text{(\u539f\u7801)} + 1000 \\; 0010 \\; \\text{(\u539f\u7801)} \\newline &amp; = 0000 \\; 0001 \\; \\text{(\u53cd\u7801)} + 1111  \\; 1101 \\; \\text{(\u53cd\u7801)} \\newline &amp; = 1111 \\; 1110 \\; \\text{(\u53cd\u7801)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(\u539f\u7801)} \\newline &amp; \\rightarrow -1 \\end{aligned} \\] <p>\u53e6\u4e00\u65b9\u9762\uff0c\u6570\u5b57\u96f6\u7684\u539f\u7801\u6709 \\(+0\\) \u548c \\(-0\\) \u4e24\u79cd\u8868\u793a\u65b9\u5f0f\u3002\u8fd9\u610f\u5473\u7740\u6570\u5b57\u96f6\u5bf9\u5e94\u4e24\u4e2a\u4e0d\u540c\u7684\u4e8c\u8fdb\u5236\u7f16\u7801\uff0c\u8fd9\u53ef\u80fd\u4f1a\u5e26\u6765\u6b67\u4e49\u3002\u6bd4\u5982\u5728\u6761\u4ef6\u5224\u65ad\u4e2d\uff0c\u5982\u679c\u6ca1\u6709\u533a\u5206\u6b63\u96f6\u548c\u8d1f\u96f6\uff0c\u5219\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5224\u65ad\u7ed3\u679c\u51fa\u9519\u3002\u800c\u5982\u679c\u6211\u4eec\u60f3\u5904\u7406\u6b63\u96f6\u548c\u8d1f\u96f6\u6b67\u4e49\uff0c\u5219\u9700\u8981\u5f15\u5165\u989d\u5916\u7684\u5224\u65ad\u64cd\u4f5c\uff0c\u8fd9\u53ef\u80fd\u4f1a\u964d\u4f4e\u8ba1\u7b97\u673a\u7684\u8fd0\u7b97\u6548\u7387\u3002</p> \\[ \\begin{aligned} +0 &amp; \\rightarrow 0000 \\; 0000 \\newline -0 &amp; \\rightarrow 1000 \\; 0000 \\end{aligned} \\] <p>\u4e0e\u539f\u7801\u4e00\u6837\uff0c\u53cd\u7801\u4e5f\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u95ee\u9898\uff0c\u56e0\u6b64\u8ba1\u7b97\u673a\u8fdb\u4e00\u6b65\u5f15\u5165\u4e86\u300c\u8865\u7801 2's complement\u300d\u3002\u6211\u4eec\u5148\u6765\u89c2\u5bdf\u4e00\u4e0b\u8d1f\u96f6\u7684\u539f\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u7684\u8f6c\u6362\u8fc7\u7a0b\uff1a</p> \\[ \\begin{aligned} -0 \\rightarrow \\; &amp; 1000 \\; 0000 \\; \\text{(\u539f\u7801)} \\newline = \\; &amp; 1111 \\; 1111 \\; \\text{(\u53cd\u7801)} \\newline = 1 \\; &amp; 0000 \\; 0000 \\; \\text{(\u8865\u7801)} \\newline \\end{aligned} \\] <p>\u5728\u8d1f\u96f6\u7684\u53cd\u7801\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u4f1a\u4ea7\u751f\u8fdb\u4f4d\uff0c\u4f46 <code>byte</code> \u7c7b\u578b\u7684\u957f\u5ea6\u53ea\u6709 8 \u4f4d\uff0c\u56e0\u6b64\u6ea2\u51fa\u5230\u7b2c 9 \u4f4d\u7684 \\(1\\) \u4f1a\u88ab\u820d\u5f03\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u8d1f\u96f6\u7684\u8865\u7801\u4e3a \\(0000 \\; 0000\\) \uff0c\u4e0e\u6b63\u96f6\u7684\u8865\u7801\u76f8\u540c\u3002\u8fd9\u610f\u5473\u7740\u5728\u8865\u7801\u8868\u793a\u4e2d\u53ea\u5b58\u5728\u4e00\u4e2a\u96f6\uff0c\u6b63\u8d1f\u96f6\u6b67\u4e49\u4ece\u800c\u5f97\u5230\u89e3\u51b3\u3002</p> <p>\u8fd8\u5269\u6700\u540e\u4e00\u4e2a\u7591\u60d1\uff1a<code>byte</code> \u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \uff0c\u591a\u51fa\u6765\u7684\u4e00\u4e2a\u8d1f\u6570 \\(-128\\) \u662f\u5982\u4f55\u5f97\u5230\u7684\u5462\uff1f\u6211\u4eec\u6ce8\u610f\u5230\uff0c\u533a\u95f4 \\([-127, +127]\\) \u5185\u7684\u6240\u6709\u6574\u6570\u90fd\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\uff0c\u5e76\u4e14\u539f\u7801\u548c\u8865\u7801\u4e4b\u95f4\u53ef\u4ee5\u4e92\u76f8\u8f6c\u6362\u3002</p> <p>\u7136\u800c\uff0c\u8865\u7801 \\(1000 \\; 0000\\) \u662f\u4e00\u4e2a\u4f8b\u5916\uff0c\u5b83\u5e76\u6ca1\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3002\u6839\u636e\u8f6c\u6362\u65b9\u6cd5\uff0c\u6211\u4eec\u5f97\u5230\u8be5\u8865\u7801\u7684\u539f\u7801\u4e3a \\(0000 \\; 0000\\) \u3002\u8fd9\u663e\u7136\u662f\u77db\u76fe\u7684\uff0c\u56e0\u4e3a\u8be5\u539f\u7801\u8868\u793a\u6570\u5b57 \\(0\\) \uff0c\u5b83\u7684\u8865\u7801\u5e94\u8be5\u662f\u81ea\u8eab\u3002\u8ba1\u7b97\u673a\u89c4\u5b9a\u8fd9\u4e2a\u7279\u6b8a\u7684\u8865\u7801 \\(1000 \\; 0000\\) \u4ee3\u8868 \\(-128\\) \u3002\u5b9e\u9645\u4e0a\uff0c\\((-1) + (-127)\\) \u5728\u8865\u7801\u4e0b\u7684\u8ba1\u7b97\u7ed3\u679c\u5c31\u662f \\(-128\\) \u3002</p> \\[ \\begin{aligned} &amp; (-127) + (-1) \\newline &amp; \\rightarrow 1111 \\; 1111 \\; \\text{(\u539f\u7801)} + 1000 \\; 0001 \\; \\text{(\u539f\u7801)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(\u53cd\u7801)} + 1111  \\; 1110 \\; \\text{(\u53cd\u7801)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(\u8865\u7801)} + 1111  \\; 1111 \\; \\text{(\u8865\u7801)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(\u8865\u7801)} \\newline &amp; \\rightarrow -128 \\end{aligned} \\] <p>\u4f60\u53ef\u80fd\u5df2\u7ecf\u53d1\u73b0\u4e86\uff0c\u4e0a\u8ff0\u6240\u6709\u8ba1\u7b97\u90fd\u662f\u52a0\u6cd5\u8fd0\u7b97\u3002\u8fd9\u6697\u793a\u7740\u4e00\u4e2a\u91cd\u8981\u4e8b\u5b9e\uff1a\u8ba1\u7b97\u673a\u5185\u90e8\u7684\u786c\u4ef6\u7535\u8def\u4e3b\u8981\u662f\u57fa\u4e8e\u52a0\u6cd5\u8fd0\u7b97\u8bbe\u8ba1\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u52a0\u6cd5\u8fd0\u7b97\u76f8\u5bf9\u4e8e\u5176\u4ed6\u8fd0\u7b97\uff08\u6bd4\u5982\u4e58\u6cd5\u3001\u9664\u6cd5\u548c\u51cf\u6cd5\uff09\u6765\u8bf4\uff0c\u786c\u4ef6\u5b9e\u73b0\u8d77\u6765\u66f4\u7b80\u5355\uff0c\u66f4\u5bb9\u6613\u8fdb\u884c\u5e76\u884c\u5316\u5904\u7406\uff0c\u8fd0\u7b97\u901f\u5ea6\u66f4\u5feb\u3002</p> <p>\u8bf7\u6ce8\u610f\uff0c\u8fd9\u5e76\u4e0d\u610f\u5473\u7740\u8ba1\u7b97\u673a\u53ea\u80fd\u505a\u52a0\u6cd5\u3002\u901a\u8fc7\u5c06\u52a0\u6cd5\u4e0e\u4e00\u4e9b\u57fa\u672c\u903b\u8f91\u8fd0\u7b97\u7ed3\u5408\uff0c\u8ba1\u7b97\u673a\u80fd\u591f\u5b9e\u73b0\u5404\u79cd\u5176\u4ed6\u7684\u6570\u5b66\u8fd0\u7b97\u3002\u4f8b\u5982\uff0c\u8ba1\u7b97\u51cf\u6cd5 \\(a - b\\) \u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u52a0\u6cd5 \\(a + (-b)\\) \uff1b\u8ba1\u7b97\u4e58\u6cd5\u548c\u9664\u6cd5\u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u591a\u6b21\u52a0\u6cd5\u6216\u51cf\u6cd5\u3002</p> <p>\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u8ba1\u7b97\u673a\u4f7f\u7528\u8865\u7801\u7684\u539f\u56e0\uff1a\u57fa\u4e8e\u8865\u7801\u8868\u793a\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u7528\u540c\u6837\u7684\u7535\u8def\u548c\u64cd\u4f5c\u6765\u5904\u7406\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\uff0c\u4e0d\u9700\u8981\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\u6765\u5904\u7406\u51cf\u6cd5\uff0c\u5e76\u4e14\u65e0\u987b\u7279\u522b\u5904\u7406\u6b63\u8d1f\u96f6\u7684\u6b67\u4e49\u95ee\u9898\u3002\u8fd9\u5927\u5927\u7b80\u5316\u4e86\u786c\u4ef6\u8bbe\u8ba1\uff0c\u63d0\u9ad8\u4e86\u8fd0\u7b97\u6548\u7387\u3002</p> <p>\u8865\u7801\u7684\u8bbe\u8ba1\u975e\u5e38\u7cbe\u5999\uff0c\u56e0\u7bc7\u5e45\u5173\u7cfb\u6211\u4eec\u5c31\u5148\u4ecb\u7ecd\u5230\u8fd9\u91cc\uff0c\u5efa\u8bae\u6709\u5174\u8da3\u7684\u8bfb\u8005\u8fdb\u4e00\u6b65\u6df1\u5165\u4e86\u89e3\u3002</p>"},{"location":"chapter_data_structure/number_encoding/#332","title":"3.3.2 \u00a0 \u6d6e\u70b9\u6570\u7f16\u7801","text":"<p>\u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\uff1a<code>int</code> \u548c <code>float</code> \u957f\u5ea6\u76f8\u540c\uff0c\u90fd\u662f 4 \u5b57\u8282 \uff0c\u4f46\u4e3a\u4ec0\u4e48 <code>float</code> \u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e <code>int</code> \uff1f\u8fd9\u975e\u5e38\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u6309\u7406\u8bf4 <code>float</code> \u9700\u8981\u8868\u793a\u5c0f\u6570\uff0c\u53d6\u503c\u8303\u56f4\u5e94\u8be5\u53d8\u5c0f\u624d\u5bf9\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8fd9\u662f\u56e0\u4e3a\u6d6e\u70b9\u6570 <code>float</code> \u91c7\u7528\u4e86\u4e0d\u540c\u7684\u8868\u793a\u65b9\u5f0f\u3002\u8bb0\u4e00\u4e2a 32 \u6bd4\u7279\u957f\u5ea6\u7684\u4e8c\u8fdb\u5236\u6570\u4e3a\uff1a</p> \\[ b_{31} b_{30} b_{29} \\ldots b_2 b_1 b_0 \\] <p>\u6839\u636e IEEE 754 \u6807\u51c6\uff0c32-bit \u957f\u5ea6\u7684 <code>float</code> \u7531\u4ee5\u4e0b\u4e09\u4e2a\u90e8\u5206\u6784\u6210\u3002</p> <ul> <li>\u7b26\u53f7\u4f4d \\(\\mathrm{S}\\) \uff1a\u5360 1 \u4f4d \uff0c\u5bf9\u5e94 \\(b_{31}\\) \u3002</li> <li>\u6307\u6570\u4f4d \\(\\mathrm{E}\\) \uff1a\u5360 8 \u4f4d \uff0c\u5bf9\u5e94 \\(b_{30} b_{29} \\ldots b_{23}\\) \u3002</li> <li>\u5206\u6570\u4f4d \\(\\mathrm{N}\\) \uff1a\u5360 23 \u4f4d \uff0c\u5bf9\u5e94 \\(b_{22} b_{21} \\ldots b_0\\) \u3002</li> </ul> <p>\u4e8c\u8fdb\u5236\u6570 <code>float</code> \u5bf9\u5e94\u503c\u7684\u8ba1\u7b97\u65b9\u6cd5\u4e3a\uff1a</p> \\[ \\text {val} = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\] <p>\u8f6c\u5316\u5230\u5341\u8fdb\u5236\u4e0b\u7684\u8ba1\u7b97\u516c\u5f0f\u4e3a\uff1a</p> \\[ \\text {val}=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\] <p>\u5176\u4e2d\u5404\u9879\u7684\u53d6\u503c\u8303\u56f4\u4e3a\uff1a</p> \\[ \\begin{aligned} \\mathrm{S} \\in &amp; \\{ 0, 1\\}, \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = &amp; (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\] <p></p> <p> \u56fe 3-5 \u00a0 IEEE 754 \u6807\u51c6\u4e0b\u7684 float \u7684\u8ba1\u7b97\u793a\u4f8b </p> <p>\u89c2\u5bdf\u56fe 3-5 \uff0c\u7ed9\u5b9a\u4e00\u4e2a\u793a\u4f8b\u6570\u636e \\(\\mathrm{S} = 0\\) \uff0c \\(\\mathrm{E} = 124\\) \uff0c\\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\) \uff0c\u5219\u6709\uff1a</p> \\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\] <p>\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u56de\u7b54\u6700\u521d\u7684\u95ee\u9898\uff1a<code>float</code> \u7684\u8868\u793a\u65b9\u5f0f\u5305\u542b\u6307\u6570\u4f4d\uff0c\u5bfc\u81f4\u5176\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e <code>int</code> \u3002\u6839\u636e\u4ee5\u4e0a\u8ba1\u7b97\uff0c<code>float</code> \u53ef\u8868\u793a\u7684\u6700\u5927\u6b63\u6570\u4e3a \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\) \uff0c\u5207\u6362\u7b26\u53f7\u4f4d\u4fbf\u53ef\u5f97\u5230\u6700\u5c0f\u8d1f\u6570\u3002</p> <p>\u5c3d\u7ba1\u6d6e\u70b9\u6570 <code>float</code> \u6269\u5c55\u4e86\u53d6\u503c\u8303\u56f4\uff0c\u4f46\u5176\u526f\u4f5c\u7528\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002\u6574\u6570\u7c7b\u578b <code>int</code> \u5c06\u5168\u90e8 32 \u6bd4\u7279\u7528\u4e8e\u8868\u793a\u6570\u5b57\uff0c\u6570\u5b57\u662f\u5747\u5300\u5206\u5e03\u7684\uff1b\u800c\u7531\u4e8e\u6307\u6570\u4f4d\u7684\u5b58\u5728\uff0c\u6d6e\u70b9\u6570 <code>float</code> \u7684\u6570\u503c\u8d8a\u5927\uff0c\u76f8\u90bb\u4e24\u4e2a\u6570\u5b57\u4e4b\u95f4\u7684\u5dee\u503c\u5c31\u4f1a\u8d8b\u5411\u8d8a\u5927\u3002</p> <p>\u5982\u8868 3-2 \u6240\u793a\uff0c\u6307\u6570\u4f4d \\(E = 0\\) \u548c \\(E = 255\\) \u5177\u6709\u7279\u6b8a\u542b\u4e49\uff0c\u7528\u4e8e\u8868\u793a\u96f6\u3001\u65e0\u7a77\u5927\u3001\\(\\mathrm{NaN}\\) \u7b49\u3002</p> <p> \u8868 3-2 \u00a0 \u6307\u6570\u4f4d\u542b\u4e49 </p> \u6307\u6570\u4f4d E \u5206\u6570\u4f4d \\(\\mathrm{N} = 0\\) \u5206\u6570\u4f4d \\(\\mathrm{N} \\ne 0\\) \u8ba1\u7b97\u516c\u5f0f \\(0\\) \\(\\pm 0\\) \u6b21\u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) \u6b63\u89c4\u6570 \u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\) <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6b21\u6b63\u89c4\u6570\u663e\u8457\u63d0\u5347\u4e86\u6d6e\u70b9\u6570\u7684\u7cbe\u5ea6\u3002\u6700\u5c0f\u6b63\u6b63\u89c4\u6570\u4e3a \\(2^{-126}\\) \uff0c\u6700\u5c0f\u6b63\u6b21\u6b63\u89c4\u6570\u4e3a \\(2^{-126} \\times 2^{-23}\\) \u3002</p> <p>\u53cc\u7cbe\u5ea6 <code>double</code> \u4e5f\u91c7\u7528\u7c7b\u4f3c\u4e8e <code>float</code> \u7684\u8868\u793a\u65b9\u6cd5\uff0c\u5728\u6b64\u4e0d\u505a\u8d58\u8ff0\u3002</p>"},{"location":"chapter_data_structure/summary/","title":"3.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_data_structure/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u4ece\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\u4e24\u4e2a\u89d2\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002\u903b\u8f91\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\uff0c\u800c\u7269\u7406\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\u3002</li> <li>\u5e38\u89c1\u7684\u903b\u8f91\u7ed3\u6784\u5305\u62ec\u7ebf\u6027\u3001\u6811\u72b6\u548c\u7f51\u72b6\u7b49\u3002\u901a\u5e38\u6211\u4eec\u6839\u636e\u903b\u8f91\u7ed3\u6784\u5c06\u6570\u636e\u7ed3\u6784\u5206\u4e3a\u7ebf\u6027\uff08\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\uff09\u548c\u975e\u7ebf\u6027\uff08\u6811\u3001\u56fe\u3001\u5806\uff09\u4e24\u79cd\u3002\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u5f53\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u88ab\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u3002\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u62e5\u6709\u5bf9\u5e94\u7684\u5185\u5b58\u5730\u5740\uff0c\u7a0b\u5e8f\u901a\u8fc7\u8fd9\u4e9b\u5185\u5b58\u5730\u5740\u8bbf\u95ee\u6570\u636e\u3002</li> <li>\u7269\u7406\u7ed3\u6784\u4e3b\u8981\u5206\u4e3a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\u3002\u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u7531\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e24\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002</li> <li>\u8ba1\u7b97\u673a\u4e2d\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\u5305\u62ec\u6574\u6570 <code>byte</code>\u3001<code>short</code>\u3001<code>int</code>\u3001<code>long</code> \uff0c\u6d6e\u70b9\u6570 <code>float</code>\u3001<code>double</code> \uff0c\u5b57\u7b26 <code>char</code> \u548c\u5e03\u5c14 <code>bool</code> \u3002\u5b83\u4eec\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5360\u7528\u7a7a\u95f4\u5927\u5c0f\u548c\u8868\u793a\u65b9\u5f0f\u3002</li> <li>\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7f16\u7801\u6570\u5b57\u7684\u4e09\u79cd\u65b9\u6cd5\uff0c\u5b83\u4eec\u4e4b\u95f4\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\u3002\u6574\u6570\u7684\u539f\u7801\u7684\u6700\u9ad8\u4f4d\u662f\u7b26\u53f7\u4f4d\uff0c\u5176\u4f59\u4f4d\u662f\u6570\u5b57\u7684\u503c\u3002</li> <li>\u6574\u6570\u5728\u8ba1\u7b97\u673a\u4e2d\u662f\u4ee5\u8865\u7801\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\u3002\u5728\u8865\u7801\u8868\u793a\u4e0b\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u5bf9\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\u4e00\u89c6\u540c\u4ec1\uff0c\u4e0d\u9700\u8981\u4e3a\u51cf\u6cd5\u64cd\u4f5c\u5355\u72ec\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\uff0c\u5e76\u4e14\u4e0d\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u7684\u95ee\u9898\u3002</li> <li>\u6d6e\u70b9\u6570\u7684\u7f16\u7801\u7531 1 \u4f4d\u7b26\u53f7\u4f4d\u30018 \u4f4d\u6307\u6570\u4f4d\u548c 23 \u4f4d\u5206\u6570\u4f4d\u6784\u6210\u3002\u7531\u4e8e\u5b58\u5728\u6307\u6570\u4f4d\uff0c\u56e0\u6b64\u6d6e\u70b9\u6570\u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e\u6574\u6570\uff0c\u4ee3\u4ef7\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002</li> <li>ASCII \u7801\u662f\u6700\u65e9\u51fa\u73b0\u7684\u82f1\u6587\u5b57\u7b26\u96c6\uff0c\u957f\u5ea6\u4e3a 1 \u5b57\u8282\uff0c\u5171\u6536\u5f55 127 \u4e2a\u5b57\u7b26\u3002GBK \u5b57\u7b26\u96c6\u662f\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u96c6\uff0c\u5171\u6536\u5f55\u4e24\u4e07\u591a\u4e2a\u6c49\u5b57\u3002Unicode \u81f4\u529b\u4e8e\u63d0\u4f9b\u4e00\u4e2a\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u6536\u5f55\u4e16\u754c\u4e0a\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u4ece\u800c\u89e3\u51b3\u7531\u4e8e\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\u4e0d\u4e00\u81f4\u800c\u5bfc\u81f4\u7684\u4e71\u7801\u95ee\u9898\u3002</li> <li>UTF-8 \u662f\u6700\u53d7\u6b22\u8fce\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\uff0c\u901a\u7528\u6027\u975e\u5e38\u597d\u3002\u5b83\u662f\u4e00\u79cd\u53d8\u957f\u7684\u7f16\u7801\u65b9\u6cd5\uff0c\u5177\u6709\u5f88\u597d\u7684\u6269\u5c55\u6027\uff0c\u6709\u6548\u63d0\u5347\u4e86\u5b58\u50a8\u7a7a\u95f4\u7684\u4f7f\u7528\u6548\u7387\u3002UTF-16 \u548c UTF-32 \u662f\u7b49\u957f\u7684\u7f16\u7801\u65b9\u6cd5\u3002\u5728\u7f16\u7801\u4e2d\u6587\u65f6\uff0cUTF-16 \u5360\u7528\u7684\u7a7a\u95f4\u6bd4 UTF-8 \u66f4\u5c0f\u3002Java \u548c C# \u7b49\u7f16\u7a0b\u8bed\u8a00\u9ed8\u8ba4\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002</li> </ul>"},{"location":"chapter_data_structure/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1f</p> <p>\u54c8\u5e0c\u8868\u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u800c\u4e3a\u4e86\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\uff0c\u6211\u4eec\u53ef\u80fd\u4f1a\u4f7f\u7528\u201c\u94fe\u5f0f\u5730\u5740\u201d\uff08\u540e\u7eed\u201c\u54c8\u5e0c\u51b2\u7a81\u201d\u7ae0\u8282\u4f1a\u8bb2\uff09\uff1a\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u6876\u6307\u5411\u4e00\u4e2a\u94fe\u8868\uff0c\u5f53\u94fe\u8868\u957f\u5ea6\u8d85\u8fc7\u4e00\u5b9a\u9608\u503c\u65f6\uff0c\u53c8\u53ef\u80fd\u88ab\u8f6c\u5316\u4e3a\u6811\uff08\u901a\u5e38\u4e3a\u7ea2\u9ed1\u6811\uff09\u3002</p> <p>\u4ece\u5b58\u50a8\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u54c8\u5e0c\u8868\u7684\u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u5176\u4e2d\u6bcf\u4e00\u4e2a\u6876\u69fd\u4f4d\u53ef\u80fd\u5305\u542b\u4e00\u4e2a\u503c\uff0c\u4e5f\u53ef\u80fd\u5305\u542b\u4e00\u4e2a\u94fe\u8868\u6216\u4e00\u68f5\u6811\u3002\u56e0\u6b64\uff0c\u54c8\u5e0c\u8868\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff08\u6570\u7ec4\u3001\u94fe\u8868\uff09\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff08\u6811\uff09\u3002</p> <p>Q\uff1a<code>char</code> \u7c7b\u578b\u7684\u957f\u5ea6\u662f 1 \u5b57\u8282\u5417\uff1f</p> <p><code>char</code> \u7c7b\u578b\u7684\u957f\u5ea6\u7531\u7f16\u7a0b\u8bed\u8a00\u91c7\u7528\u7684\u7f16\u7801\u65b9\u6cd5\u51b3\u5b9a\u3002\u4f8b\u5982\uff0cJava\u3001JavaScript\u3001TypeScript\u3001C# \u90fd\u91c7\u7528 UTF-16 \u7f16\u7801\uff08\u4fdd\u5b58 Unicode \u7801\u70b9\uff09\uff0c\u56e0\u6b64 <code>char</code> \u7c7b\u578b\u7684\u957f\u5ea6\u4e3a 2 \u5b57\u8282\u3002</p> <p>Q\uff1a\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u79f0\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d \u662f\u5426\u6709\u6b67\u4e49\uff1f\u6808\u4e5f\u53ef\u4ee5\u8fdb\u884c\u51fa\u6808\u548c\u5165\u6808\u7b49\u64cd\u4f5c\uff0c\u8fd9\u4e9b\u64cd\u4f5c\u90fd\u662f\u201c\u52a8\u6001\u201d\u7684\u3002</p> <p>\u6808\u786e\u5b9e\u53ef\u4ee5\u5b9e\u73b0\u52a8\u6001\u7684\u6570\u636e\u64cd\u4f5c\uff0c\u4f46\u6570\u636e\u7ed3\u6784\u4ecd\u7136\u662f\u201c\u9759\u6001\u201d\uff08\u957f\u5ea6\u4e0d\u53ef\u53d8\uff09\u7684\u3002\u5c3d\u7ba1\u57fa\u4e8e\u6570\u7ec4\u7684\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u52a8\u6001\u5730\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\uff0c\u4f46\u5b83\u4eec\u7684\u5bb9\u91cf\u662f\u56fa\u5b9a\u7684\u3002\u5982\u679c\u6570\u636e\u91cf\u8d85\u51fa\u4e86\u9884\u5206\u914d\u7684\u5927\u5c0f\uff0c\u5c31\u9700\u8981\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u65e7\u6570\u7ec4\u7684\u5185\u5bb9\u590d\u5236\u5230\u65b0\u6570\u7ec4\u4e2d\u3002</p> <p>Q\uff1a\u5728\u6784\u5efa\u6808\uff08\u961f\u5217\uff09\u7684\u65f6\u5019\uff0c\u672a\u6307\u5b9a\u5b83\u7684\u5927\u5c0f\uff0c\u4e3a\u4ec0\u4e48\u5b83\u4eec\u662f\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d\u5462\uff1f</p> <p>\u5728\u9ad8\u7ea7\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u65e0\u987b\u4eba\u5de5\u6307\u5b9a\u6808\uff08\u961f\u5217\uff09\u7684\u521d\u59cb\u5bb9\u91cf\uff0c\u8fd9\u4e2a\u5de5\u4f5c\u7531\u7c7b\u5185\u90e8\u81ea\u52a8\u5b8c\u6210\u3002\u4f8b\u5982\uff0cJava \u7684 <code>ArrayList</code> \u7684\u521d\u59cb\u5bb9\u91cf\u901a\u5e38\u4e3a 10\u3002\u53e6\u5916\uff0c\u6269\u5bb9\u64cd\u4f5c\u4e5f\u662f\u81ea\u52a8\u5b9e\u73b0\u7684\u3002\u8be6\u89c1\u540e\u7eed\u7684\u201c\u5217\u8868\u201d\u7ae0\u8282\u3002</p>"},{"location":"chapter_divide_and_conquer/","title":"\u7b2c 12 \u7ae0 \u00a0 \u5206\u6cbb","text":"<p>Abstract</p> <p>\u96be\u9898\u88ab\u9010\u5c42\u62c6\u89e3\uff0c\u6bcf\u4e00\u6b21\u7684\u62c6\u89e3\u90fd\u4f7f\u5b83\u53d8\u5f97\u66f4\u4e3a\u7b80\u5355\u3002</p> <p>\u5206\u800c\u6cbb\u4e4b\u63ed\u793a\u4e86\u4e00\u4e2a\u91cd\u8981\u7684\u4e8b\u5b9e\uff1a\u4ece\u7b80\u5355\u505a\u8d77\uff0c\u4e00\u5207\u90fd\u4e0d\u518d\u590d\u6742\u3002</p>"},{"location":"chapter_divide_and_conquer/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>12.1 \u00a0 \u5206\u6cbb\u7b97\u6cd5</li> <li>12.2 \u00a0 \u5206\u6cbb\u641c\u7d22\u7b56\u7565</li> <li>12.3 \u00a0 \u6784\u5efa\u6811\u95ee\u9898</li> <li>12.4 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898</li> <li>12.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_divide_and_conquer/binary_search_recur/","title":"12.2 \u00a0 \u5206\u6cbb\u641c\u7d22\u7b56\u7565","text":"<p>\u6211\u4eec\u5df2\u7ecf\u5b66\u8fc7\uff0c\u641c\u7d22\u7b97\u6cd5\u5206\u4e3a\u4e24\u5927\u7c7b\u3002</p> <ul> <li>\u66b4\u529b\u641c\u7d22\uff1a\u5b83\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</li> <li>\u81ea\u9002\u5e94\u641c\u7d22\uff1a\u5b83\u5229\u7528\u7279\u6709\u7684\u6570\u636e\u7ec4\u7ec7\u5f62\u5f0f\u6216\u5148\u9a8c\u4fe1\u606f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u7684\u641c\u7d22\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u5b9e\u73b0\u7684\uff0c\u4f8b\u5982\u4e8c\u5206\u67e5\u627e\u548c\u6811\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\u7684\u6bcf\u4e00\u6b65\u90fd\u5c06\u95ee\u9898\uff08\u5728\u6570\u7ec4\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff09\u5206\u89e3\u4e3a\u4e00\u4e2a\u5c0f\u95ee\u9898\uff08\u5728\u6570\u7ec4\u7684\u4e00\u534a\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff09\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u4e00\u76f4\u6301\u7eed\u5230\u6570\u7ec4\u4e3a\u7a7a\u6216\u627e\u5230\u76ee\u6807\u5143\u7d20\u4e3a\u6b62\u3002</li> <li>\u6811\u662f\u5206\u6cbb\u601d\u60f3\u7684\u4ee3\u8868\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u3001AVL \u6811\u3001\u5806\u7b49\u6570\u636e\u7ed3\u6784\u4e2d\uff0c\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7686\u4e3a \\(O(\\log n)\\) \u3002</li> </ul> <p>\u4e8c\u5206\u67e5\u627e\u7684\u5206\u6cbb\u7b56\u7565\u5982\u4e0b\u6240\u793a\u3002</p> <ul> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u4e8c\u5206\u67e5\u627e\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\uff08\u5728\u6570\u7ec4\u4e2d\u8fdb\u884c\u67e5\u627e\uff09\u5206\u89e3\u4e3a\u5b50\u95ee\u9898\uff08\u5728\u6570\u7ec4\u7684\u4e00\u534a\u4e2d\u8fdb\u884c\u67e5\u627e\uff09\uff0c\u8fd9\u662f\u901a\u8fc7\u6bd4\u8f83\u4e2d\u95f4\u5143\u7d20\u548c\u76ee\u6807\u5143\u7d20\u6765\u5b9e\u73b0\u7684\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u5904\u7406\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u5b83\u4e0d\u53d7\u5176\u4ed6\u5b50\u95ee\u9898\u7684\u5f71\u54cd\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u65e0\u987b\u5408\u5e76\uff1a\u4e8c\u5206\u67e5\u627e\u65e8\u5728\u67e5\u627e\u4e00\u4e2a\u7279\u5b9a\u5143\u7d20\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\u3002\u5f53\u5b50\u95ee\u9898\u5f97\u5230\u89e3\u51b3\u65f6\uff0c\u539f\u95ee\u9898\u4e5f\u4f1a\u540c\u65f6\u5f97\u5230\u89e3\u51b3\u3002</li> </ul> <p>\u5206\u6cbb\u80fd\u591f\u63d0\u5347\u641c\u7d22\u6548\u7387\uff0c\u672c\u8d28\u4e0a\u662f\u56e0\u4e3a\u66b4\u529b\u641c\u7d22\u6bcf\u8f6e\u53ea\u80fd\u6392\u9664\u4e00\u4e2a\u9009\u9879\uff0c\u800c\u5206\u6cbb\u641c\u7d22\u6bcf\u8f6e\u53ef\u4ee5\u6392\u9664\u4e00\u534a\u9009\u9879\u3002</p>"},{"location":"chapter_divide_and_conquer/binary_search_recur/#1","title":"1. \u00a0 \u57fa\u4e8e\u5206\u6cbb\u5b9e\u73b0\u4e8c\u5206\u67e5\u627e","text":"<p>\u5728\u4e4b\u524d\u7684\u7ae0\u8282\u4e2d\uff0c\u4e8c\u5206\u67e5\u627e\u662f\u57fa\u4e8e\u9012\u63a8\uff08\u8fed\u4ee3\uff09\u5b9e\u73b0\u7684\u3002\u73b0\u5728\u6211\u4eec\u57fa\u4e8e\u5206\u6cbb\uff08\u9012\u5f52\uff09\u6765\u5b9e\u73b0\u5b83\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u662f\u552f\u4e00\u7684\uff0c\u8bf7\u67e5\u627e\u5143\u7d20 <code>target</code> \u3002</p> <p>\u4ece\u5206\u6cbb\u89d2\u5ea6\uff0c\u6211\u4eec\u5c06\u641c\u7d22\u533a\u95f4 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u8bb0\u4e3a \\(f(i, j)\\) \u3002</p> <p>\u4ee5\u539f\u95ee\u9898 \\(f(0, n-1)\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u901a\u8fc7\u4ee5\u4e0b\u6b65\u9aa4\u8fdb\u884c\u4e8c\u5206\u67e5\u627e\u3002</p> <ol> <li>\u8ba1\u7b97\u641c\u7d22\u533a\u95f4 \\([i, j]\\) \u7684\u4e2d\u70b9 \\(m\\) \uff0c\u6839\u636e\u5b83\u6392\u9664\u4e00\u534a\u641c\u7d22\u533a\u95f4\u3002</li> <li>\u9012\u5f52\u6c42\u89e3\u89c4\u6a21\u51cf\u5c0f\u4e00\u534a\u7684\u5b50\u95ee\u9898\uff0c\u53ef\u80fd\u4e3a \\(f(i, m-1)\\) \u6216 \\(f(m+1, j)\\) \u3002</li> <li>\u5faa\u73af\u7b2c <code>1.</code> \u6b65\u548c\u7b2c <code>2.</code> \u6b65\uff0c\u76f4\u81f3\u627e\u5230 <code>target</code> \u6216\u533a\u95f4\u4e3a\u7a7a\u65f6\u8fd4\u56de\u3002</li> </ol> <p>\u56fe 12-4 \u5c55\u793a\u4e86\u5728\u6570\u7ec4\u4e2d\u4e8c\u5206\u67e5\u627e\u5143\u7d20 \\(6\\) \u7684\u5206\u6cbb\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 12-4 \u00a0 \u4e8c\u5206\u67e5\u627e\u7684\u5206\u6cbb\u8fc7\u7a0b </p> <p>\u5728\u5b9e\u73b0\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u9012\u5f52\u51fd\u6570 <code>dfs()</code> \u6765\u6c42\u89e3\u95ee\u9898 \\(f(i, j)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_recur.py<pre><code>def dfs(nums: list[int], target: int, i: int, j: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j)\"\"\"\n    # \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if i &gt; j:\n        return -1\n    # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    m = (i + j) // 2\n    if nums[m] &lt; target:\n        # \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j)\n    elif nums[m] &gt; target:\n        # \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1)\n    else:\n        # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m\n\ndef binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\"\"\"\n    n = len(nums)\n    # \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1)\n</code></pre> binary_search_recur.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(vector&lt;int&gt; &amp;nums, int target, int i, int j) {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    int n = nums.size();\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(int[] nums, int target, int i, int j) {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(int[] nums, int target) {\n    int n = nums.length;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint DFS(int[] nums, int target, int i, int j) {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return DFS(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return DFS(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint BinarySearch(int[] nums, int target) {\n    int n = nums.Length;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return DFS(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunc dfs(nums []int, target, i, j int) int {\n    // \u5982\u679c\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u6ca1\u6709\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if i &gt; j {\n        return -1\n    }\n    //    \u8ba1\u7b97\u7d22\u5f15\u4e2d\u70b9\n    m := i + ((j - i) &gt;&gt; 1)\n    //\u5224\u65ad\u4e2d\u70b9\u4e0e\u76ee\u6807\u5143\u7d20\u5927\u5c0f\n    if nums[m] &lt; target {\n        // \u5c0f\u4e8e\u5219\u9012\u5f52\u53f3\u534a\u6570\u7ec4\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m+1, j)\n    } else if nums[m] &gt; target {\n        // \u5c0f\u4e8e\u5219\u9012\u5f52\u5de6\u534a\u6570\u7ec4\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m-1)\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunc binarySearch(nums []int, target int) int {\n    n := len(nums)\n    return dfs(nums, target, 0, n-1)\n}\n</code></pre> binary_search_recur.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunc dfs(nums: [Int], target: Int, i: Int, j: Int) -&gt; Int {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if i &gt; j {\n        return -1\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    let m = (i + j) / 2\n    if nums[m] &lt; target {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums: nums, target: target, i: m + 1, j: j)\n    } else if nums[m] &gt; target {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums: nums, target: target, i: i, j: m - 1)\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    dfs(nums: nums, target: target, i: nums.startIndex, j: nums.endIndex - 1)\n}\n</code></pre> binary_search_recur.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunction dfs(nums, target, i, j) {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunction binarySearch(nums, target) {\n    const n = nums.length;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunction dfs(nums: number[], target: number, i: number, j: number): number {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunction binarySearch(nums: number[], target: number): number {\n    const n = nums.length;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(List&lt;int&gt; nums, int target, int i, int j) {\n  // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n  if (i &gt; j) {\n    return -1;\n  }\n  // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n  int m = (i + j) ~/ 2;\n  if (nums[m] &lt; target) {\n    // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n    return dfs(nums, target, m + 1, j);\n  } else if (nums[m] &gt; target) {\n    // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n    return dfs(nums, target, i, m - 1);\n  } else {\n    // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return m;\n  }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  int n = nums.length;\n  // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n  return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfn dfs(nums: &amp;[i32], target: i32, i: i32, j: i32) -&gt; i32 {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if i &gt; j {\n        return -1;\n    }\n    let m: i32 = (i + j) / 2;\n    if nums[m as usize] &lt; target {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if nums[m as usize] &gt; target {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let n = nums.len() as i32;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(int nums[], int target, int i, int j) {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(int nums[], int target, int numsSize) {\n    int n = numsSize;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.kt<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfun dfs(\n    nums: IntArray,\n    target: Int,\n    i: Int,\n    j: Int\n): Int {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    val m = (i + j) / 2\n    return if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        dfs(nums, target, m + 1, j)\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        dfs(nums, target, i, m - 1)\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        m\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    val n = nums.size\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.rb<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{binary_search}\n</code></pre> binary_search_recur.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{binarySearch}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/","title":"12.3 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u68f5\u4e8c\u53c9\u6811\u7684\u524d\u5e8f\u904d\u5386 <code>preorder</code> \u548c\u4e2d\u5e8f\u904d\u5386 <code>inorder</code> \uff0c\u8bf7\u4ece\u4e2d\u6784\u5efa\u4e8c\u53c9\u6811\uff0c\u8fd4\u56de\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\u3002\u5047\u8bbe\u4e8c\u53c9\u6811\u4e2d\u6ca1\u6709\u503c\u91cd\u590d\u7684\u8282\u70b9\uff08\u5982\u56fe 12-5 \u6240\u793a\uff09\u3002</p> <p></p> <p> \u56fe 12-5 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#1","title":"1. \u00a0 \u5224\u65ad\u662f\u5426\u4e3a\u5206\u6cbb\u95ee\u9898","text":"<p>\u539f\u95ee\u9898\u5b9a\u4e49\u4e3a\u4ece <code>preorder</code> \u548c <code>inorder</code> \u6784\u5efa\u4e8c\u53c9\u6811\uff0c\u662f\u4e00\u4e2a\u5178\u578b\u7684\u5206\u6cbb\u95ee\u9898\u3002</p> <ul> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u4ece\u5206\u6cbb\u7684\u89d2\u5ea6\u5207\u5165\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u539f\u95ee\u9898\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\u3001\u6784\u5efa\u53f3\u5b50\u6811\uff0c\u52a0\u4e0a\u4e00\u6b65\u64cd\u4f5c\uff1a\u521d\u59cb\u5316\u6839\u8282\u70b9\u3002\u800c\u5bf9\u4e8e\u6bcf\u68f5\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u590d\u7528\u4ee5\u4e0a\u5212\u5206\u65b9\u6cd5\uff0c\u5c06\u5176\u5212\u5206\u4e3a\u66f4\u5c0f\u7684\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u76f4\u81f3\u8fbe\u5230\u6700\u5c0f\u5b50\u95ee\u9898\uff08\u7a7a\u5b50\u6811\uff09\u65f6\u7ec8\u6b62\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u4ea4\u96c6\u3002\u5728\u6784\u5efa\u5de6\u5b50\u6811\u65f6\uff0c\u6211\u4eec\u53ea\u9700\u5173\u6ce8\u4e2d\u5e8f\u904d\u5386\u548c\u524d\u5e8f\u904d\u5386\u4e2d\u4e0e\u5de6\u5b50\u6811\u5bf9\u5e94\u7684\u90e8\u5206\u3002\u53f3\u5b50\u6811\u540c\u7406\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u4e00\u65e6\u5f97\u5230\u4e86\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u5b83\u4eec\u94fe\u63a5\u5230\u6839\u8282\u70b9\u4e0a\uff0c\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002</li> </ul>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#2","title":"2. \u00a0 \u5982\u4f55\u5212\u5206\u5b50\u6811","text":"<p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u8fd9\u9053\u9898\u53ef\u4ee5\u4f7f\u7528\u5206\u6cbb\u6765\u6c42\u89e3\uff0c\u4f46\u5982\u4f55\u901a\u8fc7\u524d\u5e8f\u904d\u5386 <code>preorder</code> \u548c\u4e2d\u5e8f\u904d\u5386 <code>inorder</code> \u6765\u5212\u5206\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u5462\uff1f</p> <p>\u6839\u636e\u5b9a\u4e49\uff0c<code>preorder</code> \u548c <code>inorder</code> \u90fd\u53ef\u4ee5\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\u3002</p> <ul> <li>\u524d\u5e8f\u904d\u5386\uff1a<code>[ \u6839\u8282\u70b9 | \u5de6\u5b50\u6811 | \u53f3\u5b50\u6811 ]</code> \uff0c\u4f8b\u5982\u56fe 12-5 \u7684\u6811\u5bf9\u5e94 <code>[ 3 | 9 | 2 1 7 ]</code> \u3002</li> <li>\u4e2d\u5e8f\u904d\u5386\uff1a<code>[ \u5de6\u5b50\u6811 | \u6839\u8282\u70b9 \uff5c \u53f3\u5b50\u6811 ]</code> \uff0c\u4f8b\u5982\u56fe 12-5 \u7684\u6811\u5bf9\u5e94 <code>[ 9 | 3 | 1 2 7 ]</code> \u3002</li> </ul> <p>\u4ee5\u4e0a\u56fe\u6570\u636e\u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u56fe 12-6 \u6240\u793a\u7684\u6b65\u9aa4\u5f97\u5230\u5212\u5206\u7ed3\u679c\u3002</p> <ol> <li>\u524d\u5e8f\u904d\u5386\u7684\u9996\u5143\u7d20 3 \u662f\u6839\u8282\u70b9\u7684\u503c\u3002</li> <li>\u67e5\u627e\u6839\u8282\u70b9 3 \u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\uff0c\u5229\u7528\u8be5\u7d22\u5f15\u53ef\u5c06 <code>inorder</code> \u5212\u5206\u4e3a <code>[ 9 | 3 \uff5c 1 2 7 ]</code> \u3002</li> <li>\u6839\u636e <code>inorder</code> \u7684\u5212\u5206\u7ed3\u679c\uff0c\u6613\u5f97\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u8282\u70b9\u6570\u91cf\u5206\u522b\u4e3a 1 \u548c 3 \uff0c\u4ece\u800c\u53ef\u5c06 <code>preorder</code> \u5212\u5206\u4e3a <code>[ 3 | 9 | 2 1 7 ]</code> \u3002</li> </ol> <p></p> <p> \u56fe 12-6 \u00a0 \u5728\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u4e2d\u5212\u5206\u5b50\u6811 </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#3","title":"3. \u00a0 \u57fa\u4e8e\u53d8\u91cf\u63cf\u8ff0\u5b50\u6811\u533a\u95f4","text":"<p>\u6839\u636e\u4ee5\u4e0a\u5212\u5206\u65b9\u6cd5\uff0c\u6211\u4eec\u5df2\u7ecf\u5f97\u5230\u6839\u8282\u70b9\u3001\u5de6\u5b50\u6811\u3001\u53f3\u5b50\u6811\u5728 <code>preorder</code> \u548c <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u3002\u800c\u4e3a\u4e86\u63cf\u8ff0\u8fd9\u4e9b\u7d22\u5f15\u533a\u95f4\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u51e0\u4e2a\u6307\u9488\u53d8\u91cf\u3002</p> <ul> <li>\u5c06\u5f53\u524d\u6811\u7684\u6839\u8282\u70b9\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15\u8bb0\u4e3a \\(i\\) \u3002</li> <li>\u5c06\u5f53\u524d\u6811\u7684\u6839\u8282\u70b9\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u8bb0\u4e3a \\(m\\) \u3002</li> <li>\u5c06\u5f53\u524d\u6811\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u8bb0\u4e3a \\([l, r]\\) \u3002</li> </ul> <p>\u5982\u8868 12-1 \u6240\u793a\uff0c\u901a\u8fc7\u4ee5\u4e0a\u53d8\u91cf\u5373\u53ef\u8868\u793a\u6839\u8282\u70b9\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15\uff0c\u4ee5\u53ca\u5b50\u6811\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u3002</p> <p> \u8868 12-1 \u00a0 \u6839\u8282\u70b9\u548c\u5b50\u6811\u5728\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u7d22\u5f15 </p> \u6839\u8282\u70b9\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15 \u5b50\u6811\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4 \u5f53\u524d\u6811 \\(i\\) \\([l, r]\\) \u5de6\u5b50\u6811 \\(i + 1\\) \\([l, m-1]\\) \u53f3\u5b50\u6811 \\(i + 1 + (m - l)\\) \\([m+1, r]\\) <p>\u8bf7\u6ce8\u610f\uff0c\u53f3\u5b50\u6811\u6839\u8282\u70b9\u7d22\u5f15\u4e2d\u7684 \\((m-l)\\) \u7684\u542b\u4e49\u662f\u201c\u5de6\u5b50\u6811\u7684\u8282\u70b9\u6570\u91cf\u201d\uff0c\u5efa\u8bae\u7ed3\u5408\u56fe 12-7 \u7406\u89e3\u3002</p> <p></p> <p> \u56fe 12-7 \u00a0 \u6839\u8282\u70b9\u548c\u5de6\u53f3\u5b50\u6811\u7684\u7d22\u5f15\u533a\u95f4\u8868\u793a </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#4","title":"4. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u4e3a\u4e86\u63d0\u5347\u67e5\u8be2 \\(m\\) \u7684\u6548\u7387\uff0c\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>hmap</code> \u6765\u5b58\u50a8\u6570\u7ec4 <code>inorder</code> \u4e2d\u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig build_tree.py<pre><code>def dfs(\n    preorder: list[int],\n    inorder_map: dict[int, int],\n    i: int,\n    l: int,\n    r: int,\n) -&gt; TreeNode | None:\n    \"\"\"\u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb\"\"\"\n    # \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if r - l &lt; 0:\n        return None\n    # \u521d\u59cb\u5316\u6839\u8282\u70b9\n    root = TreeNode(preorder[i])\n    # \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    m = inorder_map[preorder[i]]\n    # \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n    # \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n    # \u8fd4\u56de\u6839\u8282\u70b9\n    return root\n\ndef build_tree(preorder: list[int], inorder: list[int]) -&gt; TreeNode | None:\n    \"\"\"\u6784\u5efa\u4e8c\u53c9\u6811\"\"\"\n    # \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    inorder_map = {val: i for i, val in enumerate(inorder)}\n    root = dfs(preorder, inorder_map, 0, 0, len(inorder) - 1)\n    return root\n</code></pre> build_tree.cpp<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode *dfs(vector&lt;int&gt; &amp;preorder, unordered_map&lt;int, int&gt; &amp;inorderMap, int i, int l, int r) {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0)\n        return NULL;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    TreeNode *root = new TreeNode(preorder[i]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    int m = inorderMap[preorder[i]];\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode *buildTree(vector&lt;int&gt; &amp;preorder, vector&lt;int&gt; &amp;inorder) {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    unordered_map&lt;int, int&gt; inorderMap;\n    for (int i = 0; i &lt; inorder.size(); i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorder.size() - 1);\n    return root;\n}\n</code></pre> build_tree.java<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode dfs(int[] preorder, Map&lt;Integer, Integer&gt; inorderMap, int i, int l, int r) {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0)\n        return null;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    TreeNode root = new TreeNode(preorder[i]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    int m = inorderMap.get(preorder[i]);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode buildTree(int[] preorder, int[] inorder) {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    Map&lt;Integer, Integer&gt; inorderMap = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; inorder.length; i++) {\n        inorderMap.put(inorder[i], i);\n    }\n    TreeNode root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.cs<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode? DFS(int[] preorder, Dictionary&lt;int, int&gt; inorderMap, int i, int l, int r) {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0)\n        return null;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    TreeNode root = new(preorder[i]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    int m = inorderMap[preorder[i]];\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = DFS(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = DFS(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode? BuildTree(int[] preorder, int[] inorder) {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    Dictionary&lt;int, int&gt; inorderMap = [];\n    for (int i = 0; i &lt; inorder.Length; i++) {\n        inorderMap.TryAdd(inorder[i], i);\n    }\n    TreeNode? root = DFS(preorder, inorderMap, 0, 0, inorder.Length - 1);\n    return root;\n}\n</code></pre> build_tree.go<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunc dfsBuildTree(preorder []int, inorderMap map[int]int, i, l, r int) *TreeNode {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if r-l &lt; 0 {\n        return nil\n    }\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    root := NewTreeNode(preorder[i])\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    m := inorderMap[preorder[i]]\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.Left = dfsBuildTree(preorder, inorderMap, i+1, l, m-1)\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.Right = dfsBuildTree(preorder, inorderMap, i+1+m-l, m+1, r)\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunc buildTree(preorder, inorder []int) *TreeNode {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    inorderMap := make(map[int]int, len(inorder))\n    for i := 0; i &lt; len(inorder); i++ {\n        inorderMap[inorder[i]] = i\n    }\n\n    root := dfsBuildTree(preorder, inorderMap, 0, 0, len(inorder)-1)\n    return root\n}\n</code></pre> build_tree.swift<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunc dfs(preorder: [Int], inorderMap: [Int: Int], i: Int, l: Int, r: Int) -&gt; TreeNode? {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if r - l &lt; 0 {\n        return nil\n    }\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    let root = TreeNode(x: preorder[i])\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    let m = inorderMap[preorder[i]]!\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1, l: l, r: m - 1)\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1 + m - l, l: m + 1, r: r)\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunc buildTree(preorder: [Int], inorder: [Int]) -&gt; TreeNode? {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    let inorderMap = inorder.enumerated().reduce(into: [:]) { $0[$1.element] = $1.offset }\n    return dfs(preorder: preorder, inorderMap: inorderMap, i: inorder.startIndex, l: inorder.startIndex, r: inorder.endIndex - 1)\n}\n</code></pre> build_tree.js<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunction dfs(preorder, inorderMap, i, l, r) {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0) return null;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    const root = new TreeNode(preorder[i]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    const m = inorderMap.get(preorder[i]);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunction buildTree(preorder, inorder) {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    let inorderMap = new Map();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.ts<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunction dfs(\n    preorder: number[],\n    inorderMap: Map&lt;number, number&gt;,\n    i: number,\n    l: number,\n    r: number\n): TreeNode | null {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0) return null;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    const root: TreeNode = new TreeNode(preorder[i]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    const m = inorderMap.get(preorder[i]);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunction buildTree(preorder: number[], inorder: number[]): TreeNode | null {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    let inorderMap = new Map&lt;number, number&gt;();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.dart<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode? dfs(\n  List&lt;int&gt; preorder,\n  Map&lt;int, int&gt; inorderMap,\n  int i,\n  int l,\n  int r,\n) {\n  // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n  if (r - l &lt; 0) {\n    return null;\n  }\n  // \u521d\u59cb\u5316\u6839\u8282\u70b9\n  TreeNode? root = TreeNode(preorder[i]);\n  // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n  int m = inorderMap[preorder[i]]!;\n  // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n  root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n  // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n  root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n  // \u8fd4\u56de\u6839\u8282\u70b9\n  return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode? buildTree(List&lt;int&gt; preorder, List&lt;int&gt; inorder) {\n  // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n  Map&lt;int, int&gt; inorderMap = {};\n  for (int i = 0; i &lt; inorder.length; i++) {\n    inorderMap[inorder[i]] = i;\n  }\n  TreeNode? root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n  return root;\n}\n</code></pre> build_tree.rs<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfn dfs(\n    preorder: &amp;[i32],\n    inorder_map: &amp;HashMap&lt;i32, i32&gt;,\n    i: i32,\n    l: i32,\n    r: i32,\n) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if r - l &lt; 0 {\n        return None;\n    }\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    let root = TreeNode::new(preorder[i as usize]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    let m = inorder_map.get(&amp;preorder[i as usize]).unwrap();\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.borrow_mut().left = dfs(preorder, inorder_map, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.borrow_mut().right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    Some(root)\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfn build_tree(preorder: &amp;[i32], inorder: &amp;[i32]) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    let mut inorder_map: HashMap&lt;i32, i32&gt; = HashMap::new();\n    for i in 0..inorder.len() {\n        inorder_map.insert(inorder[i], i as i32);\n    }\n    let root = dfs(preorder, &amp;inorder_map, 0, 0, inorder.len() as i32 - 1);\n    root\n}\n</code></pre> build_tree.c<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode *dfs(int *preorder, int *inorderMap, int i, int l, int r, int size) {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0)\n        return NULL;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    TreeNode *root = (TreeNode *)malloc(sizeof(TreeNode));\n    root-&gt;val = preorder[i];\n    root-&gt;left = NULL;\n    root-&gt;right = NULL;\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    int m = inorderMap[preorder[i]];\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1, size);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r, size);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode *buildTree(int *preorder, int preorderSize, int *inorder, int inorderSize) {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    int *inorderMap = (int *)malloc(sizeof(int) * MAX_SIZE);\n    for (int i = 0; i &lt; inorderSize; i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorderSize - 1, inorderSize);\n    free(inorderMap);\n    return root;\n}\n</code></pre> build_tree.kt<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfun dfs(preorder: IntArray, inorderMap: Map&lt;Int?, Int?&gt;, i: Int, l: Int, r: Int): TreeNode? {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0) return null\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    val root = TreeNode(preorder[i])\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    val m = inorderMap[preorder[i]]!!\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1)\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r)\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfun buildTree(preorder: IntArray, inorder: IntArray): TreeNode? {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    val inorderMap: MutableMap&lt;Int?, Int?&gt; = HashMap()\n    for (i in inorder.indices) {\n        inorderMap[inorder[i]] = i\n    }\n    val root = dfs(preorder, inorderMap, 0, 0, inorder.size - 1)\n    return root\n}\n</code></pre> build_tree.rb<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{build_tree}\n</code></pre> build_tree.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{buildTree}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 12-8 \u5c55\u793a\u4e86\u6784\u5efa\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b\uff0c\u5404\u4e2a\u8282\u70b9\u662f\u5728\u5411\u4e0b\u201c\u9012\u201d\u7684\u8fc7\u7a0b\u4e2d\u5efa\u7acb\u7684\uff0c\u800c\u5404\u6761\u8fb9\uff08\u5f15\u7528\uff09\u662f\u5728\u5411\u4e0a\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u4e2d\u5efa\u7acb\u7684\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 12-8 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b </p> <p>\u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u5185\u7684\u524d\u5e8f\u904d\u5386 <code>preorder</code> \u548c\u4e2d\u5e8f\u904d\u5386 <code>inorder</code> \u7684\u5212\u5206\u7ed3\u679c\u5982\u56fe 12-9 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 12-9 \u00a0 \u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u4e2d\u7684\u5212\u5206\u7ed3\u679c </p> <p>\u8bbe\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u521d\u59cb\u5316\u6bcf\u4e00\u4e2a\u8282\u70b9\uff08\u6267\u884c\u4e00\u4e2a\u9012\u5f52\u51fd\u6570 <code>dfs()</code> \uff09\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u6b64\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> <p>\u54c8\u5e0c\u8868\u5b58\u50a8 <code>inorder</code> \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u9012\u5f52\u6df1\u5ea6\u8fbe\u5230 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002\u56e0\u6b64\u603b\u4f53\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/","title":"12.1 \u00a0 \u5206\u6cbb\u7b97\u6cd5","text":"<p>\u300c\u5206\u6cbb divide and conquer\u300d\uff0c\u5168\u79f0\u5206\u800c\u6cbb\u4e4b\uff0c\u662f\u4e00\u79cd\u975e\u5e38\u91cd\u8981\u4e14\u5e38\u89c1\u7684\u7b97\u6cd5\u7b56\u7565\u3002\u5206\u6cbb\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\uff0c\u5305\u62ec\u201c\u5206\u201d\u548c\u201c\u6cbb\u201d\u4e24\u4e2a\u6b65\u9aa4\u3002</p> <ol> <li>\u5206\uff08\u5212\u5206\u9636\u6bb5\uff09\uff1a\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u4e24\u4e2a\u6216\u591a\u4e2a\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u5230\u8fbe\u6700\u5c0f\u5b50\u95ee\u9898\u65f6\u7ec8\u6b62\u3002</li> <li>\u6cbb\uff08\u5408\u5e76\u9636\u6bb5\uff09\uff1a\u4ece\u5df2\u77e5\u89e3\u7684\u6700\u5c0f\u5b50\u95ee\u9898\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\uff0c\u4ece\u800c\u6784\u5efa\u51fa\u539f\u95ee\u9898\u7684\u89e3\u3002</li> </ol> <p>\u5982\u56fe 12-1 \u6240\u793a\uff0c\u201c\u5f52\u5e76\u6392\u5e8f\u201d\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u5e94\u7528\u4e4b\u4e00\u3002</p> <ol> <li>\u5206\uff1a\u9012\u5f52\u5730\u5c06\u539f\u6570\u7ec4\uff08\u539f\u95ee\u9898\uff09\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\uff09\uff0c\u76f4\u5230\u5b50\u6570\u7ec4\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\uff08\u6700\u5c0f\u5b50\u95ee\u9898\uff09\u3002</li> <li>\u6cbb\uff1a\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u6709\u5e8f\u7684\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\u8fdb\u884c\u5408\u5e76\uff0c\u4ece\u800c\u5f97\u5230\u6709\u5e8f\u7684\u539f\u6570\u7ec4\uff08\u539f\u95ee\u9898\u7684\u89e3\uff09\u3002</li> </ol> <p></p> <p> \u56fe 12-1 \u00a0 \u5f52\u5e76\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565 </p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1211","title":"12.1.1 \u00a0 \u5982\u4f55\u5224\u65ad\u5206\u6cbb\u95ee\u9898","text":"<p>\u4e00\u4e2a\u95ee\u9898\u662f\u5426\u9002\u5408\u4f7f\u7528\u5206\u6cbb\u89e3\u51b3\uff0c\u901a\u5e38\u53ef\u4ee5\u53c2\u8003\u4ee5\u4e0b\u51e0\u4e2a\u5224\u65ad\u4f9d\u636e\u3002</p> <ol> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u539f\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\u6210\u89c4\u6a21\u66f4\u5c0f\u3001\u7c7b\u4f3c\u7684\u5b50\u95ee\u9898\uff0c\u4ee5\u53ca\u80fd\u591f\u4ee5\u76f8\u540c\u65b9\u5f0f\u9012\u5f52\u5730\u8fdb\u884c\u5212\u5206\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5b50\u95ee\u9898\u4e4b\u95f4\u6ca1\u6709\u91cd\u53e0\uff0c\u4e92\u4e0d\u4f9d\u8d56\uff0c\u53ef\u4ee5\u72ec\u7acb\u89e3\u51b3\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u539f\u95ee\u9898\u7684\u89e3\u901a\u8fc7\u5408\u5e76\u5b50\u95ee\u9898\u7684\u89e3\u5f97\u6765\u3002</li> </ol> <p>\u663e\u7136\uff0c\u5f52\u5e76\u6392\u5e8f\u6ee1\u8db3\u4ee5\u4e0a\u4e09\u4e2a\u5224\u65ad\u4f9d\u636e\u3002</p> <ol> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u9012\u5f52\u5730\u5c06\u6570\u7ec4\uff08\u539f\u95ee\u9898\uff09\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\uff09\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u53ef\u4ee5\u72ec\u7acb\u5730\u8fdb\u884c\u6392\u5e8f\uff08\u5b50\u95ee\u9898\u53ef\u4ee5\u72ec\u7acb\u8fdb\u884c\u6c42\u89e3\uff09\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u4e24\u4e2a\u6709\u5e8f\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\u53ef\u4ee5\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\uff08\u539f\u95ee\u9898\u7684\u89e3\uff09\u3002</li> </ol>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1212","title":"12.1.2 \u00a0 \u901a\u8fc7\u5206\u6cbb\u63d0\u5347\u6548\u7387","text":"<p>\u5206\u6cbb\u4e0d\u4ec5\u53ef\u4ee5\u6709\u6548\u5730\u89e3\u51b3\u7b97\u6cd5\u95ee\u9898\uff0c\u5f80\u5f80\u8fd8\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002\u5728\u6392\u5e8f\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u76f8\u8f83\u4e8e\u9009\u62e9\u3001\u5192\u6ce1\u3001\u63d2\u5165\u6392\u5e8f\u66f4\u5feb\uff0c\u5c31\u662f\u56e0\u4e3a\u5b83\u4eec\u5e94\u7528\u4e86\u5206\u6cbb\u7b56\u7565\u3002</p> <p>\u90a3\u4e48\uff0c\u6211\u4eec\u4e0d\u7981\u53d1\u95ee\uff1a\u4e3a\u4ec0\u4e48\u5206\u6cbb\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\uff0c\u5176\u5e95\u5c42\u903b\u8f91\u662f\u4ec0\u4e48\uff1f\u6362\u53e5\u8bdd\u8bf4\uff0c\u5c06\u5927\u95ee\u9898\u5206\u89e3\u4e3a\u591a\u4e2a\u5b50\u95ee\u9898\u3001\u89e3\u51b3\u5b50\u95ee\u9898\u3001\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u5408\u5e76\u4e3a\u539f\u95ee\u9898\u7684\u89e3\uff0c\u8fd9\u51e0\u6b65\u7684\u6548\u7387\u4e3a\u4ec0\u4e48\u6bd4\u76f4\u63a5\u89e3\u51b3\u539f\u95ee\u9898\u7684\u6548\u7387\u66f4\u9ad8\uff1f\u8fd9\u4e2a\u95ee\u9898\u53ef\u4ee5\u4ece\u64cd\u4f5c\u6570\u91cf\u548c\u5e76\u884c\u8ba1\u7b97\u4e24\u65b9\u9762\u6765\u8ba8\u8bba\u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1","title":"1. \u00a0 \u64cd\u4f5c\u6570\u91cf\u4f18\u5316","text":"<p>\u4ee5\u201c\u5192\u6ce1\u6392\u5e8f\u201d\u4e3a\u4f8b\uff0c\u5176\u5904\u7406\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u9700\u8981 \\(O(n^2)\\) \u65f6\u95f4\u3002\u5047\u8bbe\u6211\u4eec\u6309\u7167\u56fe 12-2 \u6240\u793a\u7684\u65b9\u5f0f\uff0c\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u5219\u5212\u5206\u9700\u8981 \\(O(n)\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6bcf\u4e2a\u5b50\u6570\u7ec4\u9700\u8981 \\(O((n / 2)^2)\\) \u65f6\u95f4\uff0c\u5408\u5e76\u4e24\u4e2a\u5b50\u6570\u7ec4\u9700\u8981 \\(O(n)\\) \u65f6\u95f4\uff0c\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\uff1a</p> \\[ O(n + (\\frac{n}{2})^2 \\times 2 + n) = O(\\frac{n^2}{2} + 2n) \\] <p></p> <p> \u56fe 12-2 \u00a0 \u5212\u5206\u6570\u7ec4\u524d\u540e\u7684\u5192\u6ce1\u6392\u5e8f </p> <p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u8ba1\u7b97\u4ee5\u4e0b\u4e0d\u7b49\u5f0f\uff0c\u5176\u5de6\u8fb9\u548c\u53f3\u8fb9\u5206\u522b\u4e3a\u5212\u5206\u524d\u548c\u5212\u5206\u540e\u7684\u64cd\u4f5c\u603b\u6570\uff1a</p> \\[ \\begin{aligned} n^2 &amp; &gt; \\frac{n^2}{2} + 2n \\newline n^2 - \\frac{n^2}{2} - 2n &amp; &gt; 0 \\newline n(n - 4) &amp; &gt; 0 \\end{aligned} \\] <p>\u8fd9\u610f\u5473\u7740\u5f53 \\(n &gt; 4\\) \u65f6\uff0c\u5212\u5206\u540e\u7684\u64cd\u4f5c\u6570\u91cf\u66f4\u5c11\uff0c\u6392\u5e8f\u6548\u7387\u5e94\u8be5\u66f4\u9ad8\u3002\u8bf7\u6ce8\u610f\uff0c\u5212\u5206\u540e\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u7136\u662f\u5e73\u65b9\u9636 \\(O(n^2)\\) \uff0c\u53ea\u662f\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\u53d8\u5c0f\u4e86\u3002</p> <p>\u8fdb\u4e00\u6b65\u60f3\uff0c\u5982\u679c\u6211\u4eec\u628a\u5b50\u6570\u7ec4\u4e0d\u65ad\u5730\u518d\u4ece\u4e2d\u70b9\u5904\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\u65f6\u505c\u6b62\u5212\u5206\u5462\uff1f\u8fd9\u79cd\u601d\u8def\u5b9e\u9645\u4e0a\u5c31\u662f\u201c\u5f52\u5e76\u6392\u5e8f\u201d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p> <p>\u518d\u601d\u8003\uff0c\u5982\u679c\u6211\u4eec\u591a\u8bbe\u7f6e\u51e0\u4e2a\u5212\u5206\u70b9\uff0c\u5c06\u539f\u6570\u7ec4\u5e73\u5747\u5212\u5206\u4e3a \\(k\\) \u4e2a\u5b50\u6570\u7ec4\u5462\uff1f\u8fd9\u79cd\u60c5\u51b5\u4e0e\u201c\u6876\u6392\u5e8f\u201d\u975e\u5e38\u7c7b\u4f3c\uff0c\u5b83\u975e\u5e38\u9002\u5408\u6392\u5e8f\u6d77\u91cf\u6570\u636e\uff0c\u7406\u8bba\u4e0a\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230 \\(O(n + k)\\) \u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#2","title":"2. \u00a0 \u5e76\u884c\u8ba1\u7b97\u4f18\u5316","text":"<p>\u6211\u4eec\u77e5\u9053\uff0c\u5206\u6cbb\u751f\u6210\u7684\u5b50\u95ee\u9898\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u56e0\u6b64\u901a\u5e38\u53ef\u4ee5\u5e76\u884c\u89e3\u51b3\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5206\u6cbb\u4e0d\u4ec5\u53ef\u4ee5\u964d\u4f4e\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u8fd8\u6709\u5229\u4e8e\u64cd\u4f5c\u7cfb\u7edf\u7684\u5e76\u884c\u4f18\u5316\u3002</p> <p>\u5e76\u884c\u4f18\u5316\u5728\u591a\u6838\u6216\u591a\u5904\u7406\u5668\u7684\u73af\u5883\u4e2d\u5c24\u5176\u6709\u6548\uff0c\u56e0\u4e3a\u7cfb\u7edf\u53ef\u4ee5\u540c\u65f6\u5904\u7406\u591a\u4e2a\u5b50\u95ee\u9898\uff0c\u66f4\u52a0\u5145\u5206\u5730\u5229\u7528\u8ba1\u7b97\u8d44\u6e90\uff0c\u4ece\u800c\u663e\u8457\u51cf\u5c11\u603b\u4f53\u7684\u8fd0\u884c\u65f6\u95f4\u3002</p> <p>\u6bd4\u5982\u5728\u56fe 12-3 \u6240\u793a\u7684\u201c\u6876\u6392\u5e8f\u201d\u4e2d\uff0c\u6211\u4eec\u5c06\u6d77\u91cf\u7684\u6570\u636e\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u5219\u53ef\u6240\u6709\u6876\u7684\u6392\u5e8f\u4efb\u52a1\u5206\u6563\u5230\u5404\u4e2a\u8ba1\u7b97\u5355\u5143\uff0c\u5b8c\u6210\u540e\u518d\u5408\u5e76\u7ed3\u679c\u3002</p> <p></p> <p> \u56fe 12-3 \u00a0 \u6876\u6392\u5e8f\u7684\u5e76\u884c\u8ba1\u7b97 </p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1213","title":"12.1.3 \u00a0 \u5206\u6cbb\u5e38\u89c1\u5e94\u7528","text":"<p>\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u53ef\u4ee5\u7528\u6765\u89e3\u51b3\u8bb8\u591a\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\u3002</p> <ul> <li>\u5bfb\u627e\u6700\u8fd1\u70b9\u5bf9\uff1a\u8be5\u7b97\u6cd5\u9996\u5148\u5c06\u70b9\u96c6\u5206\u6210\u4e24\u90e8\u5206\uff0c\u7136\u540e\u5206\u522b\u627e\u51fa\u4e24\u90e8\u5206\u4e2d\u7684\u6700\u8fd1\u70b9\u5bf9\uff0c\u6700\u540e\u627e\u51fa\u8de8\u8d8a\u4e24\u90e8\u5206\u7684\u6700\u8fd1\u70b9\u5bf9\u3002</li> <li>\u5927\u6574\u6570\u4e58\u6cd5\uff1a\u4f8b\u5982 Karatsuba \u7b97\u6cd5\uff0c\u5b83\u5c06\u5927\u6574\u6570\u4e58\u6cd5\u5206\u89e3\u4e3a\u51e0\u4e2a\u8f83\u5c0f\u7684\u6574\u6570\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002</li> <li>\u77e9\u9635\u4e58\u6cd5\uff1a\u4f8b\u5982 Strassen \u7b97\u6cd5\uff0c\u5b83\u5c06\u5927\u77e9\u9635\u4e58\u6cd5\u5206\u89e3\u4e3a\u591a\u4e2a\u5c0f\u77e9\u9635\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002</li> <li>\u6c49\u8bfa\u5854\u95ee\u9898\uff1a\u6c49\u8bfa\u5854\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u9012\u5f52\u89e3\u51b3\uff0c\u8fd9\u662f\u5178\u578b\u7684\u5206\u6cbb\u7b56\u7565\u5e94\u7528\u3002</li> <li>\u6c42\u89e3\u9006\u5e8f\u5bf9\uff1a\u5728\u4e00\u4e2a\u5e8f\u5217\u4e2d\uff0c\u5982\u679c\u524d\u9762\u7684\u6570\u5b57\u5927\u4e8e\u540e\u9762\u7684\u6570\u5b57\uff0c\u90a3\u4e48\u8fd9\u4e24\u4e2a\u6570\u5b57\u6784\u6210\u4e00\u4e2a\u9006\u5e8f\u5bf9\u3002\u6c42\u89e3\u9006\u5e8f\u5bf9\u95ee\u9898\u53ef\u4ee5\u5229\u7528\u5206\u6cbb\u7684\u601d\u60f3\uff0c\u501f\u52a9\u5f52\u5e76\u6392\u5e8f\u8fdb\u884c\u6c42\u89e3\u3002</li> </ul> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u5728\u7b97\u6cd5\u548c\u6570\u636e\u7ed3\u6784\u7684\u8bbe\u8ba1\u4e2d\u5e94\u7528\u5f97\u975e\u5e38\u5e7f\u6cdb\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\uff1a\u4e8c\u5206\u67e5\u627e\u662f\u5c06\u6709\u5e8f\u6570\u7ec4\u4ece\u4e2d\u70b9\u7d22\u5f15\u5904\u5206\u4e3a\u4e24\u90e8\u5206\uff0c\u7136\u540e\u6839\u636e\u76ee\u6807\u503c\u4e0e\u4e2d\u95f4\u5143\u7d20\u503c\u6bd4\u8f83\u7ed3\u679c\uff0c\u51b3\u5b9a\u6392\u9664\u54ea\u4e00\u534a\u533a\u95f4\uff0c\u5e76\u5728\u5269\u4f59\u533a\u95f4\u6267\u884c\u76f8\u540c\u7684\u4e8c\u5206\u64cd\u4f5c\u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\uff1a\u672c\u8282\u5f00\u5934\u5df2\u4ecb\u7ecd\uff0c\u4e0d\u518d\u8d58\u8ff0\u3002</li> <li>\u5feb\u901f\u6392\u5e8f\uff1a\u5feb\u901f\u6392\u5e8f\u662f\u9009\u53d6\u4e00\u4e2a\u57fa\u51c6\u503c\uff0c\u7136\u540e\u628a\u6570\u7ec4\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u4e00\u4e2a\u5b50\u6570\u7ec4\u7684\u5143\u7d20\u6bd4\u57fa\u51c6\u503c\u5c0f\uff0c\u53e6\u4e00\u5b50\u6570\u7ec4\u7684\u5143\u7d20\u6bd4\u57fa\u51c6\u503c\u5927\uff0c\u518d\u5bf9\u8fd9\u4e24\u90e8\u5206\u8fdb\u884c\u76f8\u540c\u7684\u5212\u5206\u64cd\u4f5c\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u53ea\u5269\u4e0b\u4e00\u4e2a\u5143\u7d20\u3002</li> <li>\u6876\u6392\u5e8f\uff1a\u6876\u6392\u5e8f\u7684\u57fa\u672c\u601d\u60f3\u662f\u5c06\u6570\u636e\u5206\u6563\u5230\u591a\u4e2a\u6876\uff0c\u7136\u540e\u5bf9\u6bcf\u4e2a\u6876\u5185\u7684\u5143\u7d20\u8fdb\u884c\u6392\u5e8f\uff0c\u6700\u540e\u5c06\u5404\u4e2a\u6876\u7684\u5143\u7d20\u4f9d\u6b21\u53d6\u51fa\uff0c\u4ece\u800c\u5f97\u5230\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002</li> <li>\u6811\uff1a\u4f8b\u5982\u4e8c\u53c9\u641c\u7d22\u6811\u3001AVL \u6811\u3001\u7ea2\u9ed1\u6811\u3001B \u6811\u3001B+ \u6811\u7b49\uff0c\u5b83\u4eec\u7684\u67e5\u627e\u3001\u63d2\u5165\u548c\u5220\u9664\u7b49\u64cd\u4f5c\u90fd\u53ef\u4ee5\u89c6\u4e3a\u5206\u6cbb\u7b56\u7565\u7684\u5e94\u7528\u3002</li> <li>\u5806\uff1a\u5806\u662f\u4e00\u79cd\u7279\u6b8a\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u5176\u5404\u79cd\u64cd\u4f5c\uff0c\u5982\u63d2\u5165\u3001\u5220\u9664\u548c\u5806\u5316\uff0c\u5b9e\u9645\u4e0a\u90fd\u9690\u542b\u4e86\u5206\u6cbb\u7684\u601d\u60f3\u3002</li> <li>\u54c8\u5e0c\u8868\uff1a\u867d\u7136\u54c8\u5e0c\u8868\u5e76\u4e0d\u76f4\u63a5\u5e94\u7528\u5206\u6cbb\uff0c\u4f46\u67d0\u4e9b\u54c8\u5e0c\u51b2\u7a81\u89e3\u51b3\u65b9\u6848\u95f4\u63a5\u5e94\u7528\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u4f8b\u5982\uff0c\u94fe\u5f0f\u5730\u5740\u4e2d\u7684\u957f\u94fe\u8868\u4f1a\u88ab\u8f6c\u5316\u4e3a\u7ea2\u9ed1\u6811\uff0c\u4ee5\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002</li> </ul> <p>\u53ef\u4ee5\u770b\u51fa\uff0c\u5206\u6cbb\u662f\u4e00\u79cd\u201c\u6da6\u7269\u7ec6\u65e0\u58f0\u201d\u7684\u7b97\u6cd5\u601d\u60f3\uff0c\u9690\u542b\u5728\u5404\u79cd\u7b97\u6cd5\u4e0e\u6570\u636e\u7ed3\u6784\u4e4b\u4e2d\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/","title":"12.4 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898","text":"<p>\u5728\u5f52\u5e76\u6392\u5e8f\u548c\u6784\u5efa\u4e8c\u53c9\u6811\u4e2d\uff0c\u6211\u4eec\u90fd\u662f\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u4e24\u4e2a\u89c4\u6a21\u4e3a\u539f\u95ee\u9898\u4e00\u534a\u7684\u5b50\u95ee\u9898\u3002\u7136\u800c\u5bf9\u4e8e\u6c49\u8bfa\u5854\u95ee\u9898\uff0c\u6211\u4eec\u91c7\u7528\u4e0d\u540c\u7684\u5206\u89e3\u7b56\u7565\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e09\u6839\u67f1\u5b50\uff0c\u8bb0\u4e3a <code>A</code>\u3001<code>B</code> \u548c <code>C</code> \u3002\u8d77\u59cb\u72b6\u6001\u4e0b\uff0c\u67f1\u5b50 <code>A</code> \u4e0a\u5957\u7740 \\(n\\) \u4e2a\u5706\u76d8\uff0c\u5b83\u4eec\u4ece\u4e0a\u5230\u4e0b\u6309\u7167\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\u3002\u6211\u4eec\u7684\u4efb\u52a1\u662f\u8981\u628a\u8fd9 \\(n\\) \u4e2a\u5706\u76d8\u79fb\u5230\u67f1\u5b50 <code>C</code> \u4e0a\uff0c\u5e76\u4fdd\u6301\u5b83\u4eec\u7684\u539f\u6709\u987a\u5e8f\u4e0d\u53d8\uff08\u5982\u56fe 12-10 \u6240\u793a\uff09\u3002\u5728\u79fb\u52a8\u5706\u76d8\u7684\u8fc7\u7a0b\u4e2d\uff0c\u9700\u8981\u9075\u5b88\u4ee5\u4e0b\u89c4\u5219\u3002</p> <ol> <li>\u5706\u76d8\u53ea\u80fd\u4ece\u4e00\u6839\u67f1\u5b50\u9876\u90e8\u62ff\u51fa\uff0c\u4ece\u53e6\u4e00\u6839\u67f1\u5b50\u9876\u90e8\u653e\u5165\u3002</li> <li>\u6bcf\u6b21\u53ea\u80fd\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\u3002</li> <li>\u5c0f\u5706\u76d8\u5fc5\u987b\u65f6\u523b\u4f4d\u4e8e\u5927\u5706\u76d8\u4e4b\u4e0a\u3002</li> </ol> <p></p> <p> \u56fe 12-10 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898\u793a\u4f8b </p> <p>\u6211\u4eec\u5c06\u89c4\u6a21\u4e3a \\(i\\) \u7684\u6c49\u8bfa\u5854\u95ee\u9898\u8bb0\u4f5c \\(f(i)\\) \u3002\u4f8b\u5982 \\(f(3)\\) \u4ee3\u8868\u5c06 \\(3\\) \u4e2a\u5706\u76d8\u4ece <code>A</code> \u79fb\u52a8\u81f3 <code>C</code> \u7684\u6c49\u8bfa\u5854\u95ee\u9898\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#1","title":"1. \u00a0 \u8003\u8651\u57fa\u672c\u60c5\u51b5","text":"<p>\u5982\u56fe 12-11 \u6240\u793a\uff0c\u5bf9\u4e8e\u95ee\u9898 \\(f(1)\\) \uff0c\u5373\u5f53\u53ea\u6709\u4e00\u4e2a\u5706\u76d8\u65f6\uff0c\u6211\u4eec\u5c06\u5b83\u76f4\u63a5\u4ece <code>A</code> \u79fb\u52a8\u81f3 <code>C</code> \u5373\u53ef\u3002</p> &lt;1&gt;&lt;2&gt; <p></p> <p></p> <p> \u56fe 12-11 \u00a0 \u89c4\u6a21\u4e3a 1 \u7684\u95ee\u9898\u7684\u89e3 </p> <p>\u5982\u56fe 12-12 \u6240\u793a\uff0c\u5bf9\u4e8e\u95ee\u9898 \\(f(2)\\) \uff0c\u5373\u5f53\u6709\u4e24\u4e2a\u5706\u76d8\u65f6\uff0c\u7531\u4e8e\u8981\u65f6\u523b\u6ee1\u8db3\u5c0f\u5706\u76d8\u5728\u5927\u5706\u76d8\u4e4b\u4e0a\uff0c\u56e0\u6b64\u9700\u8981\u501f\u52a9 <code>B</code> \u6765\u5b8c\u6210\u79fb\u52a8\u3002</p> <ol> <li>\u5148\u5c06\u4e0a\u9762\u7684\u5c0f\u5706\u76d8\u4ece <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u518d\u5c06\u5927\u5706\u76d8\u4ece <code>A</code> \u79fb\u81f3 <code>C</code> \u3002</li> <li>\u6700\u540e\u5c06\u5c0f\u5706\u76d8\u4ece <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 12-12 \u00a0 \u89c4\u6a21\u4e3a 2 \u7684\u95ee\u9898\u7684\u89e3 </p> <p>\u89e3\u51b3\u95ee\u9898 \\(f(2)\\) \u7684\u8fc7\u7a0b\u53ef\u603b\u7ed3\u4e3a\uff1a\u5c06\u4e24\u4e2a\u5706\u76d8\u501f\u52a9 <code>B</code> \u4ece <code>A</code> \u79fb\u81f3 <code>C</code> \u3002\u5176\u4e2d\uff0c<code>C</code> \u79f0\u4e3a\u76ee\u6807\u67f1\u3001<code>B</code> \u79f0\u4e3a\u7f13\u51b2\u67f1\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#2","title":"2. \u00a0 \u5b50\u95ee\u9898\u5206\u89e3","text":"<p>\u5bf9\u4e8e\u95ee\u9898 \\(f(3)\\) \uff0c\u5373\u5f53\u6709\u4e09\u4e2a\u5706\u76d8\u65f6\uff0c\u60c5\u51b5\u53d8\u5f97\u7a0d\u5fae\u590d\u6742\u4e86\u4e00\u4e9b\u3002</p> <p>\u56e0\u4e3a\u5df2\u77e5 \\(f(1)\\) \u548c \\(f(2)\\) \u7684\u89e3\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ece\u5206\u6cbb\u89d2\u5ea6\u601d\u8003\uff0c\u5c06 <code>A</code> \u9876\u90e8\u7684\u4e24\u4e2a\u5706\u76d8\u770b\u4f5c\u4e00\u4e2a\u6574\u4f53\uff0c\u6267\u884c\u56fe 12-13 \u6240\u793a\u7684\u6b65\u9aa4\u3002\u8fd9\u6837\u4e09\u4e2a\u5706\u76d8\u5c31\u88ab\u987a\u5229\u5730\u4ece <code>A</code> \u79fb\u81f3 <code>C</code> \u4e86\u3002</p> <ol> <li>\u4ee4 <code>B</code> \u4e3a\u76ee\u6807\u67f1\u3001<code>C</code> \u4e3a\u7f13\u51b2\u67f1\uff0c\u5c06\u4e24\u4e2a\u5706\u76d8\u4ece <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u5c06 <code>A</code> \u4e2d\u5269\u4f59\u7684\u4e00\u4e2a\u5706\u76d8\u4ece <code>A</code> \u76f4\u63a5\u79fb\u52a8\u81f3 <code>C</code> \u3002</li> <li>\u4ee4 <code>C</code> \u4e3a\u76ee\u6807\u67f1\u3001<code>A</code> \u4e3a\u7f13\u51b2\u67f1\uff0c\u5c06\u4e24\u4e2a\u5706\u76d8\u4ece <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 12-13 \u00a0 \u89c4\u6a21\u4e3a 3 \u7684\u95ee\u9898\u7684\u89e3 </p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u6211\u4eec\u5c06\u95ee\u9898 \\(f(3)\\) \u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(2)\\) \u548c\u4e00\u4e2a\u5b50\u95ee\u9898 \\(f(1)\\) \u3002\u6309\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u4e4b\u540e\uff0c\u539f\u95ee\u9898\u968f\u4e4b\u5f97\u5230\u89e3\u51b3\u3002\u8fd9\u8bf4\u660e\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff0c\u800c\u4e14\u89e3\u53ef\u4ee5\u5408\u5e76\u3002</p> <p>\u81f3\u6b64\uff0c\u6211\u4eec\u53ef\u603b\u7ed3\u51fa\u56fe 12-14 \u6240\u793a\u7684\u89e3\u51b3\u6c49\u8bfa\u5854\u95ee\u9898\u7684\u5206\u6cbb\u7b56\u7565\uff1a\u5c06\u539f\u95ee\u9898 \\(f(n)\\) \u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(n-1)\\) \u548c\u4e00\u4e2a\u5b50\u95ee\u9898 \\(f(1)\\) \uff0c\u5e76\u6309\u7167\u4ee5\u4e0b\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u3002</p> <ol> <li>\u5c06 \\(n-1\\) \u4e2a\u5706\u76d8\u501f\u52a9 <code>C</code> \u4ece <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u5c06\u5269\u4f59 \\(1\\) \u4e2a\u5706\u76d8\u4ece <code>A</code> \u76f4\u63a5\u79fb\u81f3 <code>C</code> \u3002</li> <li>\u5c06 \\(n-1\\) \u4e2a\u5706\u76d8\u501f\u52a9 <code>A</code> \u4ece <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> <p>\u5bf9\u4e8e\u8fd9\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(n-1)\\) \uff0c\u53ef\u4ee5\u901a\u8fc7\u76f8\u540c\u7684\u65b9\u5f0f\u8fdb\u884c\u9012\u5f52\u5212\u5206\uff0c\u76f4\u81f3\u8fbe\u5230\u6700\u5c0f\u5b50\u95ee\u9898 \\(f(1)\\) \u3002\u800c \\(f(1)\\) \u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u53ea\u9700\u4e00\u6b21\u79fb\u52a8\u64cd\u4f5c\u5373\u53ef\u3002</p> <p></p> <p> \u56fe 12-14 \u00a0 \u89e3\u51b3\u6c49\u8bfa\u5854\u95ee\u9898\u7684\u5206\u6cbb\u7b56\u7565 </p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u9012\u5f52\u51fd\u6570 <code>dfs(i, src, buf, tar)</code> \uff0c\u5b83\u7684\u4f5c\u7528\u662f\u5c06\u67f1 <code>src</code> \u9876\u90e8\u7684 \\(i\\) \u4e2a\u5706\u76d8\u501f\u52a9\u7f13\u51b2\u67f1 <code>buf</code> \u79fb\u52a8\u81f3\u76ee\u6807\u67f1 <code>tar</code> \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hanota.py<pre><code>def move(src: list[int], tar: list[int]):\n    \"\"\"\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\"\"\"\n    # \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    pan = src.pop()\n    # \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.append(pan)\n\ndef dfs(i: int, src: list[int], buf: list[int], tar: list[int]):\n    \"\"\"\u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i)\"\"\"\n    # \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if i == 1:\n        move(src, tar)\n        return\n    # \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf)\n    # \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar)\n    # \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar)\n\ndef solve_hanota(A: list[int], B: list[int], C: list[int]):\n    \"\"\"\u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898\"\"\"\n    n = len(A)\n    # \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C)\n</code></pre> hanota.cpp<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;tar) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    int pan = src.back();\n    src.pop_back();\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.push_back(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;buf, vector&lt;int&gt; &amp;tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(vector&lt;int&gt; &amp;A, vector&lt;int&gt; &amp;B, vector&lt;int&gt; &amp;C) {\n    int n = A.size();\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.java<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(List&lt;Integer&gt; src, List&lt;Integer&gt; tar) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    Integer pan = src.remove(src.size() - 1);\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.add(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, List&lt;Integer&gt; src, List&lt;Integer&gt; buf, List&lt;Integer&gt; tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(List&lt;Integer&gt; A, List&lt;Integer&gt; B, List&lt;Integer&gt; C) {\n    int n = A.size();\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.cs<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid Move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    int pan = src[^1];\n    src.RemoveAt(src.Count - 1);\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.Add(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid DFS(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        Move(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    DFS(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    Move(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    DFS(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid SolveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n    int n = A.Count;\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    DFS(n, A, B, C);\n}\n</code></pre> hanota.go<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunc move(src, tar *list.List) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    pan := src.Back()\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.PushBack(pan.Value)\n    // \u79fb\u9664 src \u9876\u90e8\u5706\u76d8\n    src.Remove(pan)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunc dfsHanota(i int, src, buf, tar *list.List) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if i == 1 {\n        move(src, tar)\n        return\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfsHanota(i-1, src, tar, buf)\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar)\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfsHanota(i-1, buf, src, tar)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunc solveHanota(A, B, C *list.List) {\n    n := A.Len()\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfsHanota(n, A, B, C)\n}\n</code></pre> hanota.swift<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunc move(src: inout [Int], tar: inout [Int]) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    let pan = src.popLast()!\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.append(pan)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunc dfs(i: Int, src: inout [Int], buf: inout [Int], tar: inout [Int]) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if i == 1 {\n        move(src: &amp;src, tar: &amp;tar)\n        return\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i: i - 1, src: &amp;src, buf: &amp;tar, tar: &amp;buf)\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src: &amp;src, tar: &amp;tar)\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i: i - 1, src: &amp;buf, buf: &amp;src, tar: &amp;tar)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunc solveHanota(A: inout [Int], B: inout [Int], C: inout [Int]) {\n    let n = A.count\n    // \u5217\u8868\u5c3e\u90e8\u662f\u67f1\u5b50\u9876\u90e8\n    // \u5c06 src \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(i: n, src: &amp;A, buf: &amp;B, tar: &amp;C)\n}\n</code></pre> hanota.js<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunction move(src, tar) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    const pan = src.pop();\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunction dfs(i, src, buf, tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunction solveHanota(A, B, C) {\n    const n = A.length;\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.ts<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunction move(src: number[], tar: number[]): void {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    const pan = src.pop();\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunction dfs(i: number, src: number[], buf: number[], tar: number[]): void {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunction solveHanota(A: number[], B: number[], C: number[]): void {\n    const n = A.length;\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.dart<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n  // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n  int pan = src.removeLast();\n  // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n  tar.add(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n  // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n  if (i == 1) {\n    move(src, tar);\n    return;\n  }\n  // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n  dfs(i - 1, src, tar, buf);\n  // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n  move(src, tar);\n  // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n  dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n  int n = A.length;\n  // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n  dfs(n, A, B, C);\n}\n</code></pre> hanota.rs<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfn move_pan(src: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    let pan = src.remove(src.len() - 1);\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfn dfs(i: i32, src: &amp;mut Vec&lt;i32&gt;, buf: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if i == 1 {\n        move_pan(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move_pan(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfn solve_hanota(A: &amp;mut Vec&lt;i32&gt;, B: &amp;mut Vec&lt;i32&gt;, C: &amp;mut Vec&lt;i32&gt;) {\n    let n = A.len() as i32;\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.c<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(int *src, int *srcSize, int *tar, int *tarSize) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    int pan = src[*srcSize - 1];\n    src[*srcSize - 1] = 0;\n    (*srcSize)--;\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar[*tarSize] = pan;\n    (*tarSize)++;\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, int *src, int *srcSize, int *buf, int *bufSize, int *tar, int *tarSize) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, srcSize, tar, tarSize);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, srcSize, tar, tarSize, buf, bufSize);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, srcSize, tar, tarSize);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, bufSize, src, srcSize, tar, tarSize);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(int *A, int *ASize, int *B, int *BSize, int *C, int *CSize) {\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(*ASize, A, ASize, B, BSize, C, CSize);\n}\n</code></pre> hanota.kt<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfun move(src: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    val pan: Int = src.removeAt(src.size - 1)\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.add(pan)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfun dfs(i: Int, src: MutableList&lt;Int&gt;, buf: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, tar)\n        return\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf)\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar)\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfun solveHanota(A: MutableList&lt;Int&gt;, B: MutableList&lt;Int&gt;, C: MutableList&lt;Int&gt;) {\n    val n = A.size\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C)\n}\n</code></pre> hanota.rb<pre><code>[class]{}-[func]{move}\n\n[class]{}-[func]{dfs}\n\n[class]{}-[func]{solve_hanota}\n</code></pre> hanota.zig<pre><code>[class]{}-[func]{move}\n\n[class]{}-[func]{dfs}\n\n[class]{}-[func]{solveHanota}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 12-15 \u6240\u793a\uff0c\u6c49\u8bfa\u5854\u95ee\u9898\u5f62\u6210\u4e00\u68f5\u9ad8\u5ea6\u4e3a \\(n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u5bf9\u5e94\u4e00\u4e2a\u5f00\u542f\u7684 <code>dfs()</code> \u51fd\u6570\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> <p></p> <p> \u56fe 12-15 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898\u7684\u9012\u5f52\u6811 </p> <p>Quote</p> <p>\u6c49\u8bfa\u5854\u95ee\u9898\u6e90\u81ea\u4e00\u4e2a\u53e4\u8001\u7684\u4f20\u8bf4\u3002\u5728\u53e4\u5370\u5ea6\u7684\u4e00\u4e2a\u5bfa\u5e99\u91cc\uff0c\u50e7\u4fa3\u4eec\u6709\u4e09\u6839\u9ad8\u5927\u7684\u94bb\u77f3\u67f1\u5b50\uff0c\u4ee5\u53ca \\(64\\) \u4e2a\u5927\u5c0f\u4e0d\u4e00\u7684\u91d1\u5706\u76d8\u3002\u50e7\u4fa3\u4eec\u4e0d\u65ad\u5730\u79fb\u52a8\u5706\u76d8\uff0c\u4ed6\u4eec\u76f8\u4fe1\u5728\u6700\u540e\u4e00\u4e2a\u5706\u76d8\u88ab\u6b63\u786e\u653e\u7f6e\u7684\u90a3\u4e00\u523b\uff0c\u8fd9\u4e2a\u4e16\u754c\u5c31\u4f1a\u7ed3\u675f\u3002</p> <p>\u7136\u800c\uff0c\u5373\u4f7f\u50e7\u4fa3\u4eec\u6bcf\u79d2\u949f\u79fb\u52a8\u4e00\u6b21\uff0c\u603b\u5171\u9700\u8981\u5927\u7ea6 \\(2^{64} \\approx 1.84\u00d710^{19}\\) \u79d2\uff0c\u5408\u7ea6 \\(5850\\) \u4ebf\u5e74\uff0c\u8fdc\u8fdc\u8d85\u8fc7\u4e86\u73b0\u5728\u5bf9\u5b87\u5b99\u5e74\u9f84\u7684\u4f30\u8ba1\u3002\u6240\u4ee5\uff0c\u5018\u82e5\u8fd9\u4e2a\u4f20\u8bf4\u662f\u771f\u7684\uff0c\u6211\u4eec\u5e94\u8be5\u4e0d\u9700\u8981\u62c5\u5fc3\u4e16\u754c\u672b\u65e5\u7684\u5230\u6765\u3002</p>"},{"location":"chapter_divide_and_conquer/summary/","title":"12.5 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u5206\u6cbb\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u7b97\u6cd5\u8bbe\u8ba1\u7b56\u7565\uff0c\u5305\u62ec\u5206\uff08\u5212\u5206\uff09\u548c\u6cbb\uff08\u5408\u5e76\uff09\u4e24\u4e2a\u9636\u6bb5\uff0c\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\u3002</li> <li>\u5224\u65ad\u662f\u5426\u662f\u5206\u6cbb\u7b97\u6cd5\u95ee\u9898\u7684\u4f9d\u636e\u5305\u62ec\uff1a\u95ee\u9898\u80fd\u5426\u5206\u89e3\u3001\u5b50\u95ee\u9898\u662f\u5426\u72ec\u7acb\u3001\u5b50\u95ee\u9898\u80fd\u5426\u5408\u5e76\u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u5e94\u7528\uff0c\u5176\u9012\u5f52\u5730\u5c06\u6570\u7ec4\u5212\u5206\u4e3a\u7b49\u957f\u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u5230\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\u65f6\u5f00\u59cb\u9010\u5c42\u5408\u5e76\uff0c\u4ece\u800c\u5b8c\u6210\u6392\u5e8f\u3002</li> <li>\u5f15\u5165\u5206\u6cbb\u7b56\u7565\u5f80\u5f80\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u7b56\u7565\u51cf\u5c11\u4e86\u64cd\u4f5c\u6570\u91cf\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u540e\u6709\u5229\u4e8e\u7cfb\u7edf\u7684\u5e76\u884c\u4f18\u5316\u3002</li> <li>\u5206\u6cbb\u65e2\u53ef\u4ee5\u89e3\u51b3\u8bb8\u591a\u7b97\u6cd5\u95ee\u9898\uff0c\u4e5f\u5e7f\u6cdb\u5e94\u7528\u4e8e\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\uff0c\u5904\u5904\u53ef\u89c1\u5176\u8eab\u5f71\u3002</li> <li>\u76f8\u8f83\u4e8e\u66b4\u529b\u641c\u7d22\uff0c\u81ea\u9002\u5e94\u641c\u7d22\u6548\u7387\u66f4\u9ad8\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u7684\u641c\u7d22\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u5b9e\u73b0\u7684\u3002</li> <li>\u4e8c\u5206\u67e5\u627e\u662f\u5206\u6cbb\u7b56\u7565\u7684\u53e6\u4e00\u4e2a\u5178\u578b\u5e94\u7528\uff0c\u5b83\u4e0d\u5305\u542b\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\u7684\u6b65\u9aa4\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9012\u5f52\u5206\u6cbb\u5b9e\u73b0\u4e8c\u5206\u67e5\u627e\u3002</li> <li>\u5728\u6784\u5efa\u4e8c\u53c9\u6811\u7684\u95ee\u9898\u4e2d\uff0c\u6784\u5efa\u6811\uff08\u539f\u95ee\u9898\uff09\u53ef\u4ee5\u5212\u5206\u4e3a\u6784\u5efa\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u8fd9\u53ef\u4ee5\u901a\u8fc7\u5212\u5206\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u7684\u7d22\u5f15\u533a\u95f4\u6765\u5b9e\u73b0\u3002</li> <li>\u5728\u6c49\u8bfa\u5854\u95ee\u9898\u4e2d\uff0c\u4e00\u4e2a\u89c4\u6a21\u4e3a \\(n\\) \u7684\u95ee\u9898\u53ef\u4ee5\u5212\u5206\u4e3a\u4e24\u4e2a\u89c4\u6a21\u4e3a \\(n-1\\) \u7684\u5b50\u95ee\u9898\u548c\u4e00\u4e2a\u89c4\u6a21\u4e3a \\(1\\) \u7684\u5b50\u95ee\u9898\u3002\u6309\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u540e\uff0c\u539f\u95ee\u9898\u968f\u4e4b\u5f97\u5230\u89e3\u51b3\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/","title":"\u7b2c 14 \u7ae0 \u00a0 \u52a8\u6001\u89c4\u5212","text":"<p>Abstract</p> <p>\u5c0f\u6eaa\u6c47\u5165\u6cb3\u6d41\uff0c\u6c5f\u6cb3\u6c47\u5165\u5927\u6d77\u3002</p> <p>\u52a8\u6001\u89c4\u5212\u5c06\u5c0f\u95ee\u9898\u7684\u89e3\u6c47\u96c6\u6210\u5927\u95ee\u9898\u7684\u7b54\u6848\uff0c\u4e00\u6b65\u6b65\u5f15\u9886\u6211\u4eec\u8d70\u5411\u89e3\u51b3\u95ee\u9898\u7684\u5f7c\u5cb8\u3002</p>"},{"location":"chapter_dynamic_programming/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>14.1 \u00a0 \u521d\u63a2\u52a8\u6001\u89c4\u5212</li> <li>14.2 \u00a0 DP \u95ee\u9898\u7279\u6027</li> <li>14.3 \u00a0 DP \u89e3\u9898\u601d\u8def</li> <li>14.4 \u00a0 0-1 \u80cc\u5305\u95ee\u9898</li> <li>14.5 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898</li> <li>14.6 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898</li> <li>14.7 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_dynamic_programming/dp_problem_features/","title":"14.2 \u00a0 \u52a8\u6001\u89c4\u5212\u95ee\u9898\u7279\u6027","text":"<p>\u5728\u4e0a\u4e00\u8282\u4e2d\uff0c\u6211\u4eec\u5b66\u4e60\u4e86\u52a8\u6001\u89c4\u5212\u662f\u5982\u4f55\u901a\u8fc7\u5b50\u95ee\u9898\u5206\u89e3\u6765\u6c42\u89e3\u539f\u95ee\u9898\u7684\u3002\u5b9e\u9645\u4e0a\uff0c\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u901a\u7528\u7684\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u3001\u56de\u6eaf\u4e2d\u7684\u4fa7\u91cd\u70b9\u4e0d\u540c\u3002</p> <ul> <li>\u5206\u6cbb\u7b97\u6cd5\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\u5212\u5206\u4e3a\u591a\u4e2a\u76f8\u4e92\u72ec\u7acb\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u6700\u5c0f\u5b50\u95ee\u9898\uff0c\u5e76\u5728\u56de\u6eaf\u4e2d\u5408\u5e76\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u6700\u7ec8\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002</li> <li>\u52a8\u6001\u89c4\u5212\u4e5f\u5bf9\u95ee\u9898\u8fdb\u884c\u9012\u5f52\u5206\u89e3\uff0c\u4f46\u4e0e\u5206\u6cbb\u7b97\u6cd5\u7684\u4e3b\u8981\u533a\u522b\u662f\uff0c\u52a8\u6001\u89c4\u5212\u4e2d\u7684\u5b50\u95ee\u9898\u662f\u76f8\u4e92\u4f9d\u8d56\u7684\uff0c\u5728\u5206\u89e3\u8fc7\u7a0b\u4e2d\u4f1a\u51fa\u73b0\u8bb8\u591a\u91cd\u53e0\u5b50\u95ee\u9898\u3002</li> <li>\u56de\u6eaf\u7b97\u6cd5\u5728\u5c1d\u8bd5\u548c\u56de\u9000\u4e2d\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u7684\u89e3\uff0c\u5e76\u901a\u8fc7\u526a\u679d\u907f\u514d\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\u3002\u539f\u95ee\u9898\u7684\u89e3\u7531\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\u6784\u6210\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6bcf\u4e2a\u51b3\u7b56\u6b65\u9aa4\u4e4b\u524d\u7684\u5b50\u5e8f\u5217\u770b\u4f5c\u4e00\u4e2a\u5b50\u95ee\u9898\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u52a8\u6001\u89c4\u5212\u5e38\u7528\u6765\u6c42\u89e3\u6700\u4f18\u5316\u95ee\u9898\uff0c\u5b83\u4eec\u4e0d\u4ec5\u5305\u542b\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u8fd8\u5177\u6709\u53e6\u5916\u4e24\u5927\u7279\u6027\uff1a\u6700\u4f18\u5b50\u7ed3\u6784\u3001\u65e0\u540e\u6548\u6027\u3002</p>"},{"location":"chapter_dynamic_programming/dp_problem_features/#1421","title":"14.2.1 \u00a0 \u6700\u4f18\u5b50\u7ed3\u6784","text":"<p>\u6211\u4eec\u5bf9\u722c\u697c\u68af\u95ee\u9898\u7a0d\u4f5c\u6539\u52a8\uff0c\u4f7f\u4e4b\u66f4\u52a0\u9002\u5408\u5c55\u793a\u6700\u4f18\u5b50\u7ed3\u6784\u6982\u5ff5\u3002</p> <p>\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u6bcf\u4e00\u9636\u697c\u68af\u4e0a\u90fd\u8d34\u6709\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a\u4f60\u5728\u8be5\u53f0\u9636\u6240\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\u3002\u7ed9\u5b9a\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\u6570\u7ec4 \\(cost\\) \uff0c\u5176\u4e2d \\(cost[i]\\) \u8868\u793a\u5728\u7b2c \\(i\\) \u4e2a\u53f0\u9636\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\\(cost[0]\\) \u4e3a\u5730\u9762\uff08\u8d77\u59cb\u70b9\uff09\u3002\u8bf7\u8ba1\u7b97\u6700\u5c11\u9700\u8981\u4ed8\u51fa\u591a\u5c11\u4ee3\u4ef7\u624d\u80fd\u5230\u8fbe\u9876\u90e8\uff1f</p> <p>\u5982\u56fe 14-6 \u6240\u793a\uff0c\u82e5\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u9636\u7684\u4ee3\u4ef7\u5206\u522b\u4e3a \\(1\\)\u3001\\(10\\)\u3001\\(1\\) \uff0c\u5219\u4ece\u5730\u9762\u722c\u5230\u7b2c \\(3\\) \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7\u4e3a \\(2\\) \u3002</p> <p></p> <p> \u56fe 14-6 \u00a0 \u722c\u5230\u7b2c 3 \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7 </p> <p>\u8bbe \\(dp[i]\\) \u4e3a\u722c\u5230\u7b2c \\(i\\) \u9636\u7d2f\u8ba1\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\u7531\u4e8e\u7b2c \\(i\\) \u9636\u53ea\u53ef\u80fd\u4ece \\(i - 1\\) \u9636\u6216 \\(i - 2\\) \u9636\u8d70\u6765\uff0c\u56e0\u6b64 \\(dp[i]\\) \u53ea\u53ef\u80fd\u7b49\u4e8e \\(dp[i - 1] + cost[i]\\) \u6216 \\(dp[i - 2] + cost[i]\\) \u3002\u4e3a\u4e86\u5c3d\u53ef\u80fd\u51cf\u5c11\u4ee3\u4ef7\uff0c\u6211\u4eec\u5e94\u8be5\u9009\u62e9\u4e24\u8005\u4e2d\u8f83\u5c0f\u7684\u90a3\u4e00\u4e2a\uff1a</p> \\[ dp[i] = \\min(dp[i-1], dp[i-2]) + cost[i] \\] <p>\u8fd9\u4fbf\u53ef\u4ee5\u5f15\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u542b\u4e49\uff1a\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u662f\u4ece\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u5efa\u5f97\u6765\u7684\u3002</p> <p>\u672c\u9898\u663e\u7136\u5177\u6709\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u6211\u4eec\u4ece\u4e24\u4e2a\u5b50\u95ee\u9898\u6700\u4f18\u89e3 \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u4e2d\u6311\u9009\u51fa\u8f83\u4f18\u7684\u90a3\u4e00\u4e2a\uff0c\u5e76\u7528\u5b83\u6784\u5efa\u51fa\u539f\u95ee\u9898 \\(dp[i]\\) \u7684\u6700\u4f18\u89e3\u3002</p> <p>\u90a3\u4e48\uff0c\u4e0a\u4e00\u8282\u7684\u722c\u697c\u68af\u9898\u76ee\u6709\u6ca1\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u5462\uff1f\u5b83\u7684\u76ee\u6807\u662f\u6c42\u89e3\u65b9\u6848\u6570\u91cf\uff0c\u770b\u4f3c\u662f\u4e00\u4e2a\u8ba1\u6570\u95ee\u9898\uff0c\u4f46\u5982\u679c\u6362\u4e00\u79cd\u95ee\u6cd5\uff1a\u201c\u6c42\u89e3\u6700\u5927\u65b9\u6848\u6570\u91cf\u201d\u3002\u6211\u4eec\u610f\u5916\u5730\u53d1\u73b0\uff0c\u867d\u7136\u9898\u76ee\u4fee\u6539\u524d\u540e\u662f\u7b49\u4ef7\u7684\uff0c\u4f46\u6700\u4f18\u5b50\u7ed3\u6784\u6d6e\u73b0\u51fa\u6765\u4e86\uff1a\u7b2c \\(n\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u7b49\u4e8e\u7b2c \\(n-1\\) \u9636\u548c\u7b2c \\(n-2\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u4e4b\u548c\u3002\u6240\u4ee5\u8bf4\uff0c\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u89e3\u91ca\u65b9\u5f0f\u6bd4\u8f83\u7075\u6d3b\uff0c\u5728\u4e0d\u540c\u95ee\u9898\u4e2d\u4f1a\u6709\u4e0d\u540c\u7684\u542b\u4e49\u3002</p> <p>\u6839\u636e\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u4ee5\u53ca\u521d\u59cb\u72b6\u6001 \\(dp[1] = cost[1]\\) \u548c \\(dp[2] = cost[2]\\) \uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u52a8\u6001\u89c4\u5212\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp(cost: list[int]) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp = [0] * (n + 1)\n    # \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1], dp[2] = cost[1], cost[2]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in range(3, n + 1):\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    return dp[n]\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    vector&lt;int&gt; dp(n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint MinCostClimbingStairsDP(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.Min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDP(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp := make([]int, n+1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = min(dp[i-1], dp[i-2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDP(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = Array(repeating: 0, count: n + 1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in 3 ... n {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDP(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = new Array(n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDP(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = new Array(n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n  dp[1] = cost[1];\n  dp[2] = cost[2];\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n  }\n  return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfn min_cost_climbing_stairs_dp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    let mut dp = vec![-1; n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in 3..=n {\n        dp[i] = cmp::min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int *dp = calloc(n + 1, sizeof(int));\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = myMin(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    int res = dp[n];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfun minCostClimbingStairsDP(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    val dp = IntArray(n + 1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (i in 3..n) {\n        dp[i] = (min(dp[i - 1].toDouble(), dp[i - 2].toDouble()) + cost[i]).toInt()\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>[class]{}-[func]{min_cost_climbing_stairs_dp}\n</code></pre> min_cost_climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212\nfn minCostClimbingStairsDP(comptime cost: []i32) i32 {\n    comptime var n = cost.len - 1;\n    if (n == 1 or n == 2) {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = [_]i32{-1} ** (n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (3..n + 1) |i| {\n        dp[i] = @min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 14-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u4ee3\u7801\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 14-7 \u00a0 \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u672c\u9898\u4e5f\u53ef\u4ee5\u8fdb\u884c\u7a7a\u95f4\u4f18\u5316\uff0c\u5c06\u4e00\u7ef4\u538b\u7f29\u81f3\u96f6\u7ef4\uff0c\u4f7f\u5f97\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp_comp(cost: list[int]) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    a, b = cost[1], cost[2]\n    for i in range(3, n + 1):\n        a, b = b, min(a, b) + cost[i]\n    return b\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint MinCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.Min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDPComp(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    a, b := cost[1], cost[2]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i := 3; i &lt;= n; i++ {\n        tmp := b\n        b = min(a, tmp) + cost[i]\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDPComp(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    var (a, b) = (cost[1], cost[2])\n    for i in 3 ... n {\n        (a, b) = (b, min(a, b) + cost[i])\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDPComp(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDPComp(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  int a = cost[1], b = cost[2];\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = min(a, tmp) + cost[i];\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn min_cost_climbing_stairs_dp_comp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    };\n    let (mut a, mut b) = (cost[1], cost[2]);\n    for i in 3..=n {\n        let tmp = b;\n        b = cmp::min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    b\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = myMin(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun minCostClimbingStairsDPComp(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    var a = cost[1]\n    var b = cost[2]\n    for (i in 3..n) {\n        val tmp = b\n        b = (min(a.toDouble(), tmp.toDouble()) + cost[i]).toInt()\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>[class]{}-[func]{min_cost_climbing_stairs_dp_comp}\n</code></pre> min_cost_climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn minCostClimbingStairsDPComp(cost: []i32) i32 {\n    var n = cost.len - 1;\n    if (n == 1 or n == 2) {\n        return cost[n];\n    }\n    var a = cost[1];\n    var b = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (3..n + 1) |i| {\n        var tmp = b;\n        b = @min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/dp_problem_features/#1422","title":"14.2.2 \u00a0 \u65e0\u540e\u6548\u6027","text":"<p>\u65e0\u540e\u6548\u6027\u662f\u52a8\u6001\u89c4\u5212\u80fd\u591f\u6709\u6548\u89e3\u51b3\u95ee\u9898\u7684\u91cd\u8981\u7279\u6027\u4e4b\u4e00\uff0c\u5176\u5b9a\u4e49\u4e3a\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u786e\u5b9a\u7684\u72b6\u6001\uff0c\u5b83\u7684\u672a\u6765\u53d1\u5c55\u53ea\u4e0e\u5f53\u524d\u72b6\u6001\u6709\u5173\uff0c\u800c\u4e0e\u8fc7\u53bb\u7ecf\u5386\u7684\u6240\u6709\u72b6\u6001\u65e0\u5173\u3002</p> <p>\u4ee5\u722c\u697c\u68af\u95ee\u9898\u4e3a\u4f8b\uff0c\u7ed9\u5b9a\u72b6\u6001 \\(i\\) \uff0c\u5b83\u4f1a\u53d1\u5c55\u51fa\u72b6\u6001 \\(i+1\\) \u548c\u72b6\u6001 \\(i+2\\) \uff0c\u5206\u522b\u5bf9\u5e94\u8df3 \\(1\\) \u6b65\u548c\u8df3 \\(2\\) \u6b65\u3002\u5728\u505a\u51fa\u8fd9\u4e24\u79cd\u9009\u62e9\u65f6\uff0c\u6211\u4eec\u65e0\u987b\u8003\u8651\u72b6\u6001 \\(i\\) \u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5b83\u4eec\u5bf9\u72b6\u6001 \\(i\\) \u7684\u672a\u6765\u6ca1\u6709\u5f71\u54cd\u3002</p> <p>\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u7ed9\u722c\u697c\u68af\u95ee\u9898\u6dfb\u52a0\u4e00\u4e2a\u7ea6\u675f\uff0c\u60c5\u51b5\u5c31\u4e0d\u4e00\u6837\u4e86\u3002</p> <p>\u5e26\u7ea6\u675f\u722c\u697c\u68af</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u4f46\u4e0d\u80fd\u8fde\u7eed\u4e24\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f</p> <p>\u5982\u56fe 14-8 \u6240\u793a\uff0c\u722c\u4e0a\u7b2c \\(3\\) \u9636\u4ec5\u5269 \\(2\\) \u79cd\u53ef\u884c\u65b9\u6848\uff0c\u5176\u4e2d\u8fde\u7eed\u4e09\u6b21\u8df3 \\(1\\) \u9636\u7684\u65b9\u6848\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\uff0c\u56e0\u6b64\u88ab\u820d\u5f03\u3002</p> <p></p> <p> \u56fe 14-8 \u00a0 \u5e26\u7ea6\u675f\u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf </p> <p>\u5728\u8be5\u95ee\u9898\u4e2d\uff0c\u5982\u679c\u4e0a\u4e00\u8f6e\u662f\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\uff0c\u90a3\u4e48\u4e0b\u4e00\u8f6e\u5c31\u5fc5\u987b\u8df3 \\(2\\) \u9636\u3002\u8fd9\u610f\u5473\u7740\uff0c\u4e0b\u4e00\u6b65\u9009\u62e9\u4e0d\u80fd\u7531\u5f53\u524d\u72b6\u6001\uff08\u5f53\u524d\u6240\u5728\u697c\u68af\u9636\u6570\uff09\u72ec\u7acb\u51b3\u5b9a\uff0c\u8fd8\u548c\u524d\u4e00\u4e2a\u72b6\u6001\uff08\u4e0a\u4e00\u8f6e\u6240\u5728\u697c\u68af\u9636\u6570\uff09\u6709\u5173\u3002</p> <p>\u4e0d\u96be\u53d1\u73b0\uff0c\u6b64\u95ee\u9898\u5df2\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b \\(dp[i] = dp[i-1] + dp[i-2]\\) \u4e5f\u5931\u6548\u4e86\uff0c\u56e0\u4e3a \\(dp[i-1]\\) \u4ee3\u8868\u672c\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u4f46\u5176\u4e2d\u5305\u542b\u4e86\u8bb8\u591a\u201c\u4e0a\u4e00\u8f6e\u662f\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\u201d\u65b9\u6848\uff0c\u800c\u4e3a\u4e86\u6ee1\u8db3\u7ea6\u675f\uff0c\u6211\u4eec\u5c31\u4e0d\u80fd\u5c06 \\(dp[i-1]\\) \u76f4\u63a5\u8ba1\u5165 \\(dp[i]\\) \u4e2d\u3002</p> <p>\u4e3a\u6b64\uff0c\u6211\u4eec\u9700\u8981\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff1a\u72b6\u6001 \\([i, j]\\) \u8868\u793a\u5904\u5728\u7b2c \\(i\\) \u9636\u5e76\u4e14\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(j\\) \u9636\uff0c\u5176\u4e2d \\(j \\in \\{1, 2\\}\\) \u3002\u6b64\u72b6\u6001\u5b9a\u4e49\u6709\u6548\u5730\u533a\u5206\u4e86\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u8fd8\u662f \\(2\\) \u9636\uff0c\u6211\u4eec\u53ef\u4ee5\u636e\u6b64\u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u4ece\u4f55\u800c\u6765\u7684\u3002</p> <ul> <li>\u5f53\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u65f6\uff0c\u4e0a\u4e0a\u4e00\u8f6e\u53ea\u80fd\u9009\u62e9\u8df3 \\(2\\) \u9636\uff0c\u5373 \\(dp[i, 1]\\) \u53ea\u80fd\u4ece \\(dp[i-1, 2]\\) \u8f6c\u79fb\u8fc7\u6765\u3002</li> <li>\u5f53\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(2\\) \u9636\u65f6\uff0c\u4e0a\u4e0a\u4e00\u8f6e\u53ef\u9009\u62e9\u8df3 \\(1\\) \u9636\u6216\u8df3 \\(2\\) \u9636\uff0c\u5373 \\(dp[i, 2]\\) \u53ef\u4ee5\u4ece \\(dp[i-2, 1]\\) \u6216 \\(dp[i-2, 2]\\) \u8f6c\u79fb\u8fc7\u6765\u3002</li> </ul> <p>\u5982\u56fe 14-9 \u6240\u793a\uff0c\u5728\u8be5\u5b9a\u4e49\u4e0b\uff0c\\(dp[i, j]\\) \u8868\u793a\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u65b9\u6848\u6570\u3002\u6b64\u65f6\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ \\begin{cases} dp[i, 1] = dp[i-1, 2] \\\\ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] \\end{cases} \\] <p></p> <p> \u56fe 14-9 \u00a0 \u8003\u8651\u7ea6\u675f\u4e0b\u7684\u9012\u63a8\u5173\u7cfb </p> <p>\u6700\u7ec8\uff0c\u8fd4\u56de \\(dp[n, 1] + dp[n, 2]\\) \u5373\u53ef\uff0c\u4e24\u8005\u4e4b\u548c\u4ee3\u8868\u722c\u5230\u7b2c \\(n\\) \u9636\u7684\u65b9\u6848\u603b\u6570\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_constraint_dp.py<pre><code>def climbing_stairs_constraint_dp(n: int) -&gt; int:\n    \"\"\"\u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    if n == 1 or n == 2:\n        return 1\n    # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp = [[0] * 3 for _ in range(n + 1)]\n    # \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1], dp[1][2] = 1, 0\n    dp[2][1], dp[2][2] = 0, 1\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in range(3, n + 1):\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    return dp[n][1] + dp[n][2]\n</code></pre> climbing_stairs_constraint_dp.cpp<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(3, 0));\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.java<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[][] dp = new int[n + 1][3];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.cs<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[,] dp = new int[n + 1, 3];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1, 1] = 1;\n    dp[1, 2] = 0;\n    dp[2, 1] = 0;\n    dp[2, 2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i, 1] = dp[i - 1, 2];\n        dp[i, 2] = dp[i - 2, 1] + dp[i - 2, 2];\n    }\n    return dp[n, 1] + dp[n, 2];\n}\n</code></pre> climbing_stairs_constraint_dp.go<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsConstraintDP(n int) int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp := make([][3]int, n+1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i := 3; i &lt;= n; i++ {\n        dp[i][1] = dp[i-1][2]\n        dp[i][2] = dp[i-2][1] + dp[i-2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.swift<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsConstraintDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = Array(repeating: Array(repeating: 0, count: 3), count: n + 1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in 3 ... n {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.js<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsConstraintDP(n) {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = Array.from(new Array(n + 1), () =&gt; new Array(3));\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.ts<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsConstraintDP(n: number): number {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(3));\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.dart<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n  if (n == 1 || n == 2) {\n    return 1;\n  }\n  // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(3, 0));\n  // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n  dp[1][1] = 1;\n  dp[1][2] = 0;\n  dp[2][1] = 0;\n  dp[2][2] = 1;\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i][1] = dp[i - 1][2];\n    dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n  }\n  return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.rs<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_constraint_dp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return 1;\n    };\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    let mut dp = vec![vec![-1; 3]; n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in 3..=n {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.c<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(3, sizeof(int));\n    }\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    int res = dp[n][1] + dp[n][2];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> climbing_stairs_constraint_dp.kt<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfun climbingStairsConstraintDP(n: Int): Int {\n    if (n == 1 || n == 2) {\n        return 1\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    val dp = Array(n + 1) { IntArray(3) }\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (i in 3..n) {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.rb<pre><code>[class]{}-[func]{climbing_stairs_constraint_dp}\n</code></pre> climbing_stairs_constraint_dp.zig<pre><code>// \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\nfn climbingStairsConstraintDP(comptime n: usize) i32 {\n    if (n == 1 or n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = [_][3]i32{ [_]i32{ -1, -1, -1 } } ** (n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (3..n + 1) |i| {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5728\u4e0a\u9762\u7684\u6848\u4f8b\u4e2d\uff0c\u7531\u4e8e\u4ec5\u9700\u591a\u8003\u8651\u524d\u9762\u4e00\u4e2a\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u901a\u8fc7\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff0c\u4f7f\u5f97\u95ee\u9898\u91cd\u65b0\u6ee1\u8db3\u65e0\u540e\u6548\u6027\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u95ee\u9898\u5177\u6709\u975e\u5e38\u4e25\u91cd\u7684\u201c\u6709\u540e\u6548\u6027\u201d\u3002</p> <p>\u722c\u697c\u68af\u4e0e\u969c\u788d\u751f\u6210</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\u3002\u89c4\u5b9a\u5f53\u722c\u5230\u7b2c \\(i\\) \u9636\u65f6\uff0c\u7cfb\u7edf\u81ea\u52a8\u4f1a\u5728\u7b2c \\(2i\\) \u9636\u4e0a\u653e\u4e0a\u969c\u788d\u7269\uff0c\u4e4b\u540e\u6240\u6709\u8f6e\u90fd\u4e0d\u5141\u8bb8\u8df3\u5230\u7b2c \\(2i\\) \u9636\u4e0a\u3002\u4f8b\u5982\uff0c\u524d\u4e24\u8f6e\u5206\u522b\u8df3\u5230\u4e86\u7b2c \\(2\\)\u3001\\(3\\) \u9636\u4e0a\uff0c\u5219\u4e4b\u540e\u5c31\u4e0d\u80fd\u8df3\u5230\u7b2c \\(4\\)\u3001\\(6\\) \u9636\u4e0a\u3002\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f</p> <p>\u5728\u8fd9\u4e2a\u95ee\u9898\u4e2d\uff0c\u4e0b\u6b21\u8df3\u8dc3\u4f9d\u8d56\u8fc7\u53bb\u6240\u6709\u7684\u72b6\u6001\uff0c\u56e0\u4e3a\u6bcf\u4e00\u6b21\u8df3\u8dc3\u90fd\u4f1a\u5728\u66f4\u9ad8\u7684\u9636\u68af\u4e0a\u8bbe\u7f6e\u969c\u788d\uff0c\u5e76\u5f71\u54cd\u672a\u6765\u7684\u8df3\u8dc3\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u52a8\u6001\u89c4\u5212\u5f80\u5f80\u96be\u4ee5\u89e3\u51b3\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u590d\u6742\u7684\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff08\u4f8b\u5982\u65c5\u884c\u5546\u95ee\u9898\uff09\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9009\u62e9\u4f7f\u7528\u5176\u4ed6\u65b9\u6cd5\uff0c\u4f8b\u5982\u542f\u53d1\u5f0f\u641c\u7d22\u3001\u9057\u4f20\u7b97\u6cd5\u3001\u5f3a\u5316\u5b66\u4e60\u7b49\uff0c\u4ece\u800c\u5728\u6709\u9650\u65f6\u95f4\u5185\u5f97\u5230\u53ef\u7528\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/","title":"14.3 \u00a0 \u52a8\u6001\u89c4\u5212\u89e3\u9898\u601d\u8def","text":"<p>\u4e0a\u4e24\u8282\u4ecb\u7ecd\u4e86\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u4e3b\u8981\u7279\u5f81\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u4e00\u8d77\u63a2\u7a76\u4e24\u4e2a\u66f4\u52a0\u5b9e\u7528\u7684\u95ee\u9898\u3002</p> <ol> <li>\u5982\u4f55\u5224\u65ad\u4e00\u4e2a\u95ee\u9898\u662f\u4e0d\u662f\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff1f</li> <li>\u6c42\u89e3\u52a8\u6001\u89c4\u5212\u95ee\u9898\u8be5\u4ece\u4f55\u5904\u5165\u624b\uff0c\u5b8c\u6574\u6b65\u9aa4\u662f\u4ec0\u4e48\uff1f</li> </ol>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1431","title":"14.3.1 \u00a0 \u95ee\u9898\u5224\u65ad","text":"<p>\u603b\u7684\u6765\u8bf4\uff0c\u5982\u679c\u4e00\u4e2a\u95ee\u9898\u5305\u542b\u91cd\u53e0\u5b50\u95ee\u9898\u3001\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5e76\u6ee1\u8db3\u65e0\u540e\u6548\u6027\uff0c\u90a3\u4e48\u5b83\u901a\u5e38\u9002\u5408\u7528\u52a8\u6001\u89c4\u5212\u6c42\u89e3\u3002\u7136\u800c\uff0c\u6211\u4eec\u5f88\u96be\u4ece\u95ee\u9898\u63cf\u8ff0\u4e2d\u76f4\u63a5\u63d0\u53d6\u51fa\u8fd9\u4e9b\u7279\u6027\u3002\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f1a\u653e\u5bbd\u6761\u4ef6\uff0c\u5148\u89c2\u5bdf\u95ee\u9898\u662f\u5426\u9002\u5408\u4f7f\u7528\u56de\u6eaf\uff08\u7a77\u4e3e\uff09\u89e3\u51b3\u3002</p> <p>\u9002\u5408\u7528\u56de\u6eaf\u89e3\u51b3\u7684\u95ee\u9898\u901a\u5e38\u6ee1\u8db3\u201c\u51b3\u7b56\u6811\u6a21\u578b\u201d\uff0c\u8fd9\u79cd\u95ee\u9898\u53ef\u4ee5\u4f7f\u7528\u6811\u5f62\u7ed3\u6784\u6765\u63cf\u8ff0\uff0c\u5176\u4e2d\u6bcf\u4e00\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u51b3\u7b56\uff0c\u6bcf\u4e00\u6761\u8def\u5f84\u4ee3\u8868\u4e00\u4e2a\u51b3\u7b56\u5e8f\u5217\u3002</p> <p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u5982\u679c\u95ee\u9898\u5305\u542b\u660e\u786e\u7684\u51b3\u7b56\u6982\u5ff5\uff0c\u5e76\u4e14\u89e3\u662f\u901a\u8fc7\u4e00\u7cfb\u5217\u51b3\u7b56\u4ea7\u751f\u7684\uff0c\u90a3\u4e48\u5b83\u5c31\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\uff0c\u901a\u5e38\u53ef\u4ee5\u4f7f\u7528\u56de\u6eaf\u6765\u89e3\u51b3\u3002</p> <p>\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u52a8\u6001\u89c4\u5212\u95ee\u9898\u8fd8\u6709\u4e00\u4e9b\u5224\u65ad\u7684\u201c\u52a0\u5206\u9879\u201d\u3002</p> <ul> <li>\u95ee\u9898\u5305\u542b\u6700\u5927\uff08\u5c0f\uff09\u6216\u6700\u591a\uff08\u5c11\uff09\u7b49\u6700\u4f18\u5316\u63cf\u8ff0\u3002</li> <li>\u95ee\u9898\u7684\u72b6\u6001\u80fd\u591f\u4f7f\u7528\u4e00\u4e2a\u5217\u8868\u3001\u591a\u7ef4\u77e9\u9635\u6216\u6811\u6765\u8868\u793a\uff0c\u5e76\u4e14\u4e00\u4e2a\u72b6\u6001\u4e0e\u5176\u5468\u56f4\u7684\u72b6\u6001\u5b58\u5728\u9012\u63a8\u5173\u7cfb\u3002</li> </ul> <p>\u76f8\u5e94\u5730\uff0c\u4e5f\u5b58\u5728\u4e00\u4e9b\u201c\u51cf\u5206\u9879\u201d\u3002</p> <ul> <li>\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u800c\u4e0d\u662f\u627e\u51fa\u6700\u4f18\u89e3\u3002</li> <li>\u95ee\u9898\u63cf\u8ff0\u4e2d\u6709\u660e\u663e\u7684\u6392\u5217\u7ec4\u5408\u7684\u7279\u5f81\uff0c\u9700\u8981\u8fd4\u56de\u5177\u4f53\u7684\u591a\u4e2a\u65b9\u6848\u3002</li> </ul> <p>\u5982\u679c\u4e00\u4e2a\u95ee\u9898\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\uff0c\u5e76\u5177\u6709\u8f83\u4e3a\u660e\u663e\u7684\u201c\u52a0\u5206\u9879\u201d\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5047\u8bbe\u5b83\u662f\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff0c\u5e76\u5728\u6c42\u89e3\u8fc7\u7a0b\u4e2d\u9a8c\u8bc1\u5b83\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1432","title":"14.3.2 \u00a0 \u95ee\u9898\u6c42\u89e3\u6b65\u9aa4","text":"<p>\u52a8\u6001\u89c4\u5212\u7684\u89e3\u9898\u6d41\u7a0b\u4f1a\u56e0\u95ee\u9898\u7684\u6027\u8d28\u548c\u96be\u5ea6\u800c\u6709\u6240\u4e0d\u540c\uff0c\u4f46\u901a\u5e38\u9075\u5faa\u4ee5\u4e0b\u6b65\u9aa4\uff1a\u63cf\u8ff0\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u5efa\u7acb \\(dp\\) \u8868\uff0c\u63a8\u5bfc\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u7b49\u3002</p> <p>\u4e3a\u4e86\u66f4\u5f62\u8c61\u5730\u5c55\u793a\u89e3\u9898\u6b65\u9aa4\uff0c\u6211\u4eec\u4f7f\u7528\u4e00\u4e2a\u7ecf\u5178\u95ee\u9898\u201c\u6700\u5c0f\u8def\u5f84\u548c\u201d\u6765\u4e3e\u4f8b\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a \\(n \\times m\\) \u7684\u4e8c\u7ef4\u7f51\u683c <code>grid</code> \uff0c\u7f51\u683c\u4e2d\u7684\u6bcf\u4e2a\u5355\u5143\u683c\u5305\u542b\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a\u8be5\u5355\u5143\u683c\u7684\u4ee3\u4ef7\u3002\u673a\u5668\u4eba\u4ee5\u5de6\u4e0a\u89d2\u5355\u5143\u683c\u4e3a\u8d77\u59cb\u70b9\uff0c\u6bcf\u6b21\u53ea\u80fd\u5411\u4e0b\u6216\u8005\u5411\u53f3\u79fb\u52a8\u4e00\u6b65\uff0c\u76f4\u81f3\u5230\u8fbe\u53f3\u4e0b\u89d2\u5355\u5143\u683c\u3002\u8bf7\u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230\u53f3\u4e0b\u89d2\u7684\u6700\u5c0f\u8def\u5f84\u548c\u3002</p> <p>\u56fe 14-10 \u5c55\u793a\u4e86\u4e00\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u7f51\u683c\u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e3a \\(13\\) \u3002</p> <p></p> <p> \u56fe 14-10 \u00a0 \u6700\u5c0f\u8def\u5f84\u548c\u793a\u4f8b\u6570\u636e </p> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u672c\u9898\u7684\u6bcf\u4e00\u8f6e\u7684\u51b3\u7b56\u5c31\u662f\u4ece\u5f53\u524d\u683c\u5b50\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u3002\u8bbe\u5f53\u524d\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u4e3a \\([i, j]\\) \uff0c\u5219\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u540e\uff0c\u7d22\u5f15\u53d8\u4e3a \\([i+1, j]\\) \u6216 \\([i, j+1]\\) \u3002\u56e0\u6b64\uff0c\u72b6\u6001\u5e94\u5305\u542b\u884c\u7d22\u5f15\u548c\u5217\u7d22\u5f15\u4e24\u4e2a\u53d8\u91cf\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002</p> <p>\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u4ece\u8d77\u59cb\u70b9 \\([0, 0]\\) \u8d70\u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\uff0c\u89e3\u8bb0\u4e3a \\(dp[i, j]\\) \u3002</p> <p>\u81f3\u6b64\uff0c\u6211\u4eec\u5c31\u5f97\u5230\u4e86\u56fe 14-11 \u6240\u793a\u7684\u4e8c\u7ef4 \\(dp\\) \u77e9\u9635\uff0c\u5176\u5c3a\u5bf8\u4e0e\u8f93\u5165\u7f51\u683c \\(grid\\) \u76f8\u540c\u3002</p> <p></p> <p> \u56fe 14-11 \u00a0 \u72b6\u6001\u5b9a\u4e49\u4e0e dp \u8868 </p> <p>Note</p> <p>\u52a8\u6001\u89c4\u5212\u548c\u56de\u6eaf\u8fc7\u7a0b\u53ef\u4ee5\u63cf\u8ff0\u4e3a\u4e00\u4e2a\u51b3\u7b56\u5e8f\u5217\uff0c\u800c\u72b6\u6001\u7531\u6240\u6709\u51b3\u7b56\u53d8\u91cf\u6784\u6210\u3002\u5b83\u5e94\u5f53\u5305\u542b\u63cf\u8ff0\u89e3\u9898\u8fdb\u5ea6\u7684\u6240\u6709\u53d8\u91cf\uff0c\u5176\u5305\u542b\u4e86\u8db3\u591f\u7684\u4fe1\u606f\uff0c\u80fd\u591f\u7528\u6765\u63a8\u5bfc\u51fa\u4e0b\u4e00\u4e2a\u72b6\u6001\u3002</p> <p>\u6bcf\u4e2a\u72b6\u6001\u90fd\u5bf9\u5e94\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u6211\u4eec\u4f1a\u5b9a\u4e49\u4e00\u4e2a \\(dp\\) \u8868\u6765\u5b58\u50a8\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u72b6\u6001\u7684\u6bcf\u4e2a\u72ec\u7acb\u53d8\u91cf\u90fd\u662f \\(dp\\) \u8868\u7684\u4e00\u4e2a\u7ef4\u5ea6\u3002\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\\(dp\\) \u8868\u662f\u72b6\u6001\u548c\u5b50\u95ee\u9898\u7684\u89e3\u4e4b\u95f4\u7684\u6620\u5c04\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u5bf9\u4e8e\u72b6\u6001 \\([i, j]\\) \uff0c\u5b83\u53ea\u80fd\u4ece\u4e0a\u8fb9\u683c\u5b50 \\([i-1, j]\\) \u548c\u5de6\u8fb9\u683c\u5b50 \\([i, j-1]\\) \u8f6c\u79fb\u800c\u6765\u3002\u56e0\u6b64\u6700\u4f18\u5b50\u7ed3\u6784\u4e3a\uff1a\u5230\u8fbe \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u7531 \\([i, j-1]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e0e \\([i-1, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e2d\u8f83\u5c0f\u7684\u90a3\u4e00\u4e2a\u51b3\u5b9a\u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u63a8\u51fa\u56fe 14-12 \u6240\u793a\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a</p> \\[ dp[i, j] = \\min(dp[i-1, j], dp[i, j-1]) + grid[i, j] \\] <p></p> <p> \u56fe 14-12 \u00a0 \u6700\u4f18\u5b50\u7ed3\u6784\u4e0e\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b </p> <p>Note</p> <p>\u6839\u636e\u5b9a\u4e49\u597d\u7684 \\(dp\\) \u8868\uff0c\u601d\u8003\u539f\u95ee\u9898\u548c\u5b50\u95ee\u9898\u7684\u5173\u7cfb\uff0c\u627e\u51fa\u901a\u8fc7\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6765\u6784\u9020\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u7684\u65b9\u6cd5\uff0c\u5373\u6700\u4f18\u5b50\u7ed3\u6784\u3002</p> <p>\u4e00\u65e6\u6211\u4eec\u627e\u5230\u4e86\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528\u5b83\u6765\u6784\u5efa\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002</p> <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5728\u672c\u9898\u4e2d\uff0c\u5904\u5728\u9996\u884c\u7684\u72b6\u6001\u53ea\u80fd\u4ece\u5176\u5de6\u8fb9\u7684\u72b6\u6001\u5f97\u6765\uff0c\u5904\u5728\u9996\u5217\u7684\u72b6\u6001\u53ea\u80fd\u4ece\u5176\u4e0a\u8fb9\u7684\u72b6\u6001\u5f97\u6765\uff0c\u56e0\u6b64\u9996\u884c \\(i = 0\\) \u548c\u9996\u5217 \\(j = 0\\) \u662f\u8fb9\u754c\u6761\u4ef6\u3002</p> <p>\u5982\u56fe 14-13 \u6240\u793a\uff0c\u7531\u4e8e\u6bcf\u4e2a\u683c\u5b50\u662f\u7531\u5176\u5de6\u65b9\u683c\u5b50\u548c\u4e0a\u65b9\u683c\u5b50\u8f6c\u79fb\u800c\u6765\uff0c\u56e0\u6b64\u6211\u4eec\u4f7f\u7528\u5faa\u73af\u6765\u904d\u5386\u77e9\u9635\uff0c\u5916\u5faa\u73af\u904d\u5386\u5404\u884c\uff0c\u5185\u5faa\u73af\u904d\u5386\u5404\u5217\u3002</p> <p></p> <p> \u56fe 14-13 \u00a0 \u8fb9\u754c\u6761\u4ef6\u4e0e\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f </p> <p>Note</p> <p>\u8fb9\u754c\u6761\u4ef6\u5728\u52a8\u6001\u89c4\u5212\u4e2d\u7528\u4e8e\u521d\u59cb\u5316 \\(dp\\) \u8868\uff0c\u5728\u641c\u7d22\u4e2d\u7528\u4e8e\u526a\u679d\u3002</p> <p>\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f\u7684\u6838\u5fc3\u662f\u8981\u4fdd\u8bc1\u5728\u8ba1\u7b97\u5f53\u524d\u95ee\u9898\u7684\u89e3\u65f6\uff0c\u6240\u6709\u5b83\u4f9d\u8d56\u7684\u66f4\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u90fd\u5df2\u7ecf\u88ab\u6b63\u786e\u5730\u8ba1\u7b97\u51fa\u6765\u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u4eec\u5df2\u7ecf\u53ef\u4ee5\u76f4\u63a5\u5199\u51fa\u52a8\u6001\u89c4\u5212\u4ee3\u7801\u3002\u7136\u800c\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u4ece\u9876\u81f3\u5e95\u7684\u601d\u60f3\uff0c\u56e0\u6b64\u6309\u7167\u201c\u66b4\u529b\u641c\u7d22 \\(\\rightarrow\\) \u8bb0\u5fc6\u5316\u641c\u7d22 \\(\\rightarrow\\) \u52a8\u6001\u89c4\u5212\u201d\u7684\u987a\u5e8f\u5b9e\u73b0\u66f4\u52a0\u7b26\u5408\u601d\u7ef4\u4e60\u60ef\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"<p>\u4ece\u72b6\u6001 \\([i, j]\\) \u5f00\u59cb\u641c\u7d22\uff0c\u4e0d\u65ad\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u72b6\u6001 \\([i-1, j]\\) \u548c \\([i, j-1]\\) \uff0c\u9012\u5f52\u51fd\u6570\u5305\u62ec\u4ee5\u4e0b\u8981\u7d20\u3002</p> <ul> <li>\u9012\u5f52\u53c2\u6570\uff1a\u72b6\u6001 \\([i, j]\\) \u3002</li> <li>\u8fd4\u56de\u503c\uff1a\u4ece \\([0, 0]\\) \u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c \\(dp[i, j]\\) \u3002</li> <li>\u7ec8\u6b62\u6761\u4ef6\uff1a\u5f53 \\(i = 0\\) \u4e14 \\(j = 0\\) \u65f6\uff0c\u8fd4\u56de\u4ee3\u4ef7 \\(grid[0, 0]\\) \u3002</li> <li>\u526a\u679d\uff1a\u5f53 \\(i &lt; 0\\) \u65f6\u6216 \\(j &lt; 0\\) \u65f6\u7d22\u5f15\u8d8a\u754c\uff0c\u6b64\u65f6\u8fd4\u56de\u4ee3\u4ef7 \\(+\\infty\\) \uff0c\u4ee3\u8868\u4e0d\u53ef\u884c\u3002</li> </ul> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dfs(grid: list[list[int]], i: int, j: int) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22\"\"\"\n    # \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    up = min_path_sum_dfs(grid, i - 1, j)\n    left = min_path_sum_dfs(grid, i, j - 1)\n    # \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return min(left, up) + grid[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(vector&lt;vector&lt;int&gt;&gt; &amp;grid, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(int[][] grid, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint MinPathSumDFS(int[][] grid, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = MinPathSumDFS(grid, i - 1, j);\n    int left = MinPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return Math.Min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc minPathSumDFS(grid [][]int, i, j int) int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    up := minPathSumDFS(grid, i-1, j)\n    left := minPathSumDFS(grid, i, j-1)\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return int(math.Min(float64(left), float64(up))) + grid[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc minPathSumDFS(grid: [[Int]], i: Int, j: Int) -&gt; Int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    let up = minPathSumDFS(grid: grid, i: i - 1, j: j)\n    let left = minPathSumDFS(grid: grid, i: i, j: j - 1)\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction minPathSumDFS(grid, i, j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction minPathSumDFS(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i === 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(List&lt;List&lt;int&gt;&gt; grid, int i, int j) {\n  // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n  if (i &lt; 0 || j &lt; 0) {\n    // \u5728 Dart \u4e2d\uff0cint \u7c7b\u578b\u662f\u56fa\u5b9a\u8303\u56f4\u7684\u6574\u6570\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u65e0\u7a77\u5927\u201d\u7684\u503c\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n  int up = minPathSumDFS(grid, i - 1, j);\n  int left = minPathSumDFS(grid, i, j - 1);\n  // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n  return min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfn min_path_sum_dfs(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    let up = min_path_sum_dfs(grid, i - 1, j);\n    let left = min_path_sum_dfs(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    std::cmp::min(left, up) + grid[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(int grid[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfun minPathSumDFS(\n    grid: Array&lt;Array&lt;Int&gt;&gt;,\n    i: Int,\n    j: Int\n): Int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    val up = minPathSumDFS(grid, i - 1, j)\n    val left = minPathSumDFS(grid, i, j - 1)\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return (min(left.toDouble(), up.toDouble()) + grid[i][j]).toInt()\n}\n</code></pre> min_path_sum.rb<pre><code>[class]{}-[func]{min_path_sum_dfs}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22\nfn minPathSumDFS(grid: anytype, i: i32, j: i32) i32 {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 and j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 or j &lt; 0) {\n        return std.math.maxInt(i32);\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    var up = minPathSumDFS(grid, i - 1, j);\n    var left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 14-14 \u7ed9\u51fa\u4e86\u4ee5 \\(dp[2, 1]\\) \u4e3a\u6839\u8282\u70b9\u7684\u9012\u5f52\u6811\uff0c\u5176\u4e2d\u5305\u542b\u4e00\u4e9b\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5176\u6570\u91cf\u4f1a\u968f\u7740\u7f51\u683c <code>grid</code> \u7684\u5c3a\u5bf8\u53d8\u5927\u800c\u6025\u5267\u589e\u591a\u3002</p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9020\u6210\u91cd\u53e0\u5b50\u95ee\u9898\u7684\u539f\u56e0\u4e3a\uff1a\u5b58\u5728\u591a\u6761\u8def\u5f84\u53ef\u4ee5\u4ece\u5de6\u4e0a\u89d2\u5230\u8fbe\u67d0\u4e00\u5355\u5143\u683c\u3002</p> <p></p> <p> \u56fe 14-14 \u00a0 \u66b4\u529b\u641c\u7d22\u9012\u5f52\u6811 </p> <p>\u6bcf\u4e2a\u72b6\u6001\u90fd\u6709\u5411\u4e0b\u548c\u5411\u53f3\u4e24\u79cd\u9009\u62e9\uff0c\u4ece\u5de6\u4e0a\u89d2\u8d70\u5230\u53f3\u4e0b\u89d2\u603b\u5171\u9700\u8981 \\(m + n - 2\\) \u6b65\uff0c\u6240\u4ee5\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^{m + n})\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u8fd9\u79cd\u8ba1\u7b97\u65b9\u5f0f\u672a\u8003\u8651\u4e34\u8fd1\u7f51\u683c\u8fb9\u754c\u7684\u60c5\u51b5\uff0c\u5f53\u5230\u8fbe\u7f51\u7edc\u8fb9\u754c\u65f6\u53ea\u5269\u4e0b\u4e00\u79cd\u9009\u62e9\uff0c\u56e0\u6b64\u5b9e\u9645\u7684\u8def\u5f84\u6570\u91cf\u4f1a\u5c11\u4e00\u4e9b\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"<p>\u6211\u4eec\u5f15\u5165\u4e00\u4e2a\u548c\u7f51\u683c <code>grid</code> \u76f8\u540c\u5c3a\u5bf8\u7684\u8bb0\u5fc6\u5217\u8868 <code>mem</code> \uff0c\u7528\u4e8e\u8bb0\u5f55\u5404\u4e2a\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5e76\u5c06\u91cd\u53e0\u5b50\u95ee\u9898\u8fdb\u884c\u526a\u679d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dfs_mem(\n    grid: list[list[int]], mem: list[list[int]], i: int, j: int\n) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n    # \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][j] != -1:\n        return mem[i][j]\n    # \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n    left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n    # \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(vector&lt;vector&lt;int&gt;&gt; &amp;grid, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint MinPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = MinPathSumDFSMem(grid, mem, i - 1, j);\n    int left = MinPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = Math.Min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc minPathSumDFSMem(grid, mem [][]int, i, j int) int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    up := minPathSumDFSMem(grid, mem, i-1, j)\n    left := minPathSumDFSMem(grid, mem, i, j-1)\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = int(math.Min(float64(left), float64(up))) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc minPathSumDFSMem(grid: [[Int]], mem: inout [[Int]], i: Int, j: Int) -&gt; Int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    let up = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i - 1, j: j)\n    let left = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i, j: j - 1)\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction minPathSumDFSMem(grid, mem, i, j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] !== -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction minPathSumDFSMem(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(List&lt;List&lt;int&gt;&gt; grid, List&lt;List&lt;int&gt;&gt; mem, int i, int j) {\n  // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n  if (i &lt; 0 || j &lt; 0) {\n    // \u5728 Dart \u4e2d\uff0cint \u7c7b\u578b\u662f\u56fa\u5b9a\u8303\u56f4\u7684\u6574\u6570\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u65e0\u7a77\u5927\u201d\u7684\u503c\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n  if (mem[i][j] != -1) {\n    return mem[i][j];\n  }\n  // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n  int up = minPathSumDFSMem(grid, mem, i - 1, j);\n  int left = minPathSumDFSMem(grid, mem, i, j - 1);\n  // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n  mem[i][j] = min(left, up) + grid[i][j];\n  return mem[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn min_path_sum_dfs_mem(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i as usize][j as usize] != -1 {\n        return mem[i as usize][j as usize];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    let up = min_path_sum_dfs_mem(grid, mem, i - 1, j);\n    let left = min_path_sum_dfs_mem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];\n    mem[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(int grid[MAX_SIZE][MAX_SIZE], int mem[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfun minPathSumDFSMem(\n    grid: Array&lt;Array&lt;Int&gt;&gt;,\n    mem: Array&lt;Array&lt;Int&gt;&gt;,\n    i: Int,\n    j: Int\n): Int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j]\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    val up = minPathSumDFSMem(grid, mem, i - 1, j)\n    val left = minPathSumDFSMem(grid, mem, i, j - 1)\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = (min(left.toDouble(), up.toDouble()) + grid[i][j]).toInt()\n    return mem[i][j]\n}\n</code></pre> min_path_sum.rb<pre><code>[class]{}-[func]{min_path_sum_dfs_mem}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn minPathSumDFSMem(grid: anytype, mem: anytype, i: i32, j: i32) i32 {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 and j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 or j &lt; 0) {\n        return std.math.maxInt(i32);\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] != -1) {\n        return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    var up = minPathSumDFSMem(grid, mem, i - 1, j);\n    var left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] = @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n    return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 14-15 \u6240\u793a\uff0c\u5728\u5f15\u5165\u8bb0\u5fc6\u5316\u540e\uff0c\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\u53ea\u9700\u8ba1\u7b97\u4e00\u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u72b6\u6001\u603b\u6570\uff0c\u5373\u7f51\u683c\u5c3a\u5bf8 \\(O(nm)\\) \u3002</p> <p></p> <p> \u56fe 14-15 \u00a0 \u8bb0\u5fc6\u5316\u641c\u7d22\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"<p>\u57fa\u4e8e\u8fed\u4ee3\u5b9e\u73b0\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dp(grid: list[list[int]]) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n, m = len(grid), len(grid[0])\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * m for _ in range(n)]\n    dp[0][0] = grid[0][0]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in range(1, m):\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for i in range(1, n):\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in range(1, n):\n        for j in range(1, m):\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n    return dp[n - 1][m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n, vector&lt;int&gt;(m));\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n][m];\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint MinPathSumDP(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n, m];\n    dp[0, 0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0, j] = dp[0, j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i, 0] = dp[i - 1, 0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i, j] = Math.Min(dp[i, j - 1], dp[i - 1, j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1, m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDP(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        dp[i] = make([]int, m)\n    }\n    dp[0][0] = grid[0][0]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j := 1; j &lt; m; j++ {\n        dp[0][j] = dp[0][j-1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for i := 1; i &lt; n; i++ {\n        dp[i][0] = dp[i-1][0] + grid[i][0]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i := 1; i &lt; n; i++ {\n        for j := 1; j &lt; m; j++ {\n            dp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(dp[i-1][j]))) + grid[i][j]\n        }\n    }\n    return dp[n-1][m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDP(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: m), count: n)\n    dp[0][0] = grid[0][0]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in 1 ..&lt; m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for i in 1 ..&lt; n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in 1 ..&lt; n {\n        for j in 1 ..&lt; m {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDP(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt; n; i++) {\n        for (let j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDP(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt; n; i++) {\n        for (let j: number = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n, (i) =&gt; List.filled(m, 0));\n  dp[0][0] = grid[0][0];\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n  for (int j = 1; j &lt; m; j++) {\n    dp[0][j] = dp[0][j - 1] + grid[0][j];\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n  for (int i = 1; i &lt; n; i++) {\n    dp[i][0] = dp[i - 1][0] + grid[i][0];\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n  for (int i = 1; i &lt; n; i++) {\n    for (int j = 1; j &lt; m; j++) {\n      dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n    }\n  }\n  return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfn min_path_sum_dp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; m]; n];\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in 1..m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for i in 1..n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in 1..n {\n        for j in 1..m {\n            dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc(n * sizeof(int *));\n    for (int i = 0; i &lt; n; i++) {\n        dp[i] = calloc(m, sizeof(int));\n    }\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = myMin(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    int res = dp[n - 1][m - 1];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt; n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfun minPathSumDP(grid: Array&lt;Array&lt;Int&gt;&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n) { IntArray(m) }\n    dp[0][0] = grid[0][0]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (j in 1..&lt;m) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (i in 1..&lt;n) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (i in 1..&lt;n) {\n        for (j in 1..&lt;m) {\n            dp[i][j] =\n                (min(dp[i][j - 1].toDouble(), dp[i - 1][j].toDouble()) + grid[i][j]).toInt()\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>[class]{}-[func]{min_path_sum_dp}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212\nfn minPathSumDP(comptime grid: anytype) i32 {\n    comptime var n = grid.len;\n    comptime var m = grid[0].len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][m]i32{[_]i32{0} ** m} ** n;\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (1..m) |j| {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (1..n) |i| {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (1..n) |i| {\n        for (1..m) |j| {\n            dp[i][j] = @min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 14-16 \u5c55\u793a\u4e86\u6700\u5c0f\u8def\u5f84\u548c\u7684\u72b6\u6001\u8f6c\u79fb\u8fc7\u7a0b\uff0c\u5176\u904d\u5386\u4e86\u6574\u4e2a\u7f51\u683c\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nm)\\) \u3002</p> <p>\u6570\u7ec4 <code>dp</code> \u5927\u5c0f\u4e3a \\(n \\times m\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nm)\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-16 \u00a0 \u6700\u5c0f\u8def\u5f84\u548c\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#4","title":"4. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e\u6bcf\u4e2a\u683c\u5b50\u53ea\u4e0e\u5176\u5de6\u8fb9\u548c\u4e0a\u8fb9\u7684\u683c\u5b50\u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u53ea\u7528\u4e00\u4e2a\u5355\u884c\u6570\u7ec4\u6765\u5b9e\u73b0 \\(dp\\) \u8868\u3002</p> <p>\u8bf7\u6ce8\u610f\uff0c\u56e0\u4e3a\u6570\u7ec4 <code>dp</code> \u53ea\u80fd\u8868\u793a\u4e00\u884c\u7684\u72b6\u6001\uff0c\u6240\u4ee5\u6211\u4eec\u65e0\u6cd5\u63d0\u524d\u521d\u59cb\u5316\u9996\u5217\u72b6\u6001\uff0c\u800c\u662f\u5728\u904d\u5386\u6bcf\u884c\u65f6\u66f4\u65b0\u5b83\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dp_comp(grid: list[list[int]]) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n, m = len(grid), len(grid[0])\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * m\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for j in range(1, m):\n        dp[j] = dp[j - 1] + grid[0][j]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in range(1, n):\n        # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in range(1, m):\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n    return dp[m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(m);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[m];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint MinPathSumDPComp(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[m];\n    dp[0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.Min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDPComp(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, m)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for j := 1; j &lt; m; j++ {\n        dp[j] = dp[j-1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i := 1; i &lt; n; i++ {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j := 1; j &lt; m; j++ {\n            dp[j] = int(math.Min(float64(dp[j-1]), float64(dp[j]))) + grid[i][j]\n        }\n    }\n    return dp[m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDPComp(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: m)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for j in 1 ..&lt; m {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in 1 ..&lt; n {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in 1 ..&lt; m {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDPComp(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = new Array(m);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (let i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDPComp(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = new Array(m);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (let i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(m, 0);\n  dp[0] = grid[0][0];\n  for (int j = 1; j &lt; m; j++) {\n    dp[j] = dp[j - 1] + grid[0][j];\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n  for (int i = 1; i &lt; n; i++) {\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    dp[0] = dp[0] + grid[i][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n    for (int j = 1; j &lt; m; j++) {\n      dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n    }\n  }\n  return dp[m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn min_path_sum_dp_comp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; m];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for j in 1..m {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in 1..n {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in 1..m {\n            dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    dp[m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(m, sizeof(int));\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = myMin(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    int res = dp[m - 1];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun minPathSumDPComp(grid: Array&lt;Array&lt;Int&gt;&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(m)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for (j in 1..&lt;m) {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (i in 1..&lt;n) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (j in 1..&lt;m) {\n            dp[j] = (min(dp[j - 1].toDouble(), dp[j].toDouble()) + grid[i][j]).toInt()\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>[class]{}-[func]{min_path_sum_dp_comp}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn minPathSumDPComp(comptime grid: anytype) i32 {\n    comptime var n = grid.len;\n    comptime var m = grid[0].len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** m;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (1..m) |j| {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (1..n) |i| {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        for (1..m) |j| {\n            dp[j] = @min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/","title":"14.6 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898","text":"<p>\u7f16\u8f91\u8ddd\u79bb\uff0c\u4e5f\u79f0 Levenshtein \u8ddd\u79bb\uff0c\u6307\u4e24\u4e2a\u5b57\u7b26\u4e32\u4e4b\u95f4\u4e92\u76f8\u8f6c\u6362\u7684\u6700\u5c11\u4fee\u6539\u6b21\u6570\uff0c\u901a\u5e38\u7528\u4e8e\u5728\u4fe1\u606f\u68c0\u7d22\u548c\u81ea\u7136\u8bed\u8a00\u5904\u7406\u4e2d\u5ea6\u91cf\u4e24\u4e2a\u5e8f\u5217\u7684\u76f8\u4f3c\u5ea6\u3002</p> <p>Question</p> <p>\u8f93\u5165\u4e24\u4e2a\u5b57\u7b26\u4e32 \\(s\\) \u548c \\(t\\) \uff0c\u8fd4\u56de\u5c06 \\(s\\) \u8f6c\u6362\u4e3a \\(t\\) \u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002</p> <p>\u4f60\u53ef\u4ee5\u5728\u4e00\u4e2a\u5b57\u7b26\u4e32\u4e2d\u8fdb\u884c\u4e09\u79cd\u7f16\u8f91\u64cd\u4f5c\uff1a\u63d2\u5165\u4e00\u4e2a\u5b57\u7b26\u3001\u5220\u9664\u4e00\u4e2a\u5b57\u7b26\u3001\u5c06\u5b57\u7b26\u66ff\u6362\u4e3a\u4efb\u610f\u4e00\u4e2a\u5b57\u7b26\u3002</p> <p>\u5982\u56fe 14-27 \u6240\u793a\uff0c\u5c06 <code>kitten</code> \u8f6c\u6362\u4e3a <code>sitting</code> \u9700\u8981\u7f16\u8f91 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u6362\u64cd\u4f5c\u4e0e 1 \u6b21\u6dfb\u52a0\u64cd\u4f5c\uff1b\u5c06 <code>hello</code> \u8f6c\u6362\u4e3a <code>algo</code> \u9700\u8981 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u6362\u64cd\u4f5c\u548c 1 \u6b21\u5220\u9664\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 14-27 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u53ef\u4ee5\u5f88\u81ea\u7136\u5730\u7528\u51b3\u7b56\u6811\u6a21\u578b\u6765\u89e3\u91ca\u3002\u5b57\u7b26\u4e32\u5bf9\u5e94\u6811\u8282\u70b9\uff0c\u4e00\u8f6e\u51b3\u7b56\uff08\u4e00\u6b21\u7f16\u8f91\u64cd\u4f5c\uff09\u5bf9\u5e94\u6811\u7684\u4e00\u6761\u8fb9\u3002</p> <p>\u5982\u56fe 14-28 \u6240\u793a\uff0c\u5728\u4e0d\u9650\u5236\u64cd\u4f5c\u7684\u60c5\u51b5\u4e0b\uff0c\u6bcf\u4e2a\u8282\u70b9\u90fd\u53ef\u4ee5\u6d3e\u751f\u51fa\u8bb8\u591a\u6761\u8fb9\uff0c\u6bcf\u6761\u8fb9\u5bf9\u5e94\u4e00\u79cd\u64cd\u4f5c\uff0c\u8fd9\u610f\u5473\u7740\u4ece <code>hello</code> \u8f6c\u6362\u5230 <code>algo</code> \u6709\u8bb8\u591a\u79cd\u53ef\u80fd\u7684\u8def\u5f84\u3002</p> <p>\u4ece\u51b3\u7b56\u6811\u7684\u89d2\u5ea6\u770b\uff0c\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u8282\u70b9 <code>hello</code> \u548c\u8282\u70b9 <code>algo</code> \u4e4b\u95f4\u7684\u6700\u77ed\u8def\u5f84\u3002</p> <p></p> <p> \u56fe 14-28 \u00a0 \u57fa\u4e8e\u51b3\u7b56\u6811\u6a21\u578b\u8868\u793a\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898 </p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u6bcf\u4e00\u8f6e\u7684\u51b3\u7b56\u662f\u5bf9\u5b57\u7b26\u4e32 \\(s\\) \u8fdb\u884c\u4e00\u6b21\u7f16\u8f91\u64cd\u4f5c\u3002</p> <p>\u6211\u4eec\u5e0c\u671b\u5728\u7f16\u8f91\u64cd\u4f5c\u7684\u8fc7\u7a0b\u4e2d\uff0c\u95ee\u9898\u7684\u89c4\u6a21\u9010\u6e10\u7f29\u5c0f\uff0c\u8fd9\u6837\u624d\u80fd\u6784\u5efa\u5b50\u95ee\u9898\u3002\u8bbe\u5b57\u7b26\u4e32 \\(s\\) \u548c \\(t\\) \u7684\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) \u548c \\(m\\) \uff0c\u6211\u4eec\u5148\u8003\u8651\u4e24\u5b57\u7b26\u4e32\u5c3e\u90e8\u7684\u5b57\u7b26 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u3002</p> <ul> <li>\u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u76f8\u540c\uff0c\u6211\u4eec\u53ef\u4ee5\u8df3\u8fc7\u5b83\u4eec\uff0c\u76f4\u63a5\u8003\u8651 \\(s[n-2]\\) \u548c \\(t[m-2]\\) \u3002</li> <li>\u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u4e0d\u540c\uff0c\u6211\u4eec\u9700\u8981\u5bf9 \\(s\\) \u8fdb\u884c\u4e00\u6b21\u7f16\u8f91\uff08\u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\uff09\uff0c\u4f7f\u5f97\u4e24\u5b57\u7b26\u4e32\u5c3e\u90e8\u7684\u5b57\u7b26\u76f8\u540c\uff0c\u4ece\u800c\u53ef\u4ee5\u8df3\u8fc7\u5b83\u4eec\uff0c\u8003\u8651\u89c4\u6a21\u66f4\u5c0f\u7684\u95ee\u9898\u3002</li> </ul> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u5728\u5b57\u7b26\u4e32 \\(s\\) \u4e2d\u8fdb\u884c\u7684\u6bcf\u4e00\u8f6e\u51b3\u7b56\uff08\u7f16\u8f91\u64cd\u4f5c\uff09\uff0c\u90fd\u4f1a\u4f7f\u5f97 \\(s\\) \u548c \\(t\\) \u4e2d\u5269\u4f59\u7684\u5f85\u5339\u914d\u5b57\u7b26\u53d1\u751f\u53d8\u5316\u3002\u56e0\u6b64\uff0c\u72b6\u6001\u4e3a\u5f53\u524d\u5728 \\(s\\) \u548c \\(t\\) \u4e2d\u8003\u8651\u7684\u7b2c \\(i\\) \u548c\u7b2c \\(j\\) \u4e2a\u5b57\u7b26\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002</p> <p>\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\uff1a\u5c06 \\(s\\) \u7684\u524d \\(i\\) \u4e2a\u5b57\u7b26\u66f4\u6539\u4e3a \\(t\\) \u7684\u524d \\(j\\) \u4e2a\u5b57\u7b26\u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002</p> <p>\u81f3\u6b64\uff0c\u5f97\u5230\u4e00\u4e2a\u5c3a\u5bf8\u4e3a \\((i+1) \\times (j+1)\\) \u7684\u4e8c\u7ef4 \\(dp\\) \u8868\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u8003\u8651\u5b50\u95ee\u9898 \\(dp[i, j]\\) \uff0c\u5176\u5bf9\u5e94\u7684\u4e24\u4e2a\u5b57\u7b26\u4e32\u7684\u5c3e\u90e8\u5b57\u7b26\u4e3a \\(s[i-1]\\) \u548c \\(t[j-1]\\) \uff0c\u53ef\u6839\u636e\u4e0d\u540c\u7f16\u8f91\u64cd\u4f5c\u5206\u4e3a\u56fe 14-29 \u6240\u793a\u7684\u4e09\u79cd\u60c5\u51b5\u3002</p> <ol> <li>\u5728 \\(s[i-1]\\) \u4e4b\u540e\u6dfb\u52a0 \\(t[j-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i, j-1]\\) \u3002</li> <li>\u5220\u9664 \\(s[i-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i-1, j]\\) \u3002</li> <li>\u5c06 \\(s[i-1]\\) \u66ff\u6362\u4e3a \\(t[j-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i-1, j-1]\\) \u3002</li> </ol> <p></p> <p> \u56fe 14-29 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u72b6\u6001\u8f6c\u79fb </p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u5f97\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\\(dp[i, j]\\) \u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(dp[i, j-1]\\)\u3001\\(dp[i-1, j]\\)\u3001\\(dp[i-1, j-1]\\) \u4e09\u8005\u4e2d\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\uff0c\u518d\u52a0\u4e0a\u672c\u6b21\u7684\u7f16\u8f91\u6b65\u6570 \\(1\\) \u3002\u5bf9\u5e94\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ dp[i, j] = \\min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1 \\] <p>\u8bf7\u6ce8\u610f\uff0c\u5f53 \\(s[i-1]\\) \u548c \\(t[j-1]\\) \u76f8\u540c\u65f6\uff0c\u65e0\u987b\u7f16\u8f91\u5f53\u524d\u5b57\u7b26\uff0c\u8fd9\u79cd\u60c5\u51b5\u4e0b\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ dp[i, j] = dp[i-1, j-1] \\] <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5f53\u4e24\u5b57\u7b26\u4e32\u90fd\u4e3a\u7a7a\u65f6\uff0c\u7f16\u8f91\u6b65\u6570\u4e3a \\(0\\) \uff0c\u5373 \\(dp[0, 0] = 0\\) \u3002\u5f53 \\(s\\) \u4e3a\u7a7a\u4f46 \\(t\\) \u4e0d\u4e3a\u7a7a\u65f6\uff0c\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(t\\) \u7684\u957f\u5ea6\uff0c\u5373\u9996\u884c \\(dp[0, j] = j\\) \u3002\u5f53 \\(s\\) \u4e0d\u4e3a\u7a7a\u4f46 \\(t\\) \u4e3a\u7a7a\u65f6\uff0c\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(s\\) \u7684\u957f\u5ea6\uff0c\u5373\u9996\u5217 \\(dp[i, 0] = i\\) \u3002</p> <p>\u89c2\u5bdf\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u89e3 \\(dp[i, j]\\) \u4f9d\u8d56\u5de6\u65b9\u3001\u4e0a\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u89e3\uff0c\u56e0\u6b64\u901a\u8fc7\u4e24\u5c42\u5faa\u73af\u6b63\u5e8f\u904d\u5386\u6574\u4e2a \\(dp\\) \u8868\u5373\u53ef\u3002</p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig edit_distance.py<pre><code>def edit_distance_dp(s: str, t: str) -&gt; int:\n    \"\"\"\u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n, m = len(s), len(t)\n    dp = [[0] * (m + 1) for _ in range(n + 1)]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i in range(1, n + 1):\n        dp[i][0] = i\n    for j in range(1, m + 1):\n        dp[0][j] = j\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in range(1, n + 1):\n        for j in range(1, m + 1):\n            if s[i - 1] == t[j - 1]:\n                # \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1]\n            else:\n                # \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1\n    return dp[n][m]\n</code></pre> edit_distance.cpp<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(m + 1, 0));\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.java<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[][] dp = new int[n + 1][m + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = Math.min(Math.min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.cs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint EditDistanceDP(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[,] dp = new int[n + 1, m + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i, 0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0, j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i, j] = dp[i - 1, j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i, j] = Math.Min(Math.Min(dp[i, j - 1], dp[i - 1, j]), dp[i - 1, j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n, m];\n}\n</code></pre> edit_distance.go<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDP(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, m+1)\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i := 1; i &lt;= n; i++ {\n        dp[i][0] = i\n    }\n    for j := 1; j &lt;= m; j++ {\n        dp[0][j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= m; j++ {\n            if s[i-1] == t[j-1] {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i-1][j-1]\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = MinInt(MinInt(dp[i][j-1], dp[i-1][j]), dp[i-1][j-1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.swift<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDP(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: Array(repeating: 0, count: m + 1), count: n + 1)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i in 1 ... n {\n        dp[i][0] = i\n    }\n    for j in 1 ... m {\n        dp[0][j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in 1 ... n {\n        for j in 1 ... m {\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.js<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDP(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(m + 1).fill(0));\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.ts<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDP(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: m + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.dart<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (_) =&gt; List.filled(m + 1, 0));\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n  for (int i = 1; i &lt;= n; i++) {\n    dp[i][0] = i;\n  }\n  for (int j = 1; j &lt;= m; j++) {\n    dp[0][j] = j;\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n  for (int i = 1; i &lt;= n; i++) {\n    for (int j = 1; j &lt;= m; j++) {\n      if (s[i - 1] == t[j - 1]) {\n        // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n        dp[i][j] = dp[i - 1][j - 1];\n      } else {\n        // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n        dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n      }\n    }\n  }\n  return dp[n][m];\n}\n</code></pre> edit_distance.rs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfn edit_distance_dp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![vec![0; m + 1]; n + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i in 1..=n {\n        dp[i][0] = i as i32;\n    }\n    for j in 1..m {\n        dp[0][j] = j as i32;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in 1..=n {\n        for j in 1..=m {\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] =\n                    std::cmp::min(std::cmp::min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    dp[n][m]\n}\n</code></pre> edit_distance.c<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(char *s, char *t, int n, int m) {\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(m + 1, sizeof(int));\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = myMin(myMin(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    int res = dp[n][m];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfun editDistanceDP(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = Array(n + 1) { IntArray(m + 1) }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (i in 1..n) {\n        dp[i][0] = i\n    }\n    for (j in 1..m) {\n        dp[0][j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (i in 1..n) {\n        for (j in 1..m) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] =\n                    (min(\n                        min(dp[i][j - 1].toDouble(), dp[i - 1][j].toDouble()),\n                        dp[i - 1][j - 1].toDouble()\n                    ) + 1).toInt()\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.rb<pre><code>[class]{}-[func]{edit_distance_dp}\n</code></pre> edit_distance.zig<pre><code>// \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212\nfn editDistanceDP(comptime s: []const u8, comptime t: []const u8) i32 {\n    comptime var n = s.len;\n    comptime var m = t.len;\n    var dp = [_][m + 1]i32{[_]i32{0} ** (m + 1)} ** (n + 1);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (1..n + 1) |i| {\n        dp[i][0] = @intCast(i);\n    }\n    for (1..m + 1) |j| {\n        dp[0][j] = @intCast(j);\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (1..n + 1) |i| {\n        for (1..m + 1) |j| {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = @min(@min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 14-30 \u6240\u793a\uff0c\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u7684\u72b6\u6001\u8f6c\u79fb\u8fc7\u7a0b\u4e0e\u80cc\u5305\u95ee\u9898\u975e\u5e38\u7c7b\u4f3c\uff0c\u90fd\u53ef\u4ee5\u770b\u4f5c\u586b\u5199\u4e00\u4e2a\u4e8c\u7ef4\u7f51\u683c\u7684\u8fc7\u7a0b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-30 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#3","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e \\(dp[i,j]\\) \u662f\u7531\u4e0a\u65b9 \\(dp[i-1, j]\\)\u3001\u5de6\u65b9 \\(dp[i, j-1]\\)\u3001\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \u8f6c\u79fb\u800c\u6765\u7684\uff0c\u800c\u6b63\u5e8f\u904d\u5386\u4f1a\u4e22\u5931\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \uff0c\u5012\u5e8f\u904d\u5386\u65e0\u6cd5\u63d0\u524d\u6784\u5efa \\(dp[i, j-1]\\) \uff0c\u56e0\u6b64\u4e24\u79cd\u904d\u5386\u987a\u5e8f\u90fd\u4e0d\u53ef\u53d6\u3002</p> <p>\u4e3a\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u53d8\u91cf <code>leftup</code> \u6765\u6682\u5b58\u5de6\u4e0a\u65b9\u7684\u89e3 \\(dp[i-1, j-1]\\) \uff0c\u4ece\u800c\u53ea\u9700\u8003\u8651\u5de6\u65b9\u548c\u4e0a\u65b9\u7684\u89e3\u3002\u6b64\u65f6\u7684\u60c5\u51b5\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u76f8\u540c\uff0c\u53ef\u4f7f\u7528\u6b63\u5e8f\u904d\u5386\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig edit_distance.py<pre><code>def edit_distance_dp_comp(s: str, t: str) -&gt; int:\n    \"\"\"\u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n, m = len(s), len(t)\n    dp = [0] * (m + 1)\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in range(1, m + 1):\n        dp[j] = j\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in range(1, n + 1):\n        # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        leftup = dp[0]  # \u6682\u5b58 dp[i-1, j-1]\n        dp[0] += 1\n        # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in range(1, m + 1):\n            temp = dp[j]\n            if s[i - 1] == t[j - 1]:\n                # \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup\n            else:\n                # \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = min(dp[j - 1], dp[j], leftup) + 1\n            leftup = temp  # \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n    return dp[m]\n</code></pre> edit_distance.cpp<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;int&gt; dp(m + 1, 0);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.java<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[] dp = new int[m + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = Math.min(Math.min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.cs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint EditDistanceDPComp(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[] dp = new int[m + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = Math.Min(Math.Min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.go<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDPComp(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([]int, m+1)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j := 1; j &lt;= m; j++ {\n        dp[j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i := 1; i &lt;= n; i++ {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        leftUp := dp[0] // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j := 1; j &lt;= m; j++ {\n            temp := dp[j]\n            if s[i-1] == t[j-1] {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftUp\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = MinInt(MinInt(dp[j-1], dp[j]), leftUp) + 1\n            }\n            leftUp = temp // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.swift<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDPComp(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: 0, count: m + 1)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in 1 ... m {\n        dp[j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in 1 ... n {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        var leftup = dp[0] // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in 1 ... m {\n            let temp = dp[j]\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n            }\n            leftup = temp // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.js<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDPComp(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (let i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        let leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.ts<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDPComp(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (let i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        let leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.dart<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;int&gt; dp = List.filled(m + 1, 0);\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n  for (int j = 1; j &lt;= m; j++) {\n    dp[j] = j;\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n  for (int i = 1; i &lt;= n; i++) {\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n    dp[0] = i;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n    for (int j = 1; j &lt;= m; j++) {\n      int temp = dp[j];\n      if (s[i - 1] == t[j - 1]) {\n        // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n        dp[j] = leftup;\n      } else {\n        // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n        dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n      }\n      leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n    }\n  }\n  return dp[m];\n}\n</code></pre> edit_distance.rs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn edit_distance_dp_comp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![0; m + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in 1..m {\n        dp[j] = j as i32;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in 1..=n {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        let mut leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i as i32;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in 1..=m {\n            let temp = dp[j];\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = std::cmp::min(std::cmp::min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    dp[m]\n}\n</code></pre> edit_distance.c<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(char *s, char *t, int n, int m) {\n    int *dp = calloc(m + 1, sizeof(int));\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = myMin(myMin(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    int res = dp[m];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun editDistanceDPComp(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = IntArray(m + 1)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (j in 1..m) {\n        dp[j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (i in 1..n) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        var leftup = dp[0] // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (j in 1..m) {\n            val temp = dp[j]\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = (min(min(dp[j - 1].toDouble(), dp[j].toDouble()), leftup.toDouble()) + 1).toInt()\n            }\n            leftup = temp // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.rb<pre><code>[class]{}-[func]{edit_distance_dp_comp}\n</code></pre> edit_distance.zig<pre><code>// \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn editDistanceDPComp(comptime s: []const u8, comptime t: []const u8) i32 {\n    comptime var n = s.len;\n    comptime var m = t.len;\n    var dp = [_]i32{0} ** (m + 1);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (1..m + 1) |j| {\n        dp[j] = @intCast(j);\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (1..n + 1) |i| {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        var leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = @intCast(i);\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (1..m + 1) |j| {\n            var temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = @min(@min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/","title":"14.1 \u00a0 \u521d\u63a2\u52a8\u6001\u89c4\u5212","text":"<p>\u300c\u52a8\u6001\u89c4\u5212 dynamic programming\u300d\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u7b97\u6cd5\u8303\u5f0f\uff0c\u5b83\u5c06\u4e00\u4e2a\u95ee\u9898\u5206\u89e3\u4e3a\u4e00\u7cfb\u5217\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u5e76\u901a\u8fc7\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\u6765\u907f\u514d\u91cd\u590d\u8ba1\u7b97\uff0c\u4ece\u800c\u5927\u5e45\u63d0\u5347\u65f6\u95f4\u6548\u7387\u3002</p> <p>\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u4ece\u4e00\u4e2a\u7ecf\u5178\u4f8b\u9898\u5165\u624b\uff0c\u5148\u7ed9\u51fa\u5b83\u7684\u66b4\u529b\u56de\u6eaf\u89e3\u6cd5\uff0c\u89c2\u5bdf\u5176\u4e2d\u5305\u542b\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u518d\u9010\u6b65\u5bfc\u51fa\u66f4\u9ad8\u6548\u7684\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u3002</p> <p>\u722c\u697c\u68af</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f</p> <p>\u5982\u56fe 14-1 \u6240\u793a\uff0c\u5bf9\u4e8e\u4e00\u4e2a \\(3\\) \u9636\u697c\u68af\uff0c\u5171\u6709 \\(3\\) \u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\u3002</p> <p></p> <p> \u56fe 14-1 \u00a0 \u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf </p> <p>\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u65b9\u6848\u6570\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u56de\u6eaf\u6765\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u6027\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5c06\u722c\u697c\u68af\u60f3\u8c61\u4e3a\u4e00\u4e2a\u591a\u8f6e\u9009\u62e9\u7684\u8fc7\u7a0b\uff1a\u4ece\u5730\u9762\u51fa\u53d1\uff0c\u6bcf\u8f6e\u9009\u62e9\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u6bcf\u5f53\u5230\u8fbe\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u65b9\u6848\u6570\u91cf\u52a0 \\(1\\) \uff0c\u5f53\u8d8a\u8fc7\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u5176\u526a\u679d\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_backtrack.py<pre><code>def backtrack(choices: list[int], state: int, n: int, res: list[int]) -&gt; int:\n    \"\"\"\u56de\u6eaf\"\"\"\n    # \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if state == n:\n        res[0] += 1\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices:\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if state + choice &gt; n:\n            continue\n        # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res)\n        # \u56de\u9000\n\ndef climbing_stairs_backtrack(n: int) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\uff1a\u56de\u6eaf\"\"\"\n    choices = [1, 2]  # \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    state = 0  # \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    res = [0]  # \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res)\n    return res[0]\n</code></pre> climbing_stairs_backtrack.cpp<pre><code>/* \u56de\u6eaf */\nvoid backtrack(vector&lt;int&gt; &amp;choices, int state, int n, vector&lt;int&gt; &amp;res) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n)\n        res[0]++;\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (auto &amp;choice : choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n)\n            continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n    vector&lt;int&gt; choices = {1, 2}; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    int state = 0;                // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    vector&lt;int&gt; res = {0};        // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.java<pre><code>/* \u56de\u6eaf */\nvoid backtrack(List&lt;Integer&gt; choices, int state, int n, List&lt;Integer&gt; res) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n)\n        res.set(0, res.get(0) + 1);\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (Integer choice : choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n)\n            continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n    List&lt;Integer&gt; choices = Arrays.asList(1, 2); // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n    res.add(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.cs<pre><code>/* \u56de\u6eaf */\nvoid Backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n)\n        res[0]++;\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    foreach (int choice in choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n)\n            continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        Backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint ClimbingStairsBacktrack(int n) {\n    List&lt;int&gt; choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    List&lt;int&gt; res = [0]; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    Backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.go<pre><code>/* \u56de\u6eaf */\nfunc backtrack(choices []int, state, n int, res []int) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if state == n {\n        res[0] = res[0] + 1\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for _, choice := range choices {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if state+choice &gt; n {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state+choice, n, res)\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n int) int {\n    // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    choices := []int{1, 2}\n    // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    state := 0\n    res := make([]int, 1)\n    // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    res[0] = 0\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.swift<pre><code>/* \u56de\u6eaf */\nfunc backtrack(choices: [Int], state: Int, n: Int, res: inout [Int]) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if state == n {\n        res[0] += 1\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if state + choice &gt; n {\n            continue\n        }\n        backtrack(choices: choices, state: state + choice, n: n, res: &amp;res)\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n: Int) -&gt; Int {\n    let choices = [1, 2] // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    let state = 0 // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    var res: [Int] = []\n    res.append(0) // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices: choices, state: state, n: n, res: &amp;res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.js<pre><code>/* \u56de\u6eaf */\nfunction backtrack(choices, state, n, res) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state === n) res.set(0, res.get(0) + 1);\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n) continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n) {\n    const choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    const state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    const res = new Map();\n    res.set(0, 0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.ts<pre><code>/* \u56de\u6eaf */\nfunction backtrack(\n    choices: number[],\n    state: number,\n    n: number,\n    res: Map&lt;0, any&gt;\n): void {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state === n) res.set(0, res.get(0) + 1);\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n) continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n: number): number {\n    const choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    const state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    const res = new Map();\n    res.set(0, 0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.dart<pre><code>/* \u56de\u6eaf */\nvoid backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n  // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n  if (state == n) {\n    res[0]++;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  for (int choice in choices) {\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n    if (state + choice &gt; n) continue;\n    // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n    backtrack(choices, state + choice, n, res);\n    // \u56de\u9000\n  }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n  List&lt;int&gt; choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n  int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n  List&lt;int&gt; res = [];\n  res.add(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n  backtrack(choices, state, n, res);\n  return res[0];\n}\n</code></pre> climbing_stairs_backtrack.rs<pre><code>/* \u56de\u6eaf */\nfn backtrack(choices: &amp;[i32], state: i32, n: i32, res: &amp;mut [i32]) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if state == n {\n        res[0] = res[0] + 1;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for &amp;choice in choices {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if state + choice &gt; n {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfn climbing_stairs_backtrack(n: usize) -&gt; i32 {\n    let choices = vec![1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    let state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    let mut res = Vec::new();\n    res.push(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(&amp;choices, state, n as i32, &amp;mut res);\n    res[0]\n}\n</code></pre> climbing_stairs_backtrack.c<pre><code>/* \u56de\u6eaf */\nvoid backtrack(int *choices, int state, int n, int *res, int len) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n)\n        res[0]++;\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; len; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n)\n            continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res, len);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n    int choices[2] = {1, 2}; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    int state = 0;           // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    int *res = (int *)malloc(sizeof(int));\n    *res = 0; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    int len = sizeof(choices) / sizeof(int);\n    backtrack(choices, state, n, res, len);\n    int result = *res;\n    free(res);\n    return result;\n}\n</code></pre> climbing_stairs_backtrack.kt<pre><code>/* \u56de\u6eaf */\nfun backtrack(\n    choices: List&lt;Int&gt;,\n    state: Int,\n    n: Int,\n    res: MutableList&lt;Int&gt;\n) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n) res[0] = res[0] + 1\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (choice in choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n) continue\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res)\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfun climbingStairsBacktrack(n: Int): Int {\n    val choices = mutableListOf(1, 2) // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    val state = 0 // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    val res = ArrayList&lt;Int&gt;()\n    res.add(0) // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.rb<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{climbing_stairs_backtrack}\n</code></pre> climbing_stairs_backtrack.zig<pre><code>// \u56de\u6eaf\nfn backtrack(choices: []i32, state: i32, n: i32, res: std.ArrayList(i32)) void {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n) {\n        res.items[0] = res.items[0] + 1;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (choices) |choice| {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n// \u722c\u697c\u68af\uff1a\u56de\u6eaf\nfn climbingStairsBacktrack(n: usize) !i32 {\n    var choices = [_]i32{ 1, 2 }; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    var state: i32 = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    var res = std.ArrayList(i32).init(std.heap.page_allocator);\n    defer res.deinit();\n    try res.append(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(&amp;choices, state, @intCast(n), res);\n    return res.items[0];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1411","title":"14.1.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"<p>\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u5e76\u4e0d\u663e\u5f0f\u5730\u5bf9\u95ee\u9898\u8fdb\u884c\u62c6\u89e3\uff0c\u800c\u662f\u5c06\u6c42\u89e3\u95ee\u9898\u770b\u4f5c\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\uff0c\u901a\u8fc7\u8bd5\u63a2\u548c\u526a\u679d\uff0c\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u5c1d\u8bd5\u4ece\u95ee\u9898\u5206\u89e3\u7684\u89d2\u5ea6\u5206\u6790\u8fd9\u9053\u9898\u3002\u8bbe\u722c\u5230\u7b2c \\(i\\) \u9636\u5171\u6709 \\(dp[i]\\) \u79cd\u65b9\u6848\uff0c\u90a3\u4e48 \\(dp[i]\\) \u5c31\u662f\u539f\u95ee\u9898\uff0c\u5176\u5b50\u95ee\u9898\u5305\u62ec\uff1a</p> \\[ dp[i-1], dp[i-2], \\dots, dp[2], dp[1] \\] <p>\u7531\u4e8e\u6bcf\u8f6e\u53ea\u80fd\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u56e0\u6b64\u5f53\u6211\u4eec\u7ad9\u5728\u7b2c \\(i\\) \u9636\u697c\u68af\u4e0a\u65f6\uff0c\u4e0a\u4e00\u8f6e\u53ea\u53ef\u80fd\u7ad9\u5728\u7b2c \\(i - 1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u4e0a\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ea\u80fd\u4ece\u7b2c \\(i -1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u8fc8\u5411\u7b2c \\(i\\) \u9636\u3002</p> <p>\u7531\u6b64\u4fbf\u53ef\u5f97\u51fa\u4e00\u4e2a\u91cd\u8981\u63a8\u8bba\uff1a\u722c\u5230\u7b2c \\(i - 1\\) \u9636\u7684\u65b9\u6848\u6570\u52a0\u4e0a\u722c\u5230\u7b2c \\(i - 2\\) \u9636\u7684\u65b9\u6848\u6570\u5c31\u7b49\u4e8e\u722c\u5230\u7b2c \\(i\\) \u9636\u7684\u65b9\u6848\u6570\u3002\u516c\u5f0f\u5982\u4e0b\uff1a</p> \\[ dp[i] = dp[i-1] + dp[i-2] \\] <p>\u8fd9\u610f\u5473\u7740\u5728\u722c\u697c\u68af\u95ee\u9898\u4e2d\uff0c\u5404\u4e2a\u5b50\u95ee\u9898\u4e4b\u95f4\u5b58\u5728\u9012\u63a8\u5173\u7cfb\uff0c\u539f\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u7531\u5b50\u95ee\u9898\u7684\u89e3\u6784\u5efa\u5f97\u6765\u3002\u56fe 14-2 \u5c55\u793a\u4e86\u8be5\u9012\u63a8\u5173\u7cfb\u3002</p> <p></p> <p> \u56fe 14-2 \u00a0 \u65b9\u6848\u6570\u91cf\u9012\u63a8\u5173\u7cfb </p> <p>\u6211\u4eec\u53ef\u4ee5\u6839\u636e\u9012\u63a8\u516c\u5f0f\u5f97\u5230\u66b4\u529b\u641c\u7d22\u89e3\u6cd5\u3002\u4ee5 \\(dp[n]\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u9012\u5f52\u5730\u5c06\u4e00\u4e2a\u8f83\u5927\u95ee\u9898\u62c6\u89e3\u4e3a\u4e24\u4e2a\u8f83\u5c0f\u95ee\u9898\u7684\u548c\uff0c\u76f4\u81f3\u5230\u8fbe\u6700\u5c0f\u5b50\u95ee\u9898 \\(dp[1]\\) \u548c \\(dp[2]\\) \u65f6\u8fd4\u56de\u3002\u5176\u4e2d\uff0c\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u5373 \\(dp[1] = 1\\)\u3001\\(dp[2] = 2\\) \uff0c\u8868\u793a\u722c\u5230\u7b2c \\(1\\)\u3001\\(2\\) \u9636\u5206\u522b\u6709 \\(1\\)\u3001\\(2\\) \u79cd\u65b9\u6848\u3002</p> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u5b83\u548c\u6807\u51c6\u56de\u6eaf\u4ee3\u7801\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u4f46\u66f4\u52a0\u7b80\u6d01\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dfs.py<pre><code>def dfs(i: int) -&gt; int:\n    \"\"\"\u641c\u7d22\"\"\"\n    # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 or i == 2:\n        return i\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1) + dfs(i - 2)\n    return count\n\ndef climbing_stairs_dfs(n: int) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\uff1a\u641c\u7d22\"\"\"\n    return dfs(n)\n</code></pre> climbing_stairs_dfs.cpp<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.java<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.cs<pre><code>/* \u641c\u7d22 */\nint DFS(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1) + DFS(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint ClimbingStairsDFS(int n) {\n    return DFS(n);\n}\n</code></pre> climbing_stairs_dfs.go<pre><code>/* \u641c\u7d22 */\nfunc dfs(i int) int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfs(i-1) + dfs(i-2)\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunc climbingStairsDFS(n int) int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.swift<pre><code>/* \u641c\u7d22 */\nfunc dfs(i: Int) -&gt; Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1) + dfs(i: i - 2)\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunc climbingStairsDFS(n: Int) -&gt; Int {\n    dfs(i: n)\n}\n</code></pre> climbing_stairs_dfs.js<pre><code>/* \u641c\u7d22 */\nfunction dfs(i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunction climbingStairsDFS(n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.ts<pre><code>/* \u641c\u7d22 */\nfunction dfs(i: number): number {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunction climbingStairsDFS(n: number): number {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.dart<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n  // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n  if (i == 1 || i == 2) return i;\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1) + dfs(i - 2);\n  return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n  return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.rs<pre><code>/* \u641c\u7d22 */\nfn dfs(i: usize) -&gt; i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1) + dfs(i - 2);\n    count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfn climbing_stairs_dfs(n: usize) -&gt; i32 {\n    dfs(n)\n}\n</code></pre> climbing_stairs_dfs.c<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.kt<pre><code>/* \u641c\u7d22 */\nfun dfs(i: Int): Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2) return i\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1) + dfs(i - 2)\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfun climbingStairsDFS(n: Int): Int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.rb<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{climbing_stairs_dfs}\n</code></pre> climbing_stairs_dfs.zig<pre><code>// \u641c\u7d22\nfn dfs(i: usize) i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 or i == 2) {\n        return @intCast(i);\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    var count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n// \u722c\u697c\u68af\uff1a\u641c\u7d22\nfn climbingStairsDFS(comptime n: usize) i32 {\n    return dfs(n);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 14-3 \u5c55\u793a\u4e86\u66b4\u529b\u641c\u7d22\u5f62\u6210\u7684\u9012\u5f52\u6811\u3002\u5bf9\u4e8e\u95ee\u9898 \\(dp[n]\\) \uff0c\u5176\u9012\u5f52\u6811\u7684\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002\u6307\u6570\u9636\u5c5e\u4e8e\u7206\u70b8\u5f0f\u589e\u957f\uff0c\u5982\u679c\u6211\u4eec\u8f93\u5165\u4e00\u4e2a\u6bd4\u8f83\u5927\u7684 \\(n\\) \uff0c\u5219\u4f1a\u9677\u5165\u6f2b\u957f\u7684\u7b49\u5f85\u4e4b\u4e2d\u3002</p> <p></p> <p> \u56fe 14-3 \u00a0 \u722c\u697c\u68af\u5bf9\u5e94\u9012\u5f52\u6811 </p> <p>\u89c2\u5bdf\u56fe 14-3 \uff0c\u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u201c\u91cd\u53e0\u5b50\u95ee\u9898\u201d\u5bfc\u81f4\u7684\u3002\u4f8b\u5982 \\(dp[9]\\) \u88ab\u5206\u89e3\u4e3a \\(dp[8]\\) \u548c \\(dp[7]\\) \uff0c\\(dp[8]\\) \u88ab\u5206\u89e3\u4e3a \\(dp[7]\\) \u548c \\(dp[6]\\) \uff0c\u4e24\u8005\u90fd\u5305\u542b\u5b50\u95ee\u9898 \\(dp[7]\\) \u3002</p> <p>\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5b50\u95ee\u9898\u4e2d\u5305\u542b\u66f4\u5c0f\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5b50\u5b50\u5b59\u5b59\u65e0\u7a77\u5c3d\u4e5f\u3002\u7edd\u5927\u90e8\u5206\u8ba1\u7b97\u8d44\u6e90\u90fd\u6d6a\u8d39\u5728\u8fd9\u4e9b\u91cd\u53e0\u7684\u5b50\u95ee\u9898\u4e0a\u3002</p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1412","title":"14.1.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"<p>\u4e3a\u4e86\u63d0\u5347\u7b97\u6cd5\u6548\u7387\uff0c\u6211\u4eec\u5e0c\u671b\u6240\u6709\u7684\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u6570\u7ec4 <code>mem</code> \u6765\u8bb0\u5f55\u6bcf\u4e2a\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5e76\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u5c06\u91cd\u53e0\u5b50\u95ee\u9898\u526a\u679d\u3002</p> <ol> <li>\u5f53\u9996\u6b21\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u5c06\u5176\u8bb0\u5f55\u81f3 <code>mem[i]</code> \uff0c\u4ee5\u4fbf\u4e4b\u540e\u4f7f\u7528\u3002</li> <li>\u5f53\u518d\u6b21\u9700\u8981\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u4fbf\u53ef\u76f4\u63a5\u4ece <code>mem[i]</code> \u4e2d\u83b7\u53d6\u7ed3\u679c\uff0c\u4ece\u800c\u907f\u514d\u91cd\u590d\u8ba1\u7b97\u8be5\u5b50\u95ee\u9898\u3002</li> </ol> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dfs_mem.py<pre><code>def dfs(i: int, mem: list[int]) -&gt; int:\n    \"\"\"\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n    # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 or i == 2:\n        return i\n    # \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1:\n        return mem[i]\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    # \u8bb0\u5f55 dp[i]\n    mem[i] = count\n    return count\n\ndef climbing_stairs_dfs_mem(n: int) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n    # mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    mem = [-1] * (n + 1)\n    return dfs(n, mem)\n</code></pre> climbing_stairs_dfs_mem.cpp<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, vector&lt;int&gt; &amp;mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    vector&lt;int&gt; mem(n + 1, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.java<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int[] mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    int[] mem = new int[n + 1];\n    Arrays.fill(mem, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.cs<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint DFS(int i, int[] mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1, mem) + DFS(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint ClimbingStairsDFSMem(int n) {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    int[] mem = new int[n + 1];\n    Array.Fill(mem, -1);\n    return DFS(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.go<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc dfsMem(i int, mem []int) int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfsMem(i-1, mem) + dfsMem(i-2, mem)\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc climbingStairsDFSMem(n int) int {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    mem := make([]int, n+1)\n    for i := range mem {\n        mem[i] = -1\n    }\n    return dfsMem(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.swift<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc dfs(i: Int, mem: inout [Int]) -&gt; Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1, mem: &amp;mem) + dfs(i: i - 2, mem: &amp;mem)\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc climbingStairsDFSMem(n: Int) -&gt; Int {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    var mem = Array(repeating: -1, count: n + 1)\n    return dfs(i: n, mem: &amp;mem)\n}\n</code></pre> climbing_stairs_dfs_mem.js<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction dfs(i, mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction climbingStairsDFSMem(n) {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.ts<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction dfs(i: number, mem: number[]): number {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction climbingStairsDFSMem(n: number): number {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.dart<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, List&lt;int&gt; mem) {\n  // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n  if (i == 1 || i == 2) return i;\n  // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n  if (mem[i] != -1) return mem[i];\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n  // \u8bb0\u5f55 dp[i]\n  mem[i] = count;\n  return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n  // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n  List&lt;int&gt; mem = List.filled(n + 1, -1);\n  return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.rs<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn dfs(i: usize, mem: &amp;mut [i32]) -&gt; i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1 {\n        return mem[i];\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn climbing_stairs_dfs_mem(n: usize) -&gt; i32 {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    let mut mem = vec![-1; n + 1];\n    dfs(n, &amp;mut mem)\n}\n</code></pre> climbing_stairs_dfs_mem.c<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int *mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    int *mem = (int *)malloc((n + 1) * sizeof(int));\n    for (int i = 0; i &lt;= n; i++) {\n        mem[i] = -1;\n    }\n    int result = dfs(n, mem);\n    free(mem);\n    return result;\n}\n</code></pre> climbing_stairs_dfs_mem.kt<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfun dfs(i: Int, mem: IntArray): Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2) return i\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) return mem[i]\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfun climbingStairsDFSMem(n: Int): Int {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    val mem = IntArray(n + 1)\n    Arrays.fill(mem, -1)\n    return dfs(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.rb<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{climbing_stairs_dfs_mem}\n</code></pre> climbing_stairs_dfs_mem.zig<pre><code>// \u8bb0\u5fc6\u5316\u641c\u7d22\nfn dfs(i: usize, mem: []i32) i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 or i == 2) {\n        return @intCast(i);\n    }\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) {\n        return mem[i];\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    var count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n// \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn climbingStairsDFSMem(comptime n: usize) i32 {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    var mem = [_]i32{ -1 } ** (n + 1);\n    return dfs(n, &amp;mem);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u89c2\u5bdf\u56fe 14-4 \uff0c\u7ecf\u8fc7\u8bb0\u5fc6\u5316\u5904\u7406\u540e\uff0c\u6240\u6709\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u9700\u8ba1\u7b97\u4e00\u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u8fd9\u662f\u4e00\u4e2a\u5de8\u5927\u7684\u98de\u8dc3\u3002</p> <p></p> <p> \u56fe 14-4 \u00a0 \u8bb0\u5fc6\u5316\u641c\u7d22\u5bf9\u5e94\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1413","title":"14.1.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"<p>\u8bb0\u5fc6\u5316\u641c\u7d22\u662f\u4e00\u79cd\u201c\u4ece\u9876\u81f3\u5e95\u201d\u7684\u65b9\u6cd5\uff1a\u6211\u4eec\u4ece\u539f\u95ee\u9898\uff08\u6839\u8282\u70b9\uff09\u5f00\u59cb\uff0c\u9012\u5f52\u5730\u5c06\u8f83\u5927\u5b50\u95ee\u9898\u5206\u89e3\u4e3a\u8f83\u5c0f\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u89e3\u5df2\u77e5\u7684\u6700\u5c0f\u5b50\u95ee\u9898\uff08\u53f6\u8282\u70b9\uff09\u3002\u4e4b\u540e\uff0c\u901a\u8fc7\u56de\u6eaf\u9010\u5c42\u6536\u96c6\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u6784\u5efa\u51fa\u539f\u95ee\u9898\u7684\u89e3\u3002</p> <p>\u4e0e\u4e4b\u76f8\u53cd\uff0c\u52a8\u6001\u89c4\u5212\u662f\u4e00\u79cd\u201c\u4ece\u5e95\u81f3\u9876\u201d\u7684\u65b9\u6cd5\uff1a\u4ece\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u5f00\u59cb\uff0c\u8fed\u4ee3\u5730\u6784\u5efa\u66f4\u5927\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u76f4\u81f3\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002</p> <p>\u7531\u4e8e\u52a8\u6001\u89c4\u5212\u4e0d\u5305\u542b\u56de\u6eaf\u8fc7\u7a0b\uff0c\u56e0\u6b64\u53ea\u9700\u4f7f\u7528\u5faa\u73af\u8fed\u4ee3\u5b9e\u73b0\uff0c\u65e0\u987b\u4f7f\u7528\u9012\u5f52\u3002\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u521d\u59cb\u5316\u4e00\u4e2a\u6570\u7ec4 <code>dp</code> \u6765\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5b83\u8d77\u5230\u4e86\u4e0e\u8bb0\u5fc6\u5316\u641c\u7d22\u4e2d\u6570\u7ec4 <code>mem</code> \u76f8\u540c\u7684\u8bb0\u5f55\u4f5c\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dp.py<pre><code>def climbing_stairs_dp(n: int) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    if n == 1 or n == 2:\n        return n\n    # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp = [0] * (n + 1)\n    # \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1], dp[2] = 1, 2\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in range(3, n + 1):\n        dp[i] = dp[i - 1] + dp[i - 2]\n    return dp[n]\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    vector&lt;int&gt; dp(n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDP(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp := make([]int, n+1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1\n    dp[2] = 2\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = dp[i-1] + dp[i-2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = Array(repeating: 0, count: n + 1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1\n    dp[2] = 2\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in 3 ... n {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDP(n) {\n    if (n === 1 || n === 2) return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = new Array(n + 1).fill(-1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDP(n: number): number {\n    if (n === 1 || n === 2) return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = new Array(n + 1).fill(-1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n  if (n == 1 || n == 2) return n;\n  // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n  dp[1] = 1;\n  dp[2] = 2;\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = dp[i - 1] + dp[i - 2];\n  }\n  return dp[n];\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_dp(n: usize) -&gt; i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    let mut dp = vec![-1; n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in 3..=n {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    dp[n]\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int *dp = (int *)malloc((n + 1) * sizeof(int));\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    int result = dp[n];\n    free(dp);\n    return result;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfun climbingStairsDP(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    val dp = IntArray(n + 1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1\n    dp[2] = 2\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (i in 3..n) {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>[class]{}-[func]{climbing_stairs_dp}\n</code></pre> climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\nfn climbingStairsDP(comptime n: usize) i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (n == 1 or n == 2) {\n        return @intCast(n);\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = [_]i32{-1} ** (n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (3..n + 1) |i| {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 14-5 \u6a21\u62df\u4e86\u4ee5\u4e0a\u4ee3\u7801\u7684\u6267\u884c\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 14-5 \u00a0 \u722c\u697c\u68af\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u4e0e\u56de\u6eaf\u7b97\u6cd5\u4e00\u6837\uff0c\u52a8\u6001\u89c4\u5212\u4e5f\u4f7f\u7528\u201c\u72b6\u6001\u201d\u6982\u5ff5\u6765\u8868\u793a\u95ee\u9898\u6c42\u89e3\u7684\u7279\u5b9a\u9636\u6bb5\uff0c\u6bcf\u4e2a\u72b6\u6001\u90fd\u5bf9\u5e94\u4e00\u4e2a\u5b50\u95ee\u9898\u4ee5\u53ca\u76f8\u5e94\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002\u4f8b\u5982\uff0c\u722c\u697c\u68af\u95ee\u9898\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u5f53\u524d\u6240\u5728\u697c\u68af\u9636\u6570 \\(i\\) \u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5185\u5bb9\uff0c\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u52a8\u6001\u89c4\u5212\u7684\u5e38\u7528\u672f\u8bed\u3002</p> <ul> <li>\u5c06\u6570\u7ec4 <code>dp</code> \u79f0\u4e3a\u300c\\(dp\\) \u8868\u300d\uff0c\\(dp[i]\\) \u8868\u793a\u72b6\u6001 \\(i\\) \u5bf9\u5e94\u5b50\u95ee\u9898\u7684\u89e3\u3002</li> <li>\u5c06\u6700\u5c0f\u5b50\u95ee\u9898\u5bf9\u5e94\u7684\u72b6\u6001\uff08\u7b2c \\(1\\) \u9636\u548c\u7b2c \\(2\\) \u9636\u697c\u68af\uff09\u79f0\u4e3a\u300c\u521d\u59cb\u72b6\u6001\u300d\u3002</li> <li>\u5c06\u9012\u63a8\u516c\u5f0f \\(dp[i] = dp[i-1] + dp[i-2]\\) \u79f0\u4e3a\u300c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u300d\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1414","title":"14.1.4 \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7ec6\u5fc3\u7684\u8bfb\u8005\u53ef\u80fd\u53d1\u73b0\u4e86\uff0c\u7531\u4e8e \\(dp[i]\\) \u53ea\u4e0e \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u65e0\u987b\u4f7f\u7528\u4e00\u4e2a\u6570\u7ec4 <code>dp</code> \u6765\u5b58\u50a8\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u800c\u53ea\u9700\u4e24\u4e2a\u53d8\u91cf\u6eda\u52a8\u524d\u8fdb\u5373\u53ef\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dp.py<pre><code>def climbing_stairs_dp_comp(n: int) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    if n == 1 or n == 2:\n        return n\n    a, b = 1, 2\n    for _ in range(3, n + 1):\n        a, b = b, a + b\n    return b\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDPComp(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    a, b := 1, 2\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i := 3; i &lt;= n; i++ {\n        a, b = b, a+b\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDPComp(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    var a = 1\n    var b = 2\n    for _ in 3 ... n {\n        (a, b) = (b, a + b)\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDPComp(n) {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDPComp(n: number): number {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n  if (n == 1 || n == 2) return n;\n  int a = 1, b = 2;\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = a + b;\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_dp_comp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    let (mut a, mut b) = (1, 2);\n    for _ in 3..=n {\n        let tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    b\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun climbingStairsDPComp(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    var a = 1\n    var b = 2\n    for (i in 3..n) {\n        val tmp = b\n        b += a\n        a = tmp\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>[class]{}-[func]{climbing_stairs_dp_comp}\n</code></pre> climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn climbingStairsDPComp(comptime n: usize) i32 {\n    if (n == 1 or n == 2) {\n        return @intCast(n);\n    }\n    var a: i32 = 1;\n    var b: i32 = 2;\n    for (3..n + 1) |_| {\n        var tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u7531\u4e8e\u7701\u53bb\u4e86\u6570\u7ec4 <code>dp</code> \u5360\u7528\u7684\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002</p> <p>\u5728\u52a8\u6001\u89c4\u5212\u95ee\u9898\u4e2d\uff0c\u5f53\u524d\u72b6\u6001\u5f80\u5f80\u4ec5\u4e0e\u524d\u9762\u6709\u9650\u4e2a\u72b6\u6001\u6709\u5173\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u53ea\u4fdd\u7559\u5fc5\u8981\u7684\u72b6\u6001\uff0c\u901a\u8fc7\u201c\u964d\u7ef4\u201d\u6765\u8282\u7701\u5185\u5b58\u7a7a\u95f4\u3002\u8fd9\u79cd\u7a7a\u95f4\u4f18\u5316\u6280\u5de7\u88ab\u79f0\u4e3a\u201c\u6eda\u52a8\u53d8\u91cf\u201d\u6216\u201c\u6eda\u52a8\u6570\u7ec4\u201d\u3002</p>"},{"location":"chapter_dynamic_programming/knapsack_problem/","title":"14.4 \u00a0 0-1 \u80cc\u5305\u95ee\u9898","text":"<p>\u80cc\u5305\u95ee\u9898\u662f\u4e00\u4e2a\u975e\u5e38\u597d\u7684\u52a8\u6001\u89c4\u5212\u5165\u95e8\u9898\u76ee\uff0c\u662f\u52a8\u6001\u89c4\u5212\u4e2d\u6700\u5e38\u89c1\u7684\u95ee\u9898\u5f62\u5f0f\u3002\u5176\u5177\u6709\u5f88\u591a\u53d8\u79cd\uff0c\u4f8b\u5982 0-1 \u80cc\u5305\u95ee\u9898\u3001\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u3001\u591a\u91cd\u80cc\u5305\u95ee\u9898\u7b49\u3002</p> <p>\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5148\u6765\u6c42\u89e3\u6700\u5e38\u89c1\u7684 0-1 \u80cc\u5305\u95ee\u9898\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ea\u80fd\u9009\u62e9\u4e00\u6b21\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002</p> <p>\u89c2\u5bdf\u56fe 14-17 \uff0c\u7531\u4e8e\u7269\u54c1\u7f16\u53f7 \\(i\\) \u4ece \\(1\\) \u5f00\u59cb\u8ba1\u6570\uff0c\u6570\u7ec4\u7d22\u5f15\u4ece \\(0\\) \u5f00\u59cb\u8ba1\u6570\uff0c\u56e0\u6b64\u7269\u54c1 \\(i\\) \u5bf9\u5e94\u91cd\u91cf \\(wgt[i-1]\\) \u548c\u4ef7\u503c \\(val[i-1]\\) \u3002</p> <p></p> <p> \u56fe 14-17 \u00a0 0-1 \u80cc\u5305\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u6211\u4eec\u53ef\u4ee5\u5c06 0-1 \u80cc\u5305\u95ee\u9898\u770b\u4f5c\u4e00\u4e2a\u7531 \\(n\\) \u8f6e\u51b3\u7b56\u7ec4\u6210\u7684\u8fc7\u7a0b\uff0c\u5bf9\u4e8e\u6bcf\u4e2a\u7269\u4f53\u90fd\u6709\u4e0d\u653e\u5165\u548c\u653e\u5165\u4e24\u79cd\u51b3\u7b56\uff0c\u56e0\u6b64\u8be5\u95ee\u9898\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\u3002</p> <p>\u8be5\u95ee\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u201c\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u201d\uff0c\u56e0\u6b64\u8f83\u5927\u6982\u7387\u662f\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u95ee\u9898\u3002</p> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u5bf9\u4e8e\u6bcf\u4e2a\u7269\u54c1\u6765\u8bf4\uff0c\u4e0d\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u4e0d\u53d8\uff1b\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u51cf\u5c0f\u3002\u7531\u6b64\u53ef\u5f97\u72b6\u6001\u5b9a\u4e49\uff1a\u5f53\u524d\u7269\u54c1\u7f16\u53f7 \\(i\\) \u548c\u5269\u4f59\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u8bb0\u4e3a \\([i, c]\\) \u3002</p> <p>\u72b6\u6001 \\([i, c]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u524d \\(i\\) \u4e2a\u7269\u54c1\u5728\u5269\u4f59\u5bb9\u91cf\u4e3a \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u4ef7\u503c\uff0c\u8bb0\u4e3a \\(dp[i, c]\\) \u3002</p> <p>\u5f85\u6c42\u89e3\u7684\u662f \\(dp[n, cap]\\) \uff0c\u56e0\u6b64\u9700\u8981\u4e00\u4e2a\u5c3a\u5bf8\u4e3a \\((n+1) \\times (cap+1)\\) \u7684\u4e8c\u7ef4 \\(dp\\) \u8868\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u5f53\u6211\u4eec\u505a\u51fa\u7269\u54c1 \\(i\\) \u7684\u51b3\u7b56\u540e\uff0c\u5269\u4f59\u7684\u662f\u524d \\(i-1\\) \u4e2a\u7269\u54c1\u7684\u51b3\u7b56\uff0c\u53ef\u5206\u4e3a\u4ee5\u4e0b\u4e24\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u4e0d\u53d8\uff0c\u72b6\u6001\u53d8\u5316\u4e3a \\([i-1, c]\\) \u3002</li> <li>\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u51cf\u5c11 \\(wgt[i-1]\\) \uff0c\u4ef7\u503c\u589e\u52a0 \\(val[i-1]\\) \uff0c\u72b6\u6001\u53d8\u5316\u4e3a \\([i-1, c-wgt[i-1]]\\) \u3002</li> </ul> <p>\u4e0a\u8ff0\u5206\u6790\u5411\u6211\u4eec\u63ed\u793a\u4e86\u672c\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u6700\u5927\u4ef7\u503c \\(dp[i, c]\\) \u7b49\u4e8e\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \u548c\u653e\u5165\u7269\u54c1 \\(i\\) \u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\u3002\u7531\u6b64\u53ef\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1]) \\] <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u82e5\u5f53\u524d\u7269\u54c1\u91cd\u91cf \\(wgt[i - 1]\\) \u8d85\u51fa\u5269\u4f59\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\u3002</p> <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5f53\u65e0\u7269\u54c1\u6216\u65e0\u5269\u4f59\u80cc\u5305\u5bb9\u91cf\u65f6\u6700\u5927\u4ef7\u503c\u4e3a \\(0\\) \uff0c\u5373\u9996\u5217 \\(dp[i, 0]\\) \u548c\u9996\u884c \\(dp[0, c]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002</p> <p>\u5f53\u524d\u72b6\u6001 \\([i, c]\\) \u4ece\u4e0a\u65b9\u7684\u72b6\u6001 \\([i-1, c]\\) \u548c\u5de6\u4e0a\u65b9\u7684\u72b6\u6001 \\([i-1, c-wgt[i-1]]\\) \u8f6c\u79fb\u800c\u6765\uff0c\u56e0\u6b64\u901a\u8fc7\u4e24\u5c42\u5faa\u73af\u6b63\u5e8f\u904d\u5386\u6574\u4e2a \\(dp\\) \u8868\u5373\u53ef\u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u4eec\u63a5\u4e0b\u6765\u6309\u987a\u5e8f\u5b9e\u73b0\u66b4\u529b\u641c\u7d22\u3001\u8bb0\u5fc6\u5316\u641c\u7d22\u3001\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u3002</p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"<p>\u641c\u7d22\u4ee3\u7801\u5305\u542b\u4ee5\u4e0b\u8981\u7d20\u3002</p> <ul> <li>\u9012\u5f52\u53c2\u6570\uff1a\u72b6\u6001 \\([i, c]\\) \u3002</li> <li>\u8fd4\u56de\u503c\uff1a\u5b50\u95ee\u9898\u7684\u89e3 \\(dp[i, c]\\) \u3002</li> <li>\u7ec8\u6b62\u6761\u4ef6\uff1a\u5f53\u7269\u54c1\u7f16\u53f7\u8d8a\u754c \\(i = 0\\) \u6216\u80cc\u5305\u5269\u4f59\u5bb9\u91cf\u4e3a \\(0\\) \u65f6\uff0c\u7ec8\u6b62\u9012\u5f52\u5e76\u8fd4\u56de\u4ef7\u503c \\(0\\) \u3002</li> <li>\u526a\u679d\uff1a\u82e5\u5f53\u524d\u7269\u54c1\u91cd\u91cf\u8d85\u51fa\u80cc\u5305\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dfs(wgt: list[int], val: list[int], i: int, c: int) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22\"\"\"\n    # \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 or c == 0:\n        return 0\n    # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs(wgt, val, i - 1, c)\n    # \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    no = knapsack_dfs(wgt, val, i - 1, c)\n    yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return max(no, yes)\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return max(no, yes);\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(int[] wgt, int[] val, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint KnapsackDFS(int[] weight, int[] val, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFS(weight, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = KnapsackDFS(weight, val, i - 1, c);\n    int yes = KnapsackDFS(weight, val, i - 1, c - weight[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return Math.Max(no, yes);\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc knapsackDFS(wgt, val []int, i, c int) int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i-1] &gt; c {\n        return knapsackDFS(wgt, val, i-1, c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    no := knapsackDFS(wgt, val, i-1, c)\n    yes := knapsackDFS(wgt, val, i-1, c-wgt[i-1]) + val[i-1]\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return int(math.Max(float64(no), float64(yes)))\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc knapsackDFS(wgt: [Int], val: [Int], i: Int, c: Int) -&gt; Int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c {\n        return knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    let no = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    let yes = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return max(no, yes)\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction knapsackDFS(wgt, val, i, c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction knapsackDFS(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    i: number,\n    c: number\n): number {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(List&lt;int&gt; wgt, List&lt;int&gt; val, int i, int c) {\n  // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFS(wgt, val, i - 1, c);\n  }\n  // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n  int no = knapsackDFS(wgt, val, i - 1, c);\n  int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n  return max(no, yes);\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfn knapsack_dfs(wgt: &amp;[i32], val: &amp;[i32], i: usize, c: usize) -&gt; i32 {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    let no = knapsack_dfs(wgt, val, i - 1, c);\n    let yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    std::cmp::max(no, yes)\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(int wgt[], int val[], int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return myMax(no, yes);\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfun knapsackDFS(\n    wgt: IntArray,\n    value: IntArray,\n    i: Int,\n    c: Int\n): Int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, value, i - 1, c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    val no = knapsackDFS(wgt, value, i - 1, c)\n    val yes = knapsackDFS(wgt, value, i - 1, c - wgt[i - 1]) + value[i - 1]\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return max(no.toDouble(), yes.toDouble()).toInt()\n}\n</code></pre> knapsack.rb<pre><code>[class]{}-[func]{knapsack_dfs}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22\nfn knapsackDFS(wgt: []i32, val: []i32, i: usize, c: usize) i32 {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 or c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    var no = knapsackDFS(wgt, val, i - 1, c);\n    var yes = knapsackDFS(wgt, val, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return @max(no, yes);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 14-18 \u6240\u793a\uff0c\u7531\u4e8e\u6bcf\u4e2a\u7269\u54c1\u90fd\u4f1a\u4ea7\u751f\u4e0d\u9009\u548c\u9009\u4e24\u6761\u641c\u7d22\u5206\u652f\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002</p> <p>\u89c2\u5bdf\u9012\u5f52\u6811\uff0c\u5bb9\u6613\u53d1\u73b0\u5176\u4e2d\u5b58\u5728\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u4f8b\u5982 \\(dp[1, 10]\\) \u7b49\u3002\u800c\u5f53\u7269\u54c1\u8f83\u591a\u3001\u80cc\u5305\u5bb9\u91cf\u8f83\u5927\uff0c\u5c24\u5176\u662f\u76f8\u540c\u91cd\u91cf\u7684\u7269\u54c1\u8f83\u591a\u65f6\uff0c\u91cd\u53e0\u5b50\u95ee\u9898\u7684\u6570\u91cf\u5c06\u4f1a\u5927\u5e45\u589e\u591a\u3002</p> <p></p> <p> \u56fe 14-18 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u66b4\u529b\u641c\u7d22\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"<p>\u4e3a\u4e86\u4fdd\u8bc1\u91cd\u53e0\u5b50\u95ee\u9898\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\uff0c\u6211\u4eec\u501f\u52a9\u8bb0\u5fc6\u5217\u8868 <code>mem</code> \u6765\u8bb0\u5f55\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5176\u4e2d <code>mem[i][c]</code> \u5bf9\u5e94 \\(dp[i, c]\\) \u3002</p> <p>\u5f15\u5165\u8bb0\u5fc6\u5316\u4e4b\u540e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u5b50\u95ee\u9898\u6570\u91cf\uff0c\u4e5f\u5c31\u662f \\(O(n \\times cap)\\) \u3002\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dfs_mem(\n    wgt: list[int], val: list[int], mem: list[list[int]], i: int, c: int\n) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n    # \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 or c == 0:\n        return 0\n    # \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1:\n        return mem[i][c]\n    # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    # \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(int[] wgt, int[] val, int[][] mem, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint KnapsackDFSMem(int[] weight, int[] val, int[][] mem, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFSMem(weight, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = KnapsackDFSMem(weight, val, mem, i - 1, c);\n    int yes = KnapsackDFSMem(weight, val, mem, i - 1, c - weight[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = Math.Max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc knapsackDFSMem(wgt, val []int, mem [][]int, i, c int) int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i-1] &gt; c {\n        return knapsackDFSMem(wgt, val, mem, i-1, c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    no := knapsackDFSMem(wgt, val, mem, i-1, c)\n    yes := knapsackDFSMem(wgt, val, mem, i-1, c-wgt[i-1]) + val[i-1]\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = int(math.Max(float64(no), float64(yes)))\n    return mem[i][c]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc knapsackDFSMem(wgt: [Int], val: [Int], mem: inout [[Int]], i: Int, c: Int) -&gt; Int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c {\n        return knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    let no = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    let yes = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction knapsackDFSMem(wgt, val, mem, i, c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction knapsackDFSMem(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    c: number\n): number {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(\n  List&lt;int&gt; wgt,\n  List&lt;int&gt; val,\n  List&lt;List&lt;int&gt;&gt; mem,\n  int i,\n  int c,\n) {\n  // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n  if (mem[i][c] != -1) {\n    return mem[i][c];\n  }\n  // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFSMem(wgt, val, mem, i - 1, c);\n  }\n  // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n  int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n  int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n  mem[i][c] = max(no, yes);\n  return mem[i][c];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn knapsack_dfs_mem(wgt: &amp;[i32], val: &amp;[i32], mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: usize, c: usize) -&gt; i32 {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1 {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    let no = knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    let yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = std::cmp::max(no, yes);\n    mem[i][c]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(int wgt[], int val[], int memCols, int **mem, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = myMax(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfun knapsackDFSMem(\n    wgt: IntArray,\n    value: IntArray,\n    mem: Array&lt;IntArray&gt;,\n    i: Int,\n    c: Int\n): Int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c]\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, value, mem, i - 1, c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    val no = knapsackDFSMem(wgt, value, mem, i - 1, c)\n    val yes = knapsackDFSMem(wgt, value, mem, i - 1, c - wgt[i - 1]) + value[i - 1]\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = max(no.toDouble(), yes.toDouble()).toInt()\n    return mem[i][c]\n}\n</code></pre> knapsack.rb<pre><code>[class]{}-[func]{knapsack_dfs_mem}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn knapsackDFSMem(wgt: []i32, val: []i32, mem: anytype, i: usize, c: usize) i32 {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 or c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    var no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    var yes = knapsackDFSMem(wgt, val, mem, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = @max(no, yes);\n    return mem[i][c];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 14-19 \u5c55\u793a\u4e86\u5728\u8bb0\u5fc6\u5316\u641c\u7d22\u4e2d\u88ab\u526a\u6389\u7684\u641c\u7d22\u5206\u652f\u3002</p> <p></p> <p> \u56fe 14-19 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u8bb0\u5fc6\u5316\u641c\u7d22\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"<p>\u52a8\u6001\u89c4\u5212\u5b9e\u8d28\u4e0a\u5c31\u662f\u5728\u72b6\u6001\u8f6c\u79fb\u4e2d\u586b\u5145 \\(dp\\) \u8868\u7684\u8fc7\u7a0b\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint KnapsackDP(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (weight[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i, c] = Math.Max(dp[i - 1, c - weight[i - 1]] + val[i - 1], dp[i - 1, c]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i-1][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array(n + 1)\n        .fill(0)\n        .map(() =&gt; Array(cap + 1).fill(0));\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n        dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfn knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = std::cmp::max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1] as usize] + val[i - 1],\n                );\n            }\n        }\n    }\n    dp[n][cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = myMax(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfun knapsackDP(\n    wgt: IntArray,\n    value: IntArray,\n    cap: Int\n): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c].toDouble(), (dp[i - 1][c - wgt[i - 1]] + value[i - 1]).toDouble())\n                    .toInt()\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.rb<pre><code>[class]{}-[func]{knapsack_dp}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\nfn knapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    comptime var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..cap + 1) |c| {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = @max(dp[i - 1][c], dp[i - 1][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 14-20 \u6240\u793a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u7531\u6570\u7ec4 <code>dp</code> \u5927\u5c0f\u51b3\u5b9a\uff0c\u5373 \\(O(n \\times cap)\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-20 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#4","title":"4. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e\u6bcf\u4e2a\u72b6\u6001\u90fd\u53ea\u4e0e\u5176\u4e0a\u4e00\u884c\u7684\u72b6\u6001\u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e24\u4e2a\u6570\u7ec4\u6eda\u52a8\u524d\u8fdb\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \u3002</p> <p>\u8fdb\u4e00\u6b65\u601d\u8003\uff0c\u6211\u4eec\u80fd\u5426\u4ec5\u7528\u4e00\u4e2a\u6570\u7ec4\u5b9e\u73b0\u7a7a\u95f4\u4f18\u5316\u5462\uff1f\u89c2\u5bdf\u53ef\u77e5\uff0c\u6bcf\u4e2a\u72b6\u6001\u90fd\u662f\u7531\u6b63\u4e0a\u65b9\u6216\u5de6\u4e0a\u65b9\u7684\u683c\u5b50\u8f6c\u79fb\u8fc7\u6765\u7684\u3002\u5047\u8bbe\u53ea\u6709\u4e00\u4e2a\u6570\u7ec4\uff0c\u5f53\u5f00\u59cb\u904d\u5386\u7b2c \\(i\\) \u884c\u65f6\uff0c\u8be5\u6570\u7ec4\u5b58\u50a8\u7684\u4ecd\u7136\u662f\u7b2c \\(i-1\\) \u884c\u7684\u72b6\u6001\u3002</p> <ul> <li>\u5982\u679c\u91c7\u53d6\u6b63\u5e8f\u904d\u5386\uff0c\u90a3\u4e48\u904d\u5386\u5230 \\(dp[i, j]\\) \u65f6\uff0c\u5de6\u4e0a\u65b9 \\(dp[i-1, 1]\\) ~ \\(dp[i-1, j-1]\\) \u503c\u53ef\u80fd\u5df2\u7ecf\u88ab\u8986\u76d6\uff0c\u6b64\u65f6\u5c31\u65e0\u6cd5\u5f97\u5230\u6b63\u786e\u7684\u72b6\u6001\u8f6c\u79fb\u7ed3\u679c\u3002</li> <li>\u5982\u679c\u91c7\u53d6\u5012\u5e8f\u904d\u5386\uff0c\u5219\u4e0d\u4f1a\u53d1\u751f\u8986\u76d6\u95ee\u9898\uff0c\u72b6\u6001\u8f6c\u79fb\u53ef\u4ee5\u6b63\u786e\u8fdb\u884c\u3002</li> </ul> <p>\u56fe 14-21 \u5c55\u793a\u4e86\u5728\u5355\u4e2a\u6570\u7ec4\u4e0b\u4ece\u7b2c \\(i = 1\\) \u884c\u8f6c\u6362\u81f3\u7b2c \\(i = 2\\) \u884c\u7684\u8fc7\u7a0b\u3002\u8bf7\u601d\u8003\u6b63\u5e8f\u904d\u5386\u548c\u5012\u5e8f\u904d\u5386\u7684\u533a\u522b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-21 \u00a0 0-1 \u80cc\u5305\u7684\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u4ec5\u9700\u5c06\u6570\u7ec4 <code>dp</code> \u7684\u7b2c\u4e00\u7ef4 \\(i\\) \u76f4\u63a5\u5220\u9664\uff0c\u5e76\u4e14\u628a\u5185\u5faa\u73af\u66f4\u6539\u4e3a\u5012\u5e8f\u904d\u5386\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * (cap + 1)\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        # \u5012\u5e8f\u904d\u5386\n        for c in range(cap, 0, -1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint KnapsackDPComp(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (int c = cap; c &gt; 0; c--) {\n            if (weight[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.Max(dp[c], dp[c - weight[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, cap+1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        // \u5012\u5e8f\u904d\u5386\n        for c := cap; c &gt;= 1; c-- {\n            if wgt[i-1] &lt;= c {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: cap + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1 ... n {\n        // \u5012\u5e8f\u904d\u5386\n        for c in (1 ... cap).reversed() {\n            if wgt[i - 1] &lt;= c {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array(cap + 1).fill(0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array(cap + 1).fill(0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    // \u5012\u5e8f\u904d\u5386\n    for (int c = cap; c &gt;= 1; c--) {\n      if (wgt[i - 1] &lt;= c) {\n        // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        // \u5012\u5e8f\u904d\u5386\n        for c in (1..=cap).rev() {\n            if wgt[i - 1] &lt;= c as i32 {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(cap + 1, sizeof(int));\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun knapsackDPComp(\n    wgt: IntArray,\n    value: IntArray,\n    cap: Int\n): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(cap + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for (i in 1..n) {\n        // \u5012\u5e8f\u904d\u5386\n        for (c in cap downTo 1) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] =\n                    max(dp[c].toDouble(), (dp[c - wgt[i - 1]] + value[i - 1]).toDouble()).toInt()\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.rb<pre><code>[class]{}-[func]{knapsack_dp_comp}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn knapsackDPComp(wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (cap + 1);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        // \u5012\u5e8f\u904d\u5386\n        var c = cap;\n        while (c &gt; 0) : (c -= 1) {\n            if (wgt[i - 1] &lt; c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/summary/","title":"14.7 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u52a8\u6001\u89c4\u5212\u5bf9\u95ee\u9898\u8fdb\u884c\u5206\u89e3\uff0c\u5e76\u901a\u8fc7\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\u6765\u89c4\u907f\u91cd\u590d\u8ba1\u7b97\uff0c\u63d0\u9ad8\u8ba1\u7b97\u6548\u7387\u3002</li> <li>\u4e0d\u8003\u8651\u65f6\u95f4\u7684\u524d\u63d0\u4e0b\uff0c\u6240\u6709\u52a8\u6001\u89c4\u5212\u95ee\u9898\u90fd\u53ef\u4ee5\u7528\u56de\u6eaf\uff08\u66b4\u529b\u641c\u7d22\uff09\u8fdb\u884c\u6c42\u89e3\uff0c\u4f46\u9012\u5f52\u6811\u4e2d\u5b58\u5728\u5927\u91cf\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u6548\u7387\u6781\u4f4e\u3002\u901a\u8fc7\u5f15\u5165\u8bb0\u5fc6\u5316\u5217\u8868\uff0c\u53ef\u4ee5\u5b58\u50a8\u6240\u6709\u8ba1\u7b97\u8fc7\u7684\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u4ece\u800c\u4fdd\u8bc1\u91cd\u53e0\u5b50\u95ee\u9898\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\u3002</li> <li>\u8bb0\u5fc6\u5316\u641c\u7d22\u662f\u4e00\u79cd\u4ece\u9876\u81f3\u5e95\u7684\u9012\u5f52\u5f0f\u89e3\u6cd5\uff0c\u800c\u4e0e\u4e4b\u5bf9\u5e94\u7684\u52a8\u6001\u89c4\u5212\u662f\u4e00\u79cd\u4ece\u5e95\u81f3\u9876\u7684\u9012\u63a8\u5f0f\u89e3\u6cd5\uff0c\u5176\u5982\u540c\u201c\u586b\u5199\u8868\u683c\u201d\u4e00\u6837\u3002\u7531\u4e8e\u5f53\u524d\u72b6\u6001\u4ec5\u4f9d\u8d56\u67d0\u4e9b\u5c40\u90e8\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u6d88\u9664 \\(dp\\) \u8868\u7684\u4e00\u4e2a\u7ef4\u5ea6\uff0c\u4ece\u800c\u964d\u4f4e\u7a7a\u95f4\u590d\u6742\u5ea6\u3002</li> <li>\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u901a\u7528\u7684\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u3001\u56de\u6eaf\u4e2d\u5177\u6709\u4e0d\u540c\u7684\u6027\u8d28\u3002</li> <li>\u52a8\u6001\u89c4\u5212\u95ee\u9898\u6709\u4e09\u5927\u7279\u6027\uff1a\u91cd\u53e0\u5b50\u95ee\u9898\u3001\u6700\u4f18\u5b50\u7ed3\u6784\u3001\u65e0\u540e\u6548\u6027\u3002</li> <li>\u5982\u679c\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u53ef\u4ee5\u4ece\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u5efa\u5f97\u6765\uff0c\u5219\u5b83\u5c31\u5177\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u3002</li> <li>\u65e0\u540e\u6548\u6027\u6307\u5bf9\u4e8e\u4e00\u4e2a\u72b6\u6001\uff0c\u5176\u672a\u6765\u53d1\u5c55\u53ea\u4e0e\u8be5\u72b6\u6001\u6709\u5173\uff0c\u800c\u4e0e\u8fc7\u53bb\u7ecf\u5386\u7684\u6240\u6709\u72b6\u6001\u65e0\u5173\u3002\u8bb8\u591a\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u4e0d\u5177\u6709\u65e0\u540e\u6548\u6027\uff0c\u65e0\u6cd5\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u5feb\u901f\u6c42\u89e3\u3002</li> </ul> <p>\u80cc\u5305\u95ee\u9898</p> <ul> <li>\u80cc\u5305\u95ee\u9898\u662f\u6700\u5178\u578b\u7684\u52a8\u6001\u89c4\u5212\u95ee\u9898\u4e4b\u4e00\uff0c\u5177\u6709 0-1 \u80cc\u5305\u3001\u5b8c\u5168\u80cc\u5305\u3001\u591a\u91cd\u80cc\u5305\u7b49\u53d8\u79cd\u3002</li> <li>0-1 \u80cc\u5305\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u524d \\(i\\) \u4e2a\u7269\u54c1\u5728\u5269\u4f59\u5bb9\u91cf\u4e3a \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u4ef7\u503c\u3002\u6839\u636e\u4e0d\u653e\u5165\u80cc\u5305\u548c\u653e\u5165\u80cc\u5305\u4e24\u79cd\u51b3\u7b56\uff0c\u53ef\u5f97\u5230\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5e76\u6784\u5efa\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002\u5728\u7a7a\u95f4\u4f18\u5316\u4e2d\uff0c\u7531\u4e8e\u6bcf\u4e2a\u72b6\u6001\u4f9d\u8d56\u6b63\u4e0a\u65b9\u548c\u5de6\u4e0a\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u9700\u8981\u5012\u5e8f\u904d\u5386\u5217\u8868\uff0c\u907f\u514d\u5de6\u4e0a\u65b9\u72b6\u6001\u88ab\u8986\u76d6\u3002</li> <li>\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u6bcf\u79cd\u7269\u54c1\u7684\u9009\u53d6\u6570\u91cf\u65e0\u9650\u5236\uff0c\u56e0\u6b64\u9009\u62e9\u653e\u5165\u7269\u54c1\u7684\u72b6\u6001\u8f6c\u79fb\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u4e0d\u540c\u3002\u7531\u4e8e\u72b6\u6001\u4f9d\u8d56\u6b63\u4e0a\u65b9\u548c\u6b63\u5de6\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u5728\u7a7a\u95f4\u4f18\u5316\u4e2d\u5e94\u5f53\u6b63\u5e8f\u904d\u5386\u3002</li> <li>\u96f6\u94b1\u5151\u6362\u95ee\u9898\u662f\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u4e00\u4e2a\u53d8\u79cd\u3002\u5b83\u4ece\u6c42\u201c\u6700\u5927\u201d\u4ef7\u503c\u53d8\u4e3a\u6c42\u201c\u6700\u5c0f\u201d\u786c\u5e01\u6570\u91cf\uff0c\u56e0\u6b64\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\max()\\) \u5e94\u6539\u4e3a \\(\\min()\\) \u3002\u4ece\u8ffd\u6c42\u201c\u4e0d\u8d85\u8fc7\u201d\u80cc\u5305\u5bb9\u91cf\u5230\u8ffd\u6c42\u201c\u6070\u597d\u201d\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u4f7f\u7528 \\(amt + 1\\) \u6765\u8868\u793a\u201c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u201d\u7684\u65e0\u6548\u89e3\u3002</li> <li>\u96f6\u94b1\u5151\u6362\u95ee\u9898 II \u4ece\u6c42\u201c\u6700\u5c11\u786c\u5e01\u6570\u91cf\u201d\u6539\u4e3a\u6c42\u201c\u786c\u5e01\u7ec4\u5408\u6570\u91cf\u201d\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u76f8\u5e94\u5730\u4ece \\(\\min()\\) \u6539\u4e3a\u6c42\u548c\u8fd0\u7b97\u7b26\u3002</li> </ul> <p>\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898</p> <ul> <li>\u7f16\u8f91\u8ddd\u79bb\uff08Levenshtein \u8ddd\u79bb\uff09\u7528\u4e8e\u8861\u91cf\u4e24\u4e2a\u5b57\u7b26\u4e32\u4e4b\u95f4\u7684\u76f8\u4f3c\u5ea6\uff0c\u5176\u5b9a\u4e49\u4e3a\u4ece\u4e00\u4e2a\u5b57\u7b26\u4e32\u5230\u53e6\u4e00\u4e2a\u5b57\u7b26\u4e32\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\uff0c\u7f16\u8f91\u64cd\u4f5c\u5305\u62ec\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66ff\u6362\u3002</li> <li>\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u5c06 \\(s\\) \u7684\u524d \\(i\\) \u4e2a\u5b57\u7b26\u66f4\u6539\u4e3a \\(t\\) \u7684\u524d \\(j\\) \u4e2a\u5b57\u7b26\u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002\u5f53 \\(s[i] \\ne t[j]\\) \u65f6\uff0c\u5177\u6709\u4e09\u79cd\u51b3\u7b56\uff1a\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66ff\u6362\uff0c\u5b83\u4eec\u90fd\u6709\u76f8\u5e94\u7684\u5269\u4f59\u5b50\u95ee\u9898\u3002\u636e\u6b64\u4fbf\u53ef\u4ee5\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\u4e0e\u6784\u5efa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002\u800c\u5f53 \\(s[i] = t[j]\\) \u65f6\uff0c\u65e0\u987b\u7f16\u8f91\u5f53\u524d\u5b57\u7b26\u3002</li> <li>\u5728\u7f16\u8f91\u8ddd\u79bb\u4e2d\uff0c\u72b6\u6001\u4f9d\u8d56\u5176\u6b63\u4e0a\u65b9\u3001\u6b63\u5de6\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u7a7a\u95f4\u4f18\u5316\u540e\u6b63\u5e8f\u6216\u5012\u5e8f\u904d\u5386\u90fd\u65e0\u6cd5\u6b63\u786e\u5730\u8fdb\u884c\u72b6\u6001\u8f6c\u79fb\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u5229\u7528\u4e00\u4e2a\u53d8\u91cf\u6682\u5b58\u5de6\u4e0a\u65b9\u72b6\u6001\uff0c\u4ece\u800c\u8f6c\u5316\u5230\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7b49\u4ef7\u7684\u60c5\u51b5\uff0c\u53ef\u4ee5\u5728\u7a7a\u95f4\u4f18\u5316\u540e\u8fdb\u884c\u6b63\u5e8f\u904d\u5386\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/","title":"14.5 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898","text":"<p>\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5148\u6c42\u89e3\u53e6\u4e00\u4e2a\u5e38\u89c1\u7684\u80cc\u5305\u95ee\u9898\uff1a\u5b8c\u5168\u80cc\u5305\uff0c\u518d\u4e86\u89e3\u5b83\u7684\u4e00\u79cd\u7279\u4f8b\uff1a\u96f6\u94b1\u5151\u6362\u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1451","title":"14.5.1 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002\u793a\u4f8b\u5982\u56fe 14-22 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 14-22 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u548c 0-1 \u80cc\u5305\u95ee\u9898\u975e\u5e38\u76f8\u4f3c\uff0c\u533a\u522b\u4ec5\u5728\u4e8e\u4e0d\u9650\u5236\u7269\u54c1\u7684\u9009\u62e9\u6b21\u6570\u3002</p> <ul> <li>\u5728 0-1 \u80cc\u5305\u95ee\u9898\u4e2d\uff0c\u6bcf\u79cd\u7269\u54c1\u53ea\u6709\u4e00\u4e2a\uff0c\u56e0\u6b64\u5c06\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u540e\uff0c\u53ea\u80fd\u4ece\u524d \\(i-1\\) \u4e2a\u7269\u54c1\u4e2d\u9009\u62e9\u3002</li> <li>\u5728\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u4e2d\uff0c\u6bcf\u79cd\u7269\u54c1\u7684\u6570\u91cf\u662f\u65e0\u9650\u7684\uff0c\u56e0\u6b64\u5c06\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u540e\uff0c\u4ecd\u53ef\u4ee5\u4ece\u524d \\(i\\) \u4e2a\u7269\u54c1\u4e2d\u9009\u62e9\u3002</li> </ul> <p>\u5728\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u89c4\u5b9a\u4e0b\uff0c\u72b6\u6001 \\([i, c]\\) \u7684\u53d8\u5316\u5206\u4e3a\u4e24\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u76f8\u540c\uff0c\u8f6c\u79fb\u81f3 \\([i-1, c]\\) \u3002</li> <li>\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u4e0d\u540c\uff0c\u8f6c\u79fb\u81f3 \\([i, c-wgt[i-1]]\\) \u3002</li> </ul> <p>\u4ece\u800c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u53d8\u4e3a\uff1a</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1]) \\]"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5bf9\u6bd4\u4e24\u9053\u9898\u76ee\u7684\u4ee3\u7801\uff0c\u72b6\u6001\u8f6c\u79fb\u4e2d\u6709\u4e00\u5904\u4ece \\(i-1\\) \u53d8\u4e3a \\(i\\) \uff0c\u5176\u4f59\u5b8c\u5168\u4e00\u81f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint UnboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i, c] = Math.Max(dp[i - 1, c], dp[i, c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n        dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfn unbounded_knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i][c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = myMax(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfun unboundedKnapsackDP(\n    wgt: IntArray,\n    value: IntArray,\n    cap: Int\n): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c].toDouble(), (dp[i][c - wgt[i - 1]] + value[i - 1]).toDouble())\n                    .toInt()\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>[class]{}-[func]{unbounded_knapsack_dp}\n</code></pre> unbounded_knapsack.zig<pre><code>// \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\nfn unboundedKnapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    comptime var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..cap + 1) |c| {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = @max(dp[i - 1][c], dp[i][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e\u5f53\u524d\u72b6\u6001\u662f\u4ece\u5de6\u8fb9\u548c\u4e0a\u8fb9\u7684\u72b6\u6001\u8f6c\u79fb\u800c\u6765\u7684\uff0c\u56e0\u6b64\u7a7a\u95f4\u4f18\u5316\u540e\u5e94\u8be5\u5bf9 \\(dp\\) \u8868\u4e2d\u7684\u6bcf\u4e00\u884c\u8fdb\u884c\u6b63\u5e8f\u904d\u5386\u3002</p> <p>\u8fd9\u4e2a\u904d\u5386\u987a\u5e8f\u4e0e 0-1 \u80cc\u5305\u6b63\u597d\u76f8\u53cd\u3002\u8bf7\u501f\u52a9\u56fe 14-23 \u6765\u7406\u89e3\u4e24\u8005\u7684\u533a\u522b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-23 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u5728\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u4ee3\u7801\u5b9e\u73b0\u6bd4\u8f83\u7b80\u5355\uff0c\u4ec5\u9700\u5c06\u6570\u7ec4 <code>dp</code> \u7684\u7b2c\u4e00\u7ef4\u5220\u9664\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * (cap + 1)\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        # \u6b63\u5e8f\u904d\u5386\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint UnboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.Max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, cap+1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: cap + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n        dp[c] = dp[c];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn unbounded_knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(cap + 1, sizeof(int));\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun unboundedKnapsackDPComp(\n    wgt: IntArray,\n    value: IntArray,\n    cap: Int\n): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(cap + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] =\n                    max(dp[c].toDouble(), (dp[c - wgt[i - 1]] + value[i - 1]).toDouble()).toInt()\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>[class]{}-[func]{unbounded_knapsack_dp_comp}\n</code></pre> unbounded_knapsack.zig<pre><code>// \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn unboundedKnapsackDPComp(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    comptime var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (cap + 1);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..cap + 1) |c| {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1452","title":"14.5.2 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898","text":"<p>\u80cc\u5305\u95ee\u9898\u662f\u4e00\u5927\u7c7b\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u4ee3\u8868\uff0c\u5176\u62e5\u6709\u5f88\u591a\u53d8\u79cd\uff0c\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u80fd\u591f\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u3002\u5982\u679c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u56fe 14-24 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 14-24 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u96f6\u94b1\u5151\u6362\u53ef\u4ee5\u770b\u4f5c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u4e00\u79cd\u7279\u6b8a\u60c5\u51b5\uff0c\u4e24\u8005\u5177\u6709\u4ee5\u4e0b\u8054\u7cfb\u4e0e\u4e0d\u540c\u70b9\u3002</p> <ul> <li>\u4e24\u9053\u9898\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\uff0c\u201c\u7269\u54c1\u201d\u5bf9\u5e94\u201c\u786c\u5e01\u201d\u3001\u201c\u7269\u54c1\u91cd\u91cf\u201d\u5bf9\u5e94\u201c\u786c\u5e01\u9762\u503c\u201d\u3001\u201c\u80cc\u5305\u5bb9\u91cf\u201d\u5bf9\u5e94\u201c\u76ee\u6807\u91d1\u989d\u201d\u3002</li> <li>\u4f18\u5316\u76ee\u6807\u76f8\u53cd\uff0c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u662f\u8981\u6700\u5927\u5316\u7269\u54c1\u4ef7\u503c\uff0c\u96f6\u94b1\u5151\u6362\u95ee\u9898\u662f\u8981\u6700\u5c0f\u5316\u786c\u5e01\u6570\u91cf\u3002</li> <li>\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u662f\u6c42\u201c\u4e0d\u8d85\u8fc7\u201d\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u89e3\uff0c\u96f6\u94b1\u5151\u6362\u662f\u6c42\u201c\u6070\u597d\u201d\u51d1\u5230\u76ee\u6807\u91d1\u989d\u7684\u89e3\u3002</li> </ul> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u72b6\u6001 \\([i, a]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u524d \\(i\\) \u79cd\u786c\u5e01\u80fd\u591f\u51d1\u51fa\u91d1\u989d \\(a\\) \u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\uff0c\u8bb0\u4e3a \\(dp[i, a]\\) \u3002</p> <p>\u4e8c\u7ef4 \\(dp\\) \u8868\u7684\u5c3a\u5bf8\u4e3a \\((n+1) \\times (amt+1)\\) \u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u672c\u9898\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u5b58\u5728\u4ee5\u4e0b\u4e24\u70b9\u5dee\u5f02\u3002</p> <ul> <li>\u672c\u9898\u8981\u6c42\u6700\u5c0f\u503c\uff0c\u56e0\u6b64\u9700\u5c06\u8fd0\u7b97\u7b26 \\(\\max()\\) \u66f4\u6539\u4e3a \\(\\min()\\) \u3002</li> <li>\u4f18\u5316\u4e3b\u4f53\u662f\u786c\u5e01\u6570\u91cf\u800c\u975e\u5546\u54c1\u4ef7\u503c\uff0c\u56e0\u6b64\u5728\u9009\u4e2d\u786c\u5e01\u65f6\u6267\u884c \\(+1\\) \u5373\u53ef\u3002</li> </ul> \\[ dp[i, a] = \\min(dp[i-1, a], dp[i, a - coins[i-1]] + 1) \\] <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5f53\u76ee\u6807\u91d1\u989d\u4e3a \\(0\\) \u65f6\uff0c\u51d1\u51fa\u5b83\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u4e3a \\(0\\) \uff0c\u5373\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002</p> <p>\u5f53\u65e0\u786c\u5e01\u65f6\uff0c\u65e0\u6cd5\u51d1\u51fa\u4efb\u610f \\(&gt; 0\\) \u7684\u76ee\u6807\u91d1\u989d\uff0c\u5373\u662f\u65e0\u6548\u89e3\u3002\u4e3a\u4f7f\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\min()\\) \u51fd\u6570\u80fd\u591f\u8bc6\u522b\u5e76\u8fc7\u6ee4\u65e0\u6548\u89e3\uff0c\u6211\u4eec\u8003\u8651\u4f7f\u7528 \\(+ \\infty\\) \u6765\u8868\u793a\u5b83\u4eec\uff0c\u5373\u4ee4\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u4e8e \\(+ \\infty\\) \u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u5e76\u672a\u63d0\u4f9b \\(+ \\infty\\) \u53d8\u91cf\uff0c\u53ea\u80fd\u4f7f\u7528\u6574\u578b <code>int</code> \u7684\u6700\u5927\u503c\u6765\u4ee3\u66ff\u3002\u800c\u8fd9\u53c8\u4f1a\u5bfc\u81f4\u5927\u6570\u8d8a\u754c\uff1a\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(+ 1\\) \u64cd\u4f5c\u53ef\u80fd\u53d1\u751f\u6ea2\u51fa\u3002</p> <p>\u4e3a\u6b64\uff0c\u6211\u4eec\u91c7\u7528\u6570\u5b57 \\(amt + 1\\) \u6765\u8868\u793a\u65e0\u6548\u89e3\uff0c\u56e0\u4e3a\u51d1\u51fa \\(amt\\) \u7684\u786c\u5e01\u6570\u91cf\u6700\u591a\u4e3a \\(amt\\) \u3002\u6700\u540e\u8fd4\u56de\u524d\uff0c\u5224\u65ad \\(dp[n, amt]\\) \u662f\u5426\u7b49\u4e8e \\(amt + 1\\) \uff0c\u82e5\u662f\u5219\u8fd4\u56de \\(-1\\) \uff0c\u4ee3\u8868\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change.py<pre><code>def coin_change_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a in range(1, amt + 1):\n        dp[0][a] = MAX\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n    return dp[n][amt] if dp[n][amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][amt + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint CoinChangeDP(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, amt + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0, a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i, a] = Math.Min(dp[i - 1, a], dp[i, a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n, amt] != MAX ? dp[n, amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDP(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a := 1; a &lt;= amt; a++ {\n        dp[0][a] = max\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = int(math.Min(float64(dp[i-1][a]), float64(dp[i][a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a in 1 ... amt {\n        dp[0][a] = MAX\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDP(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n  for (int a = 1; a &lt;= amt; a++) {\n    dp[0][a] = MAX;\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n        dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfn coin_change_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a in 1..=amt {\n        dp[0][a] = max;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = std::cmp::min(dp[i - 1][a], dp[i][a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = myMin(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[n][amt] != MAX ? dp[n][amt] : -1;\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfun coinChangeDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (a in 1..amt) {\n        dp[0][a] = MAX\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a].toDouble(), (dp[i][a - coins[i - 1]] + 1).toDouble())\n                    .toInt()\n            }\n        }\n    }\n    return if (dp[n][amt] != MAX) dp[n][amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>[class]{}-[func]{coin_change_dp}\n</code></pre> coin_change.zig<pre><code>// \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212\nfn coinChangeDP(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    comptime var max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (1..amt + 1) |a| {\n        dp[0][a] = max;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = @min(dp[i - 1][a], dp[i][a - @as(usize, @intCast(coins[i - 1]))] + 1);\n            }\n        }\n    }\n    if (dp[n][amt] != max) {\n        return @intCast(dp[n][amt]);\n    } else {\n        return -1;\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 14-25 \u5c55\u793a\u4e86\u96f6\u94b1\u5151\u6362\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b\uff0c\u548c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u975e\u5e38\u76f8\u4f3c\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-25 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_1","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u96f6\u94b1\u5151\u6362\u7684\u7a7a\u95f4\u4f18\u5316\u7684\u5904\u7406\u65b9\u5f0f\u548c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u4e00\u81f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change.py<pre><code>def coin_change_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [MAX] * (amt + 1)\n    dp[0] = 0\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        # \u6b63\u5e8f\u904d\u5386\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n    return dp[amt] if dp[amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(amt + 1, MAX);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    Arrays.fill(dp, MAX);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint CoinChangeDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    Array.Fill(dp, MAX);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = Math.Min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDPComp(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, amt+1)\n    for i := 1; i &lt;= amt; i++ {\n        dp[i] = max\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        // \u5012\u5e8f\u904d\u5386\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = int(math.Min(float64(dp[a]), float64(dp[a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: MAX, count: amt + 1)\n    dp[0] = 0\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDPComp(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(amt + 1, MAX);\n  dp[0] = 0;\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n        dp[a] = dp[a];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n        dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn coin_change_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; amt + 1];\n    dp.fill(max);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = std::cmp::min(dp[a], dp[a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(amt + 1, sizeof(int));\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = myMin(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[amt] != MAX ? dp[amt] : -1;\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun coinChangeDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(amt + 1)\n    Arrays.fill(dp, MAX)\n    dp[0] = 0\n    // \u72b6\u6001\u8f6c\u79fb\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = min(dp[a].toDouble(), (dp[a - coins[i - 1]] + 1).toDouble()).toInt()\n            }\n        }\n    }\n    return if (dp[amt] != MAX) dp[amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>[class]{}-[func]{coin_change_dp_comp}\n</code></pre> coin_change.zig<pre><code>// \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn coinChangeDPComp(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    comptime var max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (amt + 1);\n    @memset(&amp;dp, max);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = @min(dp[a], dp[a - @as(usize, @intCast(coins[i - 1]))] + 1);\n            }\n        }\n    }\n    if (dp[amt] != max) {\n        return @intCast(dp[amt]);\n    } else {\n        return -1;\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1453-ii","title":"14.5.3 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898 II","text":"<p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u786c\u5e01\u7ec4\u5408\u6570\u91cf\u3002\u793a\u4f8b\u5982\u56fe 14-26 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 14-26 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898 II \u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_2","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u76f8\u6bd4\u4e8e\u4e0a\u4e00\u9898\uff0c\u672c\u9898\u76ee\u6807\u662f\u6c42\u7ec4\u5408\u6570\u91cf\uff0c\u56e0\u6b64\u5b50\u95ee\u9898\u53d8\u4e3a\uff1a\u524d \\(i\\) \u79cd\u786c\u5e01\u80fd\u591f\u51d1\u51fa\u91d1\u989d \\(a\\) \u7684\u7ec4\u5408\u6570\u91cf\u3002\u800c \\(dp\\) \u8868\u4ecd\u7136\u662f\u5c3a\u5bf8\u4e3a \\((n+1) \\times (amt + 1)\\) \u7684\u4e8c\u7ef4\u77e9\u9635\u3002</p> <p>\u5f53\u524d\u72b6\u6001\u7684\u7ec4\u5408\u6570\u91cf\u7b49\u4e8e\u4e0d\u9009\u5f53\u524d\u786c\u5e01\u4e0e\u9009\u5f53\u524d\u786c\u5e01\u8fd9\u4e24\u79cd\u51b3\u7b56\u7684\u7ec4\u5408\u6570\u91cf\u4e4b\u548c\u3002\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]] \\] <p>\u5f53\u76ee\u6807\u91d1\u989d\u4e3a \\(0\\) \u65f6\uff0c\u65e0\u987b\u9009\u62e9\u4efb\u4f55\u786c\u5e01\u5373\u53ef\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u5e94\u5c06\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u521d\u59cb\u5316\u4e3a \\(1\\) \u3002\u5f53\u65e0\u786c\u5e01\u65f6\uff0c\u65e0\u6cd5\u51d1\u51fa\u4efb\u4f55 \\(&gt;0\\) \u7684\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change_ii.py<pre><code>def coin_change_ii_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(coins)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # \u521d\u59cb\u5316\u9996\u5217\n    for i in range(n + 1):\n        dp[i][0] = 1\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n    return dp[n][amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(int[] coins, int amt) {\n    int n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][amt + 1];\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint CoinChangeIIDP(int[] coins, int amt) {\n    int n = coins.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, amt + 1];\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i, 0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i, a] = dp[i - 1, a] + dp[i, a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n, amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDP(coins []int, amt int) int {\n    n := len(coins)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // \u521d\u59cb\u5316\u9996\u5217\n    for i := 0; i &lt;= n; i++ {\n        dp[i][0] = 1\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i-1][a] + dp[i][a-coins[i-1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // \u521d\u59cb\u5316\u9996\u5217\n    for i in 0 ... n {\n        dp[i][0] = 1\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDP(coins, amt) {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // \u521d\u59cb\u5316\u9996\u5217\n  for (int i = 0; i &lt;= n; i++) {\n    dp[i][0] = 1;\n  }\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[n][amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfn coin_change_ii_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // \u521d\u59cb\u5316\u9996\u5217\n    for i in 0..=n {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[n][amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[n][amt];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfun coinChangeIIDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (i in 0..n) {\n        dp[i][0] = 1\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>[class]{}-[func]{coin_change_ii_dp}\n</code></pre> coin_change_ii.zig<pre><code>// \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212\nfn coinChangeIIDP(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (0..n + 1) |i| {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - @as(usize, @intCast(coins[i - 1]))];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_2","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7a7a\u95f4\u4f18\u5316\u5904\u7406\u65b9\u5f0f\u76f8\u540c\uff0c\u5220\u9664\u786c\u5e01\u7ef4\u5ea6\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change_ii.py<pre><code>def coin_change_ii_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(coins)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * (amt + 1)\n    dp[0] = 1\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        # \u6b63\u5e8f\u904d\u5386\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n    return dp[amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(amt + 1, 0);\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint CoinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDPComp(coins []int, amt int) int {\n    n := len(coins)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, amt+1)\n    dp[0] = 1\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        // \u5012\u5e8f\u904d\u5386\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a-coins[i-1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: amt + 1)\n    dp[0] = 1\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDPComp(coins, amt) {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(amt + 1, 0);\n  dp[0] = 1;\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n        dp[a] = dp[a];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n        dp[a] = dp[a] + dp[a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn coin_change_ii_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; amt + 1];\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(amt + 1, sizeof(int));\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[amt];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun coinChangeIIDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(amt + 1)\n    dp[0] = 1\n    // \u72b6\u6001\u8f6c\u79fb\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>[class]{}-[func]{coin_change_ii_dp_comp}\n</code></pre> coin_change_ii.zig<pre><code>// \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn coinChangeIIDPComp(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (amt + 1);\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = dp[a] + dp[a - @as(usize, @intCast(coins[i - 1]))];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_graph/","title":"\u7b2c 9 \u7ae0 \u00a0 \u56fe","text":"<p>Abstract</p> <p>\u5728\u751f\u547d\u65c5\u9014\u4e2d\uff0c\u6211\u4eec\u5c31\u50cf\u662f\u4e00\u4e2a\u4e2a\u8282\u70b9\uff0c\u88ab\u65e0\u6570\u770b\u4e0d\u89c1\u7684\u8fb9\u76f8\u8fde\u3002</p> <p>\u6bcf\u4e00\u6b21\u7684\u76f8\u8bc6\u4e0e\u76f8\u79bb\uff0c\u90fd\u5728\u8fd9\u5f20\u5de8\u5927\u7684\u7f51\u7edc\u56fe\u4e2d\u7559\u4e0b\u72ec\u7279\u7684\u5370\u8bb0\u3002</p>"},{"location":"chapter_graph/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>9.1 \u00a0 \u56fe</li> <li>9.2 \u00a0 \u56fe\u57fa\u7840\u64cd\u4f5c</li> <li>9.3 \u00a0 \u56fe\u7684\u904d\u5386</li> <li>9.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_graph/graph/","title":"9.1 \u00a0 \u56fe","text":"<p>\u300c\u56fe graph\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u7531\u300c\u9876\u70b9 vertex\u300d\u548c\u300c\u8fb9 edge\u300d\u7ec4\u6210\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u56fe \\(G\\) \u62bd\u8c61\u5730\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9 \\(V\\) \u548c\u4e00\u7ec4\u8fb9 \\(E\\) \u7684\u96c6\u5408\u3002\u4ee5\u4e0b\u793a\u4f8b\u5c55\u793a\u4e86\u4e00\u4e2a\u5305\u542b 5 \u4e2a\u9876\u70b9\u548c 7 \u6761\u8fb9\u7684\u56fe\u3002</p> \\[ \\begin{aligned} V &amp; = \\{ 1, 2, 3, 4, 5 \\} \\newline E &amp; = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G &amp; = \\{ V, E \\} \\newline \\end{aligned} \\] <p>\u5982\u679c\u5c06\u9876\u70b9\u770b\u4f5c\u8282\u70b9\uff0c\u5c06\u8fb9\u770b\u4f5c\u8fde\u63a5\u5404\u4e2a\u8282\u70b9\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u56fe\u770b\u4f5c\u4e00\u79cd\u4ece\u94fe\u8868\u62d3\u5c55\u800c\u6765\u7684\u6570\u636e\u7ed3\u6784\u3002\u5982\u56fe 9-1 \u6240\u793a\uff0c\u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u56e0\u800c\u66f4\u4e3a\u590d\u6742\u3002</p> <p></p> <p> \u56fe 9-1 \u00a0 \u94fe\u8868\u3001\u6811\u3001\u56fe\u4e4b\u95f4\u7684\u5173\u7cfb </p>"},{"location":"chapter_graph/graph/#911","title":"9.1.1 \u00a0 \u56fe\u7684\u5e38\u89c1\u7c7b\u578b\u4e0e\u672f\u8bed","text":"<p>\u6839\u636e\u8fb9\u662f\u5426\u5177\u6709\u65b9\u5411\uff0c\u53ef\u5206\u4e3a\u300c\u65e0\u5411\u56fe undirected graph\u300d\u548c\u300c\u6709\u5411\u56fe directed graph\u300d\uff0c\u5982\u56fe 9-2 \u6240\u793a\u3002</p> <ul> <li>\u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u8fb9\u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u7684\u201c\u53cc\u5411\u201d\u8fde\u63a5\u5173\u7cfb\uff0c\u4f8b\u5982\u5fae\u4fe1\u6216 QQ \u4e2d\u7684\u201c\u597d\u53cb\u5173\u7cfb\u201d\u3002</li> <li>\u5728\u6709\u5411\u56fe\u4e2d\uff0c\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u5373 \\(A \\rightarrow B\\) \u548c \\(A \\leftarrow B\\) \u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u4f8b\u5982\u5fae\u535a\u6216\u6296\u97f3\u4e0a\u7684\u201c\u5173\u6ce8\u201d\u4e0e\u201c\u88ab\u5173\u6ce8\u201d\u5173\u7cfb\u3002</li> </ul> <p></p> <p> \u56fe 9-2 \u00a0 \u6709\u5411\u56fe\u4e0e\u65e0\u5411\u56fe </p> <p>\u6839\u636e\u6240\u6709\u9876\u70b9\u662f\u5426\u8fde\u901a\uff0c\u53ef\u5206\u4e3a\u300c\u8fde\u901a\u56fe connected graph\u300d\u548c\u300c\u975e\u8fde\u901a\u56fe disconnected graph\u300d\uff0c\u5982\u56fe 9-3 \u6240\u793a\u3002</p> <ul> <li>\u5bf9\u4e8e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u53ef\u4ee5\u5230\u8fbe\u5176\u4f59\u4efb\u610f\u9876\u70b9\u3002</li> <li>\u5bf9\u4e8e\u975e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\u3002</li> </ul> <p></p> <p> \u56fe 9-3 \u00a0 \u8fde\u901a\u56fe\u4e0e\u975e\u8fde\u901a\u56fe </p> <p>\u6211\u4eec\u8fd8\u53ef\u4ee5\u4e3a\u8fb9\u6dfb\u52a0\u201c\u6743\u91cd\u201d\u53d8\u91cf\uff0c\u4ece\u800c\u5f97\u5230\u5982\u56fe 9-4 \u6240\u793a\u7684\u300c\u6709\u6743\u56fe weighted graph\u300d\u3002\u4f8b\u5982\u5728\u300a\u738b\u8005\u8363\u8000\u300b\u7b49\u624b\u6e38\u4e2d\uff0c\u7cfb\u7edf\u4f1a\u6839\u636e\u5171\u540c\u6e38\u620f\u65f6\u95f4\u6765\u8ba1\u7b97\u73a9\u5bb6\u4e4b\u95f4\u7684\u201c\u4eb2\u5bc6\u5ea6\u201d\uff0c\u8fd9\u79cd\u4eb2\u5bc6\u5ea6\u7f51\u7edc\u5c31\u53ef\u4ee5\u7528\u6709\u6743\u56fe\u6765\u8868\u793a\u3002</p> <p></p> <p> \u56fe 9-4 \u00a0 \u6709\u6743\u56fe\u4e0e\u65e0\u6743\u56fe </p> <p>\u56fe\u6570\u636e\u7ed3\u6784\u5305\u542b\u4ee5\u4e0b\u5e38\u7528\u672f\u8bed\u3002</p> <ul> <li>\u300c\u90bb\u63a5 adjacency\u300d\uff1a\u5f53\u4e24\u9876\u70b9\u4e4b\u95f4\u5b58\u5728\u8fb9\u76f8\u8fde\u65f6\uff0c\u79f0\u8fd9\u4e24\u9876\u70b9\u201c\u90bb\u63a5\u201d\u3002\u5728\u56fe 9-4 \u4e2d\uff0c\u9876\u70b9 1 \u7684\u90bb\u63a5\u9876\u70b9\u4e3a\u9876\u70b9 2\u30013\u30015\u3002</li> <li>\u300c\u8def\u5f84 path\u300d\uff1a\u4ece\u9876\u70b9 A \u5230\u9876\u70b9 B \u7ecf\u8fc7\u7684\u8fb9\u6784\u6210\u7684\u5e8f\u5217\u88ab\u79f0\u4e3a\u4ece A \u5230 B \u7684\u201c\u8def\u5f84\u201d\u3002\u5728\u56fe 9-4 \u4e2d\uff0c\u8fb9\u5e8f\u5217 1-5-2-4 \u662f\u9876\u70b9 1 \u5230\u9876\u70b9 4 \u7684\u4e00\u6761\u8def\u5f84\u3002</li> <li>\u300c\u5ea6 degree\u300d\uff1a\u4e00\u4e2a\u9876\u70b9\u62e5\u6709\u7684\u8fb9\u6570\u3002\u5bf9\u4e8e\u6709\u5411\u56fe\uff0c\u300c\u5165\u5ea6 in-degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u6307\u5411\u8be5\u9876\u70b9\uff0c\u300c\u51fa\u5ea6 out-degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u4ece\u8be5\u9876\u70b9\u6307\u51fa\u3002</li> </ul>"},{"location":"chapter_graph/graph/#912","title":"9.1.2 \u00a0 \u56fe\u7684\u8868\u793a","text":"<p>\u56fe\u7684\u5e38\u7528\u8868\u793a\u65b9\u5f0f\u5305\u62ec\u201c\u90bb\u63a5\u77e9\u9635\u201d\u548c\u201c\u90bb\u63a5\u8868\u201d\u3002\u4ee5\u4e0b\u4f7f\u7528\u65e0\u5411\u56fe\u8fdb\u884c\u4e3e\u4f8b\u3002</p>"},{"location":"chapter_graph/graph/#1","title":"1. \u00a0 \u90bb\u63a5\u77e9\u9635","text":"<p>\u8bbe\u56fe\u7684\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u300c\u90bb\u63a5\u77e9\u9635 adjacency matrix\u300d\u4f7f\u7528\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u662f\u5426\u5b58\u5728\u8fb9\u3002</p> <p>\u5982\u56fe 9-5 \u6240\u793a\uff0c\u8bbe\u90bb\u63a5\u77e9\u9635\u4e3a \\(M\\)\u3001\u9876\u70b9\u5217\u8868\u4e3a \\(V\\) \uff0c\u90a3\u4e48\u77e9\u9635\u5143\u7d20 \\(M[i, j] = 1\\) \u8868\u793a\u9876\u70b9 \\(V[i]\\) \u5230\u9876\u70b9 \\(V[j]\\) \u4e4b\u95f4\u5b58\u5728\u8fb9\uff0c\u53cd\u4e4b \\(M[i, j] = 0\\) \u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u65e0\u8fb9\u3002</p> <p></p> <p> \u56fe 9-5 \u00a0 \u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a </p> <p>\u90bb\u63a5\u77e9\u9635\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u9876\u70b9\u4e0d\u80fd\u4e0e\u81ea\u8eab\u76f8\u8fde\uff0c\u56e0\u6b64\u90bb\u63a5\u77e9\u9635\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u6ca1\u6709\u610f\u4e49\u3002</li> <li>\u5bf9\u4e8e\u65e0\u5411\u56fe\uff0c\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u7b49\u4ef7\uff0c\u6b64\u65f6\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\u3002</li> <li>\u5c06\u90bb\u63a5\u77e9\u9635\u7684\u5143\u7d20\u4ece \\(1\\) \u548c \\(0\\) \u66ff\u6362\u4e3a\u6743\u91cd\uff0c\u5219\u53ef\u8868\u793a\u6709\u6743\u56fe\u3002</li> </ul> <p>\u4f7f\u7528\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u8bbf\u95ee\u77e9\u9635\u5143\u7d20\u4ee5\u83b7\u53d6\u8fb9\uff0c\u56e0\u6b64\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002\u7136\u800c\uff0c\u77e9\u9635\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u5185\u5b58\u5360\u7528\u8f83\u591a\u3002</p>"},{"location":"chapter_graph/graph/#2","title":"2. \u00a0 \u90bb\u63a5\u8868","text":"<p>\u300c\u90bb\u63a5\u8868 adjacency list\u300d\u4f7f\u7528 \\(n\\) \u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u94fe\u8868\u8282\u70b9\u8868\u793a\u9876\u70b9\u3002\u7b2c \\(i\\) \u4e2a\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff08\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u9876\u70b9\uff09\u3002\u56fe 9-6 \u5c55\u793a\u4e86\u4e00\u4e2a\u4f7f\u7528\u90bb\u63a5\u8868\u5b58\u50a8\u7684\u56fe\u7684\u793a\u4f8b\u3002</p> <p></p> <p> \u56fe 9-6 \u00a0 \u56fe\u7684\u90bb\u63a5\u8868\u8868\u793a </p> <p>\u90bb\u63a5\u8868\u4ec5\u5b58\u50a8\u5b9e\u9645\u5b58\u5728\u7684\u8fb9\uff0c\u800c\u8fb9\u7684\u603b\u6570\u901a\u5e38\u8fdc\u5c0f\u4e8e \\(n^2\\) \uff0c\u56e0\u6b64\u5b83\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002\u7136\u800c\uff0c\u5728\u90bb\u63a5\u8868\u4e2d\u9700\u8981\u901a\u8fc7\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u5176\u65f6\u95f4\u6548\u7387\u4e0d\u5982\u90bb\u63a5\u77e9\u9635\u3002</p> <p>\u89c2\u5bdf\u56fe 9-6 \uff0c\u90bb\u63a5\u8868\u7ed3\u6784\u4e0e\u54c8\u5e0c\u8868\u4e2d\u7684\u201c\u94fe\u5f0f\u5730\u5740\u201d\u975e\u5e38\u76f8\u4f3c\uff0c\u56e0\u6b64\u6211\u4eec\u4e5f\u53ef\u4ee5\u91c7\u7528\u7c7b\u4f3c\u7684\u65b9\u6cd5\u6765\u4f18\u5316\u6548\u7387\u3002\u6bd4\u5982\u5f53\u94fe\u8868\u8f83\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u5316\u4e3a AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u6548\u7387\u4ece \\(O(n)\\) \u4f18\u5316\u81f3 \\(O(\\log n)\\) \uff1b\u8fd8\u53ef\u4ee5\u628a\u94fe\u8868\u8f6c\u6362\u4e3a\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u964d\u81f3 \\(O(1)\\) \u3002</p>"},{"location":"chapter_graph/graph/#913","title":"9.1.3 \u00a0 \u56fe\u7684\u5e38\u89c1\u5e94\u7528","text":"<p>\u5982\u8868 9-1 \u6240\u793a\uff0c\u8bb8\u591a\u73b0\u5b9e\u7cfb\u7edf\u53ef\u4ee5\u7528\u56fe\u6765\u5efa\u6a21\uff0c\u76f8\u5e94\u7684\u95ee\u9898\u4e5f\u53ef\u4ee5\u7ea6\u5316\u4e3a\u56fe\u8ba1\u7b97\u95ee\u9898\u3002</p> <p> \u8868 9-1 \u00a0 \u73b0\u5b9e\u751f\u6d3b\u4e2d\u5e38\u89c1\u7684\u56fe </p> \u9876\u70b9 \u8fb9 \u56fe\u8ba1\u7b97\u95ee\u9898 \u793e\u4ea4\u7f51\u7edc \u7528\u6237 \u597d\u53cb\u5173\u7cfb \u6f5c\u5728\u597d\u53cb\u63a8\u8350 \u5730\u94c1\u7ebf\u8def \u7ad9\u70b9 \u7ad9\u70b9\u95f4\u7684\u8fde\u901a\u6027 \u6700\u77ed\u8def\u7ebf\u63a8\u8350 \u592a\u9633\u7cfb \u661f\u4f53 \u661f\u4f53\u95f4\u7684\u4e07\u6709\u5f15\u529b\u4f5c\u7528 \u884c\u661f\u8f68\u9053\u8ba1\u7b97"},{"location":"chapter_graph/graph_operations/","title":"9.2 \u00a0 \u56fe\u7684\u57fa\u7840\u64cd\u4f5c","text":"<p>\u56fe\u7684\u57fa\u7840\u64cd\u4f5c\u53ef\u5206\u4e3a\u5bf9\u201c\u8fb9\u201d\u7684\u64cd\u4f5c\u548c\u5bf9\u201c\u9876\u70b9\u201d\u7684\u64cd\u4f5c\u3002\u5728\u201c\u90bb\u63a5\u77e9\u9635\u201d\u548c\u201c\u90bb\u63a5\u8868\u201d\u4e24\u79cd\u8868\u793a\u65b9\u6cd5\u4e0b\uff0c\u5b9e\u73b0\u65b9\u5f0f\u6709\u6240\u4e0d\u540c\u3002</p>"},{"location":"chapter_graph/graph_operations/#921","title":"9.2.1 \u00a0 \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u7684\u5b9e\u73b0","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \u7684\u65e0\u5411\u56fe\uff0c\u5219\u5404\u79cd\u64cd\u4f5c\u7684\u5b9e\u73b0\u65b9\u5f0f\u5982\u56fe 9-7 \u6240\u793a\u3002</p> <ul> <li>\u6dfb\u52a0\u6216\u5220\u9664\u8fb9\uff1a\u76f4\u63a5\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u4fee\u6539\u6307\u5b9a\u7684\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u800c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u9700\u8981\u540c\u65f6\u66f4\u65b0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002</li> <li>\u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u7684\u5c3e\u90e8\u6dfb\u52a0\u4e00\u884c\u4e00\u5217\uff0c\u5e76\u5168\u90e8\u586b \\(0\\) \u5373\u53ef\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> <li>\u5220\u9664\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u4e00\u884c\u4e00\u5217\u3002\u5f53\u5220\u9664\u9996\u884c\u9996\u5217\u65f6\u8fbe\u5230\u6700\u5dee\u60c5\u51b5\uff0c\u9700\u8981\u5c06 \\((n-1)^2\\) \u4e2a\u5143\u7d20\u201c\u5411\u5de6\u4e0a\u79fb\u52a8\u201d\uff0c\u4ece\u800c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> <li>\u521d\u59cb\u5316\uff1a\u4f20\u5165 \\(n\\) \u4e2a\u9876\u70b9\uff0c\u521d\u59cb\u5316\u957f\u5ea6\u4e3a \\(n\\) \u7684\u9876\u70b9\u5217\u8868 <code>vertices</code> \uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b\u521d\u59cb\u5316 \\(n \\times n\\) \u5927\u5c0f\u7684\u90bb\u63a5\u77e9\u9635 <code>adjMat</code> \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> </ul> \u521d\u59cb\u5316\u90bb\u63a5\u77e9\u9635\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9 <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-7 \u00a0 \u90bb\u63a5\u77e9\u9635\u7684\u521d\u59cb\u5316\u3001\u589e\u5220\u8fb9\u3001\u589e\u5220\u9876\u70b9 </p> <p>\u4ee5\u4e0b\u662f\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u7684\u5b9e\u73b0\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_adjacency_matrix.py<pre><code>class GraphAdjMat:\n    \"\"\"\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n\n    def __init__(self, vertices: list[int], edges: list[list[int]]):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        # \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n        self.vertices: list[int] = []\n        # \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n        self.adj_mat: list[list[int]] = []\n        # \u6dfb\u52a0\u9876\u70b9\n        for val in vertices:\n            self.add_vertex(val)\n        # \u6dfb\u52a0\u8fb9\n        # \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for e in edges:\n            self.add_edge(e[0], e[1])\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\n        return len(self.vertices)\n\n    def add_vertex(self, val: int):\n        \"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\n        n = self.size()\n        # \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        self.vertices.append(val)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        new_row = [0] * n\n        self.adj_mat.append(new_row)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for row in self.adj_mat:\n            row.append(0)\n\n    def remove_vertex(self, index: int):\n        \"\"\"\u5220\u9664\u9876\u70b9\"\"\"\n        if index &gt;= self.size():\n            raise IndexError()\n        # \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        self.vertices.pop(index)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        self.adj_mat.pop(index)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for row in self.adj_mat:\n            row.pop(index)\n\n    def add_edge(self, i: int, j: int):\n        \"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n        # \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        # \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        # \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        self.adj_mat[i][j] = 1\n        self.adj_mat[j][i] = 1\n\n    def remove_edge(self, i: int, j: int):\n        \"\"\"\u5220\u9664\u8fb9\"\"\"\n        # \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        # \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        self.adj_mat[i][j] = 0\n        self.adj_mat[j][i] = 0\n\n    def print(self):\n        \"\"\"\u6253\u5370\u90bb\u63a5\u77e9\u9635\"\"\"\n        print(\"\u9876\u70b9\u5217\u8868 =\", self.vertices)\n        print(\"\u90bb\u63a5\u77e9\u9635 =\")\n        print_matrix(self.adj_mat)\n</code></pre> graph_adjacency_matrix.cpp<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    vector&lt;int&gt; vertices;       // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    vector&lt;vector&lt;int&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    GraphAdjMat(const vector&lt;int&gt; &amp;vertices, const vector&lt;vector&lt;int&gt;&gt; &amp;edges) {\n        // \u6dfb\u52a0\u9876\u70b9\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (const vector&lt;int&gt; &amp;edge : edges) {\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int size() const {\n        return vertices.size();\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    void addVertex(int val) {\n        int n = size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.push_back(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        adjMat.emplace_back(vector&lt;int&gt;(n, 0));\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.push_back(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    void removeVertex(int index) {\n        if (index &gt;= size()) {\n            throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.erase(vertices.begin() + index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.erase(adjMat.begin() + index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.erase(row.begin() + index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    void addEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    void removeEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n        }\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    void print() {\n        cout &lt;&lt; \"\u9876\u70b9\u5217\u8868 = \";\n        printVector(vertices);\n        cout &lt;&lt; \"\u90bb\u63a5\u77e9\u9635 =\" &lt;&lt; endl;\n        printVectorMatrix(adjMat);\n    }\n};\n</code></pre> graph_adjacency_matrix.java<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    List&lt;Integer&gt; vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    List&lt;List&lt;Integer&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = new ArrayList&lt;&gt;();\n        this.adjMat = new ArrayList&lt;&gt;();\n        // \u6dfb\u52a0\u9876\u70b9\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (int[] e : edges) {\n            addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    public int size() {\n        return vertices.size();\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void addVertex(int val) {\n        int n = size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.add(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        List&lt;Integer&gt; newRow = new ArrayList&lt;&gt;(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.add(0);\n        }\n        adjMat.add(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.add(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void removeVertex(int index) {\n        if (index &gt;= size())\n            throw new IndexOutOfBoundsException();\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.remove(index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void addEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat.get(i).set(j, 1);\n        adjMat.get(j).set(i, 1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void removeEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        adjMat.get(i).set(j, 0);\n        adjMat.get(j).set(i, 0);\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    public void print() {\n        System.out.print(\"\u9876\u70b9\u5217\u8868 = \");\n        System.out.println(vertices);\n        System.out.println(\"\u90bb\u63a5\u77e9\u9635 =\");\n        PrintUtil.printMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.cs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    List&lt;int&gt; vertices;     // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    List&lt;List&lt;int&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u6dfb\u52a0\u9876\u70b9\n        foreach (int val in vertices) {\n            AddVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        foreach (int[] e in edges) {\n            AddEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int Size() {\n        return vertices.Count;\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void AddVertex(int val) {\n        int n = Size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.Add(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        List&lt;int&gt; newRow = new(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.Add(0);\n        }\n        adjMat.Add(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.Add(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void RemoveVertex(int index) {\n        if (index &gt;= Size())\n            throw new IndexOutOfRangeException();\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.RemoveAt(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.RemoveAt(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.RemoveAt(index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void AddEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void RemoveEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    public void Print() {\n        Console.Write(\"\u9876\u70b9\u5217\u8868 = \");\n        PrintUtil.PrintList(vertices);\n        Console.WriteLine(\"\u90bb\u63a5\u77e9\u9635 =\");\n        PrintUtil.PrintMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.go<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjMat struct {\n    // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    vertices []int\n    // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    adjMat [][]int\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n    // \u6dfb\u52a0\u9876\u70b9\n    n := len(vertices)\n    adjMat := make([][]int, n)\n    for i := range adjMat {\n        adjMat[i] = make([]int, n)\n    }\n    // \u521d\u59cb\u5316\u56fe\n    g := &amp;graphAdjMat{\n        vertices: vertices,\n        adjMat:   adjMat,\n    }\n    // \u6dfb\u52a0\u8fb9\n    // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    for i := range edges {\n        g.addEdge(edges[i][0], edges[i][1])\n    }\n    return g\n}\n\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjMat) size() int {\n    return len(g.vertices)\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjMat) addVertex(val int) {\n    n := g.size()\n    // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n    g.vertices = append(g.vertices, val)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n    newRow := make([]int, n)\n    g.adjMat = append(g.adjMat, newRow)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i], 0)\n    }\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjMat) removeVertex(index int) {\n    if index &gt;= g.size() {\n        return\n    }\n    // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n    g.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n    g.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n    }\n}\n\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) addEdge(i, j int) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n    g.adjMat[i][j] = 1\n    g.adjMat[j][i] = 1\n}\n\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    g.adjMat[i][j] = 0\n    g.adjMat[j][i] = 0\n}\n\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc (g *graphAdjMat) print() {\n    fmt.Printf(\"\\t\u9876\u70b9\u5217\u8868 = %v\\n\", g.vertices)\n    fmt.Printf(\"\\t\u90bb\u63a5\u77e9\u9635 = \\n\")\n    for i := range g.adjMat {\n        fmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n    }\n}\n</code></pre> graph_adjacency_matrix.swift<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    private var vertices: [Int] // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    private var adjMat: [[Int]] // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init(vertices: [Int], edges: [[Int]]) {\n        self.vertices = []\n        adjMat = []\n        // \u6dfb\u52a0\u9876\u70b9\n        for val in vertices {\n            addVertex(val: val)\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for e in edges {\n            addEdge(i: e[0], j: e[1])\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    func size() -&gt; Int {\n        vertices.count\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    func addVertex(val: Int) {\n        let n = size()\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.append(val)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        let newRow = Array(repeating: 0, count: n)\n        adjMat.append(newRow)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for i in adjMat.indices {\n            adjMat[i].append(0)\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    func removeVertex(index: Int) {\n        if index &gt;= size() {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.remove(at: index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.remove(at: index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for i in adjMat.indices {\n            adjMat[i].remove(at: index)\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    func addEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    func removeEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    func print() {\n        Swift.print(\"\u9876\u70b9\u5217\u8868 = \", terminator: \"\")\n        Swift.print(vertices)\n        Swift.print(\"\u90bb\u63a5\u77e9\u9635 =\")\n        PrintUtil.printMatrix(matrix: adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.js<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    constructor(vertices, edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u6dfb\u52a0\u9876\u70b9\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size() {\n        return this.vertices.length;\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(val) {\n        const n = this.size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        this.vertices.push(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        const newRow = [];\n        for (let j = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(index) {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        this.vertices.splice(index, 1);\n\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        this.adjMat.splice(index, 1);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    addEdge(i, j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    removeEdge(i, j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    print() {\n        console.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\n        console.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.ts<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    vertices: number[]; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    adjMat: number[][]; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    constructor(vertices: number[], edges: number[][]) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u6dfb\u52a0\u9876\u70b9\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size(): number {\n        return this.vertices.length;\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(val: number): void {\n        const n: number = this.size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        this.vertices.push(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        const newRow: number[] = [];\n        for (let j: number = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(index: number): void {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        this.vertices.splice(index, 1);\n\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        this.adjMat.splice(index, 1);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    addEdge(i: number, j: number): void {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    removeEdge(i: number, j: number): void {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    print(): void {\n        console.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\n        console.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.dart<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n  List&lt;int&gt; vertices = []; // \u9876\u70b9\u5143\u7d20\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n  List&lt;List&lt;int&gt;&gt; adjMat = []; //\u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  GraphAdjMat(List&lt;int&gt; vertices, List&lt;List&lt;int&gt;&gt; edges) {\n    this.vertices = [];\n    this.adjMat = [];\n    // \u6dfb\u52a0\u9876\u70b9\n    for (int val in vertices) {\n      addVertex(val);\n    }\n    // \u6dfb\u52a0\u8fb9\n    // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    for (List&lt;int&gt; e in edges) {\n      addEdge(e[0], e[1]);\n    }\n  }\n\n  /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n  int size() {\n    return vertices.length;\n  }\n\n  /* \u6dfb\u52a0\u9876\u70b9 */\n  void addVertex(int val) {\n    int n = size();\n    // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n    vertices.add(val);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n    List&lt;int&gt; newRow = List.filled(n, 0, growable: true);\n    adjMat.add(newRow);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n    for (List&lt;int&gt; row in adjMat) {\n      row.add(0);\n    }\n  }\n\n  /* \u5220\u9664\u9876\u70b9 */\n  void removeVertex(int index) {\n    if (index &gt;= size()) {\n      throw IndexError;\n    }\n    // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n    vertices.removeAt(index);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n    adjMat.removeAt(index);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n    for (List&lt;int&gt; row in adjMat) {\n      row.removeAt(index);\n    }\n  }\n\n  /* \u6dfb\u52a0\u8fb9 */\n  // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n  void addEdge(int i, int j) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n    adjMat[i][j] = 1;\n    adjMat[j][i] = 1;\n  }\n\n  /* \u5220\u9664\u8fb9 */\n  // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n  void removeEdge(int i, int j) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    adjMat[i][j] = 0;\n    adjMat[j][i] = 0;\n  }\n\n  /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n  void printAdjMat() {\n    print(\"\u9876\u70b9\u5217\u8868 = $vertices\");\n    print(\"\u90bb\u63a5\u77e9\u9635 = \");\n    printMatrix(adjMat);\n  }\n}\n</code></pre> graph_adjacency_matrix.rs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjMat {\n    // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    pub vertices: Vec&lt;i32&gt;,\n    // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    pub adj_mat: Vec&lt;Vec&lt;i32&gt;&gt;,\n}\n\nimpl GraphAdjMat {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(vertices: Vec&lt;i32&gt;, edges: Vec&lt;[usize; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjMat {\n            vertices: vec![],\n            adj_mat: vec![],\n        };\n        // \u6dfb\u52a0\u9876\u70b9\n        for val in vertices {\n            graph.add_vertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for edge in edges {\n            graph.add_edge(edge[0], edge[1])\n        }\n\n        graph\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    pub fn size(&amp;self) -&gt; usize {\n        self.vertices.len()\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    pub fn add_vertex(&amp;mut self, val: i32) {\n        let n = self.size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        self.vertices.push(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        self.adj_mat.push(vec![0; n]);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for row in &amp;mut self.adj_mat {\n            row.push(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    pub fn remove_vertex(&amp;mut self, index: usize) {\n        if index &gt;= self.size() {\n            panic!(\"index error\")\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        self.vertices.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        self.adj_mat.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for row in &amp;mut self.adj_mat {\n            row.remove(index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    pub fn add_edge(&amp;mut self, i: usize, j: usize) {\n        // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        self.adj_mat[i][j] = 1;\n        self.adj_mat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    pub fn remove_edge(&amp;mut self, i: usize, j: usize) {\n        // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        self.adj_mat[i][j] = 0;\n        self.adj_mat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    pub fn print(&amp;self) {\n        println!(\"\u9876\u70b9\u5217\u8868 = {:?}\", self.vertices);\n        println!(\"\u90bb\u63a5\u77e9\u9635 =\");\n        println!(\"[\");\n        for row in &amp;self.adj_mat {\n            println!(\"  {:?},\", row);\n        }\n        println!(\"]\")\n    }\n}\n</code></pre> graph_adjacency_matrix.c<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7ed3\u6784\u4f53 */\ntypedef struct {\n    int vertices[MAX_SIZE];\n    int adjMat[MAX_SIZE][MAX_SIZE];\n    int size;\n} GraphAdjMat;\n\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjMat *newGraphAdjMat() {\n    GraphAdjMat *graph = (GraphAdjMat *)malloc(sizeof(GraphAdjMat));\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        for (int j = 0; j &lt; MAX_SIZE; j++) {\n            graph-&gt;adjMat[i][j] = 0;\n        }\n    }\n    return graph;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjMat(GraphAdjMat *graph) {\n    free(graph);\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjMat *graph, int val) {\n    if (graph-&gt;size == MAX_SIZE) {\n        fprintf(stderr, \"\u56fe\u7684\u9876\u70b9\u6570\u91cf\u5df2\u8fbe\u6700\u5927\u503c\\n\");\n        return;\n    }\n    // \u6dfb\u52a0\u7b2c n \u4e2a\u9876\u70b9\uff0c\u5e76\u5c06\u7b2c n \u884c\u548c\u5217\u7f6e\u96f6\n    int n = graph-&gt;size;\n    graph-&gt;vertices[n] = val;\n    for (int i = 0; i &lt;= n; i++) {\n        graph-&gt;adjMat[n][i] = graph-&gt;adjMat[i][n] = 0;\n    }\n    graph-&gt;size++;\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjMat *graph, int index) {\n    if (index &lt; 0 || index &gt;= graph-&gt;size) {\n        fprintf(stderr, \"\u9876\u70b9\u7d22\u5f15\u8d8a\u754c\\n\");\n        return;\n    }\n    // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        graph-&gt;vertices[i] = graph-&gt;vertices[i + 1];\n    }\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        for (int j = 0; j &lt; graph-&gt;size; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i + 1][j];\n        }\n    }\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        for (int j = index; j &lt; graph-&gt;size - 1; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i][j + 1];\n        }\n    }\n    graph-&gt;size--;\n}\n\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 1;\n    graph-&gt;adjMat[j][i] = 1;\n}\n\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 0;\n    graph-&gt;adjMat[j][i] = 0;\n}\n\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid printGraphAdjMat(GraphAdjMat *graph) {\n    printf(\"\u9876\u70b9\u5217\u8868 = \");\n    printArray(graph-&gt;vertices, graph-&gt;size);\n    printf(\"\u90bb\u63a5\u77e9\u9635 =\\n\");\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        printArray(graph-&gt;adjMat[i], graph-&gt;size);\n    }\n}\n</code></pre> graph_adjacency_matrix.kt<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat(vertices: IntArray, edges: Array&lt;IntArray&gt;) {\n    val vertices: MutableList&lt;Int&gt; = ArrayList() // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    val adjMat: MutableList&lt;MutableList&lt;Int&gt;&gt; = ArrayList() // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    init {\n        // \u6dfb\u52a0\u9876\u70b9\n        for (vertex in vertices) {\n            addVertex(vertex)\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (edge in edges) {\n            addEdge(edge[0], edge[1])\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    fun size(): Int {\n        return vertices.size\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    fun addVertex(value: Int) {\n        val n = size()\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.add(value)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        val newRow: MutableList&lt;Int&gt; = mutableListOf()\n        for (j in 0..&lt;n) {\n            newRow.add(0)\n        }\n        adjMat.add(newRow)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (row in adjMat) {\n            row.add(0)\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    fun removeVertex(index: Int) {\n        if (index &gt;= size()) throw IndexOutOfBoundsException()\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.removeAt(index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.removeAt(index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (row in adjMat) {\n            row.removeAt(index)\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    fun addEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) throw java.lang.IndexOutOfBoundsException()\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    fun removeEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) throw java.lang.IndexOutOfBoundsException()\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    fun print() {\n        print(\"\u9876\u70b9\u5217\u8868 = \")\n        println(vertices);\n        println(\"\u90bb\u63a5\u77e9\u9635 =\");\n        printMatrix(adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.rb<pre><code>[class]{GraphAdjMat}-[func]{}\n</code></pre> graph_adjacency_matrix.zig<pre><code>[class]{GraphAdjMat}-[func]{}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_graph/graph_operations/#922","title":"9.2.2 \u00a0 \u57fa\u4e8e\u90bb\u63a5\u8868\u7684\u5b9e\u73b0","text":"<p>\u8bbe\u65e0\u5411\u56fe\u7684\u9876\u70b9\u603b\u6570\u4e3a \\(n\\)\u3001\u8fb9\u603b\u6570\u4e3a \\(m\\) \uff0c\u5219\u53ef\u6839\u636e\u56fe 9-8 \u6240\u793a\u7684\u65b9\u6cd5\u5b9e\u73b0\u5404\u79cd\u64cd\u4f5c\u3002</p> <ul> <li>\u6dfb\u52a0\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u7684\u672b\u5c3e\u6dfb\u52a0\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u4e3a\u662f\u65e0\u5411\u56fe\uff0c\u6240\u4ee5\u9700\u8981\u540c\u65f6\u6dfb\u52a0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002</li> <li>\u5220\u9664\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u4e2d\u67e5\u627e\u5e76\u5220\u9664\u6307\u5b9a\u8fb9\uff0c\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\u3002\u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u9700\u8981\u540c\u65f6\u5220\u9664\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002</li> <li>\u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u94fe\u8868\uff0c\u5e76\u5c06\u65b0\u589e\u9876\u70b9\u4f5c\u4e3a\u94fe\u8868\u5934\u8282\u70b9\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002</li> <li>\u5220\u9664\u9876\u70b9\uff1a\u9700\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5220\u9664\u5305\u542b\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002</li> <li>\u521d\u59cb\u5316\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u521b\u5efa \\(n\\) \u4e2a\u9876\u70b9\u548c \\(2m\\) \u6761\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002</li> </ul> \u521d\u59cb\u5316\u90bb\u63a5\u8868\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9 <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-8 \u00a0 \u90bb\u63a5\u8868\u7684\u521d\u59cb\u5316\u3001\u589e\u5220\u8fb9\u3001\u589e\u5220\u9876\u70b9 </p> <p>\u4ee5\u4e0b\u662f\u90bb\u63a5\u8868\u7684\u4ee3\u7801\u5b9e\u73b0\u3002\u5bf9\u6bd4\u56fe 9-8 \uff0c\u5b9e\u9645\u4ee3\u7801\u6709\u4ee5\u4e0b\u4e0d\u540c\u3002</p> <ul> <li>\u4e3a\u4e86\u65b9\u4fbf\u6dfb\u52a0\u4e0e\u5220\u9664\u9876\u70b9\uff0c\u4ee5\u53ca\u7b80\u5316\u4ee3\u7801\uff0c\u6211\u4eec\u4f7f\u7528\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u6765\u4ee3\u66ff\u94fe\u8868\u3002</li> <li>\u4f7f\u7528\u54c8\u5e0c\u8868\u6765\u5b58\u50a8\u90bb\u63a5\u8868\uff0c<code>key</code> \u4e3a\u9876\u70b9\u5b9e\u4f8b\uff0c<code>value</code> \u4e3a\u8be5\u9876\u70b9\u7684\u90bb\u63a5\u9876\u70b9\u5217\u8868\uff08\u94fe\u8868\uff09\u3002</li> </ul> <p>\u53e6\u5916\uff0c\u6211\u4eec\u5728\u90bb\u63a5\u8868\u4e2d\u4f7f\u7528 <code>Vertex</code> \u7c7b\u6765\u8868\u793a\u9876\u70b9\uff0c\u8fd9\u6837\u505a\u7684\u539f\u56e0\u662f\uff1a\u5982\u679c\u4e0e\u90bb\u63a5\u77e9\u9635\u4e00\u6837\uff0c\u7528\u5217\u8868\u7d22\u5f15\u6765\u533a\u5206\u4e0d\u540c\u9876\u70b9\uff0c\u90a3\u4e48\u5047\u8bbe\u8981\u5220\u9664\u7d22\u5f15\u4e3a \\(i\\) \u7684\u9876\u70b9\uff0c\u5219\u9700\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5c06\u6240\u6709\u5927\u4e8e \\(i\\) \u7684\u7d22\u5f15\u5168\u90e8\u51cf \\(1\\) \uff0c\u6548\u7387\u5f88\u4f4e\u3002\u800c\u5982\u679c\u6bcf\u4e2a\u9876\u70b9\u90fd\u662f\u552f\u4e00\u7684 <code>Vertex</code> \u5b9e\u4f8b\uff0c\u5220\u9664\u67d0\u4e00\u9876\u70b9\u4e4b\u540e\u5c31\u65e0\u987b\u6539\u52a8\u5176\u4ed6\u9876\u70b9\u4e86\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_adjacency_list.py<pre><code>class GraphAdjList:\n    \"\"\"\u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n\n    def __init__(self, edges: list[list[Vertex]]):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        # \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        self.adj_list = dict[Vertex, list[Vertex]]()\n        # \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for edge in edges:\n            self.add_vertex(edge[0])\n            self.add_vertex(edge[1])\n            self.add_edge(edge[0], edge[1])\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\n        return len(self.adj_list)\n\n    def add_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # \u6dfb\u52a0\u8fb9 vet1 - vet2\n        self.adj_list[vet1].append(vet2)\n        self.adj_list[vet2].append(vet1)\n\n    def remove_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"\u5220\u9664\u8fb9\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # \u5220\u9664\u8fb9 vet1 - vet2\n        self.adj_list[vet1].remove(vet2)\n        self.adj_list[vet2].remove(vet1)\n\n    def add_vertex(self, vet: Vertex):\n        \"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\n        if vet in self.adj_list:\n            return\n        # \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        self.adj_list[vet] = []\n\n    def remove_vertex(self, vet: Vertex):\n        \"\"\"\u5220\u9664\u9876\u70b9\"\"\"\n        if vet not in self.adj_list:\n            raise ValueError()\n        # \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        self.adj_list.pop(vet)\n        # \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for vertex in self.adj_list:\n            if vet in self.adj_list[vertex]:\n                self.adj_list[vertex].remove(vet)\n\n    def print(self):\n        \"\"\"\u6253\u5370\u90bb\u63a5\u8868\"\"\"\n        print(\"\u90bb\u63a5\u8868 =\")\n        for vertex in self.adj_list:\n            tmp = [v.val for v in self.adj_list[vertex]]\n            print(f\"{vertex.val}: {tmp},\")\n</code></pre> graph_adjacency_list.cpp<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n  public:\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    unordered_map&lt;Vertex *, vector&lt;Vertex *&gt;&gt; adjList;\n\n    /* \u5728 vector \u4e2d\u5220\u9664\u6307\u5b9a\u8282\u70b9 */\n    void remove(vector&lt;Vertex *&gt; &amp;vec, Vertex *vet) {\n        for (int i = 0; i &lt; vec.size(); i++) {\n            if (vec[i] == vet) {\n                vec.erase(vec.begin() + i);\n                break;\n            }\n        }\n    }\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    GraphAdjList(const vector&lt;vector&lt;Vertex *&gt;&gt; &amp;edges) {\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (const vector&lt;Vertex *&gt; &amp;edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int size() {\n        return adjList.size();\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    void addEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1].push_back(vet2);\n        adjList[vet2].push_back(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    void removeEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        remove(adjList[vet1], vet2);\n        remove(adjList[vet2], vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    void addVertex(Vertex *vet) {\n        if (adjList.count(vet))\n            return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList[vet] = vector&lt;Vertex *&gt;();\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    void removeVertex(Vertex *vet) {\n        if (!adjList.count(vet))\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.erase(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (auto &amp;adj : adjList) {\n            remove(adj.second, vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    void print() {\n        cout &lt;&lt; \"\u90bb\u63a5\u8868 =\" &lt;&lt; endl;\n        for (auto &amp;adj : adjList) {\n            const auto &amp;key = adj.first;\n            const auto &amp;vec = adj.second;\n            cout &lt;&lt; key-&gt;val &lt;&lt; \": \";\n            printVector(vetsToVals(vec));\n        }\n    }\n};\n</code></pre> graph_adjacency_list.java<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public GraphAdjList(Vertex[][] edges) {\n        this.adjList = new HashMap&lt;&gt;();\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (Vertex[] edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    public int size() {\n        return adjList.size();\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    public void addEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList.get(vet1).add(vet2);\n        adjList.get(vet2).add(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    public void removeEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList.get(vet1).remove(vet2);\n        adjList.get(vet2).remove(vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void addVertex(Vertex vet) {\n        if (adjList.containsKey(vet))\n            return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList.put(vet, new ArrayList&lt;&gt;());\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void removeVertex(Vertex vet) {\n        if (!adjList.containsKey(vet))\n            throw new IllegalArgumentException();\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.remove(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (List&lt;Vertex&gt; list : adjList.values()) {\n            list.remove(vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    public void print() {\n        System.out.println(\"\u90bb\u63a5\u8868 =\");\n        for (Map.Entry&lt;Vertex, List&lt;Vertex&gt;&gt; pair : adjList.entrySet()) {\n            List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n            for (Vertex vertex : pair.getValue())\n                tmp.add(vertex.val);\n            System.out.println(pair.getKey().val + \": \" + tmp + \",\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.cs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    public Dictionary&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* \u6784\u9020\u51fd\u6570 */\n    public GraphAdjList(Vertex[][] edges) {\n        adjList = [];\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        foreach (Vertex[] edge in edges) {\n            AddVertex(edge[0]);\n            AddVertex(edge[1]);\n            AddEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int Size() {\n        return adjList.Count;\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    public void AddEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1].Add(vet2);\n        adjList[vet2].Add(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    public void RemoveEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList[vet1].Remove(vet2);\n        adjList[vet2].Remove(vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void AddVertex(Vertex vet) {\n        if (adjList.ContainsKey(vet))\n            return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList.Add(vet, []);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void RemoveVertex(Vertex vet) {\n        if (!adjList.ContainsKey(vet))\n            throw new InvalidOperationException();\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.Remove(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        foreach (List&lt;Vertex&gt; list in adjList.Values) {\n            list.Remove(vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    public void Print() {\n        Console.WriteLine(\"\u90bb\u63a5\u8868 =\");\n        foreach (KeyValuePair&lt;Vertex, List&lt;Vertex&gt;&gt; pair in adjList) {\n            List&lt;int&gt; tmp = [];\n            foreach (Vertex vertex in pair.Value)\n                tmp.Add(vertex.val);\n            Console.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.go<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjList struct {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    adjList map[Vertex][]Vertex\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\n    g := &amp;graphAdjList{\n        adjList: make(map[Vertex][]Vertex),\n    }\n    // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n    for _, edge := range edges {\n        g.addVertex(edge[0])\n        g.addVertex(edge[1])\n        g.addEdge(edge[0], edge[1])\n    }\n    return g\n}\n\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjList) size() int {\n    return len(g.adjList)\n}\n\n/* \u6dfb\u52a0\u8fb9 */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // \u6dfb\u52a0\u8fb9 vet1 - vet2, \u6dfb\u52a0\u533f\u540d struct{},\n    g.adjList[vet1] = append(g.adjList[vet1], vet2)\n    g.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n\n/* \u5220\u9664\u8fb9 */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // \u5220\u9664\u8fb9 vet1 - vet2\n    g.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2)\n    g.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1)\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if ok {\n        return\n    }\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n    g.adjList[vet] = make([]Vertex, 0)\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if !ok {\n        panic(\"error\")\n    }\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n    delete(g.adjList, vet)\n    // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n    for v, list := range g.adjList {\n        g.adjList[v] = DeleteSliceElms(list, vet)\n    }\n}\n\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nfunc (g *graphAdjList) print() {\n    var builder strings.Builder\n    fmt.Printf(\"\u90bb\u63a5\u8868 = \\n\")\n    for k, v := range g.adjList {\n        builder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\n        for _, vet := range v {\n            builder.WriteString(strconv.Itoa(vet.Val) + \" \")\n        }\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> graph_adjacency_list.swift<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    public private(set) var adjList: [Vertex: [Vertex]]\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public init(edges: [[Vertex]]) {\n        adjList = [:]\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for edge in edges {\n            addVertex(vet: edge[0])\n            addVertex(vet: edge[1])\n            addEdge(vet1: edge[0], vet2: edge[1])\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    public func size() -&gt; Int {\n        adjList.count\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    public func addEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"\u53c2\u6570\u9519\u8bef\")\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1]?.append(vet2)\n        adjList[vet2]?.append(vet1)\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    public func removeEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"\u53c2\u6570\u9519\u8bef\")\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList[vet1]?.removeAll { $0 == vet2 }\n        adjList[vet2]?.removeAll { $0 == vet1 }\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public func addVertex(vet: Vertex) {\n        if adjList[vet] != nil {\n            return\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList[vet] = []\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public func removeVertex(vet: Vertex) {\n        if adjList[vet] == nil {\n            fatalError(\"\u53c2\u6570\u9519\u8bef\")\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.removeValue(forKey: vet)\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for key in adjList.keys {\n            adjList[key]?.removeAll { $0 == vet }\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    public func print() {\n        Swift.print(\"\u90bb\u63a5\u8868 =\")\n        for (vertex, list) in adjList {\n            let list = list.map { $0.val }\n            Swift.print(\"\\(vertex.val): \\(list),\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.js<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    adjList;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(edges) {\n        this.adjList = new Map();\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size() {\n        return this.adjList.size;\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    addEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    removeEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(vet) {\n        if (this.adjList.has(vet)) return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        this.adjList.set(vet, []);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(vet) {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        this.adjList.delete(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (const set of this.adjList.values()) {\n            const index = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    print() {\n        console.log('\u90bb\u63a5\u8868 =');\n        for (const [key, value] of this.adjList) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.ts<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    adjList: Map&lt;Vertex, Vertex[]&gt;;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(edges: Vertex[][]) {\n        this.adjList = new Map();\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size(): number {\n        return this.adjList.size;\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    addEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    removeEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(vet: Vertex): void {\n        if (this.adjList.has(vet)) return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        this.adjList.set(vet, []);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(vet: Vertex): void {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        this.adjList.delete(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (const set of this.adjList.values()) {\n            const index: number = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    print(): void {\n        console.log('\u90bb\u63a5\u8868 =');\n        for (const [key, value] of this.adjList.entries()) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.dart<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n  // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n  Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList = {};\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  GraphAdjList(List&lt;List&lt;Vertex&gt;&gt; edges) {\n    for (List&lt;Vertex&gt; edge in edges) {\n      addVertex(edge[0]);\n      addVertex(edge[1]);\n      addEdge(edge[0], edge[1]);\n    }\n  }\n\n  /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n  int size() {\n    return adjList.length;\n  }\n\n  /* \u6dfb\u52a0\u8fb9 */\n  void addEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // \u6dfb\u52a0\u8fb9 vet1 - vet2\n    adjList[vet1]!.add(vet2);\n    adjList[vet2]!.add(vet1);\n  }\n\n  /* \u5220\u9664\u8fb9 */\n  void removeEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // \u5220\u9664\u8fb9 vet1 - vet2\n    adjList[vet1]!.remove(vet2);\n    adjList[vet2]!.remove(vet1);\n  }\n\n  /* \u6dfb\u52a0\u9876\u70b9 */\n  void addVertex(Vertex vet) {\n    if (adjList.containsKey(vet)) return;\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n    adjList[vet] = [];\n  }\n\n  /* \u5220\u9664\u9876\u70b9 */\n  void removeVertex(Vertex vet) {\n    if (!adjList.containsKey(vet)) {\n      throw ArgumentError;\n    }\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n    adjList.remove(vet);\n    // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n    adjList.forEach((key, value) {\n      value.remove(vet);\n    });\n  }\n\n  /* \u6253\u5370\u90bb\u63a5\u8868 */\n  void printAdjList() {\n    print(\"\u90bb\u63a5\u8868 =\");\n    adjList.forEach((key, value) {\n      List&lt;int&gt; tmp = [];\n      for (Vertex vertex in value) {\n        tmp.add(vertex.val);\n      }\n      print(\"${key.val}: $tmp,\");\n    });\n  }\n}\n</code></pre> graph_adjacency_list.rs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    pub adj_list: HashMap&lt;Vertex, Vec&lt;Vertex&gt;&gt;,\n}\n\nimpl GraphAdjList {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(edges: Vec&lt;[Vertex; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjList {\n            adj_list: HashMap::new(),\n        };\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for edge in edges {\n            graph.add_vertex(edge[0]);\n            graph.add_vertex(edge[1]);\n            graph.add_edge(edge[0], edge[1]);\n        }\n\n        graph\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    #[allow(unused)]\n    pub fn size(&amp;self) -&gt; usize {\n        self.adj_list.len()\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    pub fn add_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n        {\n            panic!(\"value error\");\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        self.adj_list.get_mut(&amp;vet1).unwrap().push(vet2);\n        self.adj_list.get_mut(&amp;vet2).unwrap().push(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    #[allow(unused)]\n    pub fn remove_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n        {\n            panic!(\"value error\");\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        self.adj_list\n            .get_mut(&amp;vet1)\n            .unwrap()\n            .retain(|&amp;vet| vet != vet2);\n        self.adj_list\n            .get_mut(&amp;vet2)\n            .unwrap()\n            .retain(|&amp;vet| vet != vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    pub fn add_vertex(&amp;mut self, vet: Vertex) {\n        if self.adj_list.contains_key(&amp;vet) {\n            return;\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        self.adj_list.insert(vet, vec![]);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    #[allow(unused)]\n    pub fn remove_vertex(&amp;mut self, vet: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet) {\n            panic!(\"value error\");\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        self.adj_list.remove(&amp;vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for list in self.adj_list.values_mut() {\n            list.retain(|&amp;v| v != vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    pub fn print(&amp;self) {\n        println!(\"\u90bb\u63a5\u8868 =\");\n        for (vertex, list) in &amp;self.adj_list {\n            let list = list.iter().map(|vertex| vertex.val).collect::&lt;Vec&lt;i32&gt;&gt;();\n            println!(\"{}: {:?},\", vertex.val, list);\n        }\n    }\n}\n</code></pre> graph_adjacency_list.c<pre><code>/* \u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct AdjListNode {\n    Vertex *vertex;           // \u9876\u70b9\n    struct AdjListNode *next; // \u540e\u7ee7\u8282\u70b9\n} AdjListNode;\n\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* \u6dfb\u52a0\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid addEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *node = (AdjListNode *)malloc(sizeof(AdjListNode));\n    node-&gt;vertex = vet;\n    // \u5934\u63d2\u6cd5\n    node-&gt;next = head-&gt;next;\n    head-&gt;next = node;\n}\n\n/* \u5220\u9664\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid removeEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *pre = head;\n    AdjListNode *cur = head-&gt;next;\n    // \u5728\u94fe\u8868\u4e2d\u641c\u7d22 vet \u5bf9\u5e94\u8282\u70b9\n    while (cur != NULL &amp;&amp; cur-&gt;vertex != vet) {\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n    if (cur == NULL)\n        return;\n    // \u5c06 vet \u5bf9\u5e94\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\n    pre-&gt;next = cur-&gt;next;\n    // \u91ca\u653e\u5185\u5b58\n    free(cur);\n}\n\n/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntypedef struct {\n    AdjListNode *heads[MAX_SIZE]; // \u8282\u70b9\u6570\u7ec4\n    int size;                     // \u8282\u70b9\u6570\u91cf\n} GraphAdjList;\n\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjList *newGraphAdjList() {\n    GraphAdjList *graph = (GraphAdjList *)malloc(sizeof(GraphAdjList));\n    if (!graph) {\n        return NULL;\n    }\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        graph-&gt;heads[i] = NULL;\n    }\n    return graph;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjList(GraphAdjList *graph) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        AdjListNode *cur = graph-&gt;heads[i];\n        while (cur != NULL) {\n            AdjListNode *next = cur-&gt;next;\n            if (cur != graph-&gt;heads[i]) {\n                free(cur);\n            }\n            cur = next;\n        }\n        free(graph-&gt;heads[i]-&gt;vertex);\n        free(graph-&gt;heads[i]);\n    }\n    free(graph);\n}\n\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL &amp;&amp; head1 != head2);\n    // \u6dfb\u52a0\u8fb9 vet1 - vet2\n    addEdgeHelper(head1, vet2);\n    addEdgeHelper(head2, vet1);\n}\n\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL);\n    // \u5220\u9664\u8fb9 vet1 - vet2\n    removeEdgeHelper(head1, head2-&gt;vertex);\n    removeEdgeHelper(head2, head1-&gt;vertex);\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjList *graph, Vertex *vet) {\n    assert(graph != NULL &amp;&amp; graph-&gt;size &lt; MAX_SIZE);\n    AdjListNode *head = (AdjListNode *)malloc(sizeof(AdjListNode));\n    head-&gt;vertex = vet;\n    head-&gt;next = NULL;\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n    graph-&gt;heads[graph-&gt;size++] = head;\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjList *graph, Vertex *vet) {\n    AdjListNode *node = findNode(graph, vet);\n    assert(node != NULL);\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n    AdjListNode *cur = node, *pre = NULL;\n    while (cur) {\n        pre = cur;\n        cur = cur-&gt;next;\n        free(pre);\n    }\n    // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        cur = graph-&gt;heads[i];\n        pre = NULL;\n        while (cur) {\n            pre = cur;\n            cur = cur-&gt;next;\n            if (cur &amp;&amp; cur-&gt;vertex == vet) {\n                pre-&gt;next = cur-&gt;next;\n                free(cur);\n                break;\n            }\n        }\n    }\n    // \u5c06\u8be5\u9876\u70b9\u4e4b\u540e\u7684\u9876\u70b9\u5411\u524d\u79fb\u52a8\uff0c\u4ee5\u586b\u8865\u7a7a\u7f3a\n    int i;\n    for (i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i] == node)\n            break;\n    }\n    for (int j = i; j &lt; graph-&gt;size - 1; j++) {\n        graph-&gt;heads[j] = graph-&gt;heads[j + 1];\n    }\n    graph-&gt;size--;\n    free(vet);\n}\n</code></pre> graph_adjacency_list.kt<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList(edges: Array&lt;Array&lt;Vertex?&gt;&gt;) {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    val adjList: MutableMap&lt;Vertex, MutableList&lt;Vertex&gt;&gt; = HashMap()\n\n    /* \u6784\u9020\u51fd\u6570 */\n    init {\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (edge in edges) {\n            addVertex(edge[0]!!);\n            addVertex(edge[1]!!);\n            addEdge(edge[0]!!, edge[1]!!);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    fun size(): Int {\n        return adjList.size\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    fun addEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1]?.add(vet2)\n        adjList[vet2]?.add(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    fun removeEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList[vet1]?.remove(vet2);\n        adjList[vet2]?.remove(vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    fun addVertex(vet: Vertex) {\n        if (adjList.containsKey(vet))\n            return\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList[vet] = mutableListOf()\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    fun removeVertex(vet: Vertex) {\n        if (!adjList.containsKey(vet))\n            throw IllegalArgumentException()\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.remove(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (list in adjList.values) {\n            list.remove(vet)\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    fun print() {\n        println(\"\u90bb\u63a5\u8868 =\")\n        for (pair in adjList.entries) {\n            val tmp = ArrayList&lt;Int&gt;()\n            for (vertex in pair.value) {\n                tmp.add(vertex.value)\n            }\n            println(\"${pair.key.value}: $tmp,\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.rb<pre><code>[class]{GraphAdjList}-[func]{}\n</code></pre> graph_adjacency_list.zig<pre><code>[class]{GraphAdjList}-[func]{}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_graph/graph_operations/#923","title":"9.2.3 \u00a0 \u6548\u7387\u5bf9\u6bd4","text":"<p>\u8bbe\u56fe\u4e2d\u5171\u6709 \\(n\\) \u4e2a\u9876\u70b9\u548c \\(m\\) \u6761\u8fb9\uff0c\u8868 9-2 \u5bf9\u6bd4\u4e86\u90bb\u63a5\u77e9\u9635\u548c\u90bb\u63a5\u8868\u7684\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u3002</p> <p> \u8868 9-2 \u00a0 \u90bb\u63a5\u77e9\u9635\u4e0e\u90bb\u63a5\u8868\u5bf9\u6bd4 </p> \u90bb\u63a5\u77e9\u9635 \u90bb\u63a5\u8868\uff08\u94fe\u8868\uff09 \u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09 \u5224\u65ad\u662f\u5426\u90bb\u63a5 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u8fb9 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u8fb9 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u9876\u70b9 \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u9876\u70b9 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) \u5185\u5b58\u7a7a\u95f4\u5360\u7528 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\) <p>\u89c2\u5bdf\u8868 9-2 \uff0c\u4f3c\u4e4e\u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09\u7684\u65f6\u95f4\u6548\u7387\u4e0e\u7a7a\u95f4\u6548\u7387\u6700\u4f18\u3002\u4f46\u5b9e\u9645\u4e0a\uff0c\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u64cd\u4f5c\u8fb9\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u53ea\u9700\u4e00\u6b21\u6570\u7ec4\u8bbf\u95ee\u6216\u8d4b\u503c\u64cd\u4f5c\u5373\u53ef\u3002\u7efc\u5408\u6765\u770b\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u4e86\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u7684\u539f\u5219\uff0c\u800c\u90bb\u63a5\u8868\u4f53\u73b0\u4e86\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u7684\u539f\u5219\u3002</p>"},{"location":"chapter_graph/graph_traversal/","title":"9.3 \u00a0 \u56fe\u7684\u904d\u5386","text":"<p>\u6811\u4ee3\u8868\u7684\u662f\u201c\u4e00\u5bf9\u591a\u201d\u7684\u5173\u7cfb\uff0c\u800c\u56fe\u5219\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u53ef\u4ee5\u8868\u793a\u4efb\u610f\u7684\u201c\u591a\u5bf9\u591a\u201d\u5173\u7cfb\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u6811\u770b\u4f5c\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\u3002\u663e\u7136\uff0c\u6811\u7684\u904d\u5386\u64cd\u4f5c\u4e5f\u662f\u56fe\u7684\u904d\u5386\u64cd\u4f5c\u7684\u4e00\u79cd\u7279\u4f8b\u3002</p> <p>\u56fe\u548c\u6811\u90fd\u9700\u8981\u5e94\u7528\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u904d\u5386\u64cd\u4f5c\u3002\u56fe\u7684\u904d\u5386\u65b9\u5f0f\u4e5f\u53ef\u5206\u4e3a\u4e24\u79cd\uff1a\u300c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u300d\u3002</p>"},{"location":"chapter_graph/graph_traversal/#931","title":"9.3.1 \u00a0 \u5e7f\u5ea6\u4f18\u5148\u904d\u5386","text":"<p>\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u7684\u904d\u5386\u65b9\u5f0f\uff0c\u4ece\u67d0\u4e2a\u8282\u70b9\u51fa\u53d1\uff0c\u59cb\u7ec8\u4f18\u5148\u8bbf\u95ee\u8ddd\u79bb\u6700\u8fd1\u7684\u9876\u70b9\uff0c\u5e76\u4e00\u5c42\u5c42\u5411\u5916\u6269\u5f20\u3002\u5982\u56fe 9-9 \u6240\u793a\uff0c\u4ece\u5de6\u4e0a\u89d2\u9876\u70b9\u51fa\u53d1\uff0c\u9996\u5148\u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u7136\u540e\u904d\u5386\u4e0b\u4e00\u4e2a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u8bbf\u95ee\u5b8c\u6bd5\u3002</p> <p></p> <p> \u56fe 9-9 \u00a0 \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 </p>"},{"location":"chapter_graph/graph_traversal/#1","title":"1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"<p>BFS \u901a\u5e38\u501f\u52a9\u961f\u5217\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002\u961f\u5217\u5177\u6709\u201c\u5148\u5165\u5148\u51fa\u201d\u7684\u6027\u8d28\uff0c\u8fd9\u4e0e BFS \u7684\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u601d\u60f3\u5f02\u66f2\u540c\u5de5\u3002</p> <ol> <li>\u5c06\u904d\u5386\u8d77\u59cb\u9876\u70b9 <code>startVet</code> \u52a0\u5165\u961f\u5217\uff0c\u5e76\u5f00\u542f\u5faa\u73af\u3002</li> <li>\u5728\u5faa\u73af\u7684\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\uff0c\u5f39\u51fa\u961f\u9996\u9876\u70b9\u5e76\u8bb0\u5f55\u8bbf\u95ee\uff0c\u7136\u540e\u5c06\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u52a0\u5165\u5230\u961f\u5217\u5c3e\u90e8\u3002</li> <li>\u5faa\u73af\u6b65\u9aa4 <code>2.</code> \uff0c\u76f4\u5230\u6240\u6709\u9876\u70b9\u88ab\u8bbf\u95ee\u5b8c\u6bd5\u540e\u7ed3\u675f\u3002</li> </ol> <p>\u4e3a\u4e86\u9632\u6b62\u91cd\u590d\u904d\u5386\u9876\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>visited</code> \u6765\u8bb0\u5f55\u54ea\u4e9b\u8282\u70b9\u5df2\u88ab\u8bbf\u95ee\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_bfs.py<pre><code>def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\n    # \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    # \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res = []\n    # \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited = set[Vertex]([start_vet])\n    # \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    que = deque[Vertex]([start_vet])\n    # \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while len(que) &gt; 0:\n        vet = que.popleft()  # \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        # \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for adj_vet in graph.adj_list[vet]:\n            if adj_vet in visited:\n                continue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            que.append(adj_vet)  # \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adj_vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    # \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n</code></pre> graph_bfs.cpp<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector&lt;Vertex *&gt; graphBFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    vector&lt;Vertex *&gt; res;\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    unordered_set&lt;Vertex *&gt; visited = {startVet};\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    queue&lt;Vertex *&gt; que;\n    que.push(startVet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!que.empty()) {\n        Vertex *vet = que.front();\n        que.pop();          // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push_back(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (auto adjVet : graph.adjList[vet]) {\n            if (visited.count(adjVet))\n                continue;            // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            que.push(adjVet);        // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.emplace(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.java<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    visited.add(startVet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    Queue&lt;Vertex&gt; que = new LinkedList&lt;&gt;();\n    que.offer(startVet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!que.isEmpty()) {\n        Vertex vet = que.poll(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.add(vet);            // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (Vertex adjVet : graph.adjList.get(vet)) {\n            if (visited.contains(adjVet))\n                continue;        // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            que.offer(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.cs<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; GraphBFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    HashSet&lt;Vertex&gt; visited = [startVet];\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    Queue&lt;Vertex&gt; que = new();\n    que.Enqueue(startVet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (que.Count &gt; 0) {\n        Vertex vet = que.Dequeue(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.Add(vet);               // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        foreach (Vertex adjVet in graph.adjList[vet]) {\n            if (visited.Contains(adjVet)) {\n                continue;          // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.Enqueue(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.Add(adjVet);   // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.go<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res := make([]Vertex, 0)\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited := make(map[Vertex]struct{})\n    visited[startVet] = struct{}{}\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS, \u4f7f\u7528\u5207\u7247\u6a21\u62df\u961f\u5217\n    queue := make([]Vertex, 0)\n    queue = append(queue, startVet)\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    for len(queue) &gt; 0 {\n        // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        vet := queue[0]\n        queue = queue[1:]\n        // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        res = append(res, vet)\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for _, adjVet := range g.adjList[vet] {\n            _, isExist := visited[adjVet]\n            // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            if !isExist {\n                queue = append(queue, adjVet)\n                visited[adjVet] = struct{}{}\n            }\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.swift<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    var res: [Vertex] = []\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    var visited: Set&lt;Vertex&gt; = [startVet]\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    var que: [Vertex] = [startVet]\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while !que.isEmpty {\n        let vet = que.removeFirst() // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for adjVet in graph.adjList[vet] ?? [] {\n            if visited.contains(adjVet) {\n                continue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.append(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.insert(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.js<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph, startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited = new Set();\n    visited.add(startVet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    const que = [startVet];\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (que.length) {\n        const vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.ts<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res: Vertex[] = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited: Set&lt;Vertex&gt; = new Set();\n    visited.add(startVet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    const que = [startVet];\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (que.length) {\n        const vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\n            visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.dart<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n  // \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n  // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n  List&lt;Vertex&gt; res = [];\n  // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n  Set&lt;Vertex&gt; visited = {};\n  visited.add(startVet);\n  // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n  Queue&lt;Vertex&gt; que = Queue();\n  que.add(startVet);\n  // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n  while (que.isNotEmpty) {\n    Vertex vet = que.removeFirst(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n    res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (Vertex adjVet in graph.adjList[vet]!) {\n      if (visited.contains(adjVet)) {\n        continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n      }\n      que.add(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n      visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    }\n  }\n  // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n  return res;\n}\n</code></pre> graph_bfs.rs<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    let mut res = vec![];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    let mut visited = HashSet::new();\n    visited.insert(start_vet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    let mut que = VecDeque::new();\n    que.push_back(start_vet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while !que.is_empty() {\n        let vet = que.pop_front().unwrap(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n            for &amp;adj_vet in adj_vets {\n                if visited.contains(&amp;adj_vet) {\n                    continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n                }\n                que.push_back(adj_vet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n                visited.insert(adj_vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n            }\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res\n}\n</code></pre> graph_bfs.c<pre><code>/* \u8282\u70b9\u961f\u5217\u7ed3\u6784\u4f53 */\ntypedef struct {\n    Vertex *vertices[MAX_SIZE];\n    int front, rear, size;\n} Queue;\n\n/* \u6784\u9020\u51fd\u6570 */\nQueue *newQueue() {\n    Queue *q = (Queue *)malloc(sizeof(Queue));\n    q-&gt;front = q-&gt;rear = q-&gt;size = 0;\n    return q;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nint isEmpty(Queue *q) {\n    return q-&gt;size == 0;\n}\n\n/* \u5165\u961f\u64cd\u4f5c */\nvoid enqueue(Queue *q, Vertex *vet) {\n    q-&gt;vertices[q-&gt;rear] = vet;\n    q-&gt;rear = (q-&gt;rear + 1) % MAX_SIZE;\n    q-&gt;size++;\n}\n\n/* \u51fa\u961f\u64cd\u4f5c */\nVertex *dequeue(Queue *q) {\n    Vertex *vet = q-&gt;vertices[q-&gt;front];\n    q-&gt;front = (q-&gt;front + 1) % MAX_SIZE;\n    q-&gt;size--;\n    return vet;\n}\n\n/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **visited, int size, Vertex *vet) {\n    // \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\n    for (int i = 0; i &lt; size; i++) {\n        if (visited[i] == vet)\n            return 1;\n    }\n    return 0;\n}\n\n/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphBFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize, Vertex **visited, int *visitedSize) {\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    Queue *queue = newQueue();\n    enqueue(queue, startVet);\n    visited[(*visitedSize)++] = startVet;\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!isEmpty(queue)) {\n        Vertex *vet = dequeue(queue); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res[(*resSize)++] = vet;      // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        AdjListNode *node = findNode(graph, vet);\n        while (node != NULL) {\n            // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            if (!isVisited(visited, *visitedSize, node-&gt;vertex)) {\n                enqueue(queue, node-&gt;vertex);             // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n                visited[(*visitedSize)++] = node-&gt;vertex; // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n            }\n            node = node-&gt;next;\n        }\n    }\n    // \u91ca\u653e\u5185\u5b58\n    free(queue);\n}\n</code></pre> graph_bfs.kt<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfun graphBFS(graph: GraphAdjList, startVet: Vertex): List&lt;Vertex&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    val res: MutableList&lt;Vertex&gt; = ArrayList()\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    val visited: MutableSet&lt;Vertex&gt; = HashSet()\n    visited.add(startVet)\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    val que: Queue&lt;Vertex&gt; = LinkedList()\n    que.offer(startVet)\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!que.isEmpty()) {\n        val vet = que.poll() // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.add(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (adjVet in graph.adjList[vet]!!) {\n            if (visited.contains(adjVet)) continue  // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n\n            que.offer(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.rb<pre><code>[class]{}-[func]{graph_bfs}\n</code></pre> graph_bfs.zig<pre><code>[class]{}-[func]{graphBFS}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u4ee3\u7801\u76f8\u5bf9\u62bd\u8c61\uff0c\u5efa\u8bae\u5bf9\u7167\u56fe 9-10 \u6765\u52a0\u6df1\u7406\u89e3\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-10 \u00a0 \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u6b65\u9aa4 </p> <p>\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f</p> <p>\u4e0d\u552f\u4e00\u3002\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u53ea\u8981\u6c42\u6309\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u987a\u5e8f\u904d\u5386\uff0c\u800c\u591a\u4e2a\u76f8\u540c\u8ddd\u79bb\u7684\u9876\u70b9\u7684\u904d\u5386\u987a\u5e8f\u5141\u8bb8\u88ab\u4efb\u610f\u6253\u4e71\u3002\u4ee5\u56fe 9-10 \u4e3a\u4f8b\uff0c\u9876\u70b9 \\(1\\)\u3001\\(3\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\uff0c\u9876\u70b9 \\(2\\)\u3001\\(4\\)\u3001\\(6\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u4e5f\u53ef\u4ee5\u4efb\u610f\u4ea4\u6362\u3002</p>"},{"location":"chapter_graph/graph_traversal/#2","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u9876\u70b9\u90fd\u4f1a\u5165\u961f\u5e76\u51fa\u961f\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u5728\u904d\u5386\u90bb\u63a5\u9876\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002</p> <p>\u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5217\u8868 <code>res</code> \uff0c\u54c8\u5e0c\u8868 <code>visited</code> \uff0c\u961f\u5217 <code>que</code> \u4e2d\u7684\u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002</p>"},{"location":"chapter_graph/graph_traversal/#932","title":"9.3.2 \u00a0 \u6df1\u5ea6\u4f18\u5148\u904d\u5386","text":"<p>\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u518d\u56de\u5934\u7684\u904d\u5386\u65b9\u5f0f\u3002\u5982\u56fe 9-11 \u6240\u793a\uff0c\u4ece\u5de6\u4e0a\u89d2\u9876\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u5f53\u524d\u9876\u70b9\u7684\u67d0\u4e2a\u90bb\u63a5\u9876\u70b9\uff0c\u76f4\u5230\u8d70\u5230\u5c3d\u5934\u65f6\u8fd4\u56de\uff0c\u518d\u7ee7\u7eed\u8d70\u5230\u5c3d\u5934\u5e76\u8fd4\u56de\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u904d\u5386\u5b8c\u6210\u3002</p> <p></p> <p> \u56fe 9-11 \u00a0 \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386 </p>"},{"location":"chapter_graph/graph_traversal/#1_1","title":"1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"<p>\u8fd9\u79cd\u201c\u8d70\u5230\u5c3d\u5934\u518d\u8fd4\u56de\u201d\u7684\u7b97\u6cd5\u8303\u5f0f\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002\u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u5728\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u4e2d\uff0c\u6211\u4eec\u4e5f\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>visited</code> \u6765\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\uff0c\u4ee5\u907f\u514d\u91cd\u590d\u8bbf\u95ee\u9876\u70b9\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_dfs.py<pre><code>def dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n    \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570\"\"\"\n    res.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    # \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for adjVet in graph.adj_list[vet]:\n        if adjVet in visited:\n            continue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        # \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet)\n\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\n    # \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    # \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res = []\n    # \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited = set[Vertex]()\n    dfs(graph, visited, res, start_vet)\n    return res\n</code></pre> graph_dfs.cpp<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList &amp;graph, unordered_set&lt;Vertex *&gt; &amp;visited, vector&lt;Vertex *&gt; &amp;res, Vertex *vet) {\n    res.push_back(vet);   // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.emplace(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (Vertex *adjVet : graph.adjList[vet]) {\n        if (visited.count(adjVet))\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector&lt;Vertex *&gt; graphDFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    vector&lt;Vertex *&gt; res;\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    unordered_set&lt;Vertex *&gt; visited;\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.java<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList graph, Set&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (Vertex adjVet : graph.adjList.get(vet)) {\n        if (visited.contains(adjVet))\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.cs<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid DFS(GraphAdjList graph, HashSet&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.Add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.Add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    foreach (Vertex adjVet in graph.adjList[vet]) {\n        if (visited.Contains(adjVet)) {\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9                             \n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        DFS(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; GraphDFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    HashSet&lt;Vertex&gt; visited = [];\n    DFS(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.go<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n    // append \u64cd\u4f5c\u4f1a\u8fd4\u56de\u65b0\u7684\u7684\u5f15\u7528\uff0c\u5fc5\u987b\u8ba9\u539f\u5f15\u7528\u91cd\u65b0\u8d4b\u503c\u4e3a\u65b0slice\u7684\u5f15\u7528\n    *res = append(*res, vet)\n    visited[vet] = struct{}{}\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for _, adjVet := range g.adjList[vet] {\n        _, isExist := visited[adjVet]\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        if !isExist {\n            dfs(g, visited, res, adjVet)\n        }\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res := make([]Vertex, 0)\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited := make(map[Vertex]struct{})\n    dfs(g, visited, &amp;res, startVet)\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_dfs.swift<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfunc dfs(graph: GraphAdjList, visited: inout Set&lt;Vertex&gt;, res: inout [Vertex], vet: Vertex) {\n    res.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.insert(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for adjVet in graph.adjList[vet] ?? [] {\n        if visited.contains(adjVet) {\n            continue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: adjVet)\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    var res: [Vertex] = []\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    var visited: Set&lt;Vertex&gt; = []\n    dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: startVet)\n    return res\n}\n</code></pre> graph_dfs.js<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction dfs(graph, visited, res, vet) {\n    res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph, startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.ts<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfunction dfs(\n    graph: GraphAdjList,\n    visited: Set&lt;Vertex&gt;,\n    res: Vertex[],\n    vet: Vertex\n): void {\n    res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res: Vertex[] = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited: Set&lt;Vertex&gt; = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.dart<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(\n  GraphAdjList graph,\n  Set&lt;Vertex&gt; visited,\n  List&lt;Vertex&gt; res,\n  Vertex vet,\n) {\n  res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n  visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n  // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n  for (Vertex adjVet in graph.adjList[vet]!) {\n    if (visited.contains(adjVet)) {\n      continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n    }\n    // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n    dfs(graph, visited, res, adjVet);\n  }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n  // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n  List&lt;Vertex&gt; res = [];\n  // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n  Set&lt;Vertex&gt; visited = {};\n  dfs(graph, visited, res, startVet);\n  return res;\n}\n</code></pre> graph_dfs.rs<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfn dfs(graph: &amp;GraphAdjList, visited: &amp;mut HashSet&lt;Vertex&gt;, res: &amp;mut Vec&lt;Vertex&gt;, vet: Vertex) {\n    res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.insert(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n                         // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n        for &amp;adj_vet in adj_vets {\n            if visited.contains(&amp;adj_vet) {\n                continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n            dfs(graph, visited, res, adj_vet);\n        }\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    let mut res = vec![];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    let mut visited = HashSet::new();\n    dfs(&amp;graph, &amp;mut visited, &amp;mut res, start_vet);\n\n    res\n}\n</code></pre> graph_dfs.c<pre><code>/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **res, int size, Vertex *vet) {\n    // \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\n    for (int i = 0; i &lt; size; i++) {\n        if (res[i] == vet) {\n            return 1;\n        }\n    }\n    return 0;\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList *graph, Vertex **res, int *resSize, Vertex *vet) {\n    // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    res[(*resSize)++] = vet;\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    AdjListNode *node = findNode(graph, vet);\n    while (node != NULL) {\n        // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        if (!isVisited(res, *resSize, node-&gt;vertex)) {\n            // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n            dfs(graph, res, resSize, node-&gt;vertex);\n        }\n        node = node-&gt;next;\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphDFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize) {\n    dfs(graph, res, resSize, startVet);\n}\n</code></pre> graph_dfs.kt<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfun dfs(\n    graph: GraphAdjList,\n    visited: MutableSet&lt;Vertex?&gt;,\n    res: MutableList&lt;Vertex?&gt;,\n    vet: Vertex?\n) {\n    res.add(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (adjVet in graph.adjList[vet]!!) {\n        if (visited.contains(adjVet)) continue  // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet)\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfun graphDFS(\n    graph: GraphAdjList,\n    startVet: Vertex?\n): List&lt;Vertex?&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    val res: MutableList&lt;Vertex?&gt; = ArrayList()\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    val visited: MutableSet&lt;Vertex?&gt; = HashSet()\n    dfs(graph, visited, res, startVet)\n    return res\n}\n</code></pre> graph_dfs.rb<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{graph_dfs}\n</code></pre> graph_dfs.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{graphDFS}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u56fe 9-12 \u6240\u793a\u3002</p> <ul> <li>\u76f4\u865a\u7ebf\u4ee3\u8868\u5411\u4e0b\u9012\u63a8\uff0c\u8868\u793a\u5f00\u542f\u4e86\u4e00\u4e2a\u65b0\u7684\u9012\u5f52\u65b9\u6cd5\u6765\u8bbf\u95ee\u65b0\u9876\u70b9\u3002</li> <li>\u66f2\u865a\u7ebf\u4ee3\u8868\u5411\u4e0a\u56de\u6eaf\uff0c\u8868\u793a\u6b64\u9012\u5f52\u65b9\u6cd5\u5df2\u7ecf\u8fd4\u56de\uff0c\u56de\u6eaf\u5230\u4e86\u5f00\u542f\u6b64\u65b9\u6cd5\u7684\u4f4d\u7f6e\u3002</li> </ul> <p>\u4e3a\u4e86\u52a0\u6df1\u7406\u89e3\uff0c\u5efa\u8bae\u5c06\u56fe 9-12 \u4e0e\u4ee3\u7801\u7ed3\u5408\u8d77\u6765\uff0c\u5728\u8111\u4e2d\u6a21\u62df\uff08\u6216\u8005\u7528\u7b14\u753b\u4e0b\u6765\uff09\u6574\u4e2a DFS \u8fc7\u7a0b\uff0c\u5305\u62ec\u6bcf\u4e2a\u9012\u5f52\u65b9\u6cd5\u4f55\u65f6\u5f00\u542f\u3001\u4f55\u65f6\u8fd4\u56de\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-12 \u00a0 \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u6b65\u9aa4 </p> <p>\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f</p> <p>\u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5e8f\u5217\u7684\u987a\u5e8f\u4e5f\u4e0d\u662f\u552f\u4e00\u7684\u3002\u7ed9\u5b9a\u67d0\u9876\u70b9\uff0c\u5148\u5f80\u54ea\u4e2a\u65b9\u5411\u63a2\u7d22\u90fd\u53ef\u4ee5\uff0c\u5373\u90bb\u63a5\u9876\u70b9\u7684\u987a\u5e8f\u53ef\u4ee5\u4efb\u610f\u6253\u4e71\uff0c\u90fd\u662f\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002</p> <p>\u4ee5\u6811\u7684\u904d\u5386\u4e3a\u4f8b\uff0c\u201c\u6839 \\(\\rightarrow\\) \u5de6 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u53f3 \\(\\rightarrow\\) \u6839\u201d\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\uff0c\u5b83\u4eec\u5c55\u793a\u4e86\u4e09\u79cd\u904d\u5386\u4f18\u5148\u7ea7\uff0c\u7136\u800c\u8fd9\u4e09\u8005\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002</p>"},{"location":"chapter_graph/graph_traversal/#2_1","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u9876\u70b9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(1\\) \u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002</p> <p>\u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5217\u8868 <code>res</code> \uff0c\u54c8\u5e0c\u8868 <code>visited</code> \u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u6700\u5927\u4e3a \\(|V|\\) \uff0c\u56e0\u6b64\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002</p>"},{"location":"chapter_graph/summary/","title":"9.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_graph/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u56fe\u7531\u9876\u70b9\u548c\u8fb9\u7ec4\u6210\uff0c\u53ef\u4ee5\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9\u548c\u4e00\u7ec4\u8fb9\u6784\u6210\u7684\u96c6\u5408\u3002</li> <li>\u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u56e0\u800c\u66f4\u4e3a\u590d\u6742\u3002</li> <li>\u6709\u5411\u56fe\u7684\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u8fde\u901a\u56fe\u4e2d\u7684\u4efb\u610f\u9876\u70b9\u5747\u53ef\u8fbe\uff0c\u6709\u6743\u56fe\u7684\u6bcf\u6761\u8fb9\u90fd\u5305\u542b\u6743\u91cd\u53d8\u91cf\u3002</li> <li>\u90bb\u63a5\u77e9\u9635\u5229\u7528\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u6709\u8fb9\u6216\u65e0\u8fb9\u3002\u90bb\u63a5\u77e9\u9635\u5728\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u4e0a\u6548\u7387\u5f88\u9ad8\uff0c\u4f46\u7a7a\u95f4\u5360\u7528\u8f83\u591a\u3002</li> <li>\u90bb\u63a5\u8868\u4f7f\u7528\u591a\u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u7b2c \\(i\\) \u4e2a\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u3002\u90bb\u63a5\u8868\u76f8\u5bf9\u4e8e\u90bb\u63a5\u77e9\u9635\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\uff0c\u4f46\u7531\u4e8e\u9700\u8981\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u65f6\u95f4\u6548\u7387\u8f83\u4f4e\u3002</li> <li>\u5f53\u90bb\u63a5\u8868\u4e2d\u7684\u94fe\u8868\u8fc7\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u5176\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6216\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002</li> <li>\u4ece\u7b97\u6cd5\u601d\u60f3\u7684\u89d2\u5ea6\u5206\u6790\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u4e86\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff0c\u90bb\u63a5\u8868\u4f53\u73b0\u4e86\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002</li> <li>\u56fe\u53ef\u7528\u4e8e\u5efa\u6a21\u5404\u7c7b\u73b0\u5b9e\u7cfb\u7edf\uff0c\u5982\u793e\u4ea4\u7f51\u7edc\u3001\u5730\u94c1\u7ebf\u8def\u7b49\u3002</li> <li>\u6811\u662f\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\uff0c\u6811\u7684\u904d\u5386\u4e5f\u662f\u56fe\u7684\u904d\u5386\u7684\u4e00\u79cd\u7279\u4f8b\u3002</li> <li>\u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u3001\u5c42\u5c42\u6269\u5f20\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u901a\u5e38\u501f\u52a9\u961f\u5217\u5b9e\u73b0\u3002</li> <li>\u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u65f6\u518d\u56de\u6eaf\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002</li> </ul>"},{"location":"chapter_graph/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u8def\u5f84\u7684\u5b9a\u4e49\u662f\u9876\u70b9\u5e8f\u5217\u8fd8\u662f\u8fb9\u5e8f\u5217\uff1f</p> <p>\u7ef4\u57fa\u767e\u79d1\u4e0a\u4e0d\u540c\u8bed\u8a00\u7248\u672c\u7684\u5b9a\u4e49\u4e0d\u4e00\u81f4\uff1a\u82f1\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u8fb9\u5e8f\u5217\u201d\uff0c\u800c\u4e2d\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u201d\u3002\u4ee5\u4e0b\u662f\u82f1\u6587\u7248\u539f\u6587\uff1aIn graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.</p> <p>\u5728\u672c\u6587\u4e2d\uff0c\u8def\u5f84\u88ab\u89c6\u4e3a\u4e00\u4e2a\u8fb9\u5e8f\u5217\uff0c\u800c\u4e0d\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u3002\u8fd9\u662f\u56e0\u4e3a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u53ef\u80fd\u5b58\u5728\u591a\u6761\u8fb9\u8fde\u63a5\uff0c\u6b64\u65f6\u6bcf\u6761\u8fb9\u90fd\u5bf9\u5e94\u4e00\u6761\u8def\u5f84\u3002</p> <p>Q\uff1a\u975e\u8fde\u901a\u56fe\u4e2d\u662f\u5426\u4f1a\u6709\u65e0\u6cd5\u904d\u5386\u5230\u7684\u70b9\uff1f</p> <p>\u5728\u975e\u8fde\u901a\u56fe\u4e2d\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\u3002\u904d\u5386\u975e\u8fde\u901a\u56fe\u9700\u8981\u8bbe\u7f6e\u591a\u4e2a\u8d77\u70b9\uff0c\u4ee5\u904d\u5386\u5230\u56fe\u7684\u6240\u6709\u8fde\u901a\u5206\u91cf\u3002</p> <p>Q\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\uff0c\u201c\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u6240\u6709\u9876\u70b9\u201d\u7684\u9876\u70b9\u987a\u5e8f\u662f\u5426\u6709\u8981\u6c42\uff1f</p> <p>\u53ef\u4ee5\u662f\u4efb\u610f\u987a\u5e8f\u3002\u4f46\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u53ef\u80fd\u9700\u8981\u6309\u7167\u6307\u5b9a\u89c4\u5219\u6765\u6392\u5e8f\uff0c\u6bd4\u5982\u6309\u7167\u9876\u70b9\u6dfb\u52a0\u7684\u6b21\u5e8f\uff0c\u6216\u8005\u6309\u7167\u9876\u70b9\u503c\u5927\u5c0f\u7684\u987a\u5e8f\u7b49\uff0c\u8fd9\u6837\u6709\u52a9\u4e8e\u5feb\u901f\u67e5\u627e\u201c\u5e26\u6709\u67d0\u79cd\u6781\u503c\u201d\u7684\u9876\u70b9\u3002</p>"},{"location":"chapter_greedy/","title":"\u7b2c 15 \u7ae0 \u00a0 \u8d2a\u5fc3","text":"<p>Abstract</p> <p>\u5411\u65e5\u8475\u671d\u7740\u592a\u9633\u8f6c\u52a8\uff0c\u65f6\u523b\u8ffd\u6c42\u81ea\u8eab\u6210\u957f\u7684\u6700\u5927\u53ef\u80fd\u3002</p> <p>\u8d2a\u5fc3\u7b56\u7565\u5728\u4e00\u8f6e\u8f6e\u7684\u7b80\u5355\u9009\u62e9\u4e2d\uff0c\u9010\u6b65\u5bfc\u5411\u6700\u4f73\u7b54\u6848\u3002</p>"},{"location":"chapter_greedy/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>15.1 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5</li> <li>15.2 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898</li> <li>15.3 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898</li> <li>15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898</li> <li>15.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_greedy/fractional_knapsack_problem/","title":"15.2 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ea\u80fd\u9009\u62e9\u4e00\u6b21\uff0c\u4f46\u53ef\u4ee5\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u4ef7\u503c\u6839\u636e\u9009\u62e9\u7684\u91cd\u91cf\u6bd4\u4f8b\u8ba1\u7b97\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002\u793a\u4f8b\u5982\u56fe 15-3 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 15-3 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u5206\u6570\u80cc\u5305\u95ee\u9898\u548c 0-1 \u80cc\u5305\u95ee\u9898\u6574\u4f53\u4e0a\u975e\u5e38\u76f8\u4f3c\uff0c\u72b6\u6001\u5305\u542b\u5f53\u524d\u7269\u54c1 \\(i\\) \u548c\u5bb9\u91cf \\(c\\) \uff0c\u76ee\u6807\u662f\u6c42\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u6700\u5927\u4ef7\u503c\u3002</p> <p>\u4e0d\u540c\u70b9\u5728\u4e8e\uff0c\u672c\u9898\u5141\u8bb8\u53ea\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\u3002\u5982\u56fe 15-4 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5bf9\u7269\u54c1\u4efb\u610f\u5730\u8fdb\u884c\u5207\u5206\uff0c\u5e76\u6309\u7167\u91cd\u91cf\u6bd4\u4f8b\u6765\u8ba1\u7b97\u76f8\u5e94\u4ef7\u503c\u3002</p> <ol> <li>\u5bf9\u4e8e\u7269\u54c1 \\(i\\) \uff0c\u5b83\u5728\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u4ef7\u503c\u4e3a \\(val[i-1] / wgt[i-1]\\) \uff0c\u7b80\u79f0\u5355\u4f4d\u4ef7\u503c\u3002</li> <li>\u5047\u8bbe\u653e\u5165\u4e00\u90e8\u5206\u7269\u54c1 \\(i\\) \uff0c\u91cd\u91cf\u4e3a \\(w\\) \uff0c\u5219\u80cc\u5305\u589e\u52a0\u7684\u4ef7\u503c\u4e3a \\(w \\times val[i-1] / wgt[i-1]\\) \u3002</li> </ol> <p></p> <p> \u56fe 15-4 \u00a0 \u7269\u54c1\u5728\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u4ef7\u503c </p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"<p>\u6700\u5927\u5316\u80cc\u5305\u5185\u7269\u54c1\u603b\u4ef7\u503c\uff0c\u672c\u8d28\u4e0a\u662f\u6700\u5927\u5316\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u7269\u54c1\u4ef7\u503c\u3002\u7531\u6b64\u4fbf\u53ef\u63a8\u7406\u51fa\u56fe 15-5 \u6240\u793a\u7684\u8d2a\u5fc3\u7b56\u7565\u3002</p> <ol> <li>\u5c06\u7269\u54c1\u6309\u7167\u5355\u4f4d\u4ef7\u503c\u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\u3002</li> <li>\u904d\u5386\u6240\u6709\u7269\u54c1\uff0c\u6bcf\u8f6e\u8d2a\u5fc3\u5730\u9009\u62e9\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\u7684\u7269\u54c1\u3002</li> <li>\u82e5\u5269\u4f59\u80cc\u5305\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u4f7f\u7528\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u586b\u6ee1\u80cc\u5305\u3002</li> </ol> <p></p> <p> \u56fe 15-5 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u8d2a\u5fc3\u7b56\u7565 </p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u6211\u4eec\u5efa\u7acb\u4e86\u4e00\u4e2a\u7269\u54c1\u7c7b <code>Item</code> \uff0c\u4ee5\u4fbf\u5c06\u7269\u54c1\u6309\u7167\u5355\u4f4d\u4ef7\u503c\u8fdb\u884c\u6392\u5e8f\u3002\u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u5f53\u80cc\u5305\u5df2\u6ee1\u65f6\u8df3\u51fa\u5e76\u8fd4\u56de\u89e3\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig fractional_knapsack.py<pre><code>class Item:\n    \"\"\"\u7269\u54c1\"\"\"\n\n    def __init__(self, w: int, v: int):\n        self.w = w  # \u7269\u54c1\u91cd\u91cf\n        self.v = v  # \u7269\u54c1\u4ef7\u503c\n\ndef fractional_knapsack(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"\u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3\"\"\"\n    # \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    items = [Item(w, v) for w, v in zip(wgt, val)]\n    # \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    items.sort(key=lambda item: item.v / item.w, reverse=True)\n    # \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    res = 0\n    for item in items:\n        if item.w &lt;= cap:\n            # \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v\n            cap -= item.w\n        else:\n            # \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (item.v / item.w) * cap\n            # \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break\n    return res\n</code></pre> fractional_knapsack.cpp<pre><code>/* \u7269\u54c1 */\nclass Item {\n  public:\n    int w; // \u7269\u54c1\u91cd\u91cf\n    int v; // \u7269\u54c1\u4ef7\u503c\n\n    Item(int w, int v) : w(w), v(v) {\n    }\n};\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    vector&lt;Item&gt; items;\n    for (int i = 0; i &lt; wgt.size(); i++) {\n        items.push_back(Item(wgt[i], val[i]));\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    sort(items.begin(), items.end(), [](Item &amp;a, Item &amp;b) { return (double)a.v / a.w &gt; (double)b.v / b.w; });\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    double res = 0;\n    for (auto &amp;item : items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (double)item.v / item.w * cap;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.java<pre><code>/* \u7269\u54c1 */\nclass Item {\n    int w; // \u7269\u54c1\u91cd\u91cf\n    int v; // \u7269\u54c1\u4ef7\u503c\n\n    public Item(int w, int v) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    Item[] items = new Item[wgt.length];\n    for (int i = 0; i &lt; wgt.length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    Arrays.sort(items, Comparator.comparingDouble(item -&gt; -((double) item.v / item.w)));\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    double res = 0;\n    for (Item item : items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (double) item.v / item.w * cap;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.cs<pre><code>/* \u7269\u54c1 */\nclass Item(int w, int v) {\n    public int w = w; // \u7269\u54c1\u91cd\u91cf\n    public int v = v; // \u7269\u54c1\u4ef7\u503c\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble FractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    Item[] items = new Item[wgt.Length];\n    for (int i = 0; i &lt; wgt.Length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    Array.Sort(items, (x, y) =&gt; (y.v / y.w).CompareTo(x.v / x.w));\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    double res = 0;\n    foreach (Item item in items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (double)item.v / item.w * cap;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.go<pre><code>/* \u7269\u54c1 */\ntype Item struct {\n    w int // \u7269\u54c1\u91cd\u91cf\n    v int // \u7269\u54c1\u4ef7\u503c\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunc fractionalKnapsack(wgt []int, val []int, cap int) float64 {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    items := make([]Item, len(wgt))\n    for i := 0; i &lt; len(wgt); i++ {\n        items[i] = Item{wgt[i], val[i]}\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    sort.Slice(items, func(i, j int) bool {\n        return float64(items[i].v)/float64(items[i].w) &gt; float64(items[j].v)/float64(items[j].w)\n    })\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    res := 0.0\n    for _, item := range items {\n        if item.w &lt;= cap {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += float64(item.v)\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += float64(item.v) / float64(item.w) * float64(cap)\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.swift<pre><code>/* \u7269\u54c1 */\nclass Item {\n    var w: Int // \u7269\u54c1\u91cd\u91cf\n    var v: Int // \u7269\u54c1\u4ef7\u503c\n\n    init(w: Int, v: Int) {\n        self.w = w\n        self.v = v\n    }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunc fractionalKnapsack(wgt: [Int], val: [Int], cap: Int) -&gt; Double {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    var items = zip(wgt, val).map { Item(w: $0, v: $1) }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    items.sort { -(Double($0.v) / Double($0.w)) &lt; -(Double($1.v) / Double($1.w)) }\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    var res = 0.0\n    var cap = cap\n    for item in items {\n        if item.w &lt;= cap {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += Double(item.v)\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += Double(item.v) / Double(item.w) * Double(cap)\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.js<pre><code>/* \u7269\u54c1 */\nclass Item {\n    constructor(w, v) {\n        this.w = w; // \u7269\u54c1\u91cd\u91cf\n        this.v = v; // \u7269\u54c1\u4ef7\u503c\n    }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunction fractionalKnapsack(wgt, val, cap) {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    const items = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (item.v / item.w) * cap;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.ts<pre><code>/* \u7269\u54c1 */\nclass Item {\n    w: number; // \u7269\u54c1\u91cd\u91cf\n    v: number; // \u7269\u54c1\u4ef7\u503c\n\n    constructor(w: number, v: number) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunction fractionalKnapsack(wgt: number[], val: number[], cap: number): number {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    const items: Item[] = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (item.v / item.w) * cap;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.dart<pre><code>/* \u7269\u54c1 */\nclass Item {\n  int w; // \u7269\u54c1\u91cd\u91cf\n  int v; // \u7269\u54c1\u4ef7\u503c\n\n  Item(this.w, this.v);\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n  List&lt;Item&gt; items = List.generate(wgt.length, (i) =&gt; Item(wgt[i], val[i]));\n  // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n  items.sort((a, b) =&gt; (b.v / b.w).compareTo(a.v / a.w));\n  // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n  double res = 0;\n  for (Item item in items) {\n    if (item.w &lt;= cap) {\n      // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n      res += item.v;\n      cap -= item.w;\n    } else {\n      // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n      res += item.v / item.w * cap;\n      // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n      break;\n    }\n  }\n  return res;\n}\n</code></pre> fractional_knapsack.rs<pre><code>/* \u7269\u54c1 */\nstruct Item {\n    w: i32, // \u7269\u54c1\u91cd\u91cf\n    v: i32, // \u7269\u54c1\u4ef7\u503c\n}\n\nimpl Item {\n    fn new(w: i32, v: i32) -&gt; Self {\n        Self { w, v }\n    }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfn fractional_knapsack(wgt: &amp;[i32], val: &amp;[i32], mut cap: i32) -&gt; f64 {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    let mut items = wgt\n        .iter()\n        .zip(val.iter())\n        .map(|(&amp;w, &amp;v)| Item::new(w, v))\n        .collect::&lt;Vec&lt;Item&gt;&gt;();\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    items.sort_by(|a, b| {\n        (b.v as f64 / b.w as f64)\n            .partial_cmp(&amp;(a.v as f64 / a.w as f64))\n            .unwrap()\n    });\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    let mut res = 0.0;\n    for item in &amp;items {\n        if item.w &lt;= cap {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v as f64;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += item.v as f64 / item.w as f64 * cap as f64;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    res\n}\n</code></pre> fractional_knapsack.c<pre><code>/* \u7269\u54c1 */\ntypedef struct {\n    int w; // \u7269\u54c1\u91cd\u91cf\n    int v; // \u7269\u54c1\u4ef7\u503c\n} Item;\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfloat fractionalKnapsack(int wgt[], int val[], int itemCount, int cap) {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    Item *items = malloc(sizeof(Item) * itemCount);\n    for (int i = 0; i &lt; itemCount; i++) {\n        items[i] = (Item){.w = wgt[i], .v = val[i]};\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    qsort(items, (size_t)itemCount, sizeof(Item), sortByValueDensity);\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    float res = 0.0;\n    for (int i = 0; i &lt; itemCount; i++) {\n        if (items[i].w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += items[i].v;\n            cap -= items[i].w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (float)cap / items[i].w * items[i].v;\n            cap = 0;\n            break;\n        }\n    }\n    free(items);\n    return res;\n}\n</code></pre> fractional_knapsack.kt<pre><code>/* \u7269\u54c1 */\nclass Item(\n    val w: Int, // \u7269\u54c1\n    val v: Int // \u7269\u54c1\u4ef7\u503c\n)\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfun fractionalKnapsack(\n    wgt: IntArray,\n    value: IntArray,\n    c: Int\n): Double {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    var cap = c\n    val items = arrayOfNulls&lt;Item&gt;(wgt.size)\n    for (i in wgt.indices) {\n        items[i] = Item(wgt[i], value[i])\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    Arrays.sort(items, Comparator.comparingDouble { item: Item -&gt; -(item.v.toDouble() / item.w) })\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    var res = 0.0\n    for (item in items) {\n        if (item!!.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v.toDouble()\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += item.v.toDouble() / item.w * cap\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break\n        }\n    }\n    return res\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfun fractionalKnapsack(\n    wgt: IntArray,\n    value: IntArray,\n    c: Int\n): Double {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    var cap = c\n    val items = arrayOfNulls&lt;Item&gt;(wgt.size)\n    for (i in wgt.indices) {\n        items[i] = Item(wgt[i], value[i])\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    Arrays.sort(items, Comparator.comparingDouble { item: Item -&gt; -(item.v.toDouble() / item.w) })\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    var res = 0.0\n    for (item in items) {\n        if (item!!.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v.toDouble()\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += item.v.toDouble() / item.w * cap\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.rb<pre><code>[class]{Item}-[func]{}\n\n[class]{}-[func]{fractional_knapsack}\n</code></pre> fractional_knapsack.zig<pre><code>[class]{Item}-[func]{}\n\n[class]{}-[func]{fractionalKnapsack}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u9664\u6392\u5e8f\u4e4b\u5916\uff0c\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u9700\u8981\u904d\u5386\u6574\u4e2a\u7269\u54c1\u5217\u8868\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u7269\u54c1\u6570\u91cf\u3002</p> <p>\u7531\u4e8e\u521d\u59cb\u5316\u4e86\u4e00\u4e2a <code>Item</code> \u5bf9\u8c61\u5217\u8868\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"<p>\u91c7\u7528\u53cd\u8bc1\u6cd5\u3002\u5047\u8bbe\u7269\u54c1 \\(x\\) \u662f\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\u7684\u7269\u54c1\uff0c\u4f7f\u7528\u67d0\u7b97\u6cd5\u6c42\u5f97\u6700\u5927\u4ef7\u503c\u4e3a <code>res</code> \uff0c\u4f46\u8be5\u89e3\u4e2d\u4e0d\u5305\u542b\u7269\u54c1 \\(x\\) \u3002</p> <p>\u73b0\u5728\u4ece\u80cc\u5305\u4e2d\u62ff\u51fa\u5355\u4f4d\u91cd\u91cf\u7684\u4efb\u610f\u7269\u54c1\uff0c\u5e76\u66ff\u6362\u4e3a\u5355\u4f4d\u91cd\u91cf\u7684\u7269\u54c1 \\(x\\) \u3002\u7531\u4e8e\u7269\u54c1 \\(x\\) \u7684\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\uff0c\u56e0\u6b64\u66ff\u6362\u540e\u7684\u603b\u4ef7\u503c\u4e00\u5b9a\u5927\u4e8e <code>res</code> \u3002\u8fd9\u4e0e <code>res</code> \u662f\u6700\u4f18\u89e3\u77db\u76fe\uff0c\u8bf4\u660e\u6700\u4f18\u89e3\u4e2d\u5fc5\u987b\u5305\u542b\u7269\u54c1 \\(x\\) \u3002</p> <p>\u5bf9\u4e8e\u8be5\u89e3\u4e2d\u7684\u5176\u4ed6\u7269\u54c1\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u6784\u5efa\u51fa\u4e0a\u8ff0\u77db\u76fe\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u5355\u4f4d\u4ef7\u503c\u66f4\u5927\u7684\u7269\u54c1\u603b\u662f\u66f4\u4f18\u9009\u62e9\uff0c\u8fd9\u8bf4\u660e\u8d2a\u5fc3\u7b56\u7565\u662f\u6709\u6548\u7684\u3002</p> <p>\u5982\u56fe 15-6 \u6240\u793a\uff0c\u5982\u679c\u5c06\u7269\u54c1\u91cd\u91cf\u548c\u7269\u54c1\u5355\u4f4d\u4ef7\u503c\u5206\u522b\u770b\u4f5c\u4e00\u5f20\u4e8c\u7ef4\u56fe\u8868\u7684\u6a2a\u8f74\u548c\u7eb5\u8f74\uff0c\u5219\u5206\u6570\u80cc\u5305\u95ee\u9898\u53ef\u8f6c\u5316\u4e3a\u201c\u6c42\u5728\u6709\u9650\u6a2a\u8f74\u533a\u95f4\u4e0b\u56f4\u6210\u7684\u6700\u5927\u9762\u79ef\u201d\u3002\u8fd9\u4e2a\u7c7b\u6bd4\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u4ece\u51e0\u4f55\u89d2\u5ea6\u7406\u89e3\u8d2a\u5fc3\u7b56\u7565\u7684\u6709\u6548\u6027\u3002</p> <p></p> <p> \u56fe 15-6 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u51e0\u4f55\u8868\u793a </p>"},{"location":"chapter_greedy/greedy_algorithm/","title":"15.1 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5","text":"<p>\u300c\u8d2a\u5fc3\u7b97\u6cd5 greedy algorithm\u300d\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u89e3\u51b3\u4f18\u5316\u95ee\u9898\u7684\u7b97\u6cd5\uff0c\u5176\u57fa\u672c\u601d\u60f3\u662f\u5728\u95ee\u9898\u7684\u6bcf\u4e2a\u51b3\u7b56\u9636\u6bb5\uff0c\u90fd\u9009\u62e9\u5f53\u524d\u770b\u8d77\u6765\u6700\u4f18\u7684\u9009\u62e9\uff0c\u5373\u8d2a\u5fc3\u5730\u505a\u51fa\u5c40\u90e8\u6700\u4f18\u7684\u51b3\u7b56\uff0c\u4ee5\u671f\u83b7\u5f97\u5168\u5c40\u6700\u4f18\u89e3\u3002\u8d2a\u5fc3\u7b97\u6cd5\u7b80\u6d01\u4e14\u9ad8\u6548\uff0c\u5728\u8bb8\u591a\u5b9e\u9645\u95ee\u9898\u4e2d\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\u3002</p> <p>\u8d2a\u5fc3\u7b97\u6cd5\u548c\u52a8\u6001\u89c4\u5212\u90fd\u5e38\u7528\u4e8e\u89e3\u51b3\u4f18\u5316\u95ee\u9898\u3002\u5b83\u4eec\u4e4b\u95f4\u5b58\u5728\u4e00\u4e9b\u76f8\u4f3c\u4e4b\u5904\uff0c\u6bd4\u5982\u90fd\u4f9d\u8d56\u6700\u4f18\u5b50\u7ed3\u6784\u6027\u8d28\uff0c\u4f46\u5de5\u4f5c\u539f\u7406\u4e0d\u540c\u3002</p> <ul> <li>\u52a8\u6001\u89c4\u5212\u4f1a\u6839\u636e\u4e4b\u524d\u9636\u6bb5\u7684\u6240\u6709\u51b3\u7b56\u6765\u8003\u8651\u5f53\u524d\u51b3\u7b56\uff0c\u5e76\u4f7f\u7528\u8fc7\u53bb\u5b50\u95ee\u9898\u7684\u89e3\u6765\u6784\u5efa\u5f53\u524d\u5b50\u95ee\u9898\u7684\u89e3\u3002</li> <li>\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4f1a\u8003\u8651\u8fc7\u53bb\u7684\u51b3\u7b56\uff0c\u800c\u662f\u4e00\u8def\u5411\u524d\u5730\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u4e0d\u65ad\u7f29\u5c0f\u95ee\u9898\u8303\u56f4\uff0c\u76f4\u81f3\u95ee\u9898\u88ab\u89e3\u51b3\u3002</li> </ul> <p>\u6211\u4eec\u5148\u901a\u8fc7\u4f8b\u9898\u201c\u96f6\u94b1\u5151\u6362\u201d\u4e86\u89e3\u8d2a\u5fc3\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002\u8fd9\u9053\u9898\u5df2\u7ecf\u5728\u201c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u201d\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u76f8\u4fe1\u4f60\u5bf9\u5b83\u5e76\u4e0d\u964c\u751f\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u80fd\u591f\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u3002\u5982\u679c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002</p> <p>\u672c\u9898\u91c7\u53d6\u7684\u8d2a\u5fc3\u7b56\u7565\u5982\u56fe 15-1 \u6240\u793a\u3002\u7ed9\u5b9a\u76ee\u6807\u91d1\u989d\uff0c\u6211\u4eec\u8d2a\u5fc3\u5730\u9009\u62e9\u4e0d\u5927\u4e8e\u4e14\u6700\u63a5\u8fd1\u5b83\u7684\u786c\u5e01\uff0c\u4e0d\u65ad\u5faa\u73af\u8be5\u6b65\u9aa4\uff0c\u76f4\u81f3\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u4e3a\u6b62\u3002</p> <p></p> <p> \u56fe 15-1 \u00a0 \u96f6\u94b1\u5151\u6362\u7684\u8d2a\u5fc3\u7b56\u7565 </p> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change_greedy.py<pre><code>def coin_change_greedy(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3\"\"\"\n    # \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    i = len(coins) - 1\n    count = 0\n    # \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while amt &gt; 0:\n        # \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while i &gt; 0 and coins[i] &gt; amt:\n            i -= 1\n        # \u9009\u62e9 coins[i]\n        amt -= coins[i]\n        count += 1\n    # \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return count if amt == 0 else -1\n</code></pre> coin_change_greedy.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(vector&lt;int&gt; &amp;coins, int amt) {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    int i = coins.size() - 1;\n    int count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.java<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(int[] coins, int amt) {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    int i = coins.length - 1;\n    int count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.cs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint CoinChangeGreedy(int[] coins, int amt) {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    int i = coins.Length - 1;\n    int count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.go<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunc coinChangeGreedy(coins []int, amt int) int {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    i := len(coins) - 1\n    count := 0\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    for amt &gt; 0 {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        for i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i--\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i]\n        count++\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    if amt != 0 {\n        return -1\n    }\n    return count\n}\n</code></pre> coin_change_greedy.swift<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunc coinChangeGreedy(coins: [Int], amt: Int) -&gt; Int {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    var i = coins.count - 1\n    var count = 0\n    var amt = amt\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while amt &gt; 0 {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i]\n        count += 1\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt == 0 ? count : -1\n}\n</code></pre> coin_change_greedy.js<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunction coinChangeGreedy(coins, amt) {\n    // \u5047\u8bbe coins \u6570\u7ec4\u6709\u5e8f\n    let i = coins.length - 1;\n    let count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.ts<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunction coinChangeGreedy(coins: number[], amt: number): number {\n    // \u5047\u8bbe coins \u6570\u7ec4\u6709\u5e8f\n    let i = coins.length - 1;\n    let count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.dart<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(List&lt;int&gt; coins, int amt) {\n  // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n  int i = coins.length - 1;\n  int count = 0;\n  // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n  while (amt &gt; 0) {\n    // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n    while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n      i--;\n    }\n    // \u9009\u62e9 coins[i]\n    amt -= coins[i];\n    count++;\n  }\n  // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n  return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.rs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfn coin_change_greedy(coins: &amp;[i32], mut amt: i32) -&gt; i32 {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    let mut i = coins.len() - 1;\n    let mut count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while amt &gt; 0 {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count += 1;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    if amt == 0 {\n        count\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change_greedy.c<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(int *coins, int size, int amt) {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    int i = size - 1;\n    int count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.kt<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfun coinChangeGreedy(coins: IntArray, amt: Int): Int {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    var am = amt\n    var i = coins.size - 1\n    var count = 0\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (am &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; am) {\n            i--\n        }\n        // \u9009\u62e9 coins[i]\n        am -= coins[i]\n        count++\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return if (am == 0) count else -1\n}\n</code></pre> coin_change_greedy.rb<pre><code>[class]{}-[func]{coin_change_greedy}\n</code></pre> coin_change_greedy.zig<pre><code>[class]{}-[func]{coinChangeGreedy}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u4f60\u53ef\u80fd\u4f1a\u4e0d\u7531\u5730\u53d1\u51fa\u611f\u53f9\uff1aSo clean \uff01\u8d2a\u5fc3\u7b97\u6cd5\u4ec5\u7528\u7ea6\u5341\u884c\u4ee3\u7801\u5c31\u89e3\u51b3\u4e86\u96f6\u94b1\u5151\u6362\u95ee\u9898\u3002</p>"},{"location":"chapter_greedy/greedy_algorithm/#1511","title":"15.1.1 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5\u7684\u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4ec5\u64cd\u4f5c\u76f4\u63a5\u3001\u5b9e\u73b0\u7b80\u5355\uff0c\u800c\u4e14\u901a\u5e38\u6548\u7387\u4e5f\u5f88\u9ad8\u3002\u5728\u4ee5\u4e0a\u4ee3\u7801\u4e2d\uff0c\u8bb0\u786c\u5e01\u6700\u5c0f\u9762\u503c\u4e3a \\(\\min(coins)\\) \uff0c\u5219\u8d2a\u5fc3\u9009\u62e9\u6700\u591a\u5faa\u73af \\(amt / \\min(coins)\\) \u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(amt / \\min(coins))\\) \u3002\u8fd9\u6bd4\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\times amt)\\) \u5c0f\u4e86\u4e00\u4e2a\u6570\u91cf\u7ea7\u3002</p> <p>\u7136\u800c\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u786c\u5e01\u9762\u503c\u7ec4\u5408\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u5e76\u4e0d\u80fd\u627e\u5230\u6700\u4f18\u89e3\u3002\u56fe 15-2 \u7ed9\u51fa\u4e86\u4e24\u4e2a\u793a\u4f8b\u3002</p> <ul> <li>\u6b63\u4f8b \\(coins = [1, 5, 10, 20, 50, 100]\\)\uff1a\u5728\u8be5\u786c\u5e01\u7ec4\u5408\u4e0b\uff0c\u7ed9\u5b9a\u4efb\u610f \\(amt\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u90fd\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u53cd\u4f8b \\(coins = [1, 20, 50]\\)\uff1a\u5047\u8bbe \\(amt = 60\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 10\\) \u7684\u5151\u6362\u7ec4\u5408\uff0c\u5171\u8ba1 \\(11\\) \u679a\u786c\u5e01\uff0c\u4f46\u52a8\u6001\u89c4\u5212\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3 \\(20 + 20 + 20\\) \uff0c\u4ec5\u9700 \\(3\\) \u679a\u786c\u5e01\u3002</li> <li>\u53cd\u4f8b \\(coins = [1, 49, 50]\\)\uff1a\u5047\u8bbe \\(amt = 98\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 48\\) \u7684\u5151\u6362\u7ec4\u5408\uff0c\u5171\u8ba1 \\(49\\) \u679a\u786c\u5e01\uff0c\u4f46\u52a8\u6001\u89c4\u5212\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3 \\(49 + 49\\) \uff0c\u4ec5\u9700 \\(2\\) \u679a\u786c\u5e01\u3002</li> </ul> <p></p> <p> \u56fe 15-2 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5\u65e0\u6cd5\u627e\u51fa\u6700\u4f18\u89e3\u7684\u793a\u4f8b </p> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5bf9\u4e8e\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u65e0\u6cd5\u4fdd\u8bc1\u627e\u5230\u5168\u5c40\u6700\u4f18\u89e3\uff0c\u5e76\u4e14\u6709\u53ef\u80fd\u627e\u5230\u975e\u5e38\u5dee\u7684\u89e3\u3002\u5b83\u66f4\u9002\u5408\u7528\u52a8\u6001\u89c4\u5212\u89e3\u51b3\u3002</p> <p>\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u9002\u7528\u60c5\u51b5\u5206\u4ee5\u4e0b\u4e24\u79cd\u3002</p> <ol> <li>\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1a\u8d2a\u5fc3\u7b97\u6cd5\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u5f80\u5f80\u662f\u6700\u4f18\u9009\u62e9\uff0c\u56e0\u4e3a\u5b83\u5f80\u5f80\u6bd4\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u66f4\u9ad8\u6548\u3002</li> <li>\u53ef\u4ee5\u627e\u5230\u8fd1\u4f3c\u6700\u4f18\u89e3\uff1a\u8d2a\u5fc3\u7b97\u6cd5\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u4e5f\u662f\u53ef\u7528\u7684\u3002\u5bf9\u4e8e\u5f88\u591a\u590d\u6742\u95ee\u9898\u6765\u8bf4\uff0c\u5bfb\u627e\u5168\u5c40\u6700\u4f18\u89e3\u975e\u5e38\u56f0\u96be\uff0c\u80fd\u4ee5\u8f83\u9ad8\u6548\u7387\u627e\u5230\u6b21\u4f18\u89e3\u4e5f\u662f\u975e\u5e38\u4e0d\u9519\u7684\u3002</li> </ol>"},{"location":"chapter_greedy/greedy_algorithm/#1512","title":"15.1.2 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5\u7279\u6027","text":"<p>\u90a3\u4e48\u95ee\u9898\u6765\u4e86\uff0c\u4ec0\u4e48\u6837\u7684\u95ee\u9898\u9002\u5408\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u5462\uff1f\u6216\u8005\u8bf4\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1f</p> <p>\u76f8\u8f83\u4e8e\u52a8\u6001\u89c4\u5212\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u4f7f\u7528\u6761\u4ef6\u66f4\u52a0\u82db\u523b\uff0c\u5176\u4e3b\u8981\u5173\u6ce8\u95ee\u9898\u7684\u4e24\u4e2a\u6027\u8d28\u3002</p> <ul> <li>\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\uff1a\u53ea\u6709\u5f53\u5c40\u90e8\u6700\u4f18\u9009\u62e9\u59cb\u7ec8\u53ef\u4ee5\u5bfc\u81f4\u5168\u5c40\u6700\u4f18\u89e3\u65f6\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u624d\u80fd\u4fdd\u8bc1\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u5305\u542b\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u3002</li> </ul> <p>\u6700\u4f18\u5b50\u7ed3\u6784\u5df2\u7ecf\u5728\u201c\u52a8\u6001\u89c4\u5212\u201d\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u8fd9\u91cc\u4e0d\u518d\u8d58\u8ff0\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u4e00\u4e9b\u95ee\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784\u5e76\u4e0d\u660e\u663e\uff0c\u4f46\u4ecd\u7136\u53ef\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u89e3\u51b3\u3002</p> <p>\u6211\u4eec\u4e3b\u8981\u63a2\u7a76\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u7684\u5224\u65ad\u65b9\u6cd5\u3002\u867d\u7136\u5b83\u7684\u63cf\u8ff0\u770b\u4e0a\u53bb\u6bd4\u8f83\u7b80\u5355\uff0c\u4f46\u5b9e\u9645\u4e0a\u5bf9\u4e8e\u8bb8\u591a\u95ee\u9898\uff0c\u8bc1\u660e\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u5e76\u975e\u6613\u4e8b\u3002</p> <p>\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff0c\u6211\u4eec\u867d\u7136\u80fd\u591f\u5bb9\u6613\u5730\u4e3e\u51fa\u53cd\u4f8b\uff0c\u5bf9\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u8fdb\u884c\u8bc1\u4f2a\uff0c\u4f46\u8bc1\u5b9e\u7684\u96be\u5ea6\u8f83\u5927\u3002\u5982\u679c\u95ee\uff1a\u6ee1\u8db3\u4ec0\u4e48\u6761\u4ef6\u7684\u786c\u5e01\u7ec4\u5408\u53ef\u4ee5\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\uff1f\u6211\u4eec\u5f80\u5f80\u53ea\u80fd\u51ed\u501f\u76f4\u89c9\u6216\u4e3e\u4f8b\u5b50\u6765\u7ed9\u51fa\u4e00\u4e2a\u6a21\u68f1\u4e24\u53ef\u7684\u7b54\u6848\uff0c\u800c\u96be\u4ee5\u7ed9\u51fa\u4e25\u8c28\u7684\u6570\u5b66\u8bc1\u660e\u3002</p> <p>Quote</p> <p>\u6709\u4e00\u7bc7\u8bba\u6587\u7ed9\u51fa\u4e86\u4e00\u4e2a \\(O(n^3)\\) \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u7b97\u6cd5\uff0c\u7528\u4e8e\u5224\u65ad\u4e00\u4e2a\u786c\u5e01\u7ec4\u5408\u80fd\u5426\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u627e\u51fa\u4efb\u610f\u91d1\u989d\u7684\u6700\u4f18\u89e3\u3002</p> <p>Pearson, D. A polynomial-time algorithm for the change-making problem[J]. Operations Research Letters, 2005, 33(3): 231-234.</p>"},{"location":"chapter_greedy/greedy_algorithm/#1513","title":"15.1.3 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5\u89e3\u9898\u6b65\u9aa4","text":"<p>\u8d2a\u5fc3\u95ee\u9898\u7684\u89e3\u51b3\u6d41\u7a0b\u5927\u4f53\u53ef\u5206\u4e3a\u4ee5\u4e0b\u4e09\u6b65\u3002</p> <ol> <li>\u95ee\u9898\u5206\u6790\uff1a\u68b3\u7406\u4e0e\u7406\u89e3\u95ee\u9898\u7279\u6027\uff0c\u5305\u62ec\u72b6\u6001\u5b9a\u4e49\u3001\u4f18\u5316\u76ee\u6807\u548c\u7ea6\u675f\u6761\u4ef6\u7b49\u3002\u8fd9\u4e00\u6b65\u5728\u56de\u6eaf\u548c\u52a8\u6001\u89c4\u5212\u4e2d\u90fd\u6709\u6d89\u53ca\u3002</li> <li>\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\uff1a\u786e\u5b9a\u5982\u4f55\u5728\u6bcf\u4e00\u6b65\u4e2d\u505a\u51fa\u8d2a\u5fc3\u9009\u62e9\u3002\u8fd9\u4e2a\u7b56\u7565\u80fd\u591f\u5728\u6bcf\u4e00\u6b65\u51cf\u5c0f\u95ee\u9898\u7684\u89c4\u6a21\uff0c\u5e76\u6700\u7ec8\u89e3\u51b3\u6574\u4e2a\u95ee\u9898\u3002</li> <li>\u6b63\u786e\u6027\u8bc1\u660e\uff1a\u901a\u5e38\u9700\u8981\u8bc1\u660e\u95ee\u9898\u5177\u6709\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u3002\u8fd9\u4e2a\u6b65\u9aa4\u53ef\u80fd\u9700\u8981\u7528\u5230\u6570\u5b66\u8bc1\u660e\uff0c\u4f8b\u5982\u5f52\u7eb3\u6cd5\u6216\u53cd\u8bc1\u6cd5\u7b49\u3002</li> </ol> <p>\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u662f\u6c42\u89e3\u95ee\u9898\u7684\u6838\u5fc3\u6b65\u9aa4\uff0c\u4f46\u5b9e\u65bd\u8d77\u6765\u53ef\u80fd\u5e76\u4e0d\u5bb9\u6613\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u4e0d\u540c\u95ee\u9898\u7684\u8d2a\u5fc3\u7b56\u7565\u7684\u5dee\u5f02\u8f83\u5927\u3002\u5bf9\u4e8e\u8bb8\u591a\u95ee\u9898\u6765\u8bf4\uff0c\u8d2a\u5fc3\u7b56\u7565\u6bd4\u8f83\u6d45\u663e\uff0c\u6211\u4eec\u901a\u8fc7\u4e00\u4e9b\u5927\u6982\u7684\u601d\u8003\u4e0e\u5c1d\u8bd5\u5c31\u80fd\u5f97\u51fa\u3002\u800c\u5bf9\u4e8e\u4e00\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u8d2a\u5fc3\u7b56\u7565\u53ef\u80fd\u975e\u5e38\u9690\u853d\uff0c\u8fd9\u79cd\u60c5\u51b5\u5c31\u975e\u5e38\u8003\u9a8c\u4e2a\u4eba\u7684\u89e3\u9898\u7ecf\u9a8c\u4e0e\u7b97\u6cd5\u80fd\u529b\u4e86\u3002</li> <li>\u67d0\u4e9b\u8d2a\u5fc3\u7b56\u7565\u5177\u6709\u8f83\u5f3a\u7684\u8ff7\u60d1\u6027\u3002\u5f53\u6211\u4eec\u6ee1\u6000\u4fe1\u5fc3\u8bbe\u8ba1\u597d\u8d2a\u5fc3\u7b56\u7565\uff0c\u5199\u51fa\u89e3\u9898\u4ee3\u7801\u5e76\u63d0\u4ea4\u8fd0\u884c\uff0c\u5f88\u53ef\u80fd\u53d1\u73b0\u90e8\u5206\u6d4b\u8bd5\u6837\u4f8b\u65e0\u6cd5\u901a\u8fc7\u3002\u8fd9\u662f\u56e0\u4e3a\u8bbe\u8ba1\u7684\u8d2a\u5fc3\u7b56\u7565\u53ea\u662f\u201c\u90e8\u5206\u6b63\u786e\u201d\u7684\uff0c\u4e0a\u6587\u4ecb\u7ecd\u7684\u96f6\u94b1\u5151\u6362\u5c31\u662f\u4e00\u4e2a\u5178\u578b\u6848\u4f8b\u3002</li> </ul> <p>\u4e3a\u4e86\u4fdd\u8bc1\u6b63\u786e\u6027\uff0c\u6211\u4eec\u5e94\u8be5\u5bf9\u8d2a\u5fc3\u7b56\u7565\u8fdb\u884c\u4e25\u8c28\u7684\u6570\u5b66\u8bc1\u660e\uff0c\u901a\u5e38\u9700\u8981\u7528\u5230\u53cd\u8bc1\u6cd5\u6216\u6570\u5b66\u5f52\u7eb3\u6cd5\u3002</p> <p>\u7136\u800c\uff0c\u6b63\u786e\u6027\u8bc1\u660e\u4e5f\u5f88\u53ef\u80fd\u4e0d\u662f\u4e00\u4ef6\u6613\u4e8b\u3002\u5982\u82e5\u6ca1\u6709\u5934\u7eea\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9009\u62e9\u9762\u5411\u6d4b\u8bd5\u7528\u4f8b\u8fdb\u884c\u4ee3\u7801\u8c03\u8bd5\uff0c\u4e00\u6b65\u6b65\u4fee\u6539\u4e0e\u9a8c\u8bc1\u8d2a\u5fc3\u7b56\u7565\u3002</p>"},{"location":"chapter_greedy/greedy_algorithm/#1514","title":"15.1.4 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5\u5178\u578b\u4f8b\u9898","text":"<p>\u8d2a\u5fc3\u7b97\u6cd5\u5e38\u5e38\u5e94\u7528\u5728\u6ee1\u8db3\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u4f18\u5316\u95ee\u9898\u4e2d\uff0c\u4ee5\u4e0b\u5217\u4e3e\u4e86\u4e00\u4e9b\u5178\u578b\u7684\u8d2a\u5fc3\u7b97\u6cd5\u95ee\u9898\u3002</p> <ul> <li>\u786c\u5e01\u627e\u96f6\u95ee\u9898\uff1a\u5728\u67d0\u4e9b\u786c\u5e01\u7ec4\u5408\u4e0b\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u603b\u662f\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u533a\u95f4\u8c03\u5ea6\u95ee\u9898\uff1a\u5047\u8bbe\u4f60\u6709\u4e00\u4e9b\u4efb\u52a1\uff0c\u6bcf\u4e2a\u4efb\u52a1\u5728\u4e00\u6bb5\u65f6\u95f4\u5185\u8fdb\u884c\uff0c\u4f60\u7684\u76ee\u6807\u662f\u5b8c\u6210\u5c3d\u53ef\u80fd\u591a\u7684\u4efb\u52a1\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9009\u62e9\u7ed3\u675f\u65f6\u95f4\u6700\u65e9\u7684\u4efb\u52a1\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u5c31\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u5206\u6570\u80cc\u5305\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u7269\u54c1\u548c\u4e00\u4e2a\u8f7d\u91cd\u91cf\uff0c\u4f60\u7684\u76ee\u6807\u662f\u9009\u62e9\u4e00\u7ec4\u7269\u54c1\uff0c\u4f7f\u5f97\u603b\u91cd\u91cf\u4e0d\u8d85\u8fc7\u8f7d\u91cd\u91cf\uff0c\u4e14\u603b\u4ef7\u503c\u6700\u5927\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9009\u62e9\u6027\u4ef7\u6bd4\u6700\u9ad8\uff08\u4ef7\u503c / \u91cd\u91cf\uff09\u7684\u7269\u54c1\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u5728\u4e00\u4e9b\u60c5\u51b5\u4e0b\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u80a1\u7968\u4e70\u5356\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u80a1\u7968\u7684\u5386\u53f2\u4ef7\u683c\uff0c\u4f60\u53ef\u4ee5\u8fdb\u884c\u591a\u6b21\u4e70\u5356\uff0c\u4f46\u5982\u679c\u4f60\u5df2\u7ecf\u6301\u6709\u80a1\u7968\uff0c\u90a3\u4e48\u5728\u5356\u51fa\u4e4b\u524d\u4e0d\u80fd\u518d\u4e70\uff0c\u76ee\u6807\u662f\u83b7\u53d6\u6700\u5927\u5229\u6da6\u3002</li> <li>\u970d\u592b\u66fc\u7f16\u7801\uff1a\u970d\u592b\u66fc\u7f16\u7801\u662f\u4e00\u79cd\u7528\u4e8e\u65e0\u635f\u6570\u636e\u538b\u7f29\u7684\u8d2a\u5fc3\u7b97\u6cd5\u3002\u901a\u8fc7\u6784\u5efa\u970d\u592b\u66fc\u6811\uff0c\u6bcf\u6b21\u9009\u62e9\u51fa\u73b0\u9891\u7387\u6700\u4f4e\u7684\u4e24\u4e2a\u8282\u70b9\u5408\u5e76\uff0c\u6700\u540e\u5f97\u5230\u7684\u970d\u592b\u66fc\u6811\u7684\u5e26\u6743\u8def\u5f84\u957f\u5ea6\uff08\u7f16\u7801\u957f\u5ea6\uff09\u6700\u5c0f\u3002</li> <li>Dijkstra \u7b97\u6cd5\uff1a\u5b83\u662f\u4e00\u79cd\u89e3\u51b3\u7ed9\u5b9a\u6e90\u9876\u70b9\u5230\u5176\u4f59\u5404\u9876\u70b9\u7684\u6700\u77ed\u8def\u5f84\u95ee\u9898\u7684\u8d2a\u5fc3\u7b97\u6cd5\u3002</li> </ul>"},{"location":"chapter_greedy/max_capacity_problem/","title":"15.3 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898","text":"<p>Question</p> <p>\u8f93\u5165\u4e00\u4e2a\u6570\u7ec4 \\(ht\\) \uff0c\u5176\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u4ee3\u8868\u4e00\u4e2a\u5782\u76f4\u9694\u677f\u7684\u9ad8\u5ea6\u3002\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u9694\u677f\uff0c\u4ee5\u53ca\u5b83\u4eec\u4e4b\u95f4\u7684\u7a7a\u95f4\u53ef\u4ee5\u7ec4\u6210\u4e00\u4e2a\u5bb9\u5668\u3002</p> <p>\u5bb9\u5668\u7684\u5bb9\u91cf\u7b49\u4e8e\u9ad8\u5ea6\u548c\u5bbd\u5ea6\u7684\u4e58\u79ef\uff08\u9762\u79ef\uff09\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u8f83\u77ed\u7684\u9694\u677f\u51b3\u5b9a\uff0c\u5bbd\u5ea6\u662f\u4e24\u4e2a\u9694\u677f\u7684\u6570\u7ec4\u7d22\u5f15\u4e4b\u5dee\u3002</p> <p>\u8bf7\u5728\u6570\u7ec4\u4e2d\u9009\u62e9\u4e24\u4e2a\u9694\u677f\uff0c\u4f7f\u5f97\u7ec4\u6210\u7684\u5bb9\u5668\u7684\u5bb9\u91cf\u6700\u5927\uff0c\u8fd4\u56de\u6700\u5927\u5bb9\u91cf\u3002\u793a\u4f8b\u5982\u56fe 15-7 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 15-7 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u5bb9\u5668\u7531\u4efb\u610f\u4e24\u4e2a\u9694\u677f\u56f4\u6210\uff0c\u56e0\u6b64\u672c\u9898\u7684\u72b6\u6001\u4e3a\u4e24\u4e2a\u9694\u677f\u7684\u7d22\u5f15\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002</p> <p>\u6839\u636e\u9898\u610f\uff0c\u5bb9\u91cf\u7b49\u4e8e\u9ad8\u5ea6\u4e58\u4ee5\u5bbd\u5ea6\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u77ed\u677f\u51b3\u5b9a\uff0c\u5bbd\u5ea6\u662f\u4e24\u9694\u677f\u7684\u6570\u7ec4\u7d22\u5f15\u4e4b\u5dee\u3002\u8bbe\u5bb9\u91cf\u4e3a \\(cap[i, j]\\) \uff0c\u5219\u53ef\u5f97\u8ba1\u7b97\u516c\u5f0f\uff1a</p> \\[ cap[i, j] = \\min(ht[i], ht[j]) \\times (j - i) \\] <p>\u8bbe\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u4e24\u4e2a\u9694\u677f\u7684\u7ec4\u5408\u6570\u91cf\uff08\u72b6\u6001\u603b\u6570\uff09\u4e3a \\(C_n^2 = \\frac{n(n - 1)}{2}\\) \u4e2a\u3002\u6700\u76f4\u63a5\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u7a77\u4e3e\u6240\u6709\u72b6\u6001\uff0c\u4ece\u800c\u6c42\u5f97\u6700\u5927\u5bb9\u91cf\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_greedy/max_capacity_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"<p>\u8fd9\u9053\u9898\u8fd8\u6709\u66f4\u9ad8\u6548\u7387\u7684\u89e3\u6cd5\u3002\u5982\u56fe 15-8 \u6240\u793a\uff0c\u73b0\u9009\u53d6\u4e00\u4e2a\u72b6\u6001 \\([i, j]\\) \uff0c\u5176\u6ee1\u8db3\u7d22\u5f15 \\(i &lt; j\\) \u4e14\u9ad8\u5ea6 \\(ht[i] &lt; ht[j]\\) \uff0c\u5373 \\(i\\) \u4e3a\u77ed\u677f\u3001\\(j\\) \u4e3a\u957f\u677f\u3002</p> <p></p> <p> \u56fe 15-8 \u00a0 \u521d\u59cb\u72b6\u6001 </p> <p>\u5982\u56fe 15-9 \u6240\u793a\uff0c\u82e5\u6b64\u65f6\u5c06\u957f\u677f \\(j\\) \u5411\u77ed\u677f \\(i\\) \u9760\u8fd1\uff0c\u5219\u5bb9\u91cf\u4e00\u5b9a\u53d8\u5c0f\u3002</p> <p>\u8fd9\u662f\u56e0\u4e3a\u5728\u79fb\u52a8\u957f\u677f \\(j\\) \u540e\uff0c\u5bbd\u5ea6 \\(j-i\\) \u80af\u5b9a\u53d8\u5c0f\uff1b\u800c\u9ad8\u5ea6\u7531\u77ed\u677f\u51b3\u5b9a\uff0c\u56e0\u6b64\u9ad8\u5ea6\u53ea\u53ef\u80fd\u4e0d\u53d8\uff08 \\(i\\) \u4ecd\u4e3a\u77ed\u677f\uff09\u6216\u53d8\u5c0f\uff08\u79fb\u52a8\u540e\u7684 \\(j\\) \u6210\u4e3a\u77ed\u677f\uff09\u3002</p> <p></p> <p> \u56fe 15-9 \u00a0 \u5411\u5185\u79fb\u52a8\u957f\u677f\u540e\u7684\u72b6\u6001 </p> <p>\u53cd\u5411\u601d\u8003\uff0c\u6211\u4eec\u53ea\u6709\u5411\u5185\u6536\u7f29\u77ed\u677f \\(i\\) \uff0c\u624d\u6709\u53ef\u80fd\u4f7f\u5bb9\u91cf\u53d8\u5927\u3002\u56e0\u4e3a\u867d\u7136\u5bbd\u5ea6\u4e00\u5b9a\u53d8\u5c0f\uff0c\u4f46\u9ad8\u5ea6\u53ef\u80fd\u4f1a\u53d8\u5927\uff08\u79fb\u52a8\u540e\u7684\u77ed\u677f \\(i\\) \u53ef\u80fd\u4f1a\u53d8\u957f\uff09\u3002\u4f8b\u5982\u5728\u56fe 15-10 \u4e2d\uff0c\u79fb\u52a8\u77ed\u677f\u540e\u9762\u79ef\u53d8\u5927\u3002</p> <p></p> <p> \u56fe 15-10 \u00a0 \u5411\u5185\u79fb\u52a8\u77ed\u677f\u540e\u7684\u72b6\u6001 </p> <p>\u7531\u6b64\u4fbf\u53ef\u63a8\u51fa\u672c\u9898\u7684\u8d2a\u5fc3\u7b56\u7565\uff1a\u521d\u59cb\u5316\u4e24\u6307\u9488\uff0c\u4f7f\u5176\u5206\u5217\u5bb9\u5668\u4e24\u7aef\uff0c\u6bcf\u8f6e\u5411\u5185\u6536\u7f29\u77ed\u677f\u5bf9\u5e94\u7684\u6307\u9488\uff0c\u76f4\u81f3\u4e24\u6307\u9488\u76f8\u9047\u3002</p> <p>\u56fe 15-11 \u5c55\u793a\u4e86\u8d2a\u5fc3\u7b56\u7565\u7684\u6267\u884c\u8fc7\u7a0b\u3002</p> <ol> <li>\u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5206\u5217\u6570\u7ec4\u4e24\u7aef\u3002</li> <li>\u8ba1\u7b97\u5f53\u524d\u72b6\u6001\u7684\u5bb9\u91cf \\(cap[i, j]\\) \uff0c\u5e76\u66f4\u65b0\u6700\u5927\u5bb9\u91cf\u3002</li> <li>\u6bd4\u8f83\u677f \\(i\\) \u548c \u677f \\(j\\) \u7684\u9ad8\u5ea6\uff0c\u5e76\u5c06\u77ed\u677f\u5411\u5185\u79fb\u52a8\u4e00\u683c\u3002</li> <li>\u5faa\u73af\u6267\u884c\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\uff0c\u76f4\u81f3 \\(i\\) \u548c \\(j\\) \u76f8\u9047\u65f6\u7ed3\u675f\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 15-11 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898\u7684\u8d2a\u5fc3\u8fc7\u7a0b </p>"},{"location":"chapter_greedy/max_capacity_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u4ee3\u7801\u5faa\u73af\u6700\u591a \\(n\\) \u8f6e\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> <p>\u53d8\u91cf \\(i\\)\u3001\\(j\\)\u3001\\(res\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig max_capacity.py<pre><code>def max_capacity(ht: list[int]) -&gt; int:\n    \"\"\"\u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3\"\"\"\n    # \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    i, j = 0, len(ht) - 1\n    # \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    res = 0\n    # \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while i &lt; j:\n        # \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        # \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if ht[i] &lt; ht[j]:\n            i += 1\n        else:\n            j -= 1\n    return res\n</code></pre> max_capacity.cpp<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(vector&lt;int&gt; &amp;ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    int i = 0, j = ht.size() - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    int res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int cap = min(ht[i], ht[j]) * (j - i);\n        res = max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.java<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(int[] ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    int i = 0, j = ht.length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    int res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.cs<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint MaxCapacity(int[] ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    int i = 0, j = ht.Length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    int res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int cap = Math.Min(ht[i], ht[j]) * (j - i);\n        res = Math.Max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.go<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunc maxCapacity(ht []int) int {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    i, j := 0, len(ht)-1\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    res := 0\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    for i &lt; j {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        capacity := int(math.Min(float64(ht[i]), float64(ht[j]))) * (j - i)\n        res = int(math.Max(float64(res), float64(capacity)))\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if ht[i] &lt; ht[j] {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.swift<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunc maxCapacity(ht: [Int]) -&gt; Int {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    var i = ht.startIndex, j = ht.endIndex - 1\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    var res = 0\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while i &lt; j {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        let cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if ht[i] &lt; ht[j] {\n            i += 1\n        } else {\n            j -= 1\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.js<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunction maxCapacity(ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    let i = 0,\n        j = ht.length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    let res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        const cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.ts<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunction maxCapacity(ht: number[]): number {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    let i = 0,\n        j = ht.length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    let res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        const cap: number = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.dart<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(List&lt;int&gt; ht) {\n  // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n  int i = 0, j = ht.length - 1;\n  // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n  int res = 0;\n  // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n  while (i &lt; j) {\n    // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n    int cap = min(ht[i], ht[j]) * (j - i);\n    res = max(res, cap);\n    // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n    if (ht[i] &lt; ht[j]) {\n      i++;\n    } else {\n      j--;\n    }\n  }\n  return res;\n}\n</code></pre> max_capacity.rs<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfn max_capacity(ht: &amp;[i32]) -&gt; i32 {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    let mut i = 0;\n    let mut j = ht.len() - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    let mut res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while i &lt; j {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        let cap = std::cmp::min(ht[i], ht[j]) * (j - i) as i32;\n        res = std::cmp::max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if ht[i] &lt; ht[j] {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    res\n}\n</code></pre> max_capacity.c<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(int ht[], int htLength) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    int i = 0;\n    int j = htLength - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    int res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int capacity = myMin(ht[i], ht[j]) * (j - i);\n        res = myMax(res, capacity);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.kt<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfun maxCapacity(ht: IntArray): Int {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    var i = 0\n    var j = ht.size - 1\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    var res = 0\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        val cap = (min(ht[i].toDouble(), ht[j].toDouble()) * (j - i)).toInt()\n        res = max(res.toDouble(), cap.toDouble()).toInt()\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.rb<pre><code>[class]{}-[func]{max_capacity}\n</code></pre> max_capacity.zig<pre><code>[class]{}-[func]{maxCapacity}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_greedy/max_capacity_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"<p>\u4e4b\u6240\u4ee5\u8d2a\u5fc3\u6bd4\u7a77\u4e3e\u66f4\u5feb\uff0c\u662f\u56e0\u4e3a\u6bcf\u8f6e\u7684\u8d2a\u5fc3\u9009\u62e9\u90fd\u4f1a\u201c\u8df3\u8fc7\u201d\u4e00\u4e9b\u72b6\u6001\u3002</p> <p>\u6bd4\u5982\u5728\u72b6\u6001 \\(cap[i, j]\\) \u4e0b\uff0c\\(i\\) \u4e3a\u77ed\u677f\u3001\\(j\\) \u4e3a\u957f\u677f\u3002\u82e5\u8d2a\u5fc3\u5730\u5c06\u77ed\u677f \\(i\\) \u5411\u5185\u79fb\u52a8\u4e00\u683c\uff0c\u4f1a\u5bfc\u81f4\u56fe 15-12 \u6240\u793a\u7684\u72b6\u6001\u88ab\u201c\u8df3\u8fc7\u201d\u3002\u8fd9\u610f\u5473\u7740\u4e4b\u540e\u65e0\u6cd5\u9a8c\u8bc1\u8fd9\u4e9b\u72b6\u6001\u7684\u5bb9\u91cf\u5927\u5c0f\u3002</p> \\[ cap[i, i+1], cap[i, i+2], \\dots, cap[i, j-2], cap[i, j-1] \\] <p></p> <p> \u56fe 15-12 \u00a0 \u79fb\u52a8\u77ed\u677f\u5bfc\u81f4\u88ab\u8df3\u8fc7\u7684\u72b6\u6001 </p> <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u8fd9\u4e9b\u88ab\u8df3\u8fc7\u7684\u72b6\u6001\u5b9e\u9645\u4e0a\u5c31\u662f\u5c06\u957f\u677f \\(j\\) \u5411\u5185\u79fb\u52a8\u7684\u6240\u6709\u72b6\u6001\u3002\u524d\u9762\u6211\u4eec\u5df2\u7ecf\u8bc1\u660e\u5185\u79fb\u957f\u677f\u4e00\u5b9a\u4f1a\u5bfc\u81f4\u5bb9\u91cf\u53d8\u5c0f\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u88ab\u8df3\u8fc7\u7684\u72b6\u6001\u90fd\u4e0d\u53ef\u80fd\u662f\u6700\u4f18\u89e3\uff0c\u8df3\u8fc7\u5b83\u4eec\u4e0d\u4f1a\u5bfc\u81f4\u9519\u8fc7\u6700\u4f18\u89e3\u3002</p> <p>\u4ee5\u4e0a\u5206\u6790\u8bf4\u660e\uff0c\u79fb\u52a8\u77ed\u677f\u7684\u64cd\u4f5c\u662f\u201c\u5b89\u5168\u201d\u7684\uff0c\u8d2a\u5fc3\u7b56\u7565\u662f\u6709\u6548\u7684\u3002</p>"},{"location":"chapter_greedy/max_product_cutting_problem/","title":"15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570 \\(n\\) \uff0c\u5c06\u5176\u5207\u5206\u4e3a\u81f3\u5c11\u4e24\u4e2a\u6b63\u6574\u6570\u7684\u548c\uff0c\u6c42\u5207\u5206\u540e\u6240\u6709\u6574\u6570\u7684\u4e58\u79ef\u6700\u5927\u662f\u591a\u5c11\uff0c\u5982\u56fe 15-13 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 15-13 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u7684\u95ee\u9898\u5b9a\u4e49 </p> <p>\u5047\u8bbe\u6211\u4eec\u5c06 \\(n\\) \u5207\u5206\u4e3a \\(m\\) \u4e2a\u6574\u6570\u56e0\u5b50\uff0c\u5176\u4e2d\u7b2c \\(i\\) \u4e2a\u56e0\u5b50\u8bb0\u4e3a \\(n_i\\) \uff0c\u5373</p> \\[ n = \\sum_{i=1}^{m}n_i \\] <p>\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u5f97\u6240\u6709\u6574\u6570\u56e0\u5b50\u7684\u6700\u5927\u4e58\u79ef\uff0c\u5373</p> \\[ \\max(\\prod_{i=1}^{m}n_i) \\] <p>\u6211\u4eec\u9700\u8981\u601d\u8003\u7684\u662f\uff1a\u5207\u5206\u6570\u91cf \\(m\\) \u5e94\u8be5\u591a\u5927\uff0c\u6bcf\u4e2a \\(n_i\\) \u5e94\u8be5\u662f\u591a\u5c11\uff1f</p>"},{"location":"chapter_greedy/max_product_cutting_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"<p>\u6839\u636e\u7ecf\u9a8c\uff0c\u4e24\u4e2a\u6574\u6570\u7684\u4e58\u79ef\u5f80\u5f80\u6bd4\u5b83\u4eec\u7684\u52a0\u548c\u66f4\u5927\u3002\u5047\u8bbe\u4ece \\(n\\) \u4e2d\u5206\u51fa\u4e00\u4e2a\u56e0\u5b50 \\(2\\) \uff0c\u5219\u5b83\u4eec\u7684\u4e58\u79ef\u4e3a \\(2(n-2)\\) \u3002\u6211\u4eec\u5c06\u8be5\u4e58\u79ef\u4e0e \\(n\\) \u4f5c\u6bd4\u8f83\uff1a</p> \\[ \\begin{aligned} 2(n-2) &amp; \\geq n \\newline 2n - n - 4 &amp; \\geq 0 \\newline n &amp; \\geq 4 \\end{aligned} \\] <p>\u5982\u56fe 15-14 \u6240\u793a\uff0c\u5f53 \\(n \\geq 4\\) \u65f6\uff0c\u5207\u5206\u51fa\u4e00\u4e2a \\(2\\) \u540e\u4e58\u79ef\u4f1a\u53d8\u5927\uff0c\u8fd9\u8bf4\u660e\u5927\u4e8e\u7b49\u4e8e \\(4\\) \u7684\u6574\u6570\u90fd\u5e94\u8be5\u88ab\u5207\u5206\u3002</p> <p>\u8d2a\u5fc3\u7b56\u7565\u4e00\uff1a\u5982\u679c\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b \\(\\geq 4\\) \u7684\u56e0\u5b50\uff0c\u90a3\u4e48\u5b83\u5c31\u5e94\u8be5\u88ab\u7ee7\u7eed\u5207\u5206\u3002\u6700\u7ec8\u7684\u5207\u5206\u65b9\u6848\u53ea\u5e94\u51fa\u73b0 \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u8fd9\u4e09\u79cd\u56e0\u5b50\u3002</p> <p></p> <p> \u56fe 15-14 \u00a0 \u5207\u5206\u5bfc\u81f4\u4e58\u79ef\u53d8\u5927 </p> <p>\u63a5\u4e0b\u6765\u601d\u8003\u54ea\u4e2a\u56e0\u5b50\u662f\u6700\u4f18\u7684\u3002\u5728 \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u8fd9\u4e09\u4e2a\u56e0\u5b50\u4e2d\uff0c\u663e\u7136 \\(1\\) \u662f\u6700\u5dee\u7684\uff0c\u56e0\u4e3a \\(1 \\times (n-1) &lt; n\\) \u6052\u6210\u7acb\uff0c\u5373\u5207\u5206\u51fa \\(1\\) \u53cd\u800c\u4f1a\u5bfc\u81f4\u4e58\u79ef\u51cf\u5c0f\u3002</p> <p>\u5982\u56fe 15-15 \u6240\u793a\uff0c\u5f53 \\(n = 6\\) \u65f6\uff0c\u6709 \\(3 \\times 3 &gt; 2 \\times 2 \\times 2\\) \u3002\u8fd9\u610f\u5473\u7740\u5207\u5206\u51fa \\(3\\) \u6bd4\u5207\u5206\u51fa \\(2\\) \u66f4\u4f18\u3002</p> <p>\u8d2a\u5fc3\u7b56\u7565\u4e8c\uff1a\u5728\u5207\u5206\u65b9\u6848\u4e2d\uff0c\u6700\u591a\u53ea\u5e94\u5b58\u5728\u4e24\u4e2a \\(2\\) \u3002\u56e0\u4e3a\u4e09\u4e2a \\(2\\) \u603b\u662f\u53ef\u4ee5\u66ff\u6362\u4e3a\u4e24\u4e2a \\(3\\) \uff0c\u4ece\u800c\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002</p> <p></p> <p> \u56fe 15-15 \u00a0 \u6700\u4f18\u5207\u5206\u56e0\u5b50 </p> <p>\u7efc\u4e0a\u6240\u8ff0\uff0c\u53ef\u63a8\u7406\u51fa\u4ee5\u4e0b\u8d2a\u5fc3\u7b56\u7565\u3002</p> <ol> <li>\u8f93\u5165\u6574\u6570 \\(n\\) \uff0c\u4ece\u5176\u4e0d\u65ad\u5730\u5207\u5206\u51fa\u56e0\u5b50 \\(3\\) \uff0c\u76f4\u81f3\u4f59\u6570\u4e3a \\(0\\)\u3001\\(1\\)\u3001\\(2\\) \u3002</li> <li>\u5f53\u4f59\u6570\u4e3a \\(0\\) \u65f6\uff0c\u4ee3\u8868 \\(n\\) \u662f \\(3\\) \u7684\u500d\u6570\uff0c\u56e0\u6b64\u4e0d\u505a\u4efb\u4f55\u5904\u7406\u3002</li> <li>\u5f53\u4f59\u6570\u4e3a \\(2\\) \u65f6\uff0c\u4e0d\u7ee7\u7eed\u5212\u5206\uff0c\u4fdd\u7559\u3002</li> <li>\u5f53\u4f59\u6570\u4e3a \\(1\\) \u65f6\uff0c\u7531\u4e8e \\(2 \\times 2 &gt; 1 \\times 3\\) \uff0c\u56e0\u6b64\u5e94\u5c06\u6700\u540e\u4e00\u4e2a \\(3\\) \u66ff\u6362\u4e3a \\(2\\) \u3002</li> </ol>"},{"location":"chapter_greedy/max_product_cutting_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5982\u56fe 15-16 \u6240\u793a\uff0c\u6211\u4eec\u65e0\u987b\u901a\u8fc7\u5faa\u73af\u6765\u5207\u5206\u6574\u6570\uff0c\u800c\u53ef\u4ee5\u5229\u7528\u5411\u4e0b\u6574\u9664\u8fd0\u7b97\u5f97\u5230 \\(3\\) \u7684\u4e2a\u6570 \\(a\\) \uff0c\u7528\u53d6\u6a21\u8fd0\u7b97\u5f97\u5230\u4f59\u6570 \\(b\\) \uff0c\u6b64\u65f6\u6709\uff1a</p> \\[ n = 3 a + b \\] <p>\u8bf7\u6ce8\u610f\uff0c\u5bf9\u4e8e \\(n \\leq 3\\) \u7684\u8fb9\u754c\u60c5\u51b5\uff0c\u5fc5\u987b\u62c6\u5206\u51fa\u4e00\u4e2a \\(1\\) \uff0c\u4e58\u79ef\u4e3a \\(1 \\times (n - 1)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig max_product_cutting.py<pre><code>def max_product_cutting(n: int) -&gt; int:\n    \"\"\"\u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3\"\"\"\n    # \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if n &lt;= 3:\n        return 1 * (n - 1)\n    # \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    a, b = n // 3, n % 3\n    if b == 1:\n        # \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return int(math.pow(3, a - 1)) * 2 * 2\n    if b == 2:\n        # \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return int(math.pow(3, a)) * 2\n    # \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return int(math.pow(3, a))\n</code></pre> max_product_cutting.cpp<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return (int)pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return (int)pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return (int)pow(3, a);\n}\n</code></pre> max_product_cutting.java<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return (int) Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return (int) Math.pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return (int) Math.pow(3, a);\n}\n</code></pre> max_product_cutting.cs<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint MaxProductCutting(int n) {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return (int)Math.Pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return (int)Math.Pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return (int)Math.Pow(3, a);\n}\n</code></pre> max_product_cutting.go<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunc maxProductCutting(n int) int {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    a := n / 3\n    b := n % 3\n    if b == 1 {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return int(math.Pow(3, float64(a-1))) * 2 * 2\n    }\n    if b == 2 {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return int(math.Pow(3, float64(a))) * 2\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return int(math.Pow(3, float64(a)))\n}\n</code></pre> max_product_cutting.swift<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunc maxProductCutting(n: Int) -&gt; Int {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    let a = n / 3\n    let b = n % 3\n    if b == 1 {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return pow(3, a - 1) * 2 * 2\n    }\n    if b == 2 {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return pow(3, a) * 2\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return pow(3, a)\n}\n</code></pre> max_product_cutting.js<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunction maxProductCutting(n) {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    let a = Math.floor(n / 3);\n    let b = n % 3;\n    if (b === 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return Math.pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.ts<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunction maxProductCutting(n: number): number {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    let a: number = Math.floor(n / 3);\n    let b: number = n % 3;\n    if (b === 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return Math.pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.dart<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n  // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n  if (n &lt;= 3) {\n    return 1 * (n - 1);\n  }\n  // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n  int a = n ~/ 3;\n  int b = n % 3;\n  if (b == 1) {\n    // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n    return (pow(3, a - 1) * 2 * 2).toInt();\n  }\n  if (b == 2) {\n    // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return (pow(3, a) * 2).toInt();\n  }\n  // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n  return pow(3, a).toInt();\n}\n</code></pre> max_product_cutting.rs<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfn max_product_cutting(n: i32) -&gt; i32 {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if n &lt;= 3 {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    let a = n / 3;\n    let b = n % 3;\n    if b == 1 {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        3_i32.pow(a as u32 - 1) * 2 * 2\n    } else if b == 2 {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        3_i32.pow(a as u32) * 2\n    } else {\n        // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        3_i32.pow(a as u32)\n    }\n}\n</code></pre> max_product_cutting.c<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return pow(3, a);\n}\n</code></pre> max_product_cutting.kt<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfun maxProductCutting(n: Int): Int {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1)\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    val a = n / 3\n    val b = n % 3\n    if (b == 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return 3.0.pow((a - 1).toDouble()).toInt() * 2 * 2\n    }\n    if (b == 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return 3.0.pow(a.toDouble()).toInt() * 2 * 2\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return 3.0.pow(a.toDouble()).toInt()\n}\n</code></pre> max_product_cutting.rb<pre><code>[class]{}-[func]{max_product_cutting}\n</code></pre> max_product_cutting.zig<pre><code>[class]{}-[func]{maxProductCutting}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 15-16 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u7684\u8ba1\u7b97\u65b9\u6cd5 </p> <p>\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u7f16\u7a0b\u8bed\u8a00\u7684\u5e42\u8fd0\u7b97\u7684\u5b9e\u73b0\u65b9\u6cd5\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u5e38\u7528\u7684\u5e42\u8ba1\u7b97\u51fd\u6570\u6709\u4e09\u79cd\u3002</p> <ul> <li>\u8fd0\u7b97\u7b26 <code>**</code> \u548c\u51fd\u6570 <code>pow()</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log\u2061 a)\\) \u3002</li> <li>\u51fd\u6570 <code>math.pow()</code> \u5185\u90e8\u8c03\u7528 C \u8bed\u8a00\u5e93\u7684 <code>pow()</code> \u51fd\u6570\uff0c\u5176\u6267\u884c\u6d6e\u70b9\u53d6\u5e42\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</li> </ul> <p>\u53d8\u91cf \\(a\\) \u548c \\(b\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</p>"},{"location":"chapter_greedy/max_product_cutting_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"<p>\u4f7f\u7528\u53cd\u8bc1\u6cd5\uff0c\u53ea\u5206\u6790 \\(n \\geq 3\\) \u7684\u60c5\u51b5\u3002</p> <ol> <li>\u6240\u6709\u56e0\u5b50 \\(\\leq 3\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728 \\(\\geq 4\\) \u7684\u56e0\u5b50 \\(x\\) \uff0c\u90a3\u4e48\u4e00\u5b9a\u53ef\u4ee5\u5c06\u5176\u7ee7\u7eed\u5212\u5206\u4e3a \\(2(x-2)\\) \uff0c\u4ece\u800c\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</li> <li>\u5207\u5206\u65b9\u6848\u4e0d\u5305\u542b \\(1\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728\u4e00\u4e2a\u56e0\u5b50 \\(1\\) \uff0c\u90a3\u4e48\u5b83\u4e00\u5b9a\u53ef\u4ee5\u5408\u5e76\u5165\u53e6\u5916\u4e00\u4e2a\u56e0\u5b50\u4e2d\uff0c\u4ee5\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</li> <li>\u5207\u5206\u65b9\u6848\u6700\u591a\u5305\u542b\u4e24\u4e2a \\(2\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b\u4e09\u4e2a \\(2\\) \uff0c\u90a3\u4e48\u4e00\u5b9a\u53ef\u4ee5\u66ff\u6362\u4e3a\u4e24\u4e2a \\(3\\) \uff0c\u4e58\u79ef\u66f4\u5927\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</li> </ol>"},{"location":"chapter_greedy/summary/","title":"15.5 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u8d2a\u5fc3\u7b97\u6cd5\u901a\u5e38\u7528\u4e8e\u89e3\u51b3\u6700\u4f18\u5316\u95ee\u9898\uff0c\u5176\u539f\u7406\u662f\u5728\u6bcf\u4e2a\u51b3\u7b56\u9636\u6bb5\u90fd\u505a\u51fa\u5c40\u90e8\u6700\u4f18\u7684\u51b3\u7b56\uff0c\u4ee5\u671f\u83b7\u5f97\u5168\u5c40\u6700\u4f18\u89e3\u3002</li> <li>\u8d2a\u5fc3\u7b97\u6cd5\u4f1a\u8fed\u4ee3\u5730\u505a\u51fa\u4e00\u4e2a\u53c8\u4e00\u4e2a\u7684\u8d2a\u5fc3\u9009\u62e9\uff0c\u6bcf\u8f6e\u90fd\u5c06\u95ee\u9898\u8f6c\u5316\u6210\u4e00\u4e2a\u89c4\u6a21\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u5230\u95ee\u9898\u88ab\u89e3\u51b3\u3002</li> <li>\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4ec5\u5b9e\u73b0\u7b80\u5355\uff0c\u8fd8\u5177\u6709\u5f88\u9ad8\u7684\u89e3\u9898\u6548\u7387\u3002\u76f8\u6bd4\u4e8e\u52a8\u6001\u89c4\u5212\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u66f4\u4f4e\u3002</li> <li>\u5728\u96f6\u94b1\u5151\u6362\u95ee\u9898\u4e2d\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u786c\u5e01\u7ec4\u5408\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1b\u5bf9\u4e8e\u53e6\u5916\u4e00\u4e9b\u786c\u5e01\u7ec4\u5408\u5219\u4e0d\u7136\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ef\u80fd\u627e\u5230\u5f88\u5dee\u7684\u89e3\u3002</li> <li>\u9002\u5408\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u7684\u95ee\u9898\u5177\u6709\u4e24\u5927\u6027\u8d28\uff1a\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u3002\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u4ee3\u8868\u8d2a\u5fc3\u7b56\u7565\u7684\u6709\u6548\u6027\u3002</li> <li>\u5bf9\u4e8e\u67d0\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u7684\u8bc1\u660e\u5e76\u4e0d\u7b80\u5355\u3002\u76f8\u5bf9\u6765\u8bf4\uff0c\u8bc1\u4f2a\u66f4\u52a0\u5bb9\u6613\uff0c\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\u3002</li> <li>\u6c42\u89e3\u8d2a\u5fc3\u95ee\u9898\u4e3b\u8981\u5206\u4e3a\u4e09\u6b65\uff1a\u95ee\u9898\u5206\u6790\u3001\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u3001\u6b63\u786e\u6027\u8bc1\u660e\u3002\u5176\u4e2d\uff0c\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u662f\u6838\u5fc3\u6b65\u9aa4\uff0c\u6b63\u786e\u6027\u8bc1\u660e\u5f80\u5f80\u662f\u96be\u70b9\u3002</li> <li>\u5206\u6570\u80cc\u5305\u95ee\u9898\u5728 0-1 \u80cc\u5305\u7684\u57fa\u7840\u4e0a\uff0c\u5141\u8bb8\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u56e0\u6b64\u53ef\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u3002\u8d2a\u5fc3\u7b56\u7565\u7684\u6b63\u786e\u6027\u53ef\u4ee5\u4f7f\u7528\u53cd\u8bc1\u6cd5\u6765\u8bc1\u660e\u3002</li> <li>\u6700\u5927\u5bb9\u91cf\u95ee\u9898\u53ef\u4f7f\u7528\u7a77\u4e3e\u6cd5\u6c42\u89e3\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u901a\u8fc7\u8bbe\u8ba1\u8d2a\u5fc3\u7b56\u7565\uff0c\u6bcf\u8f6e\u5411\u5185\u79fb\u52a8\u77ed\u677f\uff0c\u53ef\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>\u5728\u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u5148\u540e\u63a8\u7406\u51fa\u4e24\u4e2a\u8d2a\u5fc3\u7b56\u7565\uff1a\\(\\geq 4\\) \u7684\u6574\u6570\u90fd\u5e94\u8be5\u7ee7\u7eed\u5207\u5206\uff0c\u6700\u4f18\u5207\u5206\u56e0\u5b50\u4e3a \\(3\\) \u3002\u4ee3\u7801\u4e2d\u5305\u542b\u5e42\u8fd0\u7b97\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u5e42\u8fd0\u7b97\u5b9e\u73b0\u65b9\u6cd5\uff0c\u901a\u5e38\u4e3a \\(O(1)\\) \u6216 \\(O(\\log n)\\) \u3002</li> </ul>"},{"location":"chapter_hashing/","title":"\u7b2c 6 \u7ae0 \u00a0 \u54c8\u5e0c\u8868","text":"<p>Abstract</p> <p>\u5728\u8ba1\u7b97\u673a\u4e16\u754c\u4e2d\uff0c\u54c8\u5e0c\u8868\u5982\u540c\u4e00\u4f4d\u806a\u6167\u7684\u56fe\u4e66\u7ba1\u7406\u5458\u3002</p> <p>\u4ed6\u77e5\u9053\u5982\u4f55\u8ba1\u7b97\u7d22\u4e66\u53f7\uff0c\u4ece\u800c\u53ef\u4ee5\u5feb\u901f\u627e\u5230\u76ee\u6807\u56fe\u4e66\u3002</p>"},{"location":"chapter_hashing/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>6.1 \u00a0 \u54c8\u5e0c\u8868</li> <li>6.2 \u00a0 \u54c8\u5e0c\u51b2\u7a81</li> <li>6.3 \u00a0 \u54c8\u5e0c\u7b97\u6cd5</li> <li>6.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_hashing/hash_algorithm/","title":"6.3 \u00a0 \u54c8\u5e0c\u7b97\u6cd5","text":"<p>\u524d\u4e24\u8282\u4ecb\u7ecd\u4e86\u54c8\u5e0c\u8868\u7684\u5de5\u4f5c\u539f\u7406\u548c\u54c8\u5e0c\u51b2\u7a81\u7684\u5904\u7406\u65b9\u6cd5\u3002\u7136\u800c\u65e0\u8bba\u662f\u5f00\u653e\u5bfb\u5740\u8fd8\u662f\u94fe\u5f0f\u5730\u5740\uff0c\u5b83\u4eec\u53ea\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u53d1\u751f\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\uff0c\u800c\u65e0\u6cd5\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u3002</p> <p>\u5982\u679c\u54c8\u5e0c\u51b2\u7a81\u8fc7\u4e8e\u9891\u7e41\uff0c\u54c8\u5e0c\u8868\u7684\u6027\u80fd\u5219\u4f1a\u6025\u5267\u52a3\u5316\u3002\u5982\u56fe 6-8 \u6240\u793a\uff0c\u5bf9\u4e8e\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\uff0c\u7406\u60f3\u60c5\u51b5\u4e0b\u952e\u503c\u5bf9\u5747\u5300\u5206\u5e03\u5728\u5404\u4e2a\u6876\u4e2d\uff0c\u8fbe\u5230\u6700\u4f73\u67e5\u8be2\u6548\u7387\uff1b\u6700\u5dee\u60c5\u51b5\u4e0b\u6240\u6709\u952e\u503c\u5bf9\u90fd\u5b58\u50a8\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002</p> <p></p> <p> \u56fe 6-8 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u7684\u6700\u4f73\u60c5\u51b5\u4e0e\u6700\u5dee\u60c5\u51b5 </p> <p>\u952e\u503c\u5bf9\u7684\u5206\u5e03\u60c5\u51b5\u7531\u54c8\u5e0c\u51fd\u6570\u51b3\u5b9a\u3002\u56de\u5fc6\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u6b65\u9aa4\uff0c\u5148\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u518d\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\uff1a</p> <pre><code>index = hash(key) % capacity\n</code></pre> <p>\u89c2\u5bdf\u4ee5\u4e0a\u516c\u5f0f\uff0c\u5f53\u54c8\u5e0c\u8868\u5bb9\u91cf <code>capacity</code> \u56fa\u5b9a\u65f6\uff0c\u54c8\u5e0c\u7b97\u6cd5 <code>hash()</code> \u51b3\u5b9a\u4e86\u8f93\u51fa\u503c\uff0c\u8fdb\u800c\u51b3\u5b9a\u4e86\u952e\u503c\u5bf9\u5728\u54c8\u5e0c\u8868\u4e2d\u7684\u5206\u5e03\u60c5\u51b5\u3002</p> <p>\u8fd9\u610f\u5473\u7740\uff0c\u4e3a\u4e86\u964d\u4f4e\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u6982\u7387\uff0c\u6211\u4eec\u5e94\u5f53\u5c06\u6ce8\u610f\u529b\u96c6\u4e2d\u5728\u54c8\u5e0c\u7b97\u6cd5 <code>hash()</code> \u7684\u8bbe\u8ba1\u4e0a\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#631","title":"6.3.1 \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u76ee\u6807","text":"<p>\u4e3a\u4e86\u5b9e\u73b0\u201c\u65e2\u5feb\u53c8\u7a33\u201d\u7684\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u5177\u5907\u4ee5\u4e0b\u7279\u70b9\u3002</p> <ul> <li>\u786e\u5b9a\u6027\uff1a\u5bf9\u4e8e\u76f8\u540c\u7684\u8f93\u5165\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u59cb\u7ec8\u4ea7\u751f\u76f8\u540c\u7684\u8f93\u51fa\u3002\u8fd9\u6837\u624d\u80fd\u786e\u4fdd\u54c8\u5e0c\u8868\u662f\u53ef\u9760\u7684\u3002</li> <li>\u6548\u7387\u9ad8\uff1a\u8ba1\u7b97\u54c8\u5e0c\u503c\u7684\u8fc7\u7a0b\u5e94\u8be5\u8db3\u591f\u5feb\u3002\u8ba1\u7b97\u5f00\u9500\u8d8a\u5c0f\uff0c\u54c8\u5e0c\u8868\u7684\u5b9e\u7528\u6027\u8d8a\u9ad8\u3002</li> <li>\u5747\u5300\u5206\u5e03\uff1a\u54c8\u5e0c\u7b97\u6cd5\u5e94\u4f7f\u5f97\u952e\u503c\u5bf9\u5747\u5300\u5206\u5e03\u5728\u54c8\u5e0c\u8868\u4e2d\u3002\u5206\u5e03\u8d8a\u5747\u5300\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9664\u4e86\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u54c8\u5e0c\u8868\uff0c\u8fd8\u5e7f\u6cdb\u5e94\u7528\u4e8e\u5176\u4ed6\u9886\u57df\u4e2d\u3002</p> <ul> <li>\u5bc6\u7801\u5b58\u50a8\uff1a\u4e3a\u4e86\u4fdd\u62a4\u7528\u6237\u5bc6\u7801\u7684\u5b89\u5168\uff0c\u7cfb\u7edf\u901a\u5e38\u4e0d\u4f1a\u76f4\u63a5\u5b58\u50a8\u7528\u6237\u7684\u660e\u6587\u5bc6\u7801\uff0c\u800c\u662f\u5b58\u50a8\u5bc6\u7801\u7684\u54c8\u5e0c\u503c\u3002\u5f53\u7528\u6237\u8f93\u5165\u5bc6\u7801\u65f6\uff0c\u7cfb\u7edf\u4f1a\u5bf9\u8f93\u5165\u7684\u5bc6\u7801\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u7136\u540e\u4e0e\u5b58\u50a8\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u5bc6\u7801\u5c31\u88ab\u89c6\u4e3a\u6b63\u786e\u3002</li> <li>\u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5\uff1a\u6570\u636e\u53d1\u9001\u65b9\u53ef\u4ee5\u8ba1\u7b97\u6570\u636e\u7684\u54c8\u5e0c\u503c\u5e76\u5c06\u5176\u4e00\u540c\u53d1\u9001\uff1b\u63a5\u6536\u65b9\u53ef\u4ee5\u91cd\u65b0\u8ba1\u7b97\u63a5\u6536\u5230\u7684\u6570\u636e\u7684\u54c8\u5e0c\u503c\uff0c\u5e76\u4e0e\u63a5\u6536\u5230\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u6570\u636e\u5c31\u88ab\u89c6\u4e3a\u5b8c\u6574\u3002</li> </ul> <p>\u5bf9\u4e8e\u5bc6\u7801\u5b66\u7684\u76f8\u5173\u5e94\u7528\uff0c\u4e3a\u4e86\u9632\u6b62\u4ece\u54c8\u5e0c\u503c\u63a8\u5bfc\u51fa\u539f\u59cb\u5bc6\u7801\u7b49\u9006\u5411\u5de5\u7a0b\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9700\u8981\u5177\u5907\u66f4\u9ad8\u7b49\u7ea7\u7684\u5b89\u5168\u7279\u6027\u3002</p> <ul> <li>\u5355\u5411\u6027\uff1a\u65e0\u6cd5\u901a\u8fc7\u54c8\u5e0c\u503c\u53cd\u63a8\u51fa\u5173\u4e8e\u8f93\u5165\u6570\u636e\u7684\u4efb\u4f55\u4fe1\u606f\u3002</li> <li>\u6297\u78b0\u649e\u6027\uff1a\u5e94\u5f53\u6781\u96be\u627e\u5230\u4e24\u4e2a\u4e0d\u540c\u7684\u8f93\u5165\uff0c\u4f7f\u5f97\u5b83\u4eec\u7684\u54c8\u5e0c\u503c\u76f8\u540c\u3002</li> <li>\u96ea\u5d29\u6548\u5e94\uff1a\u8f93\u5165\u7684\u5fae\u5c0f\u53d8\u5316\u5e94\u5f53\u5bfc\u81f4\u8f93\u51fa\u7684\u663e\u8457\u4e14\u4e0d\u53ef\u9884\u6d4b\u7684\u53d8\u5316\u3002</li> </ul> <p>\u8bf7\u6ce8\u610f\uff0c\u201c\u5747\u5300\u5206\u5e03\u201d\u4e0e\u201c\u6297\u78b0\u649e\u6027\u201d\u662f\u4e24\u4e2a\u72ec\u7acb\u7684\u6982\u5ff5\uff0c\u6ee1\u8db3\u5747\u5300\u5206\u5e03\u4e0d\u4e00\u5b9a\u6ee1\u8db3\u6297\u78b0\u649e\u6027\u3002\u4f8b\u5982\uff0c\u5728\u968f\u673a\u8f93\u5165 <code>key</code> \u4e0b\uff0c\u54c8\u5e0c\u51fd\u6570 <code>key % 100</code> \u53ef\u4ee5\u4ea7\u751f\u5747\u5300\u5206\u5e03\u7684\u8f93\u51fa\u3002\u7136\u800c\u8be5\u54c8\u5e0c\u7b97\u6cd5\u8fc7\u4e8e\u7b80\u5355\uff0c\u6240\u6709\u540e\u4e24\u4f4d\u76f8\u7b49\u7684 <code>key</code> \u7684\u8f93\u51fa\u90fd\u76f8\u540c\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u4ece\u54c8\u5e0c\u503c\u53cd\u63a8\u51fa\u53ef\u7528\u7684 <code>key</code> \uff0c\u4ece\u800c\u7834\u89e3\u5bc6\u7801\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#632","title":"6.3.2 \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1","text":"<p>\u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u662f\u4e00\u4e2a\u9700\u8981\u8003\u8651\u8bb8\u591a\u56e0\u7d20\u7684\u590d\u6742\u95ee\u9898\u3002\u7136\u800c\u5bf9\u4e8e\u67d0\u4e9b\u8981\u6c42\u4e0d\u9ad8\u7684\u573a\u666f\uff0c\u6211\u4eec\u4e5f\u80fd\u8bbe\u8ba1\u4e00\u4e9b\u7b80\u5355\u7684\u54c8\u5e0c\u7b97\u6cd5\u3002</p> <ul> <li>\u52a0\u6cd5\u54c8\u5e0c\uff1a\u5bf9\u8f93\u5165\u7684\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u8fdb\u884c\u76f8\u52a0\uff0c\u5c06\u5f97\u5230\u7684\u603b\u548c\u4f5c\u4e3a\u54c8\u5e0c\u503c\u3002</li> <li>\u4e58\u6cd5\u54c8\u5e0c\uff1a\u5229\u7528\u4e58\u6cd5\u7684\u4e0d\u76f8\u5173\u6027\uff0c\u6bcf\u8f6e\u4e58\u4ee5\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5404\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u54c8\u5e0c\u503c\u4e2d\u3002</li> <li>\u5f02\u6216\u54c8\u5e0c\uff1a\u5c06\u8f93\u5165\u6570\u636e\u7684\u6bcf\u4e2a\u5143\u7d20\u901a\u8fc7\u5f02\u6216\u64cd\u4f5c\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\u3002</li> <li>\u65cb\u8f6c\u54c8\u5e0c\uff1a\u5c06\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\uff0c\u6bcf\u6b21\u7d2f\u79ef\u4e4b\u524d\u90fd\u4f1a\u5bf9\u54c8\u5e0c\u503c\u8fdb\u884c\u65cb\u8f6c\u64cd\u4f5c\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig simple_hash.py<pre><code>def add_hash(key: str) -&gt; int:\n    \"\"\"\u52a0\u6cd5\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash += ord(c)\n    return hash % modulus\n\ndef mul_hash(key: str) -&gt; int:\n    \"\"\"\u4e58\u6cd5\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = 31 * hash + ord(c)\n    return hash % modulus\n\ndef xor_hash(key: str) -&gt; int:\n    \"\"\"\u5f02\u6216\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash ^= ord(c)\n    return hash % modulus\n\ndef rot_hash(key: str) -&gt; int:\n    \"\"\"\u65cb\u8f6c\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ ord(c)\n    return hash % modulus\n</code></pre> simple_hash.cpp<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (31 * hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash ^= (int)c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.java<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (31 * hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\n    int hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash ^= (int) c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n</code></pre> simple_hash.cs<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint AddHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint MulHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (31 * hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint XorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash ^= c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint RotHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.go<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (31*hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key string) int {\n    hash := 0\n    modulus := 1000000007\n    for _, b := range []byte(key) {\n        fmt.Println(int(b))\n        hash ^= int(b)\n        hash = (31*hash + int(b)) % modulus\n    }\n    return hash &amp; modulus\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ int64(b)) % modulus\n    }\n    return int(hash)\n}\n</code></pre> simple_hash.swift<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (31 * hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash ^= Int(scalar.value)\n        }\n    }\n    return hash &amp; MODULUS\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n</code></pre> simple_hash.js<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.ts<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.dart<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (31 * hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash ^= key.codeUnitAt(i);\n  }\n  return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n</code></pre> simple_hash.rs<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfn add_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfn mul_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (31 * hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfn xor_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash ^= c as i64;\n    }\n\n    (hash &amp; MODULUS) as i32\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfn rot_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n</code></pre> simple_hash.c<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (31 * hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(char *key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash ^= (unsigned char)key[i];\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (unsigned char)key[i]) % MODULUS;\n    }\n\n    return (int)hash;\n}\n</code></pre> simple_hash.kt<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfun addHash(key: String): Int {\n    var hash = 0L\n    for (c in key.toCharArray()) {\n        hash = (hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfun mulHash(key: String): Int {\n    var hash = 0L\n    for (c in key.toCharArray()) {\n        hash = (31 * hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfun xorHash(key: String): Int {\n    var hash = 0\n    for (c in key.toCharArray()) {\n        hash = hash xor c.code\n    }\n    return hash and MODULUS\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfun rotHash(key: String): Int {\n    var hash = 0L\n    for (c in key.toCharArray()) {\n        hash = ((hash shl 4) xor (hash shr 28) xor c.code.toLong()) % MODULUS\n    }\n    return hash.toInt()\n}\n</code></pre> simple_hash.rb<pre><code>[class]{}-[func]{add_hash}\n\n[class]{}-[func]{mul_hash}\n\n[class]{}-[func]{xor_hash}\n\n[class]{}-[func]{rot_hash}\n</code></pre> simple_hash.zig<pre><code>[class]{}-[func]{addHash}\n\n[class]{}-[func]{mulHash}\n\n[class]{}-[func]{xorHash}\n\n[class]{}-[func]{rotHash}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u6bcf\u79cd\u54c8\u5e0c\u7b97\u6cd5\u7684\u6700\u540e\u4e00\u6b65\u90fd\u662f\u5bf9\u5927\u8d28\u6570 \\(1000000007\\) \u53d6\u6a21\uff0c\u4ee5\u786e\u4fdd\u54c8\u5e0c\u503c\u5728\u5408\u9002\u7684\u8303\u56f4\u5185\u3002\u503c\u5f97\u601d\u8003\u7684\u662f\uff0c\u4e3a\u4ec0\u4e48\u8981\u5f3a\u8c03\u5bf9\u8d28\u6570\u53d6\u6a21\uff0c\u6216\u8005\u8bf4\u5bf9\u5408\u6570\u53d6\u6a21\u7684\u5f0a\u7aef\u662f\u4ec0\u4e48\uff1f\u8fd9\u662f\u4e00\u4e2a\u6709\u8da3\u7684\u95ee\u9898\u3002</p> <p>\u5148\u629b\u51fa\u7ed3\u8bba\uff1a\u4f7f\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u53ef\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u7684\u5747\u5300\u5206\u5e03\u3002\u56e0\u4e3a\u8d28\u6570\u4e0d\u4e0e\u5176\u4ed6\u6570\u5b57\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u53ef\u4ee5\u51cf\u5c11\u56e0\u53d6\u6a21\u64cd\u4f5c\u800c\u4ea7\u751f\u7684\u5468\u671f\u6027\u6a21\u5f0f\uff0c\u4ece\u800c\u907f\u514d\u54c8\u5e0c\u51b2\u7a81\u3002</p> <p>\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u5408\u6570 \\(9\\) \u4f5c\u4e3a\u6a21\u6570\uff0c\u5b83\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\uff0c\u90a3\u4e48\u6240\u6709\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\u7684 <code>key</code> \u90fd\u4f1a\u88ab\u6620\u5c04\u5230 \\(0\\)\u3001\\(3\\)\u3001\\(6\\) \u8fd9\u4e09\u4e2a\u54c8\u5e0c\u503c\u3002</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 9 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\dots \\} \\end{aligned} \\] <p>\u5982\u679c\u8f93\u5165 <code>key</code> \u6070\u597d\u6ee1\u8db3\u8fd9\u79cd\u7b49\u5dee\u6570\u5217\u7684\u6570\u636e\u5206\u5e03\uff0c\u90a3\u4e48\u54c8\u5e0c\u503c\u5c31\u4f1a\u51fa\u73b0\u805a\u5806\uff0c\u4ece\u800c\u52a0\u91cd\u54c8\u5e0c\u51b2\u7a81\u3002\u73b0\u5728\uff0c\u5047\u8bbe\u5c06 <code>modulus</code> \u66ff\u6362\u4e3a\u8d28\u6570 \\(13\\) \uff0c\u7531\u4e8e <code>key</code> \u548c <code>modulus</code> \u4e4b\u95f4\u4e0d\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u56e0\u6b64\u8f93\u51fa\u7684\u54c8\u5e0c\u503c\u7684\u5747\u5300\u6027\u4f1a\u660e\u663e\u63d0\u5347\u3002</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 13 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\dots \\} \\end{aligned} \\] <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5982\u679c\u80fd\u591f\u4fdd\u8bc1 <code>key</code> \u662f\u968f\u673a\u5747\u5300\u5206\u5e03\u7684\uff0c\u90a3\u4e48\u9009\u62e9\u8d28\u6570\u6216\u8005\u5408\u6570\u4f5c\u4e3a\u6a21\u6570\u90fd\u53ef\u4ee5\uff0c\u5b83\u4eec\u90fd\u80fd\u8f93\u51fa\u5747\u5300\u5206\u5e03\u7684\u54c8\u5e0c\u503c\u3002\u800c\u5f53 <code>key</code> \u7684\u5206\u5e03\u5b58\u5728\u67d0\u79cd\u5468\u671f\u6027\u65f6\uff0c\u5bf9\u5408\u6570\u53d6\u6a21\u66f4\u5bb9\u6613\u51fa\u73b0\u805a\u96c6\u73b0\u8c61\u3002</p> <p>\u603b\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u901a\u5e38\u9009\u53d6\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u5e76\u4e14\u8fd9\u4e2a\u8d28\u6570\u6700\u597d\u8db3\u591f\u5927\uff0c\u4ee5\u5c3d\u53ef\u80fd\u6d88\u9664\u5468\u671f\u6027\u6a21\u5f0f\uff0c\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#633","title":"6.3.3 \u00a0 \u5e38\u89c1\u54c8\u5e0c\u7b97\u6cd5","text":"<p>\u4e0d\u96be\u53d1\u73b0\uff0c\u4ee5\u4e0a\u4ecb\u7ecd\u7684\u7b80\u5355\u54c8\u5e0c\u7b97\u6cd5\u90fd\u6bd4\u8f83\u201c\u8106\u5f31\u201d\uff0c\u8fdc\u8fdc\u6ca1\u6709\u8fbe\u5230\u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u76ee\u6807\u3002\u4f8b\u5982\uff0c\u7531\u4e8e\u52a0\u6cd5\u548c\u5f02\u6216\u6ee1\u8db3\u4ea4\u6362\u5f8b\uff0c\u56e0\u6b64\u52a0\u6cd5\u54c8\u5e0c\u548c\u5f02\u6216\u54c8\u5e0c\u65e0\u6cd5\u533a\u5206\u5185\u5bb9\u76f8\u540c\u4f46\u987a\u5e8f\u4e0d\u540c\u7684\u5b57\u7b26\u4e32\uff0c\u8fd9\u53ef\u80fd\u4f1a\u52a0\u5267\u54c8\u5e0c\u51b2\u7a81\uff0c\u5e76\u5f15\u8d77\u4e00\u4e9b\u5b89\u5168\u95ee\u9898\u3002</p> <p>\u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u7528\u4e00\u4e9b\u6807\u51c6\u54c8\u5e0c\u7b97\u6cd5\uff0c\u4f8b\u5982 MD5\u3001SHA-1\u3001SHA-2 \u548c SHA-3 \u7b49\u3002\u5b83\u4eec\u53ef\u4ee5\u5c06\u4efb\u610f\u957f\u5ea6\u7684\u8f93\u5165\u6570\u636e\u6620\u5c04\u5230\u6052\u5b9a\u957f\u5ea6\u7684\u54c8\u5e0c\u503c\u3002</p> <p>\u8fd1\u4e00\u4e2a\u4e16\u7eaa\u4ee5\u6765\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5904\u5728\u4e0d\u65ad\u5347\u7ea7\u4e0e\u4f18\u5316\u7684\u8fc7\u7a0b\u4e2d\u3002\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u52aa\u529b\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u6027\u80fd\uff0c\u53e6\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u548c\u9ed1\u5ba2\u5219\u81f4\u529b\u4e8e\u5bfb\u627e\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u95ee\u9898\u3002\u8868 6-2 \u5c55\u793a\u4e86\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5\u3002</p> <ul> <li>MD5 \u548c SHA-1 \u5df2\u591a\u6b21\u88ab\u6210\u529f\u653b\u51fb\uff0c\u56e0\u6b64\u5b83\u4eec\u88ab\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u5f03\u7528\u3002</li> <li>SHA-2 \u7cfb\u5217\u4e2d\u7684 SHA-256 \u662f\u6700\u5b89\u5168\u7684\u54c8\u5e0c\u7b97\u6cd5\u4e4b\u4e00\uff0c\u4ecd\u672a\u51fa\u73b0\u6210\u529f\u7684\u653b\u51fb\u6848\u4f8b\uff0c\u56e0\u6b64\u5e38\u7528\u5728\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u4e2d\u3002</li> <li>SHA-3 \u76f8\u8f83 SHA-2 \u7684\u5b9e\u73b0\u5f00\u9500\u66f4\u4f4e\u3001\u8ba1\u7b97\u6548\u7387\u66f4\u9ad8\uff0c\u4f46\u76ee\u524d\u4f7f\u7528\u8986\u76d6\u5ea6\u4e0d\u5982 SHA-2 \u7cfb\u5217\u3002</li> </ul> <p> \u8868 6-2 \u00a0 \u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5 </p> MD5 SHA-1 SHA-2 SHA-3 \u63a8\u51fa\u65f6\u95f4 1992 1995 2002 2008 \u8f93\u51fa\u957f\u5ea6 128 bit 160 bit 256/512 bit 224/256/384/512 bit \u54c8\u5e0c\u51b2\u7a81 \u8f83\u591a \u8f83\u591a \u5f88\u5c11 \u5f88\u5c11 \u5b89\u5168\u7b49\u7ea7 \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u9ad8 \u9ad8 \u5e94\u7528 \u5df2\u88ab\u5f03\u7528\uff0c\u4ecd\u7528\u4e8e\u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5 \u5df2\u88ab\u5f03\u7528 \u52a0\u5bc6\u8d27\u5e01\u4ea4\u6613\u9a8c\u8bc1\u3001\u6570\u5b57\u7b7e\u540d\u7b49 \u53ef\u7528\u4e8e\u66ff\u4ee3 SHA-2"},{"location":"chapter_hashing/hash_algorithm/#634","title":"6.3.4 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u54c8\u5e0c\u503c","text":"<p>\u6211\u4eec\u77e5\u9053\uff0c\u54c8\u5e0c\u8868\u7684 <code>key</code> \u53ef\u4ee5\u662f\u6574\u6570\u3001\u5c0f\u6570\u6216\u5b57\u7b26\u4e32\u7b49\u6570\u636e\u7c7b\u578b\u3002\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u8fd9\u4e9b\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u7684\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u8c03\u7528 <code>hash()</code> \u51fd\u6570\u6765\u8ba1\u7b97\u5404\u79cd\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\u3002</p> <ul> <li>\u6574\u6570\u548c\u5e03\u5c14\u91cf\u7684\u54c8\u5e0c\u503c\u5c31\u662f\u5176\u672c\u8eab\u3002</li> <li>\u6d6e\u70b9\u6570\u548c\u5b57\u7b26\u4e32\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u8f83\u4e3a\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u8bf7\u81ea\u884c\u5b66\u4e60\u3002</li> <li>\u5143\u7ec4\u7684\u54c8\u5e0c\u503c\u662f\u5bf9\u5176\u4e2d\u6bcf\u4e00\u4e2a\u5143\u7d20\u8fdb\u884c\u54c8\u5e0c\uff0c\u7136\u540e\u5c06\u8fd9\u4e9b\u54c8\u5e0c\u503c\u7ec4\u5408\u8d77\u6765\uff0c\u5f97\u5230\u5355\u4e00\u7684\u54c8\u5e0c\u503c\u3002</li> <li>\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u57fa\u4e8e\u5176\u5185\u5b58\u5730\u5740\u751f\u6210\u3002\u901a\u8fc7\u91cd\u5199\u5bf9\u8c61\u7684\u54c8\u5e0c\u65b9\u6cd5\uff0c\u53ef\u5b9e\u73b0\u57fa\u4e8e\u5185\u5bb9\u751f\u6210\u54c8\u5e0c\u503c\u3002</li> </ul> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u54c8\u5e0c\u503c\u8ba1\u7b97\u51fd\u6570\u7684\u5b9a\u4e49\u548c\u65b9\u6cd5\u4e0d\u540c\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig built_in_hash.py<pre><code>num = 3\nhash_num = hash(num)\n# \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nbol = True\nhash_bol = hash(bol)\n# \u5e03\u5c14\u91cf True \u7684\u54c8\u5e0c\u503c\u4e3a 1\n\ndec = 3.14159\nhash_dec = hash(dec)\n# \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 326484311674566659\n\nstr = \"Hello \u7b97\u6cd5\"\nhash_str = hash(str)\n# \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 4617003410720528961\n\ntup = (12836, \"\u5c0f\u54c8\")\nhash_tup = hash(tup)\n# \u5143\u7ec4 (12836, '\u5c0f\u54c8') \u7684\u54c8\u5e0c\u503c\u4e3a 1029005403108185979\n\nobj = ListNode(0)\nhash_obj = hash(obj)\n# \u8282\u70b9\u5bf9\u8c61 &lt;ListNode object at 0x1058fd810&gt; \u7684\u54c8\u5e0c\u503c\u4e3a 274267521\n</code></pre> built_in_hash.cpp<pre><code>int num = 3;\nsize_t hashNum = hash&lt;int&gt;()(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nbool bol = true;\nsize_t hashBol = hash&lt;bool&gt;()(bol);\n// \u5e03\u5c14\u91cf 1 \u7684\u54c8\u5e0c\u503c\u4e3a 1\n\ndouble dec = 3.14159;\nsize_t hashDec = hash&lt;double&gt;()(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 4614256650576692846\n\nstring str = \"Hello \u7b97\u6cd5\";\nsize_t hashStr = hash&lt;string&gt;()(str);\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 15466937326284535026\n\n// \u5728 C++ \u4e2d\uff0c\u5185\u7f6e std:hash() \u4ec5\u63d0\u4f9b\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\n// \u6570\u7ec4\u3001\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u9700\u8981\u81ea\u884c\u5b9e\u73b0\n</code></pre> built_in_hash.java<pre><code>int num = 3;\nint hashNum = Integer.hashCode(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\n\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729\n\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -727081396\n\nObject[] arr = { 12836, \"\u5c0f\u54c8\" };\nint hashTup = Arrays.hashCode(arr);\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 1151158\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode@7dc5e7b4 \u7684\u54c8\u5e0c\u503c\u4e3a 2110121908\n</code></pre> built_in_hash.cs<pre><code>int num = 3;\nint hashNum = num.GetHashCode();\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3;\n\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1;\n\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729;\n\nstring str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.GetHashCode();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -586107568;\n\nobject[] arr = [12836, \"\u5c0f\u54c8\"];\nint hashTup = arr.GetHashCode();\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 42931033;\n\nListNode obj = new(0);\nint hashObj = obj.GetHashCode();\n// \u8282\u70b9\u5bf9\u8c61 0 \u7684\u54c8\u5e0c\u503c\u4e3a 39053774;\n</code></pre> built_in_hash.go<pre><code>// Go \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.swift<pre><code>let num = 3\nlet hashNum = num.hashValue\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 9047044699613009734\n\nlet bol = true\nlet hashBol = bol.hashValue\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a -4431640247352757451\n\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -2465384235396674631\n\nlet str = \"Hello \u7b97\u6cd5\"\nlet hashStr = str.hashValue\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -7850626797806988787\n\nlet arr = [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")]\nlet hashTup = arr.hashValue\n// \u6570\u7ec4 [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")] \u7684\u54c8\u5e0c\u503c\u4e3a -2308633508154532996\n\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode \u7684\u54c8\u5e0c\u503c\u4e3a -2434780518035996159\n</code></pre> built_in_hash.js<pre><code>// JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.ts<pre><code>// TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.dart<pre><code>int num = 3;\nint hashNum = num.hashCode;\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 34803\n\nbool bol = true;\nint hashBol = bol.hashCode;\n// \u5e03\u5c14\u503c true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\n\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 2570631074981783\n\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode;\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 468167534\n\nList arr = [12836, \"\u5c0f\u54c8\"];\nint hashArr = arr.hashCode;\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 976512528\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// \u8282\u70b9\u5bf9\u8c61 Instance of 'ListNode' \u7684\u54c8\u5e0c\u503c\u4e3a 1033450432\n</code></pre> built_in_hash.rs<pre><code>use std::collections::hash_map::DefaultHasher;\nuse std::hash::{Hash, Hasher};\n\nlet num = 3;\nlet mut num_hasher = DefaultHasher::new();\nnum.hash(&amp;mut num_hasher);\nlet hash_num = num_hasher.finish();\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 568126464209439262\n\nlet bol = true;\nlet mut bol_hasher = DefaultHasher::new();\nbol.hash(&amp;mut bol_hasher);\nlet hash_bol = bol_hasher.finish();\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 4952851536318644461\n\nlet dec: f32 = 3.14159;\nlet mut dec_hasher = DefaultHasher::new();\ndec.to_bits().hash(&amp;mut dec_hasher);\nlet hash_dec = dec_hasher.finish();\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 2566941990314602357\n\nlet str = \"Hello \u7b97\u6cd5\";\nlet mut str_hasher = DefaultHasher::new();\nstr.hash(&amp;mut str_hasher);\nlet hash_str = str_hasher.finish();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 16092673739211250988\n\nlet arr = (&amp;12836, &amp;\"\u5c0f\u54c8\");\nlet mut tup_hasher = DefaultHasher::new();\narr.hash(&amp;mut tup_hasher);\nlet hash_tup = tup_hasher.finish();\n// \u5143\u7ec4 (12836, \"\u5c0f\u54c8\") \u7684\u54c8\u5e0c\u503c\u4e3a 1885128010422702749\n\nlet node = ListNode::new(42);\nlet mut hasher = DefaultHasher::new();\nnode.borrow().val.hash(&amp;mut hasher);\nlet hash = hasher.finish();\n// \u8282\u70b9\u5bf9\u8c61 RefCell { value: ListNode { val: 42, next: None } } \u7684\u54c8\u5e0c\u503c\u4e3a15387811073369036852\n</code></pre> built_in_hash.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.kt<pre><code>val num = 3\nval hashNum = num.hashCode()\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nval bol = true\nval hashBol = bol.hashCode()\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\n\nval dec = 3.14159\nval hashDec = dec.hashCode()\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729\n\nval str = \"Hello \u7b97\u6cd5\"\nval hashStr = str.hashCode()\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -727081396\n\nval arr = arrayOf&lt;Any&gt;(12836, \"\u5c0f\u54c8\")\nval hashTup = arr.hashCode()\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 189568618\n\nval obj = ListNode(0)\nval hashObj = obj.hashCode()\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode@1d81eb93 \u7684\u54c8\u5e0c\u503c\u4e3a 495053715\n</code></pre> built_in_hash.rb<pre><code>\n</code></pre> built_in_hash.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5728\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u624d\u53ef\u4f5c\u4e3a\u54c8\u5e0c\u8868\u7684 <code>key</code> \u3002\u5047\u5982\u6211\u4eec\u5c06\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4f5c\u4e3a <code>key</code> \uff0c\u5f53\u5217\u8868\u7684\u5185\u5bb9\u53d1\u751f\u53d8\u5316\u65f6\uff0c\u5b83\u7684\u54c8\u5e0c\u503c\u4e5f\u968f\u4e4b\u6539\u53d8\uff0c\u6211\u4eec\u5c31\u65e0\u6cd5\u5728\u54c8\u5e0c\u8868\u4e2d\u67e5\u8be2\u5230\u539f\u5148\u7684 <code>value</code> \u4e86\u3002</p> <p>\u867d\u7136\u81ea\u5b9a\u4e49\u5bf9\u8c61\uff08\u6bd4\u5982\u94fe\u8868\u8282\u70b9\uff09\u7684\u6210\u5458\u53d8\u91cf\u662f\u53ef\u53d8\u7684\uff0c\u4f46\u5b83\u662f\u53ef\u54c8\u5e0c\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u901a\u5e38\u662f\u57fa\u4e8e\u5185\u5b58\u5730\u5740\u751f\u6210\u7684\uff0c\u5373\u4f7f\u5bf9\u8c61\u7684\u5185\u5bb9\u53d1\u751f\u4e86\u53d8\u5316\uff0c\u4f46\u5b83\u7684\u5185\u5b58\u5730\u5740\u4e0d\u53d8\uff0c\u54c8\u5e0c\u503c\u4ecd\u7136\u662f\u4e0d\u53d8\u7684\u3002</p> <p>\u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u53d1\u73b0\u5728\u4e0d\u540c\u63a7\u5236\u53f0\u4e2d\u8fd0\u884c\u7a0b\u5e8f\u65f6\uff0c\u8f93\u51fa\u7684\u54c8\u5e0c\u503c\u662f\u4e0d\u540c\u7684\u3002\u8fd9\u662f\u56e0\u4e3a Python \u89e3\u91ca\u5668\u5728\u6bcf\u6b21\u542f\u52a8\u65f6\uff0c\u90fd\u4f1a\u4e3a\u5b57\u7b26\u4e32\u54c8\u5e0c\u51fd\u6570\u52a0\u5165\u4e00\u4e2a\u968f\u673a\u7684\u76d0\uff08salt\uff09\u503c\u3002\u8fd9\u79cd\u505a\u6cd5\u53ef\u4ee5\u6709\u6548\u9632\u6b62 HashDoS \u653b\u51fb\uff0c\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u3002</p>"},{"location":"chapter_hashing/hash_collision/","title":"6.2 \u00a0 \u54c8\u5e0c\u51b2\u7a81","text":"<p>\u4e0a\u4e00\u8282\u63d0\u5230\uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u5165\u7a7a\u95f4\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff0c\u56e0\u6b64\u7406\u8bba\u4e0a\u54c8\u5e0c\u51b2\u7a81\u662f\u4e0d\u53ef\u907f\u514d\u7684\u3002\u6bd4\u5982\uff0c\u8f93\u5165\u7a7a\u95f4\u4e3a\u5168\u4f53\u6574\u6570\uff0c\u8f93\u51fa\u7a7a\u95f4\u4e3a\u6570\u7ec4\u5bb9\u91cf\u5927\u5c0f\uff0c\u5219\u5fc5\u7136\u6709\u591a\u4e2a\u6574\u6570\u6620\u5c04\u81f3\u540c\u4e00\u6876\u7d22\u5f15\u3002</p> <p>\u54c8\u5e0c\u51b2\u7a81\u4f1a\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u9519\u8bef\uff0c\u4e25\u91cd\u5f71\u54cd\u54c8\u5e0c\u8868\u7684\u53ef\u7528\u6027\u3002\u4e3a\u4e86\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6bcf\u5f53\u9047\u5230\u54c8\u5e0c\u51b2\u7a81\u65f6\uff0c\u6211\u4eec\u5c31\u8fdb\u884c\u54c8\u5e0c\u8868\u6269\u5bb9\uff0c\u76f4\u81f3\u51b2\u7a81\u6d88\u5931\u4e3a\u6b62\u3002\u6b64\u65b9\u6cd5\u7b80\u5355\u7c97\u66b4\u4e14\u6709\u6548\uff0c\u4f46\u6548\u7387\u592a\u4f4e\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u8981\u8fdb\u884c\u5927\u91cf\u7684\u6570\u636e\u642c\u8fd0\u4e0e\u54c8\u5e0c\u503c\u8ba1\u7b97\u3002\u4e3a\u4e86\u63d0\u5347\u6548\u7387\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u7b56\u7565\u3002</p> <ol> <li>\u6539\u826f\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u4f7f\u5f97\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\u3002</li> <li>\u4ec5\u5728\u5fc5\u8981\u65f6\uff0c\u5373\u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u624d\u6267\u884c\u6269\u5bb9\u64cd\u4f5c\u3002</li> </ol> <p>\u54c8\u5e0c\u8868\u7684\u7ed3\u6784\u6539\u826f\u65b9\u6cd5\u4e3b\u8981\u5305\u62ec\u201c\u94fe\u5f0f\u5730\u5740\u201d\u548c\u201c\u5f00\u653e\u5bfb\u5740\u201d\u3002</p>"},{"location":"chapter_hashing/hash_collision/#621","title":"6.2.1 \u00a0 \u94fe\u5f0f\u5730\u5740","text":"<p>\u5728\u539f\u59cb\u54c8\u5e0c\u8868\u4e2d\uff0c\u6bcf\u4e2a\u6876\u4ec5\u80fd\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u300c\u94fe\u5f0f\u5730\u5740 separate chaining\u300d\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u6362\u4e3a\u94fe\u8868\uff0c\u5c06\u952e\u503c\u5bf9\u4f5c\u4e3a\u94fe\u8868\u8282\u70b9\uff0c\u5c06\u6240\u6709\u53d1\u751f\u51b2\u7a81\u7684\u952e\u503c\u5bf9\u90fd\u5b58\u50a8\u5728\u540c\u4e00\u94fe\u8868\u4e2d\u3002\u56fe 6-5 \u5c55\u793a\u4e86\u4e00\u4e2a\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\u7684\u4f8b\u5b50\u3002</p> <p></p> <p> \u56fe 6-5 \u00a0 \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 </p> <p>\u57fa\u4e8e\u94fe\u5f0f\u5730\u5740\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\u7684\u64cd\u4f5c\u65b9\u6cd5\u53d1\u751f\u4e86\u4ee5\u4e0b\u53d8\u5316\u3002</p> <ul> <li>\u67e5\u8be2\u5143\u7d20\uff1a\u8f93\u5165 <code>key</code> \uff0c\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u6876\u7d22\u5f15\uff0c\u5373\u53ef\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u904d\u5386\u94fe\u8868\u5e76\u5bf9\u6bd4 <code>key</code> \u4ee5\u67e5\u627e\u76ee\u6807\u952e\u503c\u5bf9\u3002</li> <li>\u6dfb\u52a0\u5143\u7d20\uff1a\u9996\u5148\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u5c06\u8282\u70b9\uff08\u952e\u503c\u5bf9\uff09\u6dfb\u52a0\u5230\u94fe\u8868\u4e2d\u3002</li> <li>\u5220\u9664\u5143\u7d20\uff1a\u6839\u636e\u54c8\u5e0c\u51fd\u6570\u7684\u7ed3\u679c\u8bbf\u95ee\u94fe\u8868\u5934\u90e8\uff0c\u63a5\u7740\u904d\u5386\u94fe\u8868\u4ee5\u67e5\u627e\u76ee\u6807\u8282\u70b9\u5e76\u5c06\u5176\u5220\u9664\u3002</li> </ul> <p>\u94fe\u5f0f\u5730\u5740\u5b58\u5728\u4ee5\u4e0b\u5c40\u9650\u6027\u3002</p> <ul> <li>\u5360\u7528\u7a7a\u95f4\u589e\u5927\uff1a\u94fe\u8868\u5305\u542b\u8282\u70b9\u6307\u9488\uff0c\u5b83\u76f8\u6bd4\u6570\u7ec4\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u67e5\u8be2\u6548\u7387\u964d\u4f4e\uff1a\u56e0\u4e3a\u9700\u8981\u7ebf\u6027\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u5bf9\u5e94\u5143\u7d20\u3002</li> </ul> <p>\u4ee5\u4e0b\u4ee3\u7801\u7ed9\u51fa\u4e86\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\u7684\u7b80\u5355\u5b9e\u73b0\uff0c\u9700\u8981\u6ce8\u610f\u4e24\u70b9\u3002</p> <ul> <li>\u4f7f\u7528\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4ee3\u66ff\u94fe\u8868\uff0c\u4ece\u800c\u7b80\u5316\u4ee3\u7801\u3002\u5728\u8fd9\u79cd\u8bbe\u5b9a\u4e0b\uff0c\u54c8\u5e0c\u8868\uff08\u6570\u7ec4\uff09\u5305\u542b\u591a\u4e2a\u6876\uff0c\u6bcf\u4e2a\u6876\u90fd\u662f\u4e00\u4e2a\u5217\u8868\u3002</li> <li>\u4ee5\u4e0b\u5b9e\u73b0\u5305\u542b\u54c8\u5e0c\u8868\u6269\u5bb9\u65b9\u6cd5\u3002\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(\\frac{2}{3}\\) \u65f6\uff0c\u6211\u4eec\u5c06\u54c8\u5e0c\u8868\u6269\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map_chaining.py<pre><code>class HashMapChaining:\n    \"\"\"\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\n        self.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\n        self.load_thres = 2.0 / 3.0  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        self.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\n        self.buckets = [[] for _ in range(self.capacity)]  # \u6876\u6570\u7ec4\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\n        return self.size / self.capacity\n\n    def get(self, key: int) -&gt; str | None:\n        \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for pair in bucket:\n            if pair.key == key:\n                return pair.val\n        # \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n        # \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for pair in bucket:\n            if pair.key == key:\n                pair.val = val\n                return\n        # \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        pair = Pair(key, val)\n        bucket.append(pair)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for pair in bucket:\n            if pair.key == key:\n                bucket.remove(pair)\n                self.size -= 1\n                break\n\n    def extend(self):\n        \"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n        # \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        buckets = self.buckets\n        # \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio\n        self.buckets = [[] for _ in range(self.capacity)]\n        self.size = 0\n        # \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for bucket in buckets:\n            for pair in bucket:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n        for bucket in self.buckets:\n            res = []\n            for pair in bucket:\n                res.append(str(pair.key) + \" -&gt; \" + pair.val)\n            print(res)\n</code></pre> hash_map_chaining.cpp<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n  private:\n    int size;                       // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity;                   // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres;               // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio;                // \u6269\u5bb9\u500d\u6570\n    vector&lt;vector&lt;Pair *&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    HashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3.0), extendRatio(2) {\n        buckets.resize(capacity);\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~HashMapChaining() {\n        for (auto &amp;bucket : buckets) {\n            for (Pair *pair : bucket) {\n                // \u91ca\u653e\u5185\u5b58\n                delete pair;\n            }\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double loadFactor() {\n        return (double)size / (double)capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    string get(int key) {\n        int index = hashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                return pair-&gt;val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n        return \"\";\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                pair-&gt;val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        buckets[index].push_back(new Pair(key, val));\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        auto &amp;bucket = buckets[index];\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (int i = 0; i &lt; bucket.size(); i++) {\n            if (bucket[i]-&gt;key == key) {\n                Pair *tmp = bucket[i];\n                bucket.erase(bucket.begin() + i); // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n                delete tmp;                       // \u91ca\u653e\u5185\u5b58\n                size--;\n                return;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        vector&lt;vector&lt;Pair *&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets.clear();\n        buckets.resize(capacity);\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (auto &amp;bucket : bucketsTmp) {\n            for (Pair *pair : bucket) {\n                put(pair-&gt;key, pair-&gt;val);\n                // \u91ca\u653e\u5185\u5b58\n                delete pair;\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (auto &amp;bucket : buckets) {\n            cout &lt;&lt; \"[\";\n            for (Pair *pair : bucket) {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; \", \";\n            }\n            cout &lt;&lt; \"]\\n\";\n        }\n    }\n};\n</code></pre> hash_map_chaining.java<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio; // \u6269\u5bb9\u500d\u6570\n    List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    String get(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, String val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        Pair pair = new Pair(key, val);\n        bucket.add(pair);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (List&lt;Pair&gt; bucket : bucketsTmp) {\n            for (Pair pair : bucket) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (List&lt;Pair&gt; bucket : buckets) {\n            List&lt;String&gt; res = new ArrayList&lt;&gt;();\n            for (Pair pair : bucket) {\n                res.add(pair.key + \" -&gt; \" + pair.val);\n            }\n            System.out.println(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.cs<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio; // \u6269\u5bb9\u500d\u6570\n    List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        int index = HashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        buckets[index].Add(new Pair(key, val));\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        foreach (Pair pair in buckets[index].ToList()) {\n            if (pair.key == key) {\n                buckets[index].Remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void Extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        foreach (List&lt;Pair&gt; bucket in bucketsTmp) {\n            foreach (Pair pair in bucket) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void Print() {\n        foreach (List&lt;Pair&gt; bucket in buckets) {\n            List&lt;string&gt; res = [];\n            foreach (Pair pair in bucket) {\n                res.Add(pair.key + \" -&gt; \" + pair.val);\n            }\n            foreach (string kv in res) {\n                Console.WriteLine(kv);\n            }\n        }\n    }\n}\n</code></pre> hash_map_chaining.go<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntype hashMapChaining struct {\n    size        int      // \u952e\u503c\u5bf9\u6570\u91cf\n    capacity    int      // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    loadThres   float64  // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    extendRatio int      // \u6269\u5bb9\u500d\u6570\n    buckets     [][]pair // \u6876\u6570\u7ec4\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapChaining() *hashMapChaining {\n    buckets := make([][]pair, 4)\n    for i := 0; i &lt; 4; i++ {\n        buckets[i] = make([]pair, 0)\n    }\n    return &amp;hashMapChaining{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     buckets,\n    }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapChaining) hashFunc(key int) int {\n    return key % m.capacity\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapChaining) loadFactor() float64 {\n    return float64(m.size) / float64(m.capacity)\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapChaining) get(key int) string {\n    idx := m.hashFunc(key)\n    bucket := m.buckets[idx]\n    // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    for _, p := range bucket {\n        if p.key == key {\n            return p.val\n        }\n    }\n    // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n    return \"\"\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapChaining) put(key int, val string) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if m.loadFactor() &gt; m.loadThres {\n        m.extend()\n    }\n    idx := m.hashFunc(key)\n    // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n    for i := range m.buckets[idx] {\n        if m.buckets[idx][i].key == key {\n            m.buckets[idx][i].val = val\n            return\n        }\n    }\n    // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n    p := pair{\n        key: key,\n        val: val,\n    }\n    m.buckets[idx] = append(m.buckets[idx], p)\n    m.size += 1\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapChaining) remove(key int) {\n    idx := m.hashFunc(key)\n    // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n    for i, p := range m.buckets[idx] {\n        if p.key == key {\n            // \u5207\u7247\u5220\u9664\n            m.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\n            m.size -= 1\n            break\n        }\n    }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    tmpBuckets := make([][]pair, len(m.buckets))\n    for i := 0; i &lt; len(m.buckets); i++ {\n        tmpBuckets[i] = make([]pair, len(m.buckets[i]))\n        copy(tmpBuckets[i], m.buckets[i])\n    }\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    m.capacity *= m.extendRatio\n    m.buckets = make([][]pair, m.capacity)\n    for i := 0; i &lt; m.capacity; i++ {\n        m.buckets[i] = make([]pair, 0)\n    }\n    m.size = 0\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for _, bucket := range tmpBuckets {\n        for _, p := range bucket {\n            m.put(p.key, p.val)\n        }\n    }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) print() {\n    var builder strings.Builder\n\n    for _, bucket := range m.buckets {\n        builder.WriteString(\"[\")\n        for _, p := range bucket {\n            builder.WriteString(strconv.Itoa(p.key) + \" -&gt; \" + p.val + \" \")\n        }\n        builder.WriteString(\"]\")\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> hash_map_chaining.swift<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n    var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    var loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    var extendRatio: Int // \u6269\u5bb9\u500d\u6570\n    var buckets: [[Pair]] // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: [], count: capacity)\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for pair in bucket {\n            if pair.key == key {\n                return pair.val\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de nil\n        return nil\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val\n                return\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        let pair = Pair(key: key, val: val)\n        buckets[index].append(pair)\n        size += 1\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (pairIndex, pair) in bucket.enumerated() {\n            if pair.key == key {\n                buckets[index].remove(at: pairIndex)\n                size -= 1\n                break\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    func extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        buckets = Array(repeating: [], count: capacity)\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for bucket in bucketsTmp {\n            for pair in bucket {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    func print() {\n        for bucket in buckets {\n            let res = bucket.map { \"\\($0.key) -&gt; \\($0.val)\" }\n            Swift.print(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.js<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    #size; // \u952e\u503c\u5bf9\u6570\u91cf\n    #capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    #loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    #extendRatio; // \u6269\u5bb9\u500d\u6570\n    #buckets; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key) {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key, val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key) {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    #extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print() {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.ts<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    #size: number; // \u952e\u503c\u5bf9\u6570\u91cf\n    #capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    #loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    #extendRatio: number; // \u6269\u5bb9\u500d\u6570\n    #buckets: Pair[][]; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key: number): number {\n        return key % this.#capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    #loadFactor(): number {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key: number): string | null {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key: number, val: string): void {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key: number): void {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    #extend(): void {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print(): void {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.dart<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n  late int size; // \u952e\u503c\u5bf9\u6570\u91cf\n  late int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n  late double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n  late int extendRatio; // \u6269\u5bb9\u500d\u6570\n  late List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  HashMapChaining() {\n    size = 0;\n    capacity = 4;\n    loadThres = 2.0 / 3.0;\n    extendRatio = 2;\n    buckets = List.generate(capacity, (_) =&gt; []);\n  }\n\n  /* \u54c8\u5e0c\u51fd\u6570 */\n  int hashFunc(int key) {\n    return key % capacity;\n  }\n\n  /* \u8d1f\u8f7d\u56e0\u5b50 */\n  double loadFactor() {\n    return size / capacity;\n  }\n\n  /* \u67e5\u8be2\u64cd\u4f5c */\n  String? get(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        return pair.val;\n      }\n    }\n    // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n    return null;\n  }\n\n  /* \u6dfb\u52a0\u64cd\u4f5c */\n  void put(int key, String val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor() &gt; loadThres) {\n      extend();\n    }\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        pair.val = val;\n        return;\n      }\n    }\n    // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n    Pair pair = Pair(key, val);\n    bucket.add(pair);\n    size++;\n  }\n\n  /* \u5220\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        bucket.remove(pair);\n        size--;\n        break;\n      }\n    }\n  }\n\n  /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n  void extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    capacity *= extendRatio;\n    buckets = List.generate(capacity, (_) =&gt; []);\n    size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (List&lt;Pair&gt; bucket in bucketsTmp) {\n      for (Pair pair in bucket) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* \u6253\u5370\u54c8\u5e0c\u8868 */\n  void printHashMap() {\n    for (List&lt;Pair&gt; bucket in buckets) {\n      List&lt;String&gt; res = [];\n      for (Pair pair in bucket) {\n        res.add(\"${pair.key} -&gt; ${pair.val}\");\n      }\n      print(res);\n    }\n  }\n}\n</code></pre> hash_map_chaining.rs<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapChaining {\n    size: i32,\n    capacity: i32,\n    load_thres: f32,\n    extend_ratio: i32,\n    buckets: Vec&lt;Vec&lt;Pair&gt;&gt;,\n}\n\nimpl HashMapChaining {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![vec![]; 4],\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % self.capacity as usize\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fn load_factor(&amp;self) -&gt; f32 {\n        self.size as f32 / self.capacity as f32\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fn remove(&amp;mut self, key: i32) -&gt; Option&lt;String&gt; {\n        let index = self.hash_func(key);\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for i in 0..bucket.len() {\n            if bucket[i].key == key {\n                let pair = bucket.remove(i);\n                self.size -= 1;\n                return Some(pair.val);\n            }\n        }\n\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de None\n        None\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fn extend(&amp;mut self) {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let buckets_tmp = std::mem::replace(&amp;mut self.buckets, vec![]);\n\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![Vec::new(); self.capacity as usize];\n        self.size = 0;\n\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for bucket in buckets_tmp {\n            for pair in bucket {\n                self.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fn print(&amp;self) {\n        for bucket in &amp;self.buckets {\n            let mut res = Vec::new();\n            for pair in bucket {\n                res.push(format!(\"{} -&gt; {}\", pair.key, pair.val));\n            }\n            println!(\"{:?}\", res);\n        }\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n\n        let index = self.hash_func(key);\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val.clone();\n                return;\n            }\n        }\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        let pair = Pair {\n            key,\n            val: val.clone(),\n        };\n        bucket.push(pair);\n        self.size += 1;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        let index = self.hash_func(key);\n        let bucket = &amp;self.buckets[index];\n\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for pair in bucket {\n            if pair.key == key {\n                return Some(&amp;pair.val);\n            }\n        }\n\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de None\n        None\n    }\n}\n</code></pre> hash_map_chaining.c<pre><code>/* \u94fe\u8868\u8282\u70b9 */\ntypedef struct Node {\n    Pair *pair;\n    struct Node *next;\n} Node;\n\n/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\n    int size;         // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity;     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio;  // \u6269\u5bb9\u500d\u6570\n    Node **buckets;   // \u6876\u6570\u7ec4\n} HashMapChaining;\n\n/* \u6784\u9020\u51fd\u6570 */\nHashMapChaining *newHashMapChaining() {\n    HashMapChaining *hashMap = (HashMapChaining *)malloc(sizeof(HashMapChaining));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    return hashMap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapChaining(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        while (cur) {\n            Node *tmp = cur;\n            cur = cur-&gt;next;\n            free(tmp-&gt;pair);\n            free(tmp);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap);\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapChaining *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapChaining *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            return cur-&gt;pair-&gt;val;\n        }\n        cur = cur-&gt;next;\n    }\n    return \"\"; // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapChaining *hashMap, int key, const char *val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    int index = hashFunc(hashMap, key);\n    // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            strcpy(cur-&gt;pair-&gt;val, val); // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n            return;\n        }\n        cur = cur-&gt;next;\n    }\n    // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n    Pair *newPair = (Pair *)malloc(sizeof(Pair));\n    newPair-&gt;key = key;\n    strcpy(newPair-&gt;val, val);\n    Node *newNode = (Node *)malloc(sizeof(Node));\n    newNode-&gt;pair = newPair;\n    newNode-&gt;next = hashMap-&gt;buckets[index];\n    hashMap-&gt;buckets[index] = newNode;\n    hashMap-&gt;size++;\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapChaining *hashMap) {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    int oldCapacity = hashMap-&gt;capacity;\n    Node **oldBuckets = hashMap-&gt;buckets;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    hashMap-&gt;size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Node *cur = oldBuckets[i];\n        while (cur) {\n            put(hashMap, cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            Node *temp = cur;\n            cur = cur-&gt;next;\n            // \u91ca\u653e\u5185\u5b58\n            free(temp-&gt;pair);\n            free(temp);\n        }\n    }\n\n    free(oldBuckets);\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    Node *cur = hashMap-&gt;buckets[index];\n    Node *pre = NULL;\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n            if (pre) {\n                pre-&gt;next = cur-&gt;next;\n            } else {\n                hashMap-&gt;buckets[index] = cur-&gt;next;\n            }\n            // \u91ca\u653e\u5185\u5b58\n            free(cur-&gt;pair);\n            free(cur);\n            hashMap-&gt;size--;\n            return;\n        }\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        printf(\"[\");\n        while (cur) {\n            printf(\"%d -&gt; %s, \", cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            cur = cur-&gt;next;\n        }\n        printf(\"]\\n\");\n    }\n}\n</code></pre> hash_map_chaining.kt<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining() {\n    var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n    var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    val loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    val extendRatio: Int // \u6269\u5bb9\u500d\u6570\n    var buckets: MutableList&lt;MutableList&lt;Pair&gt;&gt; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = ArrayList(capacity)\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (pair in bucket) {\n            if (pair.key == key) return pair.value\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (pair in bucket) {\n            if (pair.key == key) {\n                pair.value = value\n                return\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        val pair = Pair(key, value)\n        bucket.add(pair)\n        size++\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (pair in bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair)\n                size--\n                break\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fun extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        val bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        // mutablelist \u65e0\u56fa\u5b9a\u5927\u5c0f\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (bucket in bucketsTmp) {\n            for (pair in bucket) {\n                put(pair.key, pair.value)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (bucket in buckets) {\n            val res = mutableListOf&lt;String&gt;()\n            for (pair in bucket) {\n                val k = pair.key\n                val v = pair.value\n                res.add(\"$k -&gt; $v\")\n            }\n            println(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.rb<pre><code>[class]{HashMapChaining}-[func]{}\n</code></pre> hash_map_chaining.zig<pre><code>[class]{HashMapChaining}-[func]{}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5f53\u94fe\u8868\u5f88\u957f\u65f6\uff0c\u67e5\u8be2\u6548\u7387 \\(O(n)\\) \u5f88\u5dee\u3002\u6b64\u65f6\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u201cAVL \u6811\u201d\u6216\u201c\u7ea2\u9ed1\u6811\u201d\uff0c\u4ece\u800c\u5c06\u67e5\u8be2\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002</p>"},{"location":"chapter_hashing/hash_collision/#622","title":"6.2.2 \u00a0 \u5f00\u653e\u5bfb\u5740","text":"<p>\u300c\u5f00\u653e\u5bfb\u5740 open addressing\u300d\u4e0d\u5f15\u5165\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u800c\u662f\u901a\u8fc7\u201c\u591a\u6b21\u63a2\u6d4b\u201d\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\uff0c\u63a2\u6d4b\u65b9\u5f0f\u4e3b\u8981\u5305\u62ec\u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u548c\u591a\u6b21\u54c8\u5e0c\u7b49\u3002</p> <p>\u4e0b\u9762\u4ee5\u7ebf\u6027\u63a2\u6d4b\u4e3a\u4f8b\uff0c\u4ecb\u7ecd\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\u7684\u5de5\u4f5c\u673a\u5236\u3002</p>"},{"location":"chapter_hashing/hash_collision/#1","title":"1. \u00a0 \u7ebf\u6027\u63a2\u6d4b","text":"<p>\u7ebf\u6027\u63a2\u6d4b\u91c7\u7528\u56fa\u5b9a\u6b65\u957f\u7684\u7ebf\u6027\u641c\u7d22\u6765\u8fdb\u884c\u63a2\u6d4b\uff0c\u5176\u64cd\u4f5c\u65b9\u6cd5\u4e0e\u666e\u901a\u54c8\u5e0c\u8868\u6709\u6240\u4e0d\u540c\u3002</p> <ul> <li>\u63d2\u5165\u5143\u7d20\uff1a\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u82e5\u53d1\u73b0\u6876\u5185\u5df2\u6709\u5143\u7d20\uff0c\u5219\u4ece\u51b2\u7a81\u4f4d\u7f6e\u5411\u540e\u7ebf\u6027\u904d\u5386\uff08\u6b65\u957f\u901a\u5e38\u4e3a \\(1\\) \uff09\uff0c\u76f4\u81f3\u627e\u5230\u7a7a\u6876\uff0c\u5c06\u5143\u7d20\u63d2\u5165\u5176\u4e2d\u3002</li> <li>\u67e5\u627e\u5143\u7d20\uff1a\u82e5\u53d1\u73b0\u54c8\u5e0c\u51b2\u7a81\uff0c\u5219\u4f7f\u7528\u76f8\u540c\u6b65\u957f\u5411\u540e\u8fdb\u884c\u7ebf\u6027\u904d\u5386\uff0c\u76f4\u5230\u627e\u5230\u5bf9\u5e94\u5143\u7d20\uff0c\u8fd4\u56de <code>value</code> \u5373\u53ef\uff1b\u5982\u679c\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u76ee\u6807\u5143\u7d20\u4e0d\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8fd4\u56de <code>None</code> \u3002</li> </ul> <p>\u56fe 6-6 \u5c55\u793a\u4e86\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u7684\u952e\u503c\u5bf9\u5206\u5e03\u3002\u6839\u636e\u6b64\u54c8\u5e0c\u51fd\u6570\uff0c\u6700\u540e\u4e24\u4f4d\u76f8\u540c\u7684 <code>key</code> \u90fd\u4f1a\u88ab\u6620\u5c04\u5230\u76f8\u540c\u7684\u6876\u3002\u800c\u901a\u8fc7\u7ebf\u6027\u63a2\u6d4b\uff0c\u5b83\u4eec\u88ab\u4f9d\u6b21\u5b58\u50a8\u5728\u8be5\u6876\u4ee5\u53ca\u4e4b\u4e0b\u7684\u6876\u4e2d\u3002</p> <p></p> <p> \u56fe 6-6 \u00a0 \u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u7684\u952e\u503c\u5bf9\u5206\u5e03 </p> <p>\u7136\u800c\uff0c\u7ebf\u6027\u63a2\u6d4b\u5bb9\u6613\u4ea7\u751f\u201c\u805a\u96c6\u73b0\u8c61\u201d\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u6570\u7ec4\u4e2d\u8fde\u7eed\u88ab\u5360\u7528\u7684\u4f4d\u7f6e\u8d8a\u957f\uff0c\u8fd9\u4e9b\u8fde\u7eed\u4f4d\u7f6e\u53d1\u751f\u54c8\u5e0c\u51b2\u7a81\u7684\u53ef\u80fd\u6027\u8d8a\u5927\uff0c\u4ece\u800c\u8fdb\u4e00\u6b65\u4fc3\u4f7f\u8be5\u4f4d\u7f6e\u7684\u805a\u5806\u751f\u957f\uff0c\u5f62\u6210\u6076\u6027\u5faa\u73af\uff0c\u6700\u7ec8\u5bfc\u81f4\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u6548\u7387\u52a3\u5316\u3002</p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6211\u4eec\u4e0d\u80fd\u5728\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\u4e2d\u76f4\u63a5\u5220\u9664\u5143\u7d20\u3002\u8fd9\u662f\u56e0\u4e3a\u5220\u9664\u5143\u7d20\u4f1a\u5728\u6570\u7ec4\u5185\u4ea7\u751f\u4e00\u4e2a\u7a7a\u6876 <code>None</code> \uff0c\u800c\u5f53\u67e5\u8be2\u5143\u7d20\u65f6\uff0c\u7ebf\u6027\u63a2\u6d4b\u5230\u8be5\u7a7a\u6876\u5c31\u4f1a\u8fd4\u56de\uff0c\u56e0\u6b64\u5728\u8be5\u7a7a\u6876\u4e4b\u4e0b\u7684\u5143\u7d20\u90fd\u65e0\u6cd5\u518d\u88ab\u8bbf\u95ee\u5230\uff0c\u7a0b\u5e8f\u53ef\u80fd\u8bef\u5224\u8fd9\u4e9b\u5143\u7d20\u4e0d\u5b58\u5728\uff0c\u5982\u56fe 6-7 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 6-7 \u00a0 \u5728\u5f00\u653e\u5bfb\u5740\u4e2d\u5220\u9664\u5143\u7d20\u5bfc\u81f4\u7684\u67e5\u8be2\u95ee\u9898 </p> <p>\u4e3a\u4e86\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u300c\u61d2\u5220\u9664 lazy deletion\u300d\u673a\u5236\uff1a\u5b83\u4e0d\u76f4\u63a5\u4ece\u54c8\u5e0c\u8868\u4e2d\u79fb\u9664\u5143\u7d20\uff0c\u800c\u662f\u5229\u7528\u4e00\u4e2a\u5e38\u91cf <code>TOMBSTONE</code> \u6765\u6807\u8bb0\u8fd9\u4e2a\u6876\u3002\u5728\u8be5\u673a\u5236\u4e0b\uff0c<code>None</code> \u548c <code>TOMBSTONE</code> \u90fd\u4ee3\u8868\u7a7a\u6876\uff0c\u90fd\u53ef\u4ee5\u653e\u7f6e\u952e\u503c\u5bf9\u3002\u4f46\u4e0d\u540c\u7684\u662f\uff0c\u7ebf\u6027\u63a2\u6d4b\u5230 <code>TOMBSTONE</code> \u65f6\u5e94\u8be5\u7ee7\u7eed\u904d\u5386\uff0c\u56e0\u4e3a\u5176\u4e4b\u4e0b\u53ef\u80fd\u8fd8\u5b58\u5728\u952e\u503c\u5bf9\u3002</p> <p>\u7136\u800c\uff0c\u61d2\u5220\u9664\u53ef\u80fd\u4f1a\u52a0\u901f\u54c8\u5e0c\u8868\u7684\u6027\u80fd\u9000\u5316\u3002\u8fd9\u662f\u56e0\u4e3a\u6bcf\u6b21\u5220\u9664\u64cd\u4f5c\u90fd\u4f1a\u4ea7\u751f\u4e00\u4e2a\u5220\u9664\u6807\u8bb0\uff0c\u968f\u7740 <code>TOMBSTONE</code> \u7684\u589e\u52a0\uff0c\u641c\u7d22\u65f6\u95f4\u4e5f\u4f1a\u589e\u52a0\uff0c\u56e0\u4e3a\u7ebf\u6027\u63a2\u6d4b\u53ef\u80fd\u9700\u8981\u8df3\u8fc7\u591a\u4e2a <code>TOMBSTONE</code> \u624d\u80fd\u627e\u5230\u76ee\u6807\u5143\u7d20\u3002</p> <p>\u4e3a\u6b64\uff0c\u8003\u8651\u5728\u7ebf\u6027\u63a2\u6d4b\u4e2d\u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a <code>TOMBSTONE</code> \u7684\u7d22\u5f15\uff0c\u5e76\u5c06\u641c\u7d22\u5230\u7684\u76ee\u6807\u5143\u7d20\u4e0e\u8be5 <code>TOMBSTONE</code> \u4ea4\u6362\u4f4d\u7f6e\u3002\u8fd9\u6837\u505a\u7684\u597d\u5904\u662f\u5f53\u6bcf\u6b21\u67e5\u8be2\u6216\u6dfb\u52a0\u5143\u7d20\u65f6\uff0c\u5143\u7d20\u4f1a\u88ab\u79fb\u52a8\u81f3\u8ddd\u79bb\u7406\u60f3\u4f4d\u7f6e\uff08\u63a2\u6d4b\u8d77\u59cb\u70b9\uff09\u66f4\u8fd1\u7684\u6876\uff0c\u4ece\u800c\u4f18\u5316\u67e5\u8be2\u6548\u7387\u3002</p> <p>\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u4e2a\u5305\u542b\u61d2\u5220\u9664\u7684\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u3002\u4e3a\u4e86\u66f4\u52a0\u5145\u5206\u5730\u4f7f\u7528\u54c8\u5e0c\u8868\u7684\u7a7a\u95f4\uff0c\u6211\u4eec\u5c06\u54c8\u5e0c\u8868\u770b\u4f5c\u4e00\u4e2a\u201c\u73af\u5f62\u6570\u7ec4\u201d\uff0c\u5f53\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map_open_addressing.py<pre><code>class HashMapOpenAddressing:\n    \"\"\"\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\n        self.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\n        self.load_thres = 2.0 / 3.0  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        self.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\n        self.buckets: list[Pair | None] = [None] * self.capacity  # \u6876\u6570\u7ec4\n        self.TOMBSTONE = Pair(-1, \"-1\")  # \u5220\u9664\u6807\u8bb0\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\n        return self.size / self.capacity\n\n    def find_bucket(self, key: int) -&gt; int:\n        \"\"\"\u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\"\"\"\n        index = self.hash_func(key)\n        first_tombstone = -1\n        # \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while self.buckets[index] is not None:\n            # \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if self.buckets[index].key == key:\n                # \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if first_tombstone != -1:\n                    self.buckets[first_tombstone] = self.buckets[index]\n                    self.buckets[index] = self.TOMBSTONE\n                    return first_tombstone  # \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                return index  # \u8fd4\u56de\u6876\u7d22\u5f15\n            # \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if first_tombstone == -1 and self.buckets[index] is self.TOMBSTONE:\n                first_tombstone = index\n            # \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % self.capacity\n        # \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return index if first_tombstone == -1 else first_tombstone\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n        # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            return self.buckets[index].val\n        # \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n        # \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index].val = val\n            return\n        # \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        self.buckets[index] = Pair(key, val)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n        # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index] = self.TOMBSTONE\n            self.size -= 1\n\n    def extend(self):\n        \"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n        # \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        buckets_tmp = self.buckets\n        # \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio\n        self.buckets = [None] * self.capacity\n        self.size = 0\n        # \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for pair in buckets_tmp:\n            if pair not in [None, self.TOMBSTONE]:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n        for pair in self.buckets:\n            if pair is None:\n                print(\"None\")\n            elif pair is self.TOMBSTONE:\n                print(\"TOMBSTONE\")\n            else:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> hash_map_open_addressing.cpp<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n  private:\n    int size;                             // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity = 4;                     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    const double loadThres = 2.0 / 3.0;     // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    const int extendRatio = 2;            // \u6269\u5bb9\u500d\u6570\n    vector&lt;Pair *&gt; buckets;               // \u6876\u6570\u7ec4\n    Pair *TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    HashMapOpenAddressing() : size(0), buckets(capacity, nullptr) {\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~HashMapOpenAddressing() {\n        for (Pair *pair : buckets) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                delete pair;\n            }\n        }\n        delete TOMBSTONE;\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double loadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != nullptr) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index]-&gt;key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    string get(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]-&gt;val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n        return \"\";\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]-&gt;val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            delete buckets[index];\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        vector&lt;Pair *&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = vector&lt;Pair *&gt;(capacity, nullptr);\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (Pair *pair : bucketsTmp) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                put(pair-&gt;key, pair-&gt;val);\n                delete pair;\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (Pair *pair : buckets) {\n            if (pair == nullptr) {\n                cout &lt;&lt; \"nullptr\" &lt;&lt; endl;\n            } else if (pair == TOMBSTONE) {\n                cout &lt;&lt; \"TOMBSTONE\" &lt;&lt; endl;\n            } else {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; endl;\n            }\n        }\n    }\n};\n</code></pre> hash_map_open_addressing.java<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    private int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    private int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    private final double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    private final int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n    private Pair[] buckets; // \u6876\u6570\u7ec4\n    private final Pair TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    private double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    private int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index].key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public String get(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void put(int key, String val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void remove(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    private void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        Pair[] bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (Pair pair : bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void print() {\n        for (Pair pair : buckets) {\n            if (pair == null) {\n                System.out.println(\"null\");\n            } else if (pair == TOMBSTONE) {\n                System.out.println(\"TOMBSTONE\");\n            } else {\n                System.out.println(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.cs<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n    Pair[] buckets; // \u6876\u6570\u7ec4\n    Pair TOMBSTONE = new(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    int FindBucket(int key) {\n        int index = HashFunc(key);\n        int firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index].key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public string? Get(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void Extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        Pair[] bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        foreach (Pair pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void Print() {\n        foreach (Pair pair in buckets) {\n            if (pair == null) {\n                Console.WriteLine(\"null\");\n            } else if (pair == TOMBSTONE) {\n                Console.WriteLine(\"TOMBSTONE\");\n            } else {\n                Console.WriteLine(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.go<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntype hashMapOpenAddressing struct {\n    size        int     // \u952e\u503c\u5bf9\u6570\u91cf\n    capacity    int     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    loadThres   float64 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    extendRatio int     // \u6269\u5bb9\u500d\u6570\n    buckets     []*pair // \u6876\u6570\u7ec4\n    TOMBSTONE   *pair   // \u5220\u9664\u6807\u8bb0\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\n    return &amp;hashMapOpenAddressing{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     make([]*pair, 4),\n        TOMBSTONE:   &amp;pair{-1, \"-1\"},\n    }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (h *hashMapOpenAddressing) hashFunc(key int) int {\n    return key % h.capacity // \u6839\u636e\u952e\u8ba1\u7b97\u54c8\u5e0c\u503c\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (h *hashMapOpenAddressing) loadFactor() float64 {\n    return float64(h.size) / float64(h.capacity) // \u8ba1\u7b97\u5f53\u524d\u8d1f\u8f7d\u56e0\u5b50\n}\n\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nfunc (h *hashMapOpenAddressing) findBucket(key int) int {\n    index := h.hashFunc(key) // \u83b7\u53d6\u521d\u59cb\u7d22\u5f15\n    firstTombstone := -1     // \u8bb0\u5f55\u9047\u5230\u7684\u7b2c\u4e00\u4e2aTOMBSTONE\u7684\u4f4d\u7f6e\n    for h.buckets[index] != nil {\n        if h.buckets[index].key == key {\n            if firstTombstone != -1 {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                h.buckets[firstTombstone] = h.buckets[index]\n                h.buckets[index] = h.TOMBSTONE\n                return firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n            }\n            return index // \u8fd4\u56de\u627e\u5230\u7684\u7d22\u5f15\n        }\n        if firstTombstone == -1 &amp;&amp; h.buckets[index] == h.TOMBSTONE {\n            firstTombstone = index // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\u7684\u4f4d\u7f6e\n        }\n        index = (index + 1) % h.capacity // \u7ebf\u6027\u63a2\u6d4b\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n    if firstTombstone != -1 {\n        return firstTombstone\n    }\n    return index\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) get(key int) string {\n    index := h.findBucket(key) // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        return h.buckets[index].val // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    }\n    return \"\" // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de \"\"\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) put(key int, val string) {\n    if h.loadFactor() &gt; h.loadThres {\n        h.extend() // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    }\n    index := h.findBucket(key) // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] == nil || h.buckets[index] == h.TOMBSTONE {\n        h.buckets[index] = &amp;pair{key, val} // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        h.size++\n    } else {\n        h.buckets[index].val = val // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val\n    }\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) remove(key int) {\n    index := h.findBucket(key) // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        h.buckets[index] = h.TOMBSTONE // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        h.size--\n    }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (h *hashMapOpenAddressing) extend() {\n    oldBuckets := h.buckets               // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    h.capacity *= h.extendRatio           // \u66f4\u65b0\u5bb9\u91cf\n    h.buckets = make([]*pair, h.capacity) // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    h.size = 0                            // \u91cd\u7f6e\u5927\u5c0f\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for _, pair := range oldBuckets {\n        if pair != nil &amp;&amp; pair != h.TOMBSTONE {\n            h.put(pair.key, pair.val)\n        }\n    }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (h *hashMapOpenAddressing) print() {\n    for _, pair := range h.buckets {\n        if pair == nil {\n            fmt.Println(\"nil\")\n        } else if pair == h.TOMBSTONE {\n            fmt.Println(\"TOMBSTONE\")\n        } else {\n            fmt.Printf(\"%d -&gt; %s\\n\", pair.key, pair.val)\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.swift<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n    var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    var loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    var extendRatio: Int // \u6269\u5bb9\u500d\u6570\n    var buckets: [Pair?] // \u6876\u6570\u7ec4\n    var TOMBSTONE: Pair // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: nil, count: capacity)\n        TOMBSTONE = Pair(key: -1, val: \"-1\")\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    func findBucket(key: Int) -&gt; Int {\n        var index = hashFunc(key: key)\n        var firstTombstone = -1\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while buckets[index] != nil {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if buckets[index]!.key == key {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if firstTombstone != -1 {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE {\n                firstTombstone = index\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            return buckets[index]!.val\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return nil\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index]!.val = val\n            return\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = Pair(key: key, val: val)\n        size += 1\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index] = TOMBSTONE\n            size -= 1\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    func extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        buckets = Array(repeating: nil, count: capacity)\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for pair in bucketsTmp {\n            if let pair, pair != TOMBSTONE {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    func print() {\n        for pair in buckets {\n            if pair == nil {\n                Swift.print(\"null\")\n            } else if pair == TOMBSTONE {\n                Swift.print(\"TOMBSTONE\")\n            } else {\n                Swift.print(\"\\(pair!.key) -&gt; \\(pair!.val)\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.js<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    #size; // \u952e\u503c\u5bf9\u6570\u91cf\n    #capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    #loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    #extendRatio; // \u6269\u5bb9\u500d\u6570\n    #buckets; // \u6876\u6570\u7ec4\n    #TOMBSTONE; // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\n        this.#capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n        this.#loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        this.#extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n        this.#buckets = Array(this.#capacity).fill(null); // \u6876\u6570\u7ec4\n        this.#TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    #findBucket(key) {\n        let index = this.#hashFunc(key);\n        let firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (this.#buckets[index] !== null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (this.#buckets[index].key === key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone !== -1) {\n                    this.#buckets[firstTombstone] = this.#buckets[index];\n                    this.#buckets[index] = this.#TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.#buckets[index] === this.#TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % this.#capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            return this.#buckets[index].val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key, val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        this.#buckets[index] = new Pair(key, val);\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index] = this.#TOMBSTONE;\n            this.#size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    #extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = Array(this.#capacity).fill(null);\n        this.#size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.#TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print() {\n        for (const pair of this.#buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.#TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.ts<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    private size: number; // \u952e\u503c\u5bf9\u6570\u91cf\n    private capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    private loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    private extendRatio: number; // \u6269\u5bb9\u500d\u6570\n    private buckets: Array&lt;Pair | null&gt;; // \u6876\u6570\u7ec4\n    private TOMBSTONE: Pair; // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\n        this.capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n        this.loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        this.extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n        this.buckets = Array(this.capacity).fill(null); // \u6876\u6570\u7ec4\n        this.TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private hashFunc(key: number): number {\n        return key % this.capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    private loadFactor(): number {\n        return this.size / this.capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    private findBucket(key: number): number {\n        let index = this.hashFunc(key);\n        let firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (this.buckets[index] !== null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (this.buckets[index]!.key === key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone !== -1) {\n                    this.buckets[firstTombstone] = this.buckets[index];\n                    this.buckets[index] = this.TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.buckets[index] === this.TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % this.capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key: number): string | null {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            return this.buckets[index]!.val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key: number, val: string): void {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.loadFactor() &gt; this.loadThres) {\n            this.extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index]!.val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        this.buckets[index] = new Pair(key, val);\n        this.size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key: number): void {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index] = this.TOMBSTONE;\n            this.size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    private extend(): void {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.capacity *= this.extendRatio;\n        this.buckets = Array(this.capacity).fill(null);\n        this.size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print(): void {\n        for (const pair of this.buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.dart<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n  late int _size; // \u952e\u503c\u5bf9\u6570\u91cf\n  int _capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n  double _loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n  int _extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n  late List&lt;Pair?&gt; _buckets; // \u6876\u6570\u7ec4\n  Pair _TOMBSTONE = Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  HashMapOpenAddressing() {\n    _size = 0;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n  }\n\n  /* \u54c8\u5e0c\u51fd\u6570 */\n  int hashFunc(int key) {\n    return key % _capacity;\n  }\n\n  /* \u8d1f\u8f7d\u56e0\u5b50 */\n  double loadFactor() {\n    return _size / _capacity;\n  }\n\n  /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n  int findBucket(int key) {\n    int index = hashFunc(key);\n    int firstTombstone = -1;\n    // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n    while (_buckets[index] != null) {\n      // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n      if (_buckets[index]!.key == key) {\n        // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n        if (firstTombstone != -1) {\n          _buckets[firstTombstone] = _buckets[index];\n          _buckets[index] = _TOMBSTONE;\n          return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n        }\n        return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n      }\n      // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n      if (firstTombstone == -1 &amp;&amp; _buckets[index] == _TOMBSTONE) {\n        firstTombstone = index;\n      }\n      // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n      index = (index + 1) % _capacity;\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n    return firstTombstone == -1 ? index : firstTombstone;\n  }\n\n  /* \u67e5\u8be2\u64cd\u4f5c */\n  String? get(int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      return _buckets[index]!.val;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n    return null;\n  }\n\n  /* \u6dfb\u52a0\u64cd\u4f5c */\n  void put(int key, String val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor() &gt; _loadThres) {\n      extend();\n    }\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index]!.val = val;\n      return;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n    _buckets[index] = new Pair(key, val);\n    _size++;\n  }\n\n  /* \u5220\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index] = _TOMBSTONE;\n      _size--;\n    }\n  }\n\n  /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n  void extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    List&lt;Pair?&gt; bucketsTmp = _buckets;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    _capacity *= _extendRatio;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n    _size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (Pair? pair in bucketsTmp) {\n      if (pair != null &amp;&amp; pair != _TOMBSTONE) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* \u6253\u5370\u54c8\u5e0c\u8868 */\n  void printHashMap() {\n    for (Pair? pair in _buckets) {\n      if (pair == null) {\n        print(\"null\");\n      } else if (pair == _TOMBSTONE) {\n        print(\"TOMBSTONE\");\n      } else {\n        print(\"${pair.key} -&gt; ${pair.val}\");\n      }\n    }\n  }\n}\n</code></pre> hash_map_open_addressing.rs<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapOpenAddressing {\n    size: usize,                // \u952e\u503c\u5bf9\u6570\u91cf\n    capacity: usize,            // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    load_thres: f64,            // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    extend_ratio: usize,        // \u6269\u5bb9\u500d\u6570\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;, // \u6876\u6570\u7ec4\n    TOMBSTONE: Option&lt;Pair&gt;,    // \u5220\u9664\u6807\u8bb0\n}\n\nimpl HashMapOpenAddressing {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![None; 4],\n            TOMBSTONE: Some(Pair {\n                key: -1,\n                val: \"-1\".to_string(),\n            }),\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        (key % self.capacity as i32) as usize\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fn load_factor(&amp;self) -&gt; f64 {\n        self.size as f64 / self.capacity as f64\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    fn find_bucket(&amp;mut self, key: i32) -&gt; usize {\n        let mut index = self.hash_func(key);\n        let mut first_tombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while self.buckets[index].is_some() {\n            // \u82e5\u9047\u5230 key\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if self.buckets[index].as_ref().unwrap().key == key {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u5efa\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n                if first_tombstone != -1 {\n                    self.buckets[first_tombstone as usize] = self.buckets[index].take();\n                    self.buckets[index] = self.TOMBSTONE.clone();\n                    return first_tombstone as usize; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if first_tombstone == -1 &amp;&amp; self.buckets[index] == self.TOMBSTONE {\n                first_tombstone = index as i32;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % self.capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        if first_tombstone == -1 {\n            index\n        } else {\n            first_tombstone as usize\n        }\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fn get(&amp;mut self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            return self.buckets[index].as_ref().map(|pair| &amp;pair.val as &amp;str);\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        None\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index].as_mut().unwrap().val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        self.buckets[index] = Some(Pair { key, val });\n        self.size += 1;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fn remove(&amp;mut self, key: i32) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index] = self.TOMBSTONE.clone();\n            self.size -= 1;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fn extend(&amp;mut self) {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let buckets_tmp = self.buckets.clone();\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![None; self.capacity];\n        self.size = 0;\n\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for pair in buckets_tmp {\n            if pair.is_none() || pair == self.TOMBSTONE {\n                continue;\n            }\n            let pair = pair.unwrap();\n\n            self.put(pair.key, pair.val);\n        }\n    }\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fn print(&amp;self) {\n        for pair in &amp;self.buckets {\n            if pair.is_none() {\n                println!(\"null\");\n            } else if pair == &amp;self.TOMBSTONE {\n                println!(\"TOMBSTONE\");\n            } else {\n                let pair = pair.as_ref().unwrap();\n                println!(\"{} -&gt; {}\", pair.key, pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.c<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\n    int size;         // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity;     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio;  // \u6269\u5bb9\u500d\u6570\n    Pair **buckets;   // \u6876\u6570\u7ec4\n    Pair *TOMBSTONE;  // \u5220\u9664\u6807\u8bb0\n} HashMapOpenAddressing;\n\n/* \u6784\u9020\u51fd\u6570 */\nHashMapOpenAddressing *newHashMapOpenAddressing() {\n    HashMapOpenAddressing *hashMap = (HashMapOpenAddressing *)malloc(sizeof(HashMapOpenAddressing));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Pair **)malloc(sizeof(Pair *) * hashMap-&gt;capacity);\n    hashMap-&gt;TOMBSTONE = (Pair *)malloc(sizeof(Pair));\n    hashMap-&gt;TOMBSTONE-&gt;key = -1;\n    hashMap-&gt;TOMBSTONE-&gt;val = \"-1\";\n\n    return hashMap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapOpenAddressing(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapOpenAddressing *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapOpenAddressing *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nint findBucket(HashMapOpenAddressing *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    int firstTombstone = -1;\n    // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n    while (hashMap-&gt;buckets[index] != NULL) {\n        // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        if (hashMap-&gt;buckets[index]-&gt;key == key) {\n            // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n            if (firstTombstone != -1) {\n                hashMap-&gt;buckets[firstTombstone] = hashMap-&gt;buckets[index];\n                hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n                return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n            }\n            return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n        }\n        // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n        if (firstTombstone == -1 &amp;&amp; hashMap-&gt;buckets[index] == hashMap-&gt;TOMBSTONE) {\n            firstTombstone = index;\n        }\n        // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n        index = (index + 1) % hashMap-&gt;capacity;\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n    return firstTombstone == -1 ? index : firstTombstone;\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapOpenAddressing *hashMap, int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        return hashMap-&gt;buckets[index]-&gt;val;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n    return \"\";\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapOpenAddressing *hashMap, int key, char *val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        free(hashMap-&gt;buckets[index]-&gt;val);\n        hashMap-&gt;buckets[index]-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n        strcpy(hashMap-&gt;buckets[index]-&gt;val, val);\n        hashMap-&gt;buckets[index]-&gt;val[strlen(val)] = '\\0';\n        return;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n    Pair *pair = (Pair *)malloc(sizeof(Pair));\n    pair-&gt;key = key;\n    pair-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n    strcpy(pair-&gt;val, val);\n    pair-&gt;val[strlen(val)] = '\\0';\n\n    hashMap-&gt;buckets[index] = pair;\n    hashMap-&gt;size++;\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapOpenAddressing *hashMap, int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        Pair *pair = hashMap-&gt;buckets[index];\n        free(pair-&gt;val);\n        free(pair);\n        hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n        hashMap-&gt;size--;\n    }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapOpenAddressing *hashMap) {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    Pair **bucketsTmp = hashMap-&gt;buckets;\n    int oldCapacity = hashMap-&gt;capacity;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Pair **)malloc(sizeof(Pair *) * hashMap-&gt;capacity);\n    hashMap-&gt;size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Pair *pair = bucketsTmp[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            put(hashMap, pair-&gt;key, pair-&gt;val);\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(bucketsTmp);\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair == NULL) {\n            printf(\"NULL\\n\");\n        } else if (pair == hashMap-&gt;TOMBSTONE) {\n            printf(\"TOMBSTONE\\n\");\n        } else {\n            printf(\"%d -&gt; %s\\n\", pair-&gt;key, pair-&gt;val);\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.kt<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    private var size: Int = 0 // \u952e\u503c\u5bf9\u6570\u91cf\n    private var capacity = 4 // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    private val loadThres: Double = 2.0 / 3.0 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    private val extendRatio = 2 // \u6269\u5bb9\u500d\u6570\n    private var buckets: Array&lt;Pair?&gt; // \u6876\u6570\u7ec4\n    private val TOMBSTONE = Pair(-1, \"-1\") // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init {\n        buckets = arrayOfNulls(capacity)\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    fun findBucket(key: Int): Int {\n        var index = hashFunc(key)\n        var firstTombstone = -1\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index]?.key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return if (firstTombstone == -1) index else firstTombstone\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]?.value\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]!!.value = value\n            return\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = Pair(key, value)\n        size++\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE\n            size--\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fun extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        val bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        buckets = arrayOfNulls(capacity)\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair.value)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (pair in buckets) {\n            if (pair == null) {\n                println(\"null\")\n            } else if (pair == TOMBSTONE) {\n                println(\"TOMESTOME\")\n            } else {\n                println(\"${pair.key} -&gt; ${pair.value}\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.rb<pre><code>[class]{HashMapOpenAddressing}-[func]{}\n</code></pre> hash_map_open_addressing.zig<pre><code>[class]{HashMapOpenAddressing}-[func]{}\n</code></pre>"},{"location":"chapter_hashing/hash_collision/#2","title":"2. \u00a0 \u5e73\u65b9\u63a2\u6d4b","text":"<p>\u5e73\u65b9\u63a2\u6d4b\u4e0e\u7ebf\u6027\u63a2\u6d4b\u7c7b\u4f3c\uff0c\u90fd\u662f\u5f00\u653e\u5bfb\u5740\u7684\u5e38\u89c1\u7b56\u7565\u4e4b\u4e00\u3002\u5f53\u53d1\u751f\u51b2\u7a81\u65f6\uff0c\u5e73\u65b9\u63a2\u6d4b\u4e0d\u662f\u7b80\u5355\u5730\u8df3\u8fc7\u4e00\u4e2a\u56fa\u5b9a\u7684\u6b65\u6570\uff0c\u800c\u662f\u8df3\u8fc7\u201c\u63a2\u6d4b\u6b21\u6570\u7684\u5e73\u65b9\u201d\u7684\u6b65\u6570\uff0c\u5373 \\(1, 4, 9, \\dots\\) \u6b65\u3002</p> <p>\u5e73\u65b9\u63a2\u6d4b\u4e3b\u8981\u5177\u6709\u4ee5\u4e0b\u4f18\u52bf\u3002</p> <ul> <li>\u5e73\u65b9\u63a2\u6d4b\u901a\u8fc7\u8df3\u8fc7\u63a2\u6d4b\u6b21\u6570\u5e73\u65b9\u7684\u8ddd\u79bb\uff0c\u8bd5\u56fe\u7f13\u89e3\u7ebf\u6027\u63a2\u6d4b\u7684\u805a\u96c6\u6548\u5e94\u3002</li> <li>\u5e73\u65b9\u63a2\u6d4b\u4f1a\u8df3\u8fc7\u66f4\u5927\u7684\u8ddd\u79bb\u6765\u5bfb\u627e\u7a7a\u4f4d\u7f6e\uff0c\u6709\u52a9\u4e8e\u6570\u636e\u5206\u5e03\u5f97\u66f4\u52a0\u5747\u5300\u3002</li> </ul> <p>\u7136\u800c\uff0c\u5e73\u65b9\u63a2\u6d4b\u5e76\u4e0d\u662f\u5b8c\u7f8e\u7684\u3002</p> <ul> <li>\u4ecd\u7136\u5b58\u5728\u805a\u96c6\u73b0\u8c61\uff0c\u5373\u67d0\u4e9b\u4f4d\u7f6e\u6bd4\u5176\u4ed6\u4f4d\u7f6e\u66f4\u5bb9\u6613\u88ab\u5360\u7528\u3002</li> <li>\u7531\u4e8e\u5e73\u65b9\u7684\u589e\u957f\uff0c\u5e73\u65b9\u63a2\u6d4b\u53ef\u80fd\u4e0d\u4f1a\u63a2\u6d4b\u6574\u4e2a\u54c8\u5e0c\u8868\uff0c\u8fd9\u610f\u5473\u7740\u5373\u4f7f\u54c8\u5e0c\u8868\u4e2d\u6709\u7a7a\u6876\uff0c\u5e73\u65b9\u63a2\u6d4b\u4e5f\u53ef\u80fd\u65e0\u6cd5\u8bbf\u95ee\u5230\u5b83\u3002</li> </ul>"},{"location":"chapter_hashing/hash_collision/#3","title":"3. \u00a0 \u591a\u6b21\u54c8\u5e0c","text":"<p>\u987e\u540d\u601d\u4e49\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\)\u3001\\(f_2(x)\\)\u3001\\(f_3(x)\\)\u3001\\(\\dots\\) \u8fdb\u884c\u63a2\u6d4b\u3002</p> <ul> <li>\u63d2\u5165\u5143\u7d20\uff1a\u82e5\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\) \u51fa\u73b0\u51b2\u7a81\uff0c\u5219\u5c1d\u8bd5 \\(f_2(x)\\) \uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u5230\u627e\u5230\u7a7a\u4f4d\u540e\u63d2\u5165\u5143\u7d20\u3002</li> <li>\u67e5\u627e\u5143\u7d20\uff1a\u5728\u76f8\u540c\u7684\u54c8\u5e0c\u51fd\u6570\u987a\u5e8f\u4e0b\u8fdb\u884c\u67e5\u627e\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u65f6\u8fd4\u56de\uff1b\u82e5\u9047\u5230\u7a7a\u4f4d\u6216\u5df2\u5c1d\u8bd5\u6240\u6709\u54c8\u5e0c\u51fd\u6570\uff0c\u8bf4\u660e\u54c8\u5e0c\u8868\u4e2d\u4e0d\u5b58\u5728\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de <code>None</code> \u3002</li> </ul> <p>\u4e0e\u7ebf\u6027\u63a2\u6d4b\u76f8\u6bd4\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u4f1a\u5e26\u6765\u989d\u5916\u7684\u8ba1\u7b97\u91cf\u3002</p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u548c\u591a\u6b21\u54c8\u5e0c\uff09\u54c8\u5e0c\u8868\u90fd\u5b58\u5728\u201c\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u201d\u7684\u95ee\u9898\u3002</p>"},{"location":"chapter_hashing/hash_collision/#623","title":"6.2.3 \u00a0 \u7f16\u7a0b\u8bed\u8a00\u7684\u9009\u62e9","text":"<p>\u5404\u79cd\u7f16\u7a0b\u8bed\u8a00\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u54c8\u5e0c\u8868\u5b9e\u73b0\u7b56\u7565\uff0c\u4e0b\u9762\u4e3e\u51e0\u4e2a\u4f8b\u5b50\u3002</p> <ul> <li>Python \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002\u5b57\u5178 <code>dict</code> \u4f7f\u7528\u4f2a\u968f\u673a\u6570\u8fdb\u884c\u63a2\u6d4b\u3002</li> <li>Java \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002\u81ea JDK 1.8 \u4ee5\u6765\uff0c\u5f53 <code>HashMap</code> \u5185\u6570\u7ec4\u957f\u5ea6\u8fbe\u5230 64 \u4e14\u94fe\u8868\u957f\u5ea6\u8fbe\u5230 8 \u65f6\uff0c\u94fe\u8868\u4f1a\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u4ee5\u63d0\u5347\u67e5\u627e\u6027\u80fd\u3002</li> <li>Go \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002Go \u89c4\u5b9a\u6bcf\u4e2a\u6876\u6700\u591a\u5b58\u50a8 8 \u4e2a\u952e\u503c\u5bf9\uff0c\u8d85\u51fa\u5bb9\u91cf\u5219\u8fde\u63a5\u4e00\u4e2a\u6ea2\u51fa\u6876\uff1b\u5f53\u6ea2\u51fa\u6876\u8fc7\u591a\u65f6\uff0c\u4f1a\u6267\u884c\u4e00\u6b21\u7279\u6b8a\u7684\u7b49\u91cf\u6269\u5bb9\u64cd\u4f5c\uff0c\u4ee5\u786e\u4fdd\u6027\u80fd\u3002</li> </ul>"},{"location":"chapter_hashing/hash_map/","title":"6.1 \u00a0 \u54c8\u5e0c\u8868","text":"<p>\u300c\u54c8\u5e0c\u8868 hash table\u300d\uff0c\u53c8\u79f0\u300c\u6563\u5217\u8868\u300d\uff0c\u5b83\u901a\u8fc7\u5efa\u7acb\u952e <code>key</code> \u4e0e\u503c <code>value</code> \u4e4b\u95f4\u7684\u6620\u5c04\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u8be2\u3002\u5177\u4f53\u800c\u8a00\uff0c\u6211\u4eec\u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u4e00\u4e2a\u952e <code>key</code> \uff0c\u5219\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u5bf9\u5e94\u7684\u503c <code>value</code> \u3002</p> <p>\u5982\u56fe 6-1 \u6240\u793a\uff0c\u7ed9\u5b9a \\(n\\) \u4e2a\u5b66\u751f\uff0c\u6bcf\u4e2a\u5b66\u751f\u90fd\u6709\u201c\u59d3\u540d\u201d\u548c\u201c\u5b66\u53f7\u201d\u4e24\u9879\u6570\u636e\u3002\u5047\u5982\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u201c\u8f93\u5165\u4e00\u4e2a\u5b66\u53f7\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u59d3\u540d\u201d\u7684\u67e5\u8be2\u529f\u80fd\uff0c\u5219\u53ef\u4ee5\u91c7\u7528\u56fe 6-1 \u6240\u793a\u7684\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002</p> <p></p> <p> \u56fe 6-1 \u00a0 \u54c8\u5e0c\u8868\u7684\u62bd\u8c61\u8868\u793a </p> <p>\u9664\u54c8\u5e0c\u8868\u5916\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u4e5f\u53ef\u4ee5\u5b9e\u73b0\u67e5\u8be2\u529f\u80fd\uff0c\u5b83\u4eec\u7684\u6548\u7387\u5bf9\u6bd4\u5982\u8868 6-1 \u6240\u793a\u3002</p> <ul> <li>\u6dfb\u52a0\u5143\u7d20\uff1a\u4ec5\u9700\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u6570\u7ec4\uff08\u94fe\u8868\uff09\u7684\u5c3e\u90e8\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002</li> <li>\u67e5\u8be2\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\uff08\u94fe\u8868\uff09\u662f\u4e71\u5e8f\u7684\uff0c\u56e0\u6b64\u9700\u8981\u904d\u5386\u5176\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> <li>\u5220\u9664\u5143\u7d20\uff1a\u9700\u8981\u5148\u67e5\u8be2\u5230\u5143\u7d20\uff0c\u518d\u4ece\u6570\u7ec4\uff08\u94fe\u8868\uff09\u4e2d\u5220\u9664\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> </ul> <p> \u8868 6-1 \u00a0 \u5143\u7d20\u67e5\u8be2\u6548\u7387\u5bf9\u6bd4 </p> \u6570\u7ec4 \u94fe\u8868 \u54c8\u5e0c\u8868 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u5728\u54c8\u5e0c\u8868\u4e2d\u8fdb\u884c\u589e\u5220\u67e5\u6539\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f \\(O(1)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</p>"},{"location":"chapter_hashing/hash_map/#611","title":"6.1.1 \u00a0 \u54c8\u5e0c\u8868\u5e38\u7528\u64cd\u4f5c","text":"<p>\u54c8\u5e0c\u8868\u7684\u5e38\u89c1\u64cd\u4f5c\u5305\u62ec\uff1a\u521d\u59cb\u5316\u3001\u67e5\u8be2\u64cd\u4f5c\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u548c\u5220\u9664\u952e\u503c\u5bf9\u7b49\uff0c\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map.py<pre><code># \u521d\u59cb\u5316\u54c8\u5e0c\u8868\nhmap: dict = {}\n\n# \u6dfb\u52a0\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u5570\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9e2d\"\n\n# \u67e5\u8be2\u64cd\u4f5c\n# \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname: str = hmap[15937]\n\n# \u5220\u9664\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nhmap.pop(10583)\n</code></pre> hash_map.cpp<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nunordered_map&lt;int, string&gt; map;\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.erase(10583);\n</code></pre> hash_map.java<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.put(12836, \"\u5c0f\u54c8\");   \nmap.put(15937, \"\u5c0f\u5570\");   \nmap.put(16750, \"\u5c0f\u7b97\");   \nmap.put(13276, \"\u5c0f\u6cd5\");\nmap.put(10583, \"\u5c0f\u9e2d\");\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map.get(15937);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n</code></pre> hash_map.cs<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nDictionary&lt;int, string&gt; map = new() {\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    // \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\n    { 12836, \"\u5c0f\u54c8\" },\n    { 15937, \"\u5c0f\u5570\" },\n    { 16750, \"\u5c0f\u7b97\" },\n    { 13276, \"\u5c0f\u6cd5\" },\n    { 10583, \"\u5c0f\u9e2d\" }\n};\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.Remove(10583);\n</code></pre> hash_map_test.go<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nhmap := make(map[int]string)\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u5570\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9e2d\"\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname := hmap[15937]\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\ndelete(hmap, 10583)\n</code></pre> hash_map.swift<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nvar map: [Int: String] = [:]\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u5570\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9e2d\"\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map[15937]!\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.removeValue(forKey: 10583)\n</code></pre> hash_map.js<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\n</code></pre> hash_map.ts<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map&lt;number, string&gt;();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\nconsole.info('\\n\u6dfb\u52a0\u5b8c\u6210\u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -&gt; Value');\nconsole.info(map);\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\nconsole.info('\\n\u8f93\u5165\u5b66\u53f7 15937 \uff0c\u67e5\u8be2\u5230\u59d3\u540d ' + name);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\nconsole.info('\\n\u5220\u9664 10583 \u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -&gt; Value');\nconsole.info(map);\n</code></pre> hash_map.dart<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap&lt;int, String&gt; map = {};\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map[15937];\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n</code></pre> hash_map.rs<pre><code>use std::collections::HashMap;\n\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nlet mut map: HashMap&lt;i32, String&gt; = HashMap::new();\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.insert(12836, \"\u5c0f\u54c8\".to_string());\nmap.insert(15937, \"\u5c0f\u5570\".to_string());\nmap.insert(16750, \"\u5c0f\u7b97\".to_string());\nmap.insert(13279, \"\u5c0f\u6cd5\".to_string());\nmap.insert(10583, \"\u5c0f\u9e2d\".to_string());\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet _name: Option&lt;&amp;String&gt; = map.get(&amp;15937);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nlet _removed_value: Option&lt;String&gt; = map.remove(&amp;10583);\n</code></pre> hash_map.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n</code></pre> hash_map.kt<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nval map = HashMap&lt;Int,String&gt;()\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u5570\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9e2d\"\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nval name = map[15937]\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583)\n</code></pre> hash_map.rb<pre><code>\n</code></pre> hash_map.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u54c8\u5e0c\u8868\u6709\u4e09\u79cd\u5e38\u7528\u7684\u904d\u5386\u65b9\u5f0f\uff1a\u904d\u5386\u952e\u503c\u5bf9\u3001\u904d\u5386\u952e\u548c\u904d\u5386\u503c\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map.py<pre><code># \u904d\u5386\u54c8\u5e0c\u8868\n# \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor key, value in hmap.items():\n    print(key, \"-&gt;\", value)\n# \u5355\u72ec\u904d\u5386\u952e key\nfor key in hmap.keys():\n    print(key)\n# \u5355\u72ec\u904d\u5386\u503c value\nfor value in hmap.values():\n    print(value)\n</code></pre> hash_map.cpp<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor (auto kv: map) {\n    cout &lt;&lt; kv.first &lt;&lt; \" -&gt; \" &lt;&lt; kv.second &lt;&lt; endl;\n}\n// \u4f7f\u7528\u8fed\u4ee3\u5668\u904d\u5386 key-&gt;value\nfor (auto iter = map.begin(); iter != map.end(); iter++) {\n    cout &lt;&lt; iter-&gt;first &lt;&lt; \"-&gt;\" &lt;&lt; iter-&gt;second &lt;&lt; endl;\n}\n</code></pre> hash_map.java<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor (Map.Entry &lt;Integer, String&gt; kv: map.entrySet()) {\n    System.out.println(kv.getKey() + \" -&gt; \" + kv.getValue());\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (int key: map.keySet()) {\n    System.out.println(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (String val: map.values()) {\n    System.out.println(val);\n}\n</code></pre> hash_map.cs<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nforeach (var kv in map) {\n    Console.WriteLine(kv.Key + \" -&gt; \" + kv.Value);\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nforeach (int key in map.Keys) {\n    Console.WriteLine(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nforeach (string val in map.Values) {\n    Console.WriteLine(val);\n}\n</code></pre> hash_map_test.go<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor key, value := range hmap {\n    fmt.Println(key, \"-&gt;\", value)\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor key := range hmap {\n    fmt.Println(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor _, value := range hmap {\n    fmt.Println(value)\n}\n</code></pre> hash_map.swift<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nfor (key, value) in map {\n    print(\"\\(key) -&gt; \\(value)\")\n}\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys {\n    print(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values {\n    print(value)\n}\n</code></pre> hash_map.js<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.ts<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.dart<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nmap.forEach((key, value) {\n  print('$key -&gt; $value');\n});\n\n// \u5355\u72ec\u904d\u5386\u952e Key\nmap.keys.forEach((key) {\n  print(key);\n});\n\n// \u5355\u72ec\u904d\u5386\u503c Value\nmap.values.forEach((value) {\n  print(value);\n});\n</code></pre> hash_map.rs<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nfor (key, value) in &amp;map {\n    println!(\"{key} -&gt; {value}\");\n}\n\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys() {\n    println!(\"{key}\"); \n}\n\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values() {\n    println!(\"{value}\");\n}\n</code></pre> hash_map.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n</code></pre> hash_map.kt<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor ((key, value) in map) {\n    println(\"$key -&gt; $value\")\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (key in map.keys) {\n    println(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (_val in map.values) {\n    println(_val)\n}\n</code></pre> hash_map.rb<pre><code>\n</code></pre> hash_map.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_hashing/hash_map/#612","title":"6.1.2 \u00a0 \u54c8\u5e0c\u8868\u7b80\u5355\u5b9e\u73b0","text":"<p>\u6211\u4eec\u5148\u8003\u8651\u6700\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u4ec5\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u5b9e\u73b0\u54c8\u5e0c\u8868\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5c06\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u7a7a\u4f4d\u79f0\u4e3a\u300c\u6876 bucket\u300d\uff0c\u6bcf\u4e2a\u6876\u53ef\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u56e0\u6b64\uff0c\u67e5\u8be2\u64cd\u4f5c\u5c31\u662f\u627e\u5230 <code>key</code> \u5bf9\u5e94\u7684\u6876\uff0c\u5e76\u5728\u6876\u4e2d\u83b7\u53d6 <code>value</code> \u3002</p> <p>\u90a3\u4e48\uff0c\u5982\u4f55\u57fa\u4e8e <code>key</code> \u5b9a\u4f4d\u5bf9\u5e94\u7684\u6876\u5462\uff1f\u8fd9\u662f\u901a\u8fc7\u300c\u54c8\u5e0c\u51fd\u6570 hash function\u300d\u5b9e\u73b0\u7684\u3002\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u8f93\u5165\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u8f93\u51fa\u7a7a\u95f4\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8f93\u5165\u7a7a\u95f4\u662f\u6240\u6709 <code>key</code> \uff0c\u8f93\u51fa\u7a7a\u95f4\u662f\u6240\u6709\u6876\uff08\u6570\u7ec4\u7d22\u5f15\uff09\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u8f93\u5165\u4e00\u4e2a <code>key</code> \uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u8be5 <code>key</code> \u5bf9\u5e94\u7684\u952e\u503c\u5bf9\u5728\u6570\u7ec4\u4e2d\u7684\u5b58\u50a8\u4f4d\u7f6e\u3002</p> <p>\u8f93\u5165\u4e00\u4e2a <code>key</code> \uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u8fc7\u7a0b\u5206\u4e3a\u4ee5\u4e0b\u4e24\u6b65\u3002</p> <ol> <li>\u901a\u8fc7\u67d0\u79cd\u54c8\u5e0c\u7b97\u6cd5 <code>hash()</code> \u8ba1\u7b97\u5f97\u5230\u54c8\u5e0c\u503c\u3002</li> <li>\u5c06\u54c8\u5e0c\u503c\u5bf9\u6876\u6570\u91cf\uff08\u6570\u7ec4\u957f\u5ea6\uff09<code>capacity</code> \u53d6\u6a21\uff0c\u4ece\u800c\u83b7\u53d6\u8be5 <code>key</code> \u5bf9\u5e94\u7684\u6570\u7ec4\u7d22\u5f15 <code>index</code> \u3002</li> </ol> <pre><code>index = hash(key) % capacity\n</code></pre> <p>\u968f\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5229\u7528 <code>index</code> \u5728\u54c8\u5e0c\u8868\u4e2d\u8bbf\u95ee\u5bf9\u5e94\u7684\u6876\uff0c\u4ece\u800c\u83b7\u53d6 <code>value</code> \u3002</p> <p>\u8bbe\u6570\u7ec4\u957f\u5ea6 <code>capacity = 100</code>\u3001\u54c8\u5e0c\u7b97\u6cd5 <code>hash(key) = key</code> \uff0c\u6613\u5f97\u54c8\u5e0c\u51fd\u6570\u4e3a <code>key % 100</code> \u3002\u56fe 6-2 \u4ee5 <code>key</code> \u5b66\u53f7\u548c <code>value</code> \u59d3\u540d\u4e3a\u4f8b\uff0c\u5c55\u793a\u4e86\u54c8\u5e0c\u51fd\u6570\u7684\u5de5\u4f5c\u539f\u7406\u3002</p> <p></p> <p> \u56fe 6-2 \u00a0 \u54c8\u5e0c\u51fd\u6570\u5de5\u4f5c\u539f\u7406 </p> <p>\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u4e2a\u7b80\u5355\u54c8\u5e0c\u8868\u3002\u5176\u4e2d\uff0c\u6211\u4eec\u5c06 <code>key</code> \u548c <code>value</code> \u5c01\u88c5\u6210\u4e00\u4e2a\u7c7b <code>Pair</code> \uff0c\u4ee5\u8868\u793a\u952e\u503c\u5bf9\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_hash_map.py<pre><code>class Pair:\n    \"\"\"\u952e\u503c\u5bf9\"\"\"\n\n    def __init__(self, key: int, val: str):\n        self.key = key\n        self.val = val\n\nclass ArrayHashMap:\n    \"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        # \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        self.buckets: list[Pair | None] = [None] * 100\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n        index = key % 100\n        return index\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n        index: int = self.hash_func(key)\n        pair: Pair = self.buckets[index]\n        if pair is None:\n            return None\n        return pair.val\n\n    def put(self, key: int, val: str):\n        \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n        pair = Pair(key, val)\n        index: int = self.hash_func(key)\n        self.buckets[index] = pair\n\n    def remove(self, key: int):\n        \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n        index: int = self.hash_func(key)\n        # \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\n        self.buckets[index] = None\n\n    def entry_set(self) -&gt; list[Pair]:\n        \"\"\"\u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\"\"\"\n        result: list[Pair] = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair)\n        return result\n\n    def key_set(self) -&gt; list[int]:\n        \"\"\"\u83b7\u53d6\u6240\u6709\u952e\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.key)\n        return result\n\n    def value_set(self) -&gt; list[str]:\n        \"\"\"\u83b7\u53d6\u6240\u6709\u503c\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.val)\n        return result\n\n    def print(self):\n        \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n        for pair in self.buckets:\n            if pair is not None:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> array_hash_map.cpp<pre><code>/* \u952e\u503c\u5bf9 */\nstruct Pair {\n  public:\n    int key;\n    string val;\n    Pair(int key, string val) {\n        this-&gt;key = key;\n        this-&gt;val = val;\n    }\n};\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n  private:\n    vector&lt;Pair *&gt; buckets;\n\n  public:\n    ArrayHashMap() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = vector&lt;Pair *&gt;(100);\n    }\n\n    ~ArrayHashMap() {\n        // \u91ca\u653e\u5185\u5b58\n        for (const auto &amp;bucket : buckets) {\n            delete bucket;\n        }\n        buckets.clear();\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    string get(int key) {\n        int index = hashFunc(key);\n        Pair *pair = buckets[index];\n        if (pair == nullptr)\n            return \"\";\n        return pair-&gt;val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, string val) {\n        Pair *pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        // \u91ca\u653e\u5185\u5b58\u5e76\u7f6e\u4e3a nullptr\n        delete buckets[index];\n        buckets[index] = nullptr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    vector&lt;Pair *&gt; pairSet() {\n        vector&lt;Pair *&gt; pairSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                pairSet.push_back(pair);\n            }\n        }\n        return pairSet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    vector&lt;int&gt; keySet() {\n        vector&lt;int&gt; keySet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                keySet.push_back(pair-&gt;key);\n            }\n        }\n        return keySet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    vector&lt;string&gt; valueSet() {\n        vector&lt;string&gt; valueSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                valueSet.push_back(pair-&gt;val);\n            }\n        }\n        return valueSet;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (Pair *kv : pairSet()) {\n            cout &lt;&lt; kv-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; kv-&gt;val &lt;&lt; endl;\n        }\n    }\n};\n</code></pre> array_hash_map.java<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair {\n    public int key;\n    public String val;\n\n    public Pair(int key, String val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private List&lt;Pair&gt; buckets;\n\n    public ArrayHashMap() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = new ArrayList&lt;&gt;();\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.add(null);\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public String get(int key) {\n        int index = hashFunc(key);\n        Pair pair = buckets.get(index);\n        if (pair == null)\n            return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void put(int key, String val) {\n        Pair pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets.set(index, pair);\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void remove(int key) {\n        int index = hashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets.set(index, null);\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    public List&lt;Pair&gt; pairSet() {\n        List&lt;Pair&gt; pairSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                pairSet.add(pair);\n        }\n        return pairSet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    public List&lt;Integer&gt; keySet() {\n        List&lt;Integer&gt; keySet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                keySet.add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    public List&lt;String&gt; valueSet() {\n        List&lt;String&gt; valueSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                valueSet.add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void print() {\n        for (Pair kv : pairSet()) {\n            System.out.println(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.cs<pre><code>/* \u952e\u503c\u5bf9 int-&gt;string */\nclass Pair(int key, string val) {\n    public int key = key;\n    public string val = val;\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    List&lt;Pair?&gt; buckets;\n    public ArrayHashMap() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = [];\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.Add(null);\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int HashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        Pair? pair = buckets[index];\n        if (pair == null) return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        Pair pair = new(key, val);\n        int index = HashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = null;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    public List&lt;Pair&gt; PairSet() {\n        List&lt;Pair&gt; pairSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                pairSet.Add(pair);\n        }\n        return pairSet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    public List&lt;int&gt; KeySet() {\n        List&lt;int&gt; keySet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                keySet.Add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    public List&lt;string&gt; ValueSet() {\n        List&lt;string&gt; valueSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                valueSet.Add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void Print() {\n        foreach (Pair kv in PairSet()) {\n            Console.WriteLine(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.go<pre><code>/* \u952e\u503c\u5bf9 */\ntype pair struct {\n    key int\n    val string\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntype arrayHashMap struct {\n    buckets []*pair\n}\n\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nfunc newArrayHashMap() *arrayHashMap {\n    // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n    buckets := make([]*pair, 100)\n    return &amp;arrayHashMap{buckets: buckets}\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (a *arrayHashMap) hashFunc(key int) int {\n    index := key % 100\n    return index\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (a *arrayHashMap) get(key int) string {\n    index := a.hashFunc(key)\n    pair := a.buckets[index]\n    if pair == nil {\n        return \"Not Found\"\n    }\n    return pair.val\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (a *arrayHashMap) put(key int, val string) {\n    pair := &amp;pair{key: key, val: val}\n    index := a.hashFunc(key)\n    a.buckets[index] = pair\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (a *arrayHashMap) remove(key int) {\n    index := a.hashFunc(key)\n    // \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\n    a.buckets[index] = nil\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e\u5bf9 */\nfunc (a *arrayHashMap) pairSet() []*pair {\n    var pairs []*pair\n    for _, pair := range a.buckets {\n        if pair != nil {\n            pairs = append(pairs, pair)\n        }\n    }\n    return pairs\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc (a *arrayHashMap) keySet() []int {\n    var keys []int\n    for _, pair := range a.buckets {\n        if pair != nil {\n            keys = append(keys, pair.key)\n        }\n    }\n    return keys\n}\n\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc (a *arrayHashMap) valueSet() []string {\n    var values []string\n    for _, pair := range a.buckets {\n        if pair != nil {\n            values = append(values, pair.val)\n        }\n    }\n    return values\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (a *arrayHashMap) print() {\n    for _, pair := range a.buckets {\n        if pair != nil {\n            fmt.Println(pair.key, \"-&gt;\", pair.val)\n        }\n    }\n}\n</code></pre> array_hash_map.swift<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair: Equatable {\n    public var key: Int\n    public var val: String\n\n    public init(key: Int, val: String) {\n        self.key = key\n        self.val = val\n    }\n\n    public static func == (lhs: Pair, rhs: Pair) -&gt; Bool {\n        lhs.key == rhs.key &amp;&amp; lhs.val == rhs.val\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private var buckets: [Pair?]\n\n    init() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = Array(repeating: nil, count: 100)\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private func hashFunc(key: Int) -&gt; Int {\n        let index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let pair = buckets[index]\n        return pair?.val\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        let pair = Pair(key: key, val: val)\n        let index = hashFunc(key: key)\n        buckets[index] = pair\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        // \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = nil\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    func pairSet() -&gt; [Pair] {\n        buckets.compactMap { $0 }\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    func keySet() -&gt; [Int] {\n        buckets.compactMap { $0?.key }\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    func valueSet() -&gt; [String] {\n        buckets.compactMap { $0?.val }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    func print() {\n        for pair in pairSet() {\n            Swift.print(\"\\(pair.key) -&gt; \\(pair.val)\")\n        }\n    }\n}\n</code></pre> array_hash_map.js<pre><code>/* \u952e\u503c\u5bf9 Number -&gt; String */\nclass Pair {\n    constructor(key, val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    #buckets;\n    constructor() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        this.#buckets = new Array(100).fill(null);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key) {\n        return key % 100;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key) {\n        let index = this.#hashFunc(key);\n        let pair = this.#buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    set(key, val) {\n        let index = this.#hashFunc(key);\n        this.#buckets[index] = new Pair(key, val);\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    delete(key) {\n        let index = this.#hashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        this.#buckets[index] = null;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    entries() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    keys() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    values() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.ts<pre><code>/* \u952e\u503c\u5bf9 Number -&gt; String */\nclass Pair {\n    public key: number;\n    public val: string;\n\n    constructor(key: number, val: string) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private readonly buckets: (Pair | null)[];\n\n    constructor() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        this.buckets = new Array(100).fill(null);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private hashFunc(key: number): number {\n        return key % 100;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public get(key: number): string | null {\n        let index = this.hashFunc(key);\n        let pair = this.buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public set(key: number, val: string) {\n        let index = this.hashFunc(key);\n        this.buckets[index] = new Pair(key, val);\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public delete(key: number) {\n        let index = this.hashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        this.buckets[index] = null;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    public entries(): (Pair | null)[] {\n        let arr: (Pair | null)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    public keys(): (number | undefined)[] {\n        let arr: (number | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    public values(): (string | undefined)[] {\n        let arr: (string | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.dart<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair {\n  int key;\n  String val;\n  Pair(this.key, this.val);\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n  late List&lt;Pair?&gt; _buckets;\n\n  ArrayHashMap() {\n    // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n    _buckets = List.filled(100, null);\n  }\n\n  /* \u54c8\u5e0c\u51fd\u6570 */\n  int _hashFunc(int key) {\n    final int index = key % 100;\n    return index;\n  }\n\n  /* \u67e5\u8be2\u64cd\u4f5c */\n  String? get(int key) {\n    final int index = _hashFunc(key);\n    final Pair? pair = _buckets[index];\n    if (pair == null) {\n      return null;\n    }\n    return pair.val;\n  }\n\n  /* \u6dfb\u52a0\u64cd\u4f5c */\n  void put(int key, String val) {\n    final Pair pair = Pair(key, val);\n    final int index = _hashFunc(key);\n    _buckets[index] = pair;\n  }\n\n  /* \u5220\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    final int index = _hashFunc(key);\n    _buckets[index] = null;\n  }\n\n  /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n  List&lt;Pair&gt; pairSet() {\n    List&lt;Pair&gt; pairSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        pairSet.add(pair);\n      }\n    }\n    return pairSet;\n  }\n\n  /* \u83b7\u53d6\u6240\u6709\u952e */\n  List&lt;int&gt; keySet() {\n    List&lt;int&gt; keySet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        keySet.add(pair.key);\n      }\n    }\n    return keySet;\n  }\n\n  /* \u83b7\u53d6\u6240\u6709\u503c */\n  List&lt;String&gt; values() {\n    List&lt;String&gt; valueSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        valueSet.add(pair.val);\n      }\n    }\n    return valueSet;\n  }\n\n  /* \u6253\u5370\u54c8\u5e0c\u8868 */\n  void printHashMap() {\n    for (final Pair kv in pairSet()) {\n      print(\"${kv.key} -&gt; ${kv.val}\");\n    }\n  }\n}\n</code></pre> array_hash_map.rs<pre><code>/* \u952e\u503c\u5bf9 */\n#[derive(Debug, Clone, PartialEq)]\npub struct Pair {\n    pub key: i32,\n    pub val: String,\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\npub struct ArrayHashMap {\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;,\n}\n\nimpl ArrayHashMap {\n    pub fn new() -&gt; ArrayHashMap {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        Self {\n            buckets: vec![None; 100],\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % 100\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    pub fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;String&gt; {\n        let index = self.hash_func(key);\n        self.buckets[index].as_ref().map(|pair| &amp;pair.val)\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    pub fn put(&amp;mut self, key: i32, val: &amp;str) {\n        let index = self.hash_func(key);\n        self.buckets[index] = Some(Pair {\n            key,\n            val: val.to_string(),\n        });\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    pub fn remove(&amp;mut self, key: i32) {\n        let index = self.hash_func(key);\n        // \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\n        self.buckets[index] = None;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    pub fn entry_set(&amp;self) -&gt; Vec&lt;&amp;Pair&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref())\n            .collect()\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    pub fn key_set(&amp;self) -&gt; Vec&lt;&amp;i32&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.key))\n            .collect()\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    pub fn value_set(&amp;self) -&gt; Vec&lt;&amp;String&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.val))\n            .collect()\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    pub fn print(&amp;self) {\n        for pair in self.entry_set() {\n            println!(\"{} -&gt; {}\", pair.key, pair.val);\n        }\n    }\n}\n</code></pre> array_hash_map.c<pre><code>/* \u952e\u503c\u5bf9 int-&gt;string */\ntypedef struct {\n    int key;\n    char *val;\n} Pair;\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntypedef struct {\n    Pair *buckets[HASHTABLE_CAPACITY];\n} ArrayHashMap;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayHashMap *newArrayHashMap() {\n    ArrayHashMap *hmap = malloc(sizeof(ArrayHashMap));\n    return hmap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayHashMap(ArrayHashMap *hmap) {\n    for (int i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            free(hmap-&gt;buckets[i]-&gt;val);\n            free(hmap-&gt;buckets[i]);\n        }\n    }\n    free(hmap);\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(ArrayHashMap *hmap, const int key, const char *val) {\n    Pair *Pair = malloc(sizeof(Pair));\n    Pair-&gt;key = key;\n    Pair-&gt;val = malloc(strlen(val) + 1);\n    strcpy(Pair-&gt;val, val);\n\n    int index = hashFunc(key);\n    hmap-&gt;buckets[index] = Pair;\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(ArrayHashMap *hmap, const int key) {\n    int index = hashFunc(key);\n    free(hmap-&gt;buckets[index]-&gt;val);\n    free(hmap-&gt;buckets[index]);\n    hmap-&gt;buckets[index] = NULL;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nvoid pairSet(ArrayHashMap *hmap, MapSet *set) {\n    Pair *entries;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    entries = malloc(sizeof(Pair) * total);\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            entries[index].key = hmap-&gt;buckets[i]-&gt;key;\n            entries[index].val = malloc(strlen(hmap-&gt;buckets[i]-&gt;val) + 1);\n            strcpy(entries[index].val, hmap-&gt;buckets[i]-&gt;val);\n            index++;\n        }\n    }\n    set-&gt;set = entries;\n    set-&gt;len = total;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e */\nvoid keySet(ArrayHashMap *hmap, MapSet *set) {\n    int *keys;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    keys = malloc(total * sizeof(int));\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            keys[index] = hmap-&gt;buckets[i]-&gt;key;\n            index++;\n        }\n    }\n    set-&gt;set = keys;\n    set-&gt;len = total;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvoid valueSet(ArrayHashMap *hmap, MapSet *set) {\n    char **vals;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    vals = malloc(total * sizeof(char *));\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            vals[index] = hmap-&gt;buckets[i]-&gt;val;\n            index++;\n        }\n    }\n    set-&gt;set = vals;\n    set-&gt;len = total;\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(ArrayHashMap *hmap) {\n    int i;\n    MapSet set;\n    pairSet(hmap, &amp;set);\n    Pair *entries = (Pair *)set.set;\n    for (i = 0; i &lt; set.len; i++) {\n        printf(\"%d -&gt; %s\\n\", entries[i].key, entries[i].val);\n    }\n    free(set.set);\n}\n</code></pre> array_hash_map.kt<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair(\n    var key: Int,\n    var value: String\n)\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private val buckets = arrayOfNulls&lt;Pair&gt;(100)\n\n    init {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        for (i in 0..&lt;100) {\n            buckets[i] = null\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        val index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val pair = buckets[index] ?: return null\n        return pair.value\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        val pair = Pair(key, value)\n        val index = hashFunc(key)\n        buckets[index] = pair\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = null\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    fun pairSet(): MutableList&lt;Pair&gt; {\n        val pairSet = ArrayList&lt;Pair&gt;()\n        for (pair in buckets) {\n            if (pair != null) pairSet.add(pair)\n        }\n        return pairSet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    fun keySet(): MutableList&lt;Int&gt; {\n        val keySet = ArrayList&lt;Int&gt;()\n        for (pair in buckets) {\n            if (pair != null) keySet.add(pair.key)\n        }\n        return keySet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    fun valueSet(): MutableList&lt;String&gt; {\n        val valueSet = ArrayList&lt;String&gt;()\n        for (pair in buckets) {\n            pair?.let { valueSet.add(it.value) }\n        }\n        return valueSet\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (kv in pairSet()) {\n            val key = kv.key\n            val value = kv.value\n            println(\"${key}-&gt;${value}\")\n        }\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private val buckets = arrayOfNulls&lt;Pair&gt;(100)\n\n    init {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        for (i in 0..&lt;100) {\n            buckets[i] = null\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        val index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val pair = buckets[index] ?: return null\n        return pair.value\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        val pair = Pair(key, value)\n        val index = hashFunc(key)\n        buckets[index] = pair\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = null\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    fun pairSet(): MutableList&lt;Pair&gt; {\n        val pairSet = ArrayList&lt;Pair&gt;()\n        for (pair in buckets) {\n            if (pair != null) pairSet.add(pair)\n        }\n        return pairSet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    fun keySet(): MutableList&lt;Int&gt; {\n        val keySet = ArrayList&lt;Int&gt;()\n        for (pair in buckets) {\n            if (pair != null) keySet.add(pair.key)\n        }\n        return keySet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    fun valueSet(): MutableList&lt;String&gt; {\n        val valueSet = ArrayList&lt;String&gt;()\n        for (pair in buckets) {\n            pair?.let { valueSet.add(it.value) }\n        }\n        return valueSet\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (kv in pairSet()) {\n            val key = kv.key\n            val value = kv.value\n            println(\"${key}-&gt;${value}\")\n        }\n    }\n}\n</code></pre> array_hash_map.rb<pre><code>[class]{Pair}-[func]{}\n\n[class]{ArrayHashMap}-[func]{}\n</code></pre> array_hash_map.zig<pre><code>// \u952e\u503c\u5bf9\nconst Pair = struct {\n    key: usize = undefined,\n    val: []const u8 = undefined,\n\n   pub fn init(key: usize, val: []const u8) Pair {\n        return Pair {\n            .key = key,\n            .val = val,\n        };\n    }\n};\n\n// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\nfn ArrayHashMap(comptime T: type) type {\n    return struct {\n        bucket: ?std.ArrayList(?T) = null,\n        mem_allocator: std.mem.Allocator = undefined,\n\n        const Self = @This();\n\n        // \u6784\u9020\u51fd\u6570\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            self.mem_allocator = allocator;\n            // \u521d\u59cb\u5316\u4e00\u4e2a\u957f\u5ea6\u4e3a 100 \u7684\u6876\uff08\u6570\u7ec4\uff09\n            self.bucket = std.ArrayList(?T).init(self.mem_allocator);\n            var i: i32 = 0;\n            while (i &lt; 100) : (i += 1) {\n                try self.bucket.?.append(null);\n            }\n        }\n\n        // \u6790\u6784\u51fd\u6570\n        pub fn deinit(self: *Self) void {\n            if (self.bucket != null) self.bucket.?.deinit();\n        }\n\n        // \u54c8\u5e0c\u51fd\u6570\n        fn hashFunc(key: usize) usize {\n            var index = key % 100;\n            return index;\n        }\n\n        // \u67e5\u8be2\u64cd\u4f5c\n        pub fn get(self: *Self, key: usize) []const u8 {\n            var index = hashFunc(key);\n            var pair = self.bucket.?.items[index];\n            return pair.?.val;\n        }\n\n        // \u6dfb\u52a0\u64cd\u4f5c\n        pub fn put(self: *Self, key: usize, val: []const u8) !void {\n            var pair = Pair.init(key, val);\n            var index = hashFunc(key);\n            self.bucket.?.items[index] = pair;\n        }\n\n        // \u5220\u9664\u64cd\u4f5c\n        pub fn remove(self: *Self, key: usize) !void {\n            var index = hashFunc(key);\n            // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n            self.bucket.?.items[index] = null;\n        }       \n\n        // \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\n        pub fn pairSet(self: *Self) !std.ArrayList(T) {\n            var entry_set = std.ArrayList(T).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try entry_set.append(item.?);\n            }\n            return entry_set;\n        }  \n\n        // \u83b7\u53d6\u6240\u6709\u952e\n        pub fn keySet(self: *Self) !std.ArrayList(usize) {\n            var key_set = std.ArrayList(usize).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try key_set.append(item.?.key);\n            }\n            return key_set;\n        }  \n\n        // \u83b7\u53d6\u6240\u6709\u503c\n        pub fn valueSet(self: *Self) !std.ArrayList([]const u8) {\n            var value_set = std.ArrayList([]const u8).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try value_set.append(item.?.val);\n            }\n            return value_set;\n        }\n\n        // \u6253\u5370\u54c8\u5e0c\u8868\n        pub fn print(self: *Self) !void {\n            var entry_set = try self.pairSet();\n            defer entry_set.deinit();\n            for (entry_set.items) |item| {\n                std.debug.print(\"{} -&gt; {s}\\n\", .{item.key, item.val});\n            }\n        }\n    };\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_hashing/hash_map/#613","title":"6.1.3 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u4e0e\u6269\u5bb9","text":"<p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u6240\u6709 <code>key</code> \u6784\u6210\u7684\u8f93\u5165\u7a7a\u95f4\u6620\u5c04\u5230\u6570\u7ec4\u6240\u6709\u7d22\u5f15\u6784\u6210\u7684\u8f93\u51fa\u7a7a\u95f4\uff0c\u800c\u8f93\u5165\u7a7a\u95f4\u5f80\u5f80\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7406\u8bba\u4e0a\u4e00\u5b9a\u5b58\u5728\u201c\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u76f8\u540c\u8f93\u51fa\u201d\u7684\u60c5\u51b5\u3002</p> <p>\u5bf9\u4e8e\u4e0a\u8ff0\u793a\u4f8b\u4e2d\u7684\u54c8\u5e0c\u51fd\u6570\uff0c\u5f53\u8f93\u5165\u7684 <code>key</code> \u540e\u4e24\u4f4d\u76f8\u540c\u65f6\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u51fa\u7ed3\u679c\u4e5f\u76f8\u540c\u3002\u4f8b\u5982\uff0c\u67e5\u8be2\u5b66\u53f7\u4e3a 12836 \u548c 20336 \u7684\u4e24\u4e2a\u5b66\u751f\u65f6\uff0c\u6211\u4eec\u5f97\u5230\uff1a</p> <pre><code>12836 % 100 = 36\n20336 % 100 = 36\n</code></pre> <p>\u5982\u56fe 6-3 \u6240\u793a\uff0c\u4e24\u4e2a\u5b66\u53f7\u6307\u5411\u4e86\u540c\u4e00\u4e2a\u59d3\u540d\uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002\u6211\u4eec\u5c06\u8fd9\u79cd\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u540c\u4e00\u8f93\u51fa\u7684\u60c5\u51b5\u79f0\u4e3a\u300c\u54c8\u5e0c\u51b2\u7a81 hash collision\u300d\u3002</p> <p></p> <p> \u56fe 6-3 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u793a\u4f8b </p> <p>\u5bb9\u6613\u60f3\u5230\uff0c\u54c8\u5e0c\u8868\u5bb9\u91cf \\(n\\) \u8d8a\u5927\uff0c\u591a\u4e2a <code>key</code> \u88ab\u5206\u914d\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\uff0c\u51b2\u7a81\u5c31\u8d8a\u5c11\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002</p> <p>\u5982\u56fe 6-4 \u6240\u793a\uff0c\u6269\u5bb9\u524d\u952e\u503c\u5bf9 <code>(136, A)</code> \u548c <code>(236, D)</code> \u53d1\u751f\u51b2\u7a81\uff0c\u6269\u5bb9\u540e\u51b2\u7a81\u6d88\u5931\u3002</p> <p></p> <p> \u56fe 6-4 \u00a0 \u54c8\u5e0c\u8868\u6269\u5bb9 </p> <p>\u7c7b\u4f3c\u4e8e\u6570\u7ec4\u6269\u5bb9\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u5c06\u6240\u6709\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u8fc1\u79fb\u81f3\u65b0\u54c8\u5e0c\u8868\uff0c\u975e\u5e38\u8017\u65f6\uff1b\u5e76\u4e14\u7531\u4e8e\u54c8\u5e0c\u8868\u5bb9\u91cf <code>capacity</code> \u6539\u53d8\uff0c\u6211\u4eec\u9700\u8981\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u6765\u91cd\u65b0\u8ba1\u7b97\u6240\u6709\u952e\u503c\u5bf9\u7684\u5b58\u50a8\u4f4d\u7f6e\uff0c\u8fd9\u8fdb\u4e00\u6b65\u589e\u52a0\u4e86\u6269\u5bb9\u8fc7\u7a0b\u7684\u8ba1\u7b97\u5f00\u9500\u3002\u4e3a\u6b64\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u9884\u7559\u8db3\u591f\u5927\u7684\u54c8\u5e0c\u8868\u5bb9\u91cf\uff0c\u9632\u6b62\u9891\u7e41\u6269\u5bb9\u3002</p> <p>\u300c\u8d1f\u8f7d\u56e0\u5b50 load factor\u300d\u662f\u54c8\u5e0c\u8868\u7684\u4e00\u4e2a\u91cd\u8981\u6982\u5ff5\uff0c\u5176\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u7684\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u7528\u4e8e\u8861\u91cf\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u4e5f\u5e38\u4f5c\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u89e6\u53d1\u6761\u4ef6\u3002\u4f8b\u5982\u5728 Java \u4e2d\uff0c\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(0.75\\) \u65f6\uff0c\u7cfb\u7edf\u4f1a\u5c06\u54c8\u5e0c\u8868\u6269\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002</p>"},{"location":"chapter_hashing/summary/","title":"6.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_hashing/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u8f93\u5165 <code>key</code> \uff0c\u54c8\u5e0c\u8868\u80fd\u591f\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u67e5\u8be2\u5230 <code>value</code> \uff0c\u6548\u7387\u975e\u5e38\u9ad8\u3002</li> <li>\u5e38\u89c1\u7684\u54c8\u5e0c\u8868\u64cd\u4f5c\u5305\u62ec\u67e5\u8be2\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u3001\u5220\u9664\u952e\u503c\u5bf9\u548c\u904d\u5386\u54c8\u5e0c\u8868\u7b49\u3002</li> <li>\u54c8\u5e0c\u51fd\u6570\u5c06 <code>key</code> \u6620\u5c04\u4e3a\u6570\u7ec4\u7d22\u5f15\uff0c\u4ece\u800c\u8bbf\u95ee\u5bf9\u5e94\u6876\u5e76\u83b7\u53d6 <code>value</code> \u3002</li> <li>\u4e24\u4e2a\u4e0d\u540c\u7684 <code>key</code> \u53ef\u80fd\u5728\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u540e\u5f97\u5230\u76f8\u540c\u7684\u6570\u7ec4\u7d22\u5f15\uff0c\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u51fa\u9519\uff0c\u8fd9\u79cd\u73b0\u8c61\u88ab\u79f0\u4e3a\u54c8\u5e0c\u51b2\u7a81\u3002</li> <li>\u54c8\u5e0c\u8868\u5bb9\u91cf\u8d8a\u5927\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002\u56e0\u6b64\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\u3002\u4e0e\u6570\u7ec4\u6269\u5bb9\u7c7b\u4f3c\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u64cd\u4f5c\u7684\u5f00\u9500\u5f88\u5927\u3002</li> <li>\u8d1f\u8f7d\u56e0\u5b50\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u4e2d\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u53cd\u6620\u4e86\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u5e38\u7528\u4f5c\u89e6\u53d1\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u6761\u4ef6\u3002</li> <li>\u94fe\u5f0f\u5730\u5740\u901a\u8fc7\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u5316\u4e3a\u94fe\u8868\uff0c\u5c06\u6240\u6709\u51b2\u7a81\u5143\u7d20\u5b58\u50a8\u5728\u540c\u4e00\u4e2a\u94fe\u8868\u4e2d\u3002\u7136\u800c\uff0c\u94fe\u8868\u8fc7\u957f\u4f1a\u964d\u4f4e\u67e5\u8be2\u6548\u7387\uff0c\u53ef\u4ee5\u901a\u8fc7\u8fdb\u4e00\u6b65\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6765\u63d0\u9ad8\u6548\u7387\u3002</li> <li>\u5f00\u653e\u5bfb\u5740\u901a\u8fc7\u591a\u6b21\u63a2\u6d4b\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\u3002\u7ebf\u6027\u63a2\u6d4b\u4f7f\u7528\u56fa\u5b9a\u6b65\u957f\uff0c\u7f3a\u70b9\u662f\u4e0d\u80fd\u5220\u9664\u5143\u7d20\uff0c\u4e14\u5bb9\u6613\u4ea7\u751f\u805a\u96c6\u3002\u591a\u6b21\u54c8\u5e0c\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u8fdb\u884c\u63a2\u6d4b\uff0c\u76f8\u8f83\u7ebf\u6027\u63a2\u6d4b\u66f4\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u589e\u52a0\u4e86\u8ba1\u7b97\u91cf\u3002</li> <li>\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u54c8\u5e0c\u8868\u5b9e\u73b0\u3002\u4f8b\u5982\uff0cJava \u7684 <code>HashMap</code> \u4f7f\u7528\u94fe\u5f0f\u5730\u5740\uff0c\u800c Python \u7684 <code>Dict</code> \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002</li> <li>\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5e0c\u671b\u54c8\u5e0c\u7b97\u6cd5\u5177\u6709\u786e\u5b9a\u6027\u3001\u9ad8\u6548\u7387\u548c\u5747\u5300\u5206\u5e03\u7684\u7279\u70b9\u3002\u5728\u5bc6\u7801\u5b66\u4e2d\uff0c\u54c8\u5e0c\u7b97\u6cd5\u8fd8\u5e94\u8be5\u5177\u5907\u6297\u78b0\u649e\u6027\u548c\u96ea\u5d29\u6548\u5e94\u3002</li> <li>\u54c8\u5e0c\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u5747\u5300\u5206\u5e03\uff0c\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002</li> <li>\u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5\u5305\u62ec MD5\u3001SHA-1\u3001SHA-2 \u548c SHA-3 \u7b49\u3002MD5 \u5e38\u7528\u4e8e\u6821\u9a8c\u6587\u4ef6\u5b8c\u6574\u6027\uff0cSHA-2 \u5e38\u7528\u4e8e\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u3002</li> <li>\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u662f\u53ef\u54c8\u5e0c\u7684\u3002</li> </ul>"},{"location":"chapter_hashing/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u662f \\(O(n)\\) \uff1f</p> <p>\u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u9000\u5316\u81f3 \\(O(n)\\) \u3002\u5f53\u54c8\u5e0c\u51fd\u6570\u8bbe\u8ba1\u5f97\u6bd4\u8f83\u597d\u3001\u5bb9\u91cf\u8bbe\u7f6e\u6bd4\u8f83\u5408\u7406\u3001\u51b2\u7a81\u6bd4\u8f83\u5e73\u5747\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u6211\u4eec\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u54c8\u5e0c\u8868\u65f6\uff0c\u901a\u5e38\u8ba4\u4e3a\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002</p> <p>Q\uff1a\u4e3a\u4ec0\u4e48\u4e0d\u4f7f\u7528\u54c8\u5e0c\u51fd\u6570 \\(f(x) = x\\) \u5462\uff1f\u8fd9\u6837\u5c31\u4e0d\u4f1a\u6709\u51b2\u7a81\u4e86\u3002</p> <p>\u5728 \\(f(x) = x\\) \u54c8\u5e0c\u51fd\u6570\u4e0b\uff0c\u6bcf\u4e2a\u5143\u7d20\u5bf9\u5e94\u552f\u4e00\u7684\u6876\u7d22\u5f15\uff0c\u8fd9\u4e0e\u6570\u7ec4\u7b49\u4ef7\u3002\u7136\u800c\uff0c\u8f93\u5165\u7a7a\u95f4\u901a\u5e38\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff08\u6570\u7ec4\u957f\u5ea6\uff09\uff0c\u56e0\u6b64\u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u54c8\u5e0c\u8868\u7684\u76ee\u6807\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u72b6\u6001\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u7a7a\u95f4\uff0c\u5e76\u63d0\u4f9b \\(O(1)\\) \u7684\u67e5\u8be2\u6548\u7387\u3002</p> <p>Q\uff1a\u54c8\u5e0c\u8868\u5e95\u5c42\u5b9e\u73b0\u662f\u6570\u7ec4\u3001\u94fe\u8868\u3001\u4e8c\u53c9\u6811\uff0c\u4f46\u4e3a\u4ec0\u4e48\u6548\u7387\u53ef\u4ee5\u6bd4\u5b83\u4eec\u66f4\u9ad8\u5462\uff1f</p> <p>\u9996\u5148\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\uff0c\u4f46\u7a7a\u95f4\u6548\u7387\u53d8\u4f4e\u4e86\u3002\u54c8\u5e0c\u8868\u6709\u76f8\u5f53\u4e00\u90e8\u5206\u5185\u5b58\u672a\u4f7f\u7528\u3002</p> <p>\u5176\u6b21\uff0c\u53ea\u662f\u5728\u7279\u5b9a\u4f7f\u7528\u573a\u666f\u4e0b\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\u4e86\u3002\u5982\u679c\u4e00\u4e2a\u529f\u80fd\u80fd\u591f\u5728\u76f8\u540c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u4f7f\u7528\u6570\u7ec4\u6216\u94fe\u8868\u5b9e\u73b0\uff0c\u90a3\u4e48\u901a\u5e38\u6bd4\u54c8\u5e0c\u8868\u66f4\u5feb\u3002\u8fd9\u662f\u56e0\u4e3a\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u9700\u8981\u5f00\u9500\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u6570\u9879\u66f4\u5927\u3002</p> <p>\u6700\u540e\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u80fd\u53d1\u751f\u52a3\u5316\u3002\u4f8b\u5982\u5728\u94fe\u5f0f\u5730\u5740\u4e2d\uff0c\u6211\u4eec\u91c7\u53d6\u5728\u94fe\u8868\u6216\u7ea2\u9ed1\u6811\u4e2d\u6267\u884c\u67e5\u627e\u64cd\u4f5c\uff0c\u4ecd\u7136\u6709\u9000\u5316\u81f3 \\(O(n)\\) \u65f6\u95f4\u7684\u98ce\u9669\u3002</p> <p>Q\uff1a\u591a\u6b21\u54c8\u5e0c\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\u5417\uff1f\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u8fd8\u80fd\u518d\u6b21\u4f7f\u7528\u5417\uff1f</p> <p>\u591a\u6b21\u54c8\u5e0c\u662f\u5f00\u653e\u5bfb\u5740\u7684\u4e00\u79cd\uff0c\u5f00\u653e\u5bfb\u5740\u6cd5\u90fd\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\uff0c\u9700\u8981\u901a\u8fc7\u6807\u8bb0\u5220\u9664\u3002\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u53ef\u4ee5\u518d\u6b21\u4f7f\u7528\u3002\u5f53\u5c06\u65b0\u5143\u7d20\u63d2\u5165\u54c8\u5e0c\u8868\uff0c\u5e76\u4e14\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u4f4d\u7f6e\u65f6\uff0c\u8be5\u4f4d\u7f6e\u53ef\u4ee5\u88ab\u65b0\u5143\u7d20\u4f7f\u7528\u3002\u8fd9\u6837\u505a\u65e2\u80fd\u4fdd\u6301\u54c8\u5e0c\u8868\u7684\u63a2\u6d4b\u5e8f\u5217\u4e0d\u53d8\uff0c\u53c8\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u7684\u7a7a\u95f4\u4f7f\u7528\u7387\u3002</p> <p>Q\uff1a\u4e3a\u4ec0\u4e48\u5728\u7ebf\u6027\u63a2\u6d4b\u4e2d\uff0c\u67e5\u627e\u5143\u7d20\u7684\u65f6\u5019\u4f1a\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u5462\uff1f</p> <p>\u67e5\u627e\u7684\u65f6\u5019\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u5bf9\u5e94\u7684\u6876\u548c\u952e\u503c\u5bf9\uff0c\u53d1\u73b0 <code>key</code> \u4e0d\u5339\u914d\uff0c\u8fd9\u5c31\u4ee3\u8868\u6709\u54c8\u5e0c\u51b2\u7a81\u3002\u56e0\u6b64\uff0c\u7ebf\u6027\u63a2\u6d4b\u6cd5\u4f1a\u6839\u636e\u9884\u5148\u8bbe\u5b9a\u7684\u6b65\u957f\u4f9d\u6b21\u5411\u4e0b\u67e5\u627e\uff0c\u76f4\u81f3\u627e\u5230\u6b63\u786e\u7684\u952e\u503c\u5bf9\u6216\u65e0\u6cd5\u627e\u5230\u8df3\u51fa\u4e3a\u6b62\u3002</p> <p>Q\uff1a\u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u6269\u5bb9\u80fd\u591f\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\uff1f</p> <p>\u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d6\u6a21\uff08\u53d6\u4f59\uff09\uff0c\u8ba9\u8f93\u51fa\u503c\u843d\u5728\u6570\u7ec4\u7d22\u5f15\u8303\u56f4\u5185\uff1b\u5728\u6269\u5bb9\u540e\uff0c\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d1\u751f\u53d8\u5316\uff0c\u800c <code>key</code> \u5bf9\u5e94\u7684\u7d22\u5f15\u4e5f\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002\u539f\u5148\u843d\u5728\u540c\u4e00\u4e2a\u6876\u7684\u591a\u4e2a <code>key</code> \uff0c\u5728\u6269\u5bb9\u540e\u53ef\u80fd\u4f1a\u88ab\u5206\u914d\u5230\u591a\u4e2a\u6876\u4e2d\uff0c\u4ece\u800c\u5b9e\u73b0\u54c8\u5e0c\u51b2\u7a81\u7684\u7f13\u89e3\u3002</p>"},{"location":"chapter_heap/","title":"\u7b2c 8 \u7ae0 \u00a0 \u5806","text":"<p>Abstract</p> <p>\u5806\u5c31\u50cf\u662f\u5c71\u5cb3\u5cf0\u5ce6\uff0c\u5c42\u53e0\u8d77\u4f0f\u3001\u5f62\u6001\u5404\u5f02\u3002</p> <p>\u5ea7\u5ea7\u5c71\u5cf0\u9ad8\u4f4e\u9519\u843d\uff0c\u800c\u6700\u9ad8\u7684\u5c71\u5cf0\u603b\u662f\u6700\u5148\u6620\u5165\u773c\u5e18\u3002</p>"},{"location":"chapter_heap/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>8.1 \u00a0 \u5806</li> <li>8.2 \u00a0 \u5efa\u5806\u64cd\u4f5c</li> <li>8.3 \u00a0 Top-k \u95ee\u9898</li> <li>8.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_heap/build_heap/","title":"8.2 \u00a0 \u5efa\u5806\u64cd\u4f5c","text":"<p>\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u4f7f\u7528\u4e00\u4e2a\u5217\u8868\u7684\u6240\u6709\u5143\u7d20\u6765\u6784\u5efa\u4e00\u4e2a\u5806\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u88ab\u79f0\u4e3a\u201c\u5efa\u5806\u64cd\u4f5c\u201d\u3002</p>"},{"location":"chapter_heap/build_heap/#821","title":"8.2.1 \u00a0 \u501f\u52a9\u5165\u5806\u64cd\u4f5c\u5b9e\u73b0","text":"<p>\u6211\u4eec\u9996\u5148\u521b\u5efa\u4e00\u4e2a\u7a7a\u5806\uff0c\u7136\u540e\u904d\u5386\u5217\u8868\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e2a\u5143\u7d20\u6267\u884c\u201c\u5165\u5806\u64cd\u4f5c\u201d\uff0c\u5373\u5148\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u5806\u7684\u5c3e\u90e8\uff0c\u518d\u5bf9\u8be5\u5143\u7d20\u6267\u884c\u201c\u4ece\u5e95\u81f3\u9876\u201d\u5806\u5316\u3002</p> <p>\u6bcf\u5f53\u4e00\u4e2a\u5143\u7d20\u5165\u5806\uff0c\u5806\u7684\u957f\u5ea6\u5c31\u52a0\u4e00\u3002\u7531\u4e8e\u8282\u70b9\u662f\u4ece\u9876\u5230\u5e95\u4f9d\u6b21\u88ab\u6dfb\u52a0\u8fdb\u4e8c\u53c9\u6811\u7684\uff0c\u56e0\u6b64\u5806\u662f\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u6784\u5efa\u7684\u3002</p> <p>\u8bbe\u5143\u7d20\u6570\u91cf\u4e3a \\(n\\) \uff0c\u6bcf\u4e2a\u5143\u7d20\u7684\u5165\u5806\u64cd\u4f5c\u4f7f\u7528 \\(O(\\log{n})\\) \u65f6\u95f4\uff0c\u56e0\u6b64\u8be5\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p>"},{"location":"chapter_heap/build_heap/#822","title":"8.2.2 \u00a0 \u901a\u8fc7\u904d\u5386\u5806\u5316\u5b9e\u73b0","text":"<p>\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9e\u73b0\u4e00\u79cd\u66f4\u4e3a\u9ad8\u6548\u7684\u5efa\u5806\u65b9\u6cd5\uff0c\u5171\u5206\u4e3a\u4e24\u6b65\u3002</p> <ol> <li>\u5c06\u5217\u8868\u6240\u6709\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u5730\u6dfb\u52a0\u5230\u5806\u4e2d\uff0c\u6b64\u65f6\u5806\u7684\u6027\u8d28\u5c1a\u672a\u5f97\u5230\u6ee1\u8db3\u3002</li> <li>\u5012\u5e8f\u904d\u5386\u5806\uff08\u5c42\u5e8f\u904d\u5386\u7684\u5012\u5e8f\uff09\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e2a\u975e\u53f6\u8282\u70b9\u6267\u884c\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u3002</li> </ol> <p>\u6bcf\u5f53\u5806\u5316\u4e00\u4e2a\u8282\u70b9\u540e\uff0c\u4ee5\u8be5\u8282\u70b9\u4e3a\u6839\u8282\u70b9\u7684\u5b50\u6811\u5c31\u5f62\u6210\u4e00\u4e2a\u5408\u6cd5\u7684\u5b50\u5806\u3002\u800c\u7531\u4e8e\u662f\u5012\u5e8f\u904d\u5386\uff0c\u56e0\u6b64\u5806\u662f\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u6784\u5efa\u7684\u3002</p> <p>\u4e4b\u6240\u4ee5\u9009\u62e9\u5012\u5e8f\u904d\u5386\uff0c\u662f\u56e0\u4e3a\u8fd9\u6837\u80fd\u591f\u4fdd\u8bc1\u5f53\u524d\u8282\u70b9\u4e4b\u4e0b\u7684\u5b50\u6811\u5df2\u7ecf\u662f\u5408\u6cd5\u7684\u5b50\u5806\uff0c\u8fd9\u6837\u5806\u5316\u5f53\u524d\u8282\u70b9\u624d\u662f\u6709\u6548\u7684\u3002</p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u7531\u4e8e\u53f6\u8282\u70b9\u6ca1\u6709\u5b50\u8282\u70b9\uff0c\u56e0\u6b64\u5b83\u4eec\u5929\u7136\u5c31\u662f\u5408\u6cd5\u7684\u5b50\u5806\uff0c\u65e0\u987b\u5806\u5316\u3002\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u6700\u540e\u4e00\u4e2a\u975e\u53f6\u8282\u70b9\u662f\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u7684\u7236\u8282\u70b9\uff0c\u6211\u4eec\u4ece\u5b83\u5f00\u59cb\u5012\u5e8f\u904d\u5386\u5e76\u6267\u884c\u5806\u5316\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def __init__(self, nums: list[int]):\n    \"\"\"\u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\"\"\"\n    # \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    self.max_heap = nums\n    # \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in range(self.parent(self.size() - 1), -1, -1):\n        self.sift_down(i)\n</code></pre> my_heap.cpp<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(vector&lt;int&gt; nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = nums;\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List&lt;Integer&gt; nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = new ArrayList&lt;&gt;(nums);\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(IEnumerable&lt;int&gt; nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = new List&lt;int&gt;(nums);\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    var size = Parent(this.Size() - 1);\n    for (int i = size; i &gt;= 0; i--) {\n        SiftDown(i);\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nfunc newMaxHeap(nums []any) *maxHeap {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    h := &amp;maxHeap{data: nums}\n    for i := h.parent(len(h.data) - 1); i &gt;= 0; i-- {\n        // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n        h.siftDown(i)\n    }\n    return h\n}\n</code></pre> my_heap.swift<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\ninit(nums: [Int]) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = nums\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in (0 ... parent(i: size() - 1)).reversed() {\n        siftDown(i: i)\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    this.#maxHeap = nums === undefined ? [] : [...nums];\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = this.#parent(this.size() - 1); i &gt;= 0; i--) {\n        this.#siftDown(i);\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums?: number[]) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    this.maxHeap = nums === undefined ? [] : [...nums];\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = this.parent(this.size() - 1); i &gt;= 0; i--) {\n        this.siftDown(i);\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List&lt;int&gt; nums) {\n  // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n  _maxHeap = nums;\n  // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n  for (int i = _parent(size() - 1); i &gt;= 0; i--) {\n    siftDown(i);\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nfn new(nums: Vec&lt;i32&gt;) -&gt; Self {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    let mut heap = MaxHeap { max_heap: nums };\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in (0..=Self::parent(heap.size() - 1)).rev() {\n        heap.sift_down(i);\n    }\n    heap\n}\n</code></pre> my_heap.c<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nMaxHeap *newMaxHeap(int nums[], int size) {\n    // \u6240\u6709\u5143\u7d20\u5165\u5806\n    MaxHeap *maxHeap = (MaxHeap *)malloc(sizeof(MaxHeap));\n    maxHeap-&gt;size = size;\n    memcpy(maxHeap-&gt;data, nums, size * sizeof(int));\n    for (int i = parent(maxHeap, size - 1); i &gt;= 0; i--) {\n        // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n        siftDown(maxHeap, i);\n    }\n    return maxHeap;\n}\n</code></pre> my_heap.kt<pre><code>/* \u5927\u9876\u5806 */\nclass MaxHeap(nums: List&lt;Int&gt;?) {\n    // \u4f7f\u7528\u5217\u8868\u800c\u975e\u6570\u7ec4\uff0c\u8fd9\u6837\u65e0\u987b\u8003\u8651\u6269\u5bb9\u95ee\u9898\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    private val maxHeap = ArrayList(nums!!)\n\n    /* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\n    init {\n        // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n        for (i in parent(size() - 1) downTo 0) {\n            siftDown(i)\n        }\n    }\n\n    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    private fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    private fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    private fun parent(i: Int): Int {\n        return (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u4ea4\u6362\u5143\u7d20 */\n    private fun swap(i: Int, j: Int) {\n        maxHeap[i] = maxHeap[j].also { maxHeap[j] = maxHeap[i] }\n    }\n\n    /* \u83b7\u53d6\u5806\u5927\u5c0f */\n    fun size(): Int {\n        return maxHeap.size\n    }\n\n    /* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        /* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\n        return size() == 0\n    }\n\n    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\n    fun peek(): Int {\n        return maxHeap[0]\n    }\n\n    /* \u5143\u7d20\u5165\u5806 */\n    fun push(value: Int) {\n        // \u6dfb\u52a0\u8282\u70b9\n        maxHeap.add(value)\n        // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n        siftUp(size() - 1)\n    }\n\n    /* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n    private fun siftUp(it: Int) {\n        // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n        var i = it\n        while (true) {\n            // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n            val p = parent(i)\n            // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n            if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n            // \u4ea4\u6362\u4e24\u8282\u70b9\n            swap(i, p)\n            // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n            i = p\n        }\n    }\n\n    /* \u5143\u7d20\u51fa\u5806 */\n    fun pop(): Int {\n        // \u5224\u7a7a\u5904\u7406\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        swap(0, size() - 1)\n        // \u5220\u9664\u8282\u70b9\n        val value = maxHeap.removeAt(size() - 1)\n        // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n        siftDown(0)\n        // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n        return value\n    }\n\n    /* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n    private fun siftDown(it: Int) {\n        // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n        var i = it\n        while (true) {\n            // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n            val l = left(i)\n            val r = right(i)\n            var ma = i\n            if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n            if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n            // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n            if (ma == i) break\n            // \u4ea4\u6362\u4e24\u8282\u70b9\n            swap(i, ma)\n            // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n            i = ma\n        }\n    }\n\n    /* \u6253\u5370\u5806\uff08\u4e8c\u53c9\u6811\uff09 */\n    fun print() {\n        val queue = PriorityQueue { a: Int, b: Int -&gt; b - a }\n        queue.addAll(maxHeap)\n        printHeap(queue)\n    }\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{__init__}\n</code></pre> my_heap.zig<pre><code>// \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\nfn init(self: *Self, allocator: std.mem.Allocator, nums: []const T) !void {\n    if (self.max_heap != null) return;\n    self.max_heap = std.ArrayList(T).init(allocator);\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    try self.max_heap.?.appendSlice(nums);\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    var i: usize = parent(self.size() - 1) + 1;\n    while (i &gt; 0) : (i -= 1) {\n        try self.siftDown(i - 1);\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_heap/build_heap/#823","title":"8.2.3 \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>\u4e0b\u9762\uff0c\u6211\u4eec\u6765\u5c1d\u8bd5\u63a8\u7b97\u7b2c\u4e8c\u79cd\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <ul> <li>\u5047\u8bbe\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u5219\u53f6\u8282\u70b9\u6570\u91cf\u4e3a \\((n + 1) / 2\\) \uff0c\u5176\u4e2d \\(/\\) \u4e3a\u5411\u4e0b\u6574\u9664\u3002\u56e0\u6b64\u9700\u8981\u5806\u5316\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\((n - 1) / 2\\) \u3002</li> <li>\u5728\u4ece\u9876\u81f3\u5e95\u5806\u5316\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e2a\u8282\u70b9\u6700\u591a\u5806\u5316\u5230\u53f6\u8282\u70b9\uff0c\u56e0\u6b64\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u4e3a\u4e8c\u53c9\u6811\u9ad8\u5ea6 \\(\\log n\\) \u3002</li> </ul> <p>\u5c06\u4e0a\u8ff0\u4e24\u8005\u76f8\u4e58\uff0c\u53ef\u5f97\u5230\u5efa\u5806\u8fc7\u7a0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u4f46\u8fd9\u4e2a\u4f30\u7b97\u7ed3\u679c\u5e76\u4e0d\u51c6\u786e\uff0c\u56e0\u4e3a\u6211\u4eec\u6ca1\u6709\u8003\u8651\u5230\u4e8c\u53c9\u6811\u5e95\u5c42\u8282\u70b9\u6570\u91cf\u8fdc\u591a\u4e8e\u9876\u5c42\u8282\u70b9\u7684\u6027\u8d28\u3002</p> <p>\u63a5\u4e0b\u6765\u6211\u4eec\u6765\u8fdb\u884c\u66f4\u4e3a\u51c6\u786e\u7684\u8ba1\u7b97\u3002\u4e3a\u4e86\u964d\u4f4e\u8ba1\u7b97\u96be\u5ea6\uff0c\u5047\u8bbe\u7ed9\u5b9a\u4e00\u4e2a\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \u3001\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u201c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u201d\uff0c\u8be5\u5047\u8bbe\u4e0d\u4f1a\u5f71\u54cd\u8ba1\u7b97\u7ed3\u679c\u7684\u6b63\u786e\u6027\u3002</p> <p></p> <p> \u56fe 8-5 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u5404\u5c42\u8282\u70b9\u6570\u91cf </p> <p>\u5982\u56fe 8-5 \u6240\u793a\uff0c\u8282\u70b9\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u7684\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u7b49\u4e8e\u8be5\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u800c\u8be5\u8ddd\u79bb\u6b63\u662f\u201c\u8282\u70b9\u9ad8\u5ea6\u201d\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5bf9\u5404\u5c42\u7684\u201c\u8282\u70b9\u6570\u91cf \\(\\times\\) \u8282\u70b9\u9ad8\u5ea6\u201d\u6c42\u548c\uff0c\u5f97\u5230\u6240\u6709\u8282\u70b9\u7684\u5806\u5316\u8fed\u4ee3\u6b21\u6570\u7684\u603b\u548c\u3002</p> \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{(h-1)}\\times1 \\] <p>\u5316\u7b80\u4e0a\u5f0f\u9700\u8981\u501f\u52a9\u4e2d\u5b66\u7684\u6570\u5217\u77e5\u8bc6\uff0c\u5148\u5c06 \\(T(h)\\) \u4e58\u4ee5 \\(2\\) \uff0c\u5f97\u5230\uff1a</p> \\[ \\begin{aligned} T(h) &amp; = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{h-1}\\times1 \\newline 2 T(h) &amp; = 2^1h + 2^2(h-1) + 2^3(h-2) + \\dots + 2^{h}\\times1 \\newline \\end{aligned} \\] <p>\u4f7f\u7528\u9519\u4f4d\u76f8\u51cf\u6cd5\uff0c\u7528\u4e0b\u5f0f \\(2 T(h)\\) \u51cf\u53bb\u4e0a\u5f0f \\(T(h)\\) \uff0c\u53ef\u5f97\uff1a</p> \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\dots + 2^{h-1} + 2^h \\] <p>\u89c2\u5bdf\u4e0a\u5f0f\uff0c\u53d1\u73b0 \\(T(h)\\) \u662f\u4e00\u4e2a\u7b49\u6bd4\u6570\u5217\uff0c\u53ef\u76f4\u63a5\u4f7f\u7528\u6c42\u548c\u516c\u5f0f\uff0c\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\uff1a</p> \\[ \\begin{aligned} T(h) &amp; = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline &amp; = 2^{h+1} - h - 2 \\newline &amp; = O(2^h) \\end{aligned} \\] <p>\u8fdb\u4e00\u6b65\uff0c\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n = 2^{h+1} - 1\\) \uff0c\u6613\u5f97\u590d\u6742\u5ea6\u4e3a \\(O(2^h) = O(n)\\) \u3002\u4ee5\u4e0a\u63a8\u7b97\u8868\u660e\uff0c\u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</p>"},{"location":"chapter_heap/heap/","title":"8.1 \u00a0 \u5806","text":"<p>\u300c\u5806 heap\u300d\u662f\u4e00\u79cd\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u4e3b\u8981\u53ef\u5206\u4e3a\u4e24\u79cd\u7c7b\u578b\uff0c\u5982\u56fe 8-1 \u6240\u793a\u3002</p> <ul> <li>\u300c\u5c0f\u9876\u5806 min heap\u300d\uff1a\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\leq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\u3002</li> <li>\u300c\u5927\u9876\u5806 max heap\u300d\uff1a\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\geq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\u3002</li> </ul> <p></p> <p> \u56fe 8-1 \u00a0 \u5c0f\u9876\u5806\u4e0e\u5927\u9876\u5806 </p> <p>\u5806\u4f5c\u4e3a\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u6700\u5e95\u5c42\u8282\u70b9\u9760\u5de6\u586b\u5145\uff0c\u5176\u4ed6\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u3002</li> <li>\u6211\u4eec\u5c06\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\u79f0\u4e3a\u201c\u5806\u9876\u201d\uff0c\u5c06\u5e95\u5c42\u6700\u9760\u53f3\u7684\u8282\u70b9\u79f0\u4e3a\u201c\u5806\u5e95\u201d\u3002</li> <li>\u5bf9\u4e8e\u5927\u9876\u5806\uff08\u5c0f\u9876\u5806\uff09\uff0c\u5806\u9876\u5143\u7d20\uff08\u6839\u8282\u70b9\uff09\u7684\u503c\u662f\u6700\u5927\uff08\u6700\u5c0f\uff09\u7684\u3002</li> </ul>"},{"location":"chapter_heap/heap/#811","title":"8.1.1 \u00a0 \u5806\u7684\u5e38\u7528\u64cd\u4f5c","text":"<p>\u9700\u8981\u6307\u51fa\u7684\u662f\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u662f\u300c\u4f18\u5148\u961f\u5217 priority queue\u300d\uff0c\u8fd9\u662f\u4e00\u79cd\u62bd\u8c61\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5b9a\u4e49\u4e3a\u5177\u6709\u4f18\u5148\u7ea7\u6392\u5e8f\u7684\u961f\u5217\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u5806\u901a\u5e38\u7528\u4e8e\u5b9e\u73b0\u4f18\u5148\u961f\u5217\uff0c\u5927\u9876\u5806\u76f8\u5f53\u4e8e\u5143\u7d20\u6309\u4ece\u5927\u5230\u5c0f\u7684\u987a\u5e8f\u51fa\u961f\u7684\u4f18\u5148\u961f\u5217\u3002\u4ece\u4f7f\u7528\u89d2\u5ea6\u6765\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u201c\u4f18\u5148\u961f\u5217\u201d\u548c\u201c\u5806\u201d\u770b\u4f5c\u7b49\u4ef7\u7684\u6570\u636e\u7ed3\u6784\u3002\u56e0\u6b64\uff0c\u672c\u4e66\u5bf9\u4e24\u8005\u4e0d\u505a\u7279\u522b\u533a\u5206\uff0c\u7edf\u4e00\u79f0\u4f5c\u201c\u5806\u201d\u3002</p> <p>\u5806\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u8868 8-1 \uff0c\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002</p> <p> \u8868 8-1 \u00a0 \u5806\u7684\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 <code>push()</code> \u5143\u7d20\u5165\u5806 \\(O(\\log n)\\) <code>pop()</code> \u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) <code>peek()</code> \u8bbf\u95ee\u5806\u9876\u5143\u7d20\uff08\u5bf9\u4e8e\u5927 / \u5c0f\u9876\u5806\u5206\u522b\u4e3a\u6700\u5927 / \u5c0f\u503c\uff09 \\(O(1)\\) <code>size()</code> \u83b7\u53d6\u5806\u7684\u5143\u7d20\u6570\u91cf \\(O(1)\\) <code>isEmpty()</code> \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a \\(O(1)\\) <p>\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u5806\u7c7b\uff08\u6216\u4f18\u5148\u961f\u5217\u7c7b\uff09\u3002</p> <p>\u7c7b\u4f3c\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e2d\u7684\u201c\u4ece\u5c0f\u5230\u5927\u6392\u5217\u201d\u548c\u201c\u4ece\u5927\u5230\u5c0f\u6392\u5217\u201d\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u8bbe\u7f6e\u4e00\u4e2a <code>flag</code> \u6216\u4fee\u6539 <code>Comparator</code> \u5b9e\u73b0\u201c\u5c0f\u9876\u5806\u201d\u4e0e\u201c\u5927\u9876\u5806\u201d\u4e4b\u95f4\u7684\u8f6c\u6362\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig heap.py<pre><code># \u521d\u59cb\u5316\u5c0f\u9876\u5806\nmin_heap, flag = [], 1\n# \u521d\u59cb\u5316\u5927\u9876\u5806\nmax_heap, flag = [], -1\n\n# Python \u7684 heapq \u6a21\u5757\u9ed8\u8ba4\u5b9e\u73b0\u5c0f\u9876\u5806\n# \u8003\u8651\u5c06\u201c\u5143\u7d20\u53d6\u8d1f\u201d\u540e\u518d\u5165\u5806\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5c06\u5927\u5c0f\u5173\u7cfb\u98a0\u5012\uff0c\u4ece\u800c\u5b9e\u73b0\u5927\u9876\u5806\n# \u5728\u672c\u793a\u4f8b\u4e2d\uff0cflag = 1 \u65f6\u5bf9\u5e94\u5c0f\u9876\u5806\uff0cflag = -1 \u65f6\u5bf9\u5e94\u5927\u9876\u5806\n\n# \u5143\u7d20\u5165\u5806\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n\n# \u83b7\u53d6\u5806\u9876\u5143\u7d20\npeek: int = flag * max_heap[0] # 5\n\n# \u5806\u9876\u5143\u7d20\u51fa\u5806\n# \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n\n# \u83b7\u53d6\u5806\u5927\u5c0f\nsize: int = len(max_heap)\n\n# \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = not max_heap\n\n# \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\nmin_heap: list[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n</code></pre> heap.cpp<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap;\n// \u521d\u59cb\u5316\u5927\u9876\u5806\npriority_queue&lt;int, vector&lt;int&gt;, less&lt;int&gt;&gt; maxHeap;\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.top(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.empty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nvector&lt;int&gt; input{1, 3, 2, 5, 4};\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap(input.begin(), input.end());\n</code></pre> heap.java<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nQueue&lt;Integer&gt; minHeap = new PriorityQueue&lt;&gt;();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nQueue&lt;Integer&gt; maxHeap = new PriorityQueue&lt;&gt;((a, b) -&gt; b - a);\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.peek(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll(); // 5\npeek = maxHeap.poll(); // 4\npeek = maxHeap.poll(); // 3\npeek = maxHeap.poll(); // 2\npeek = maxHeap.poll(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = maxHeap.isEmpty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue&lt;&gt;(Arrays.asList(1, 3, 2, 5, 4));\n</code></pre> heap.cs<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nPriorityQueue&lt;int, int&gt; minHeap = new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nPriorityQueue&lt;int, int&gt; maxHeap = new(Comparer&lt;int&gt;.Create((x, y) =&gt; y - x));\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.Peek();//5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.Count;\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.Count == 0;\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue&lt;int, int&gt;([(1, 1), (3, 3), (2, 2), (5, 5), (4, 4)]);\n</code></pre> heap.go<pre><code>// Go \u8bed\u8a00\u4e2d\u53ef\u4ee5\u901a\u8fc7\u5b9e\u73b0 heap.Interface \u6765\u6784\u5efa\u6574\u6570\u5927\u9876\u5806\n// \u5b9e\u73b0 heap.Interface \u9700\u8981\u540c\u65f6\u5b9e\u73b0 sort.Interface\ntype intHeap []any\n\n// Push heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u63a8\u5165\u5143\u7d20\u5230\u5806\nfunc (h *intHeap) Push(x any) {\n    // Push \u548c Pop \u4f7f\u7528 pointer receiver \u4f5c\u4e3a\u53c2\u6570\n    // \u56e0\u4e3a\u5b83\u4eec\u4e0d\u4ec5\u4f1a\u5bf9\u5207\u7247\u7684\u5185\u5bb9\u8fdb\u884c\u8c03\u6574\uff0c\u8fd8\u4f1a\u4fee\u6539\u5207\u7247\u7684\u957f\u5ea6\u3002\n    *h = append(*h, x.(int))\n}\n\n// Pop heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u5f39\u51fa\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Pop() any {\n    // \u5f85\u51fa\u5806\u5143\u7d20\u5b58\u653e\u5728\u6700\u540e\n    last := (*h)[len(*h)-1]\n    *h = (*h)[:len(*h)-1]\n    return last\n}\n\n// Len sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Len() int {\n    return len(*h)\n}\n\n// Less sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Less(i, j int) bool {\n    // \u5982\u679c\u5b9e\u73b0\u5c0f\u9876\u5806\uff0c\u5219\u9700\u8981\u8c03\u6574\u4e3a\u5c0f\u4e8e\u53f7\n    return (*h)[i].(int) &gt; (*h)[j].(int)\n}\n\n// Swap sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Swap(i, j int) {\n    (*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n\n// Top \u83b7\u53d6\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Top() any {\n    return (*h)[0]\n}\n\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n    /* \u521d\u59cb\u5316\u5806 */\n    // \u521d\u59cb\u5316\u5927\u9876\u5806\n    maxHeap := &amp;intHeap{}\n    heap.Init(maxHeap)\n    /* \u5143\u7d20\u5165\u5806 */\n    // \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u6dfb\u52a0\u5143\u7d20\n    heap.Push(maxHeap, 1)\n    heap.Push(maxHeap, 3)\n    heap.Push(maxHeap, 2)\n    heap.Push(maxHeap, 4)\n    heap.Push(maxHeap, 5)\n\n    /* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\n    top := maxHeap.Top()\n    fmt.Printf(\"\u5806\u9876\u5143\u7d20\u4e3a %d\\n\", top)\n\n    /* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n    // \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u79fb\u9664\u5143\u7d20\n    heap.Pop(maxHeap) // 5\n    heap.Pop(maxHeap) // 4\n    heap.Pop(maxHeap) // 3\n    heap.Pop(maxHeap) // 2\n    heap.Pop(maxHeap) // 1\n\n    /* \u83b7\u53d6\u5806\u5927\u5c0f */\n    size := len(*maxHeap)\n    fmt.Printf(\"\u5806\u5143\u7d20\u6570\u91cf\u4e3a %d\\n\", size)\n\n    /* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\n    isEmpty := len(*maxHeap) == 0\n    fmt.Printf(\"\u5806\u662f\u5426\u4e3a\u7a7a %t\\n\", isEmpty)\n}\n</code></pre> heap.swift<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// Swift \u7684 Heap \u7c7b\u578b\u540c\u65f6\u652f\u6301\u6700\u5927\u5806\u548c\u6700\u5c0f\u5806\uff0c\u4e14\u9700\u8981\u5f15\u5165 swift-collections\nvar heap = Heap&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u5806 */\nheap.insert(1)\nheap.insert(3)\nheap.insert(2)\nheap.insert(5)\nheap.insert(4)\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nvar peek = heap.max()!\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\npeek = heap.removeMax() // 5\npeek = heap.removeMax() // 4\npeek = heap.removeMax() // 3\npeek = heap.removeMax() // 2\npeek = heap.removeMax() // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nlet size = heap.count\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = heap.isEmpty\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nlet heap2 = Heap([1, 3, 2, 5, 4])\n</code></pre> heap.js<pre><code>// JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.ts<pre><code>// TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.dart<pre><code>// Dart \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.rs<pre><code>use std::collections::BinaryHeap;\nuse std::cmp::Reverse;\n\n/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nlet mut min_heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\nlet mut max_heap = BinaryHeap::new();\n\n/* \u5143\u7d20\u5165\u5806 */\nmax_heap.push(1);\nmax_heap.push(3);\nmax_heap.push(2);\nmax_heap.push(5);\nmax_heap.push(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nlet peek = max_heap.peek().unwrap();  // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nlet peek = max_heap.pop().unwrap();   // 5\nlet peek = max_heap.pop().unwrap();   // 4\nlet peek = max_heap.pop().unwrap();   // 3\nlet peek = max_heap.pop().unwrap();   // 2\nlet peek = max_heap.pop().unwrap();   // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nlet size = max_heap.len();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = max_heap.is_empty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nlet min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]);\n</code></pre> heap.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.kt<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nvar minHeap = PriorityQueue&lt;Int&gt;()\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nval maxHeap = PriorityQueue { a: Int, b: Int -&gt; b - a }\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1)\nmaxHeap.offer(3)\nmaxHeap.offer(2)\nmaxHeap.offer(5)\nmaxHeap.offer(4)\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nvar peek = maxHeap.peek() // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll() // 5\npeek = maxHeap.poll() // 4\npeek = maxHeap.poll() // 3\npeek = maxHeap.poll() // 2\npeek = maxHeap.poll() // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nval size = maxHeap.size\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nval isEmpty = maxHeap.isEmpty()\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = PriorityQueue(mutableListOf(1, 3, 2, 5, 4))\n</code></pre> heap.rb<pre><code>\n</code></pre> heap.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_heap/heap/#812","title":"8.1.2 \u00a0 \u5806\u7684\u5b9e\u73b0","text":"<p>\u4e0b\u6587\u5b9e\u73b0\u7684\u662f\u5927\u9876\u5806\u3002\u82e5\u8981\u5c06\u5176\u8f6c\u6362\u4e3a\u5c0f\u9876\u5806\uff0c\u53ea\u9700\u5c06\u6240\u6709\u5927\u5c0f\u903b\u8f91\u5224\u65ad\u53d6\u9006\uff08\u4f8b\u5982\uff0c\u5c06 \\(\\geq\\) \u66ff\u6362\u4e3a \\(\\leq\\) \uff09\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002</p>"},{"location":"chapter_heap/heap/#1","title":"1. \u00a0 \u5806\u7684\u5b58\u50a8\u4e0e\u8868\u793a","text":"<p>\u201c\u4e8c\u53c9\u6811\u201d\u7ae0\u8282\u8bb2\u8fc7\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u7531\u4e8e\u5806\u6b63\u662f\u4e00\u79cd\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u56e0\u6b64\u6211\u4eec\u5c06\u91c7\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002</p> <p>\u5f53\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u4e8c\u53c9\u6811\u65f6\uff0c\u5143\u7d20\u4ee3\u8868\u8282\u70b9\u503c\uff0c\u7d22\u5f15\u4ee3\u8868\u8282\u70b9\u5728\u4e8c\u53c9\u6811\u4e2d\u7684\u4f4d\u7f6e\u3002\u8282\u70b9\u6307\u9488\u901a\u8fc7\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u6765\u5b9e\u73b0\u3002</p> <p>\u5982\u56fe 8-2 \u6240\u793a\uff0c\u7ed9\u5b9a\u7d22\u5f15 \\(i\\) \uff0c\u5176\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\u4e3a \\(2i + 2\\) \uff0c\u7236\u8282\u70b9\u7684\u7d22\u5f15\u4e3a \\((i - 1) / 2\\)\uff08\u5411\u4e0b\u6574\u9664\uff09\u3002\u5f53\u7d22\u5f15\u8d8a\u754c\u65f6\uff0c\u8868\u793a\u7a7a\u8282\u70b9\u6216\u8282\u70b9\u4e0d\u5b58\u5728\u3002</p> <p></p> <p> \u56fe 8-2 \u00a0 \u5806\u7684\u8868\u793a\u4e0e\u5b58\u50a8 </p> <p>\u6211\u4eec\u53ef\u4ee5\u5c06\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def left(self, i: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n    return 2 * i + 1\n\ndef right(self, i: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n    return 2 * i + 2\n\ndef parent(self, i: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n    return (i - 1) // 2  # \u5411\u4e0b\u6574\u9664\n</code></pre> my_heap.cpp<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.java<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.cs<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint Left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint Right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint Parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.go<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) left(i int) int {\n    return 2*i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) right(i int) int {\n    return 2*i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) parent(i int) int {\n    // \u5411\u4e0b\u6574\u9664\n    return (i - 1) / 2\n}\n</code></pre> my_heap.swift<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc left(i: Int) -&gt; Int {\n    2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc right(i: Int) -&gt; Int {\n    2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc parent(i: Int) -&gt; Int {\n    (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.js<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n#left(i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n#right(i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n#parent(i) {\n    return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.ts<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nleft(i: number): number {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nright(i: number): number {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nparent(i: number): number {\n    return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.dart<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint _left(int i) {\n  return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint _right(int i) {\n  return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint _parent(int i) {\n  return (i - 1) ~/ 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.rs<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfn left(i: usize) -&gt; usize {\n    2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfn right(i: usize) -&gt; usize {\n    2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfn parent(i: usize) -&gt; usize {\n    (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.c<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint left(MaxHeap *maxHeap, int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint right(MaxHeap *maxHeap, int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint parent(MaxHeap *maxHeap, int i) {\n    return (i - 1) / 2;\n}\n</code></pre> my_heap.kt<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfun left(i: Int): Int {\n    return 2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfun right(i: Int): Int {\n    return 2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfun parent(i: Int): Int {\n    return (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{left}\n\n[class]{MaxHeap}-[func]{right}\n\n[class]{MaxHeap}-[func]{parent}\n</code></pre> my_heap.zig<pre><code>// \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\nfn left(i: usize) usize {\n    return 2 * i + 1;\n}\n\n// \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\nfn right(i: usize) usize {\n    return 2 * i + 2;\n}\n\n// \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15\nfn parent(i: usize) usize {\n    // return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n    return @divFloor(i - 1, 2);\n}\n</code></pre>"},{"location":"chapter_heap/heap/#2","title":"2. \u00a0 \u8bbf\u95ee\u5806\u9876\u5143\u7d20","text":"<p>\u5806\u9876\u5143\u7d20\u5373\u4e3a\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u4e5f\u5c31\u662f\u5217\u8868\u7684\u9996\u4e2a\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def peek(self) -&gt; int:\n    \"\"\"\u8bbf\u95ee\u5806\u9876\u5143\u7d20\"\"\"\n    return self.max_heap[0]\n</code></pre> my_heap.cpp<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.java<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n    return maxHeap.get(0);\n}\n</code></pre> my_heap.cs<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint Peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.go<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc (h *maxHeap) peek() any {\n    return h.data[0]\n}\n</code></pre> my_heap.swift<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc peek() -&gt; Int {\n    maxHeap[0]\n}\n</code></pre> my_heap.js<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek() {\n    return this.#maxHeap[0];\n}\n</code></pre> my_heap.ts<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek(): number {\n    return this.maxHeap[0];\n}\n</code></pre> my_heap.dart<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n  return _maxHeap[0];\n}\n</code></pre> my_heap.rs<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfn peek(&amp;self) -&gt; Option&lt;i32&gt; {\n    self.max_heap.first().copied()\n}\n</code></pre> my_heap.c<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek(MaxHeap *maxHeap) {\n    return maxHeap-&gt;data[0];\n}\n</code></pre> my_heap.kt<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfun peek(): Int {\n    return maxHeap[0]\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{peek}\n</code></pre> my_heap.zig<pre><code>// \u8bbf\u95ee\u5806\u9876\u5143\u7d20\nfn peek(self: *Self) T {\n    return self.max_heap.?.items[0];\n}  \n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_heap/heap/#3","title":"3. \u00a0 \u5143\u7d20\u5165\u5806","text":"<p>\u7ed9\u5b9a\u5143\u7d20 <code>val</code> \uff0c\u6211\u4eec\u9996\u5148\u5c06\u5176\u6dfb\u52a0\u5230\u5806\u5e95\u3002\u6dfb\u52a0\u4e4b\u540e\uff0c\u7531\u4e8e <code>val</code> \u53ef\u80fd\u5927\u4e8e\u5806\u4e2d\u5176\u4ed6\u5143\u7d20\uff0c\u5806\u7684\u6210\u7acb\u6761\u4ef6\u53ef\u80fd\u5df2\u88ab\u7834\u574f\uff0c\u56e0\u6b64\u9700\u8981\u4fee\u590d\u4ece\u63d2\u5165\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u8fd9\u4e2a\u64cd\u4f5c\u88ab\u79f0\u4e3a\u300c\u5806\u5316 heapify\u300d\u3002</p> <p>\u8003\u8651\u4ece\u5165\u5806\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u6267\u884c\u5806\u5316\u3002\u5982\u56fe 8-3 \u6240\u793a\uff0c\u6211\u4eec\u6bd4\u8f83\u63d2\u5165\u8282\u70b9\u4e0e\u5176\u7236\u8282\u70b9\u7684\u503c\uff0c\u5982\u679c\u63d2\u5165\u8282\u70b9\u66f4\u5927\uff0c\u5219\u5c06\u5b83\u4eec\u4ea4\u6362\u3002\u7136\u540e\u7ee7\u7eed\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u4ece\u5e95\u81f3\u9876\u4fee\u590d\u5806\u4e2d\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u76f4\u81f3\u8d8a\u8fc7\u6839\u8282\u70b9\u6216\u9047\u5230\u65e0\u987b\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 8-3 \u00a0 \u5143\u7d20\u5165\u5806\u6b65\u9aa4 </p> <p>\u8bbe\u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u6811\u7684\u9ad8\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u7531\u6b64\u53ef\u77e5\uff0c\u5806\u5316\u64cd\u4f5c\u7684\u5faa\u73af\u8f6e\u6570\u6700\u591a\u4e3a \\(O(\\log n)\\) \uff0c\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def push(self, val: int):\n    \"\"\"\u5143\u7d20\u5165\u5806\"\"\"\n    # \u6dfb\u52a0\u8282\u70b9\n    self.max_heap.append(val)\n    # \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    self.sift_up(self.size() - 1)\n\ndef sift_up(self, i: int):\n    \"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\"\"\"\n    while True:\n        # \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        p = self.parent(i)\n        # \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if p &lt; 0 or self.max_heap[i] &lt;= self.max_heap[p]:\n            break\n        # \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, p)\n        # \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.push_back(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(maxHeap[i], maxHeap[p]);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.add(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap.get(i) &lt;= maxHeap.get(p))\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid Push(int val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.Add(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    SiftUp(Size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid SiftUp(int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = Parent(i);\n        // \u82e5\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        Swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunc (h *maxHeap) push(val any) {\n    // \u6dfb\u52a0\u8282\u70b9\n    h.data = append(h.data, val)\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    h.siftUp(len(h.data) - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc (h *maxHeap) siftUp(i int) {\n    for true {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        p := h.parent(i)\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if p &lt; 0 || h.data[i].(int) &lt;= h.data[p].(int) {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        h.swap(i, p)\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunc push(val: Int) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.append(val)\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(i: size() - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc siftUp(i: Int) {\n    var i = i\n    while true {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        let p = parent(i: i)\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if p &lt; 0 || maxHeap[i] &lt;= maxHeap[p] {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i: i, j: p)\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u5165\u5806 */\npush(val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    this.#maxHeap.push(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    this.#siftUp(this.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n#siftUp(i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        const p = this.#parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || this.#maxHeap[i] &lt;= this.#maxHeap[p]) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.#swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u5165\u5806 */\npush(val: number): void {\n    // \u6dfb\u52a0\u8282\u70b9\n    this.maxHeap.push(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    this.siftUp(this.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nsiftUp(i: number): void {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        const p = this.parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || this.maxHeap[i] &lt;= this.maxHeap[p]) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n  // \u6dfb\u52a0\u8282\u70b9\n  _maxHeap.add(val);\n  // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n  siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n  while (true) {\n    // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n    int p = _parent(i);\n    // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n    if (p &lt; 0 || _maxHeap[i] &lt;= _maxHeap[p]) {\n      break;\n    }\n    // \u4ea4\u6362\u4e24\u8282\u70b9\n    _swap(i, p);\n    // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n    i = p;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfn push(&amp;mut self, val: i32) {\n    // \u6dfb\u52a0\u8282\u70b9\n    self.max_heap.push(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    self.sift_up(self.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfn sift_up(&amp;mut self, mut i: usize) {\n    loop {\n        // \u8282\u70b9 i \u5df2\u7ecf\u662f\u5806\u9876\u8282\u70b9\u4e86\uff0c\u7ed3\u675f\u5806\u5316\n        if i == 0 {\n            break;\n        }\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        let p = Self::parent(i);\n        // \u5f53\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if self.max_heap[i] &lt;= self.max_heap[p] {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(MaxHeap *maxHeap, int val) {\n    // \u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c\u4e0d\u5e94\u8be5\u6dfb\u52a0\u8fd9\u4e48\u591a\u8282\u70b9\n    if (maxHeap-&gt;size == MAX_SIZE) {\n        printf(\"heap is full!\");\n        return;\n    }\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap-&gt;data[maxHeap-&gt;size] = val;\n    maxHeap-&gt;size++;\n\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(maxHeap, maxHeap-&gt;size - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = parent(maxHeap, i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap-&gt;data[i] &lt;= maxHeap-&gt;data[p]) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(maxHeap, i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfun push(value: Int) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.add(value)\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(size() - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfun siftUp(it: Int) {\n    // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n    var i = it\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        val p = parent(i)\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, p)\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{push}\n\n[class]{MaxHeap}-[func]{sift_up}\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u5165\u5806\nfn push(self: *Self, val: T) !void {\n    // \u6dfb\u52a0\u8282\u70b9\n    try self.max_heap.?.append(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    try self.siftUp(self.size() - 1);\n}  \n\n// \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\nfn siftUp(self: *Self, i_: usize) !void {\n    var i = i_;\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        var p = parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 or self.max_heap.?.items[i] &lt;= self.max_heap.?.items[p]) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        try self.swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_heap/heap/#4","title":"4. \u00a0 \u5806\u9876\u5143\u7d20\u51fa\u5806","text":"<p>\u5806\u9876\u5143\u7d20\u662f\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u5373\u5217\u8868\u9996\u5143\u7d20\u3002\u5982\u679c\u6211\u4eec\u76f4\u63a5\u4ece\u5217\u8868\u4e2d\u5220\u9664\u9996\u5143\u7d20\uff0c\u90a3\u4e48\u4e8c\u53c9\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u7d22\u5f15\u90fd\u4f1a\u53d1\u751f\u53d8\u5316\uff0c\u8fd9\u5c06\u4f7f\u5f97\u540e\u7eed\u4f7f\u7528\u5806\u5316\u8fdb\u884c\u4fee\u590d\u53d8\u5f97\u56f0\u96be\u3002\u4e3a\u4e86\u5c3d\u91cf\u51cf\u5c11\u5143\u7d20\u7d22\u5f15\u7684\u53d8\u52a8\uff0c\u6211\u4eec\u91c7\u7528\u4ee5\u4e0b\u64cd\u4f5c\u6b65\u9aa4\u3002</p> <ol> <li>\u4ea4\u6362\u5806\u9876\u5143\u7d20\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff09\u3002</li> <li>\u4ea4\u6362\u5b8c\u6210\u540e\uff0c\u5c06\u5806\u5e95\u4ece\u5217\u8868\u4e2d\u5220\u9664\uff08\u6ce8\u610f\uff0c\u7531\u4e8e\u5df2\u7ecf\u4ea4\u6362\uff0c\u56e0\u6b64\u5b9e\u9645\u4e0a\u5220\u9664\u7684\u662f\u539f\u6765\u7684\u5806\u9876\u5143\u7d20\uff09\u3002</li> <li>\u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u6267\u884c\u5806\u5316\u3002</li> </ol> <p>\u5982\u56fe 8-4 \u6240\u793a\uff0c\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u7684\u64cd\u4f5c\u65b9\u5411\u4e0e\u201c\u4ece\u5e95\u81f3\u9876\u5806\u5316\u201d\u76f8\u53cd\uff0c\u6211\u4eec\u5c06\u6839\u8282\u70b9\u7684\u503c\u4e0e\u5176\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u503c\u8fdb\u884c\u6bd4\u8f83\uff0c\u5c06\u6700\u5927\u7684\u5b50\u8282\u70b9\u4e0e\u6839\u8282\u70b9\u4ea4\u6362\u3002\u7136\u540e\u5faa\u73af\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\u6216\u9047\u5230\u65e0\u987b\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 8-4 \u00a0 \u5806\u9876\u5143\u7d20\u51fa\u5806\u6b65\u9aa4 </p> <p>\u4e0e\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u76f8\u4f3c\uff0c\u5806\u9876\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4e3a \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def pop(self) -&gt; int:\n    \"\"\"\u5143\u7d20\u51fa\u5806\"\"\"\n    # \u5224\u7a7a\u5904\u7406\n    if self.is_empty():\n        raise IndexError(\"\u5806\u4e3a\u7a7a\")\n    # \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    self.swap(0, self.size() - 1)\n    # \u5220\u9664\u8282\u70b9\n    val = self.max_heap.pop()\n    # \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    self.sift_down(0)\n    # \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val\n\ndef sift_down(self, i: int):\n    \"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\n    while True:\n        # \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        l, r, ma = self.left(i), self.right(i), i\n        if l &lt; self.size() and self.max_heap[l] &gt; self.max_heap[ma]:\n            ma = l\n        if r &lt; self.size() and self.max_heap[r] &gt; self.max_heap[ma]:\n            ma = r\n        # \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i:\n            break\n        # \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, ma)\n        # \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nvoid pop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty()) {\n        throw out_of_range(\"\u5806\u4e3a\u7a7a\");\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(maxHeap[0], maxHeap[size() - 1]);\n    // \u5220\u9664\u8282\u70b9\n    maxHeap.pop_back();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(0);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        swap(maxHeap[i], maxHeap[ma]);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty())\n        throw new IndexOutOfBoundsException();\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(0, size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    int val = maxHeap.remove(size() - 1);\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap.get(l) &gt; maxHeap.get(ma))\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap.get(r) &gt; maxHeap.get(ma))\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint Pop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (IsEmpty())\n        throw new IndexOutOfRangeException();\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    Swap(0, Size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    int val = maxHeap.Last();\n    maxHeap.RemoveAt(Size() - 1);\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    SiftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid SiftDown(int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = Left(i), r = Right(i), ma = i;\n        if (l &lt; Size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; Size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // \u82e5\u201c\u8282\u70b9 i \u6700\u5927\u201d\u6216\u201c\u8d8a\u8fc7\u53f6\u8282\u70b9\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\n        if (ma == i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        Swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfunc (h *maxHeap) pop() any {\n    // \u5224\u7a7a\u5904\u7406\n    if h.isEmpty() {\n        fmt.Println(\"error\")\n        return nil\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    h.swap(0, h.size()-1)\n    // \u5220\u9664\u8282\u70b9\n    val := h.data[len(h.data)-1]\n    h.data = h.data[:len(h.data)-1]\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    h.siftDown(0)\n\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc (h *maxHeap) siftDown(i int) {\n    for true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\n        l, r, max := h.left(i), h.right(i), i\n        if l &lt; h.size() &amp;&amp; h.data[l].(int) &gt; h.data[max].(int) {\n            max = l\n        }\n        if r &lt; h.size() &amp;&amp; h.data[r].(int) &gt; h.data[max].(int) {\n            max = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if max == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        h.swap(i, max)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = max\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfunc pop() -&gt; Int {\n    // \u5224\u7a7a\u5904\u7406\n    if isEmpty() {\n        fatalError(\"\u5806\u4e3a\u7a7a\")\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(i: 0, j: size() - 1)\n    // \u5220\u9664\u8282\u70b9\n    let val = maxHeap.remove(at: size() - 1)\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(i: 0)\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(i: Int) {\n    var i = i\n    while true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = left(i: i)\n        let r = right(i: i)\n        var ma = i\n        if l &lt; size(), maxHeap[l] &gt; maxHeap[ma] {\n            ma = l\n        }\n        if r &lt; size(), maxHeap[r] &gt; maxHeap[ma] {\n            ma = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i: i, j: ma)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u51fa\u5806 */\npop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (this.isEmpty()) throw new Error('\u5806\u4e3a\u7a7a');\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    this.#swap(0, this.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    const val = this.#maxHeap.pop();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    this.#siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n#siftDown(i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        const l = this.#left(i),\n            r = this.#right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.#maxHeap[l] &gt; this.#maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.#maxHeap[r] &gt; this.#maxHeap[ma]) ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.#swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u51fa\u5806 */\npop(): number {\n    // \u5224\u7a7a\u5904\u7406\n    if (this.isEmpty()) throw new RangeError('Heap is empty.');\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    this.swap(0, this.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    const val = this.maxHeap.pop();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    this.siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nsiftDown(i: number): void {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        const l = this.left(i),\n            r = this.right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.maxHeap[l] &gt; this.maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.maxHeap[r] &gt; this.maxHeap[ma]) ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n  // \u5224\u7a7a\u5904\u7406\n  if (isEmpty()) throw Exception('\u5806\u4e3a\u7a7a');\n  // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n  _swap(0, size() - 1);\n  // \u5220\u9664\u8282\u70b9\n  int val = _maxHeap.removeLast();\n  // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n  siftDown(0);\n  // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n  return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n  while (true) {\n    // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n    int l = _left(i);\n    int r = _right(i);\n    int ma = i;\n    if (l &lt; size() &amp;&amp; _maxHeap[l] &gt; _maxHeap[ma]) ma = l;\n    if (r &lt; size() &amp;&amp; _maxHeap[r] &gt; _maxHeap[ma]) ma = r;\n    // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n    if (ma == i) break;\n    // \u4ea4\u6362\u4e24\u8282\u70b9\n    _swap(i, ma);\n    // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n    i = ma;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfn pop(&amp;mut self) -&gt; i32 {\n    // \u5224\u7a7a\u5904\u7406\n    if self.is_empty() {\n        panic!(\"index out of bounds\");\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    self.swap(0, self.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    let val = self.max_heap.remove(self.size() - 1);\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    self.sift_down(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfn sift_down(&amp;mut self, mut i: usize) {\n    loop {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let (l, r, mut ma) = (Self::left(i), Self::right(i), i);\n        if l &lt; self.size() &amp;&amp; self.max_heap[l] &gt; self.max_heap[ma] {\n            ma = l;\n        }\n        if r &lt; self.size() &amp;&amp; self.max_heap[r] &gt; self.max_heap[ma] {\n            ma = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop(MaxHeap *maxHeap) {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty(maxHeap)) {\n        printf(\"heap is empty!\");\n        return INT_MAX;\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(maxHeap, 0, size(maxHeap) - 1);\n    // \u5220\u9664\u8282\u70b9\n    int val = maxHeap-&gt;data[maxHeap-&gt;size - 1];\n    maxHeap-&gt;size--;\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(maxHeap, 0);\n\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\n        int l = left(maxHeap, i);\n        int r = right(maxHeap, i);\n        int max = i;\n        if (l &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[l] &gt; maxHeap-&gt;data[max]) {\n            max = l;\n        }\n        if (r &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[r] &gt; maxHeap-&gt;data[max]) {\n            max = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (max == i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(maxHeap, i, max);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = max;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfun pop(): Int {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty()) throw IndexOutOfBoundsException()\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(0, size() - 1)\n    // \u5220\u9664\u8282\u70b9\n    val value = maxHeap.removeAt(size() - 1)\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(0)\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return value\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfun siftDown(it: Int) {\n    // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n    var i = it\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        val l = left(i)\n        val r = right(i)\n        var ma = i\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) break\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, ma)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{pop}\n\n[class]{MaxHeap}-[func]{sift_down}\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u51fa\u5806\nfn pop(self: *Self) !T {\n    // \u5224\u65ad\u5904\u7406\n    if (self.isEmpty()) unreachable;\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    try self.swap(0, self.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    var val = self.max_heap.?.pop();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    try self.siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n} \n\n// \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\nfn siftDown(self: *Self, i_: usize) !void {\n    var i = i_;\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        var l = left(i);\n        var r = right(i);\n        var ma = i;\n        if (l &lt; self.size() and self.max_heap.?.items[l] &gt; self.max_heap.?.items[ma]) ma = l;\n        if (r &lt; self.size() and self.max_heap.?.items[r] &gt; self.max_heap.?.items[ma]) ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        try self.swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_heap/heap/#813","title":"8.1.3 \u00a0 \u5806\u7684\u5e38\u89c1\u5e94\u7528","text":"<ul> <li>\u4f18\u5148\u961f\u5217\uff1a\u5806\u901a\u5e38\u4f5c\u4e3a\u5b9e\u73b0\u4f18\u5148\u961f\u5217\u7684\u9996\u9009\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \uff0c\u800c\u5efa\u961f\u64cd\u4f5c\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e9b\u64cd\u4f5c\u90fd\u975e\u5e38\u9ad8\u6548\u3002</li> <li>\u5806\u6392\u5e8f\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u5b83\u4eec\u5efa\u7acb\u4e00\u4e2a\u5806\uff0c\u7136\u540e\u4e0d\u65ad\u5730\u6267\u884c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\uff0c\u4ece\u800c\u5f97\u5230\u6709\u5e8f\u6570\u636e\u3002\u7136\u800c\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u4f7f\u7528\u4e00\u79cd\u66f4\u4f18\u96c5\u7684\u65b9\u5f0f\u5b9e\u73b0\u5806\u6392\u5e8f\uff0c\u8be6\u89c1\u201c\u5806\u6392\u5e8f\u201d\u7ae0\u8282\u3002</li> <li>\u83b7\u53d6\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff1a\u8fd9\u662f\u4e00\u4e2a\u7ecf\u5178\u7684\u7b97\u6cd5\u95ee\u9898\uff0c\u540c\u65f6\u4e5f\u662f\u4e00\u79cd\u5178\u578b\u5e94\u7528\uff0c\u4f8b\u5982\u9009\u62e9\u70ed\u5ea6\u524d 10 \u7684\u65b0\u95fb\u4f5c\u4e3a\u5fae\u535a\u70ed\u641c\uff0c\u9009\u53d6\u9500\u91cf\u524d 10 \u7684\u5546\u54c1\u7b49\u3002</li> </ul>"},{"location":"chapter_heap/summary/","title":"8.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_heap/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u5806\u662f\u4e00\u68f5\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u6839\u636e\u6210\u7acb\u6761\u4ef6\u53ef\u5206\u4e3a\u5927\u9876\u5806\u548c\u5c0f\u9876\u5806\u3002\u5927\uff08\u5c0f\uff09\u9876\u5806\u7684\u5806\u9876\u5143\u7d20\u662f\u6700\u5927\uff08\u5c0f\uff09\u7684\u3002</li> <li>\u4f18\u5148\u961f\u5217\u7684\u5b9a\u4e49\u662f\u5177\u6709\u51fa\u961f\u4f18\u5148\u7ea7\u7684\u961f\u5217\uff0c\u901a\u5e38\u4f7f\u7528\u5806\u6765\u5b9e\u73b0\u3002</li> <li>\u5806\u7684\u5e38\u7528\u64cd\u4f5c\u53ca\u5176\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5305\u62ec\uff1a\u5143\u7d20\u5165\u5806 \\(O(\\log n)\\)\u3001\u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) \u548c\u8bbf\u95ee\u5806\u9876\u5143\u7d20 \\(O(1)\\) \u7b49\u3002</li> <li>\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u8868\u793a\uff0c\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002</li> <li>\u5806\u5316\u64cd\u4f5c\u7528\u4e8e\u7ef4\u62a4\u5806\u7684\u6027\u8d28\uff0c\u5728\u5165\u5806\u548c\u51fa\u5806\u64cd\u4f5c\u4e2d\u90fd\u4f1a\u7528\u5230\u3002</li> <li>\u8f93\u5165 \\(n\\) \u4e2a\u5143\u7d20\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</li> <li>Top-k \u662f\u4e00\u4e2a\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\uff0c\u53ef\u4ee5\u4f7f\u7528\u5806\u6570\u636e\u7ed3\u6784\u9ad8\u6548\u89e3\u51b3\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002</li> </ul>"},{"location":"chapter_heap/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u6570\u636e\u7ed3\u6784\u7684\u201c\u5806\u201d\u4e0e\u5185\u5b58\u7ba1\u7406\u7684\u201c\u5806\u201d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\u5417\uff1f</p> <p>\u4e24\u8005\u4e0d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\uff0c\u53ea\u662f\u78b0\u5de7\u90fd\u53eb\u201c\u5806\u201d\u3002\u8ba1\u7b97\u673a\u7cfb\u7edf\u5185\u5b58\u4e2d\u7684\u5806\u662f\u52a8\u6001\u5185\u5b58\u5206\u914d\u7684\u4e00\u90e8\u5206\uff0c\u7a0b\u5e8f\u5728\u8fd0\u884c\u65f6\u53ef\u4ee5\u4f7f\u7528\u5b83\u6765\u5b58\u50a8\u6570\u636e\u3002\u7a0b\u5e8f\u53ef\u4ee5\u8bf7\u6c42\u4e00\u5b9a\u91cf\u7684\u5806\u5185\u5b58\uff0c\u7528\u4e8e\u5b58\u50a8\u5982\u5bf9\u8c61\u548c\u6570\u7ec4\u7b49\u590d\u6742\u7ed3\u6784\u3002\u5f53\u8fd9\u4e9b\u6570\u636e\u4e0d\u518d\u9700\u8981\u65f6\uff0c\u7a0b\u5e8f\u9700\u8981\u91ca\u653e\u8fd9\u4e9b\u5185\u5b58\uff0c\u4ee5\u9632\u6b62\u5185\u5b58\u6cc4\u6f0f\u3002\u76f8\u8f83\u4e8e\u6808\u5185\u5b58\uff0c\u5806\u5185\u5b58\u7684\u7ba1\u7406\u548c\u4f7f\u7528\u9700\u8981\u66f4\u8c28\u614e\uff0c\u4f7f\u7528\u4e0d\u5f53\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5185\u5b58\u6cc4\u6f0f\u548c\u91ce\u6307\u9488\u7b49\u95ee\u9898\u3002</p>"},{"location":"chapter_heap/top_k/","title":"8.3 \u00a0 Top-k \u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u65e0\u5e8f\u6570\u7ec4 <code>nums</code> \uff0c\u8bf7\u8fd4\u56de\u6570\u7ec4\u4e2d\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u3002</p> <p>\u5bf9\u4e8e\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u5148\u4ecb\u7ecd\u4e24\u79cd\u601d\u8def\u6bd4\u8f83\u76f4\u63a5\u7684\u89e3\u6cd5\uff0c\u518d\u4ecb\u7ecd\u6548\u7387\u66f4\u9ad8\u7684\u5806\u89e3\u6cd5\u3002</p>"},{"location":"chapter_heap/top_k/#831","title":"8.3.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u904d\u5386\u9009\u62e9","text":"<p>\u6211\u4eec\u53ef\u4ee5\u8fdb\u884c\u56fe 8-6 \u6240\u793a\u7684 \\(k\\) \u8f6e\u904d\u5386\uff0c\u5206\u522b\u5728\u6bcf\u8f6e\u4e2d\u63d0\u53d6\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(\\dots\\)\u3001\\(k\\) \u5927\u7684\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nk)\\) \u3002</p> <p>\u6b64\u65b9\u6cd5\u53ea\u9002\u7528\u4e8e \\(k \\ll n\\) \u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u5f53 \\(k\\) \u4e0e \\(n\\) \u6bd4\u8f83\u63a5\u8fd1\u65f6\uff0c\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411\u4e8e \\(O(n^2)\\) \uff0c\u975e\u5e38\u8017\u65f6\u3002</p> <p></p> <p> \u56fe 8-6 \u00a0 \u904d\u5386\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 </p> <p>Tip</p> <p>\u5f53 \\(k = n\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5b8c\u6574\u7684\u6709\u5e8f\u5e8f\u5217\uff0c\u6b64\u65f6\u7b49\u4ef7\u4e8e\u201c\u9009\u62e9\u6392\u5e8f\u201d\u7b97\u6cd5\u3002</p>"},{"location":"chapter_heap/top_k/#832","title":"8.3.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u6392\u5e8f","text":"<p>\u5982\u56fe 8-7 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u5bf9\u6570\u7ec4 <code>nums</code> \u8fdb\u884c\u6392\u5e8f\uff0c\u518d\u8fd4\u56de\u6700\u53f3\u8fb9\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p> <p>\u663e\u7136\uff0c\u8be5\u65b9\u6cd5\u201c\u8d85\u989d\u201d\u5b8c\u6210\u4efb\u52a1\u4e86\uff0c\u56e0\u4e3a\u6211\u4eec\u53ea\u9700\u627e\u51fa\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u5373\u53ef\uff0c\u800c\u4e0d\u9700\u8981\u6392\u5e8f\u5176\u4ed6\u5143\u7d20\u3002</p> <p></p> <p> \u56fe 8-7 \u00a0 \u6392\u5e8f\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 </p>"},{"location":"chapter_heap/top_k/#833","title":"8.3.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u5806","text":"<p>\u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u5806\u66f4\u52a0\u9ad8\u6548\u5730\u89e3\u51b3 Top-k \u95ee\u9898\uff0c\u6d41\u7a0b\u5982\u56fe 8-8 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5176\u5806\u9876\u5143\u7d20\u6700\u5c0f\u3002</li> <li>\u5148\u5c06\u6570\u7ec4\u7684\u524d \\(k\\) \u4e2a\u5143\u7d20\u4f9d\u6b21\u5165\u5806\u3002</li> <li>\u4ece\u7b2c \\(k + 1\\) \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\uff0c\u5e76\u5c06\u5f53\u524d\u5143\u7d20\u5165\u5806\u3002</li> <li>\u904d\u5386\u5b8c\u6210\u540e\uff0c\u5806\u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 8-8 \u00a0 \u57fa\u4e8e\u5806\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 </p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig top_k.py<pre><code>def top_k_heap(nums: list[int], k: int) -&gt; list[int]:\n    \"\"\"\u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\"\"\"\n    # \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    heap = []\n    # \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for i in range(k):\n        heapq.heappush(heap, nums[i])\n    # \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for i in range(k, len(nums)):\n        # \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if nums[i] &gt; heap[0]:\n            heapq.heappop(heap)\n            heapq.heappush(heap, nums[i])\n    return heap\n</code></pre> top_k.cpp<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; topKHeap(vector&lt;int&gt; &amp;nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    priority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; heap;\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        heap.push(nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; nums.size(); i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.top()) {\n            heap.pop();\n            heap.push(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.java<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nQueue&lt;Integer&gt; topKHeap(int[] nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    Queue&lt;Integer&gt; heap = new PriorityQueue&lt;Integer&gt;();\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        heap.offer(nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; nums.length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll();\n            heap.offer(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.cs<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nPriorityQueue&lt;int, int&gt; TopKHeap(int[] nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    PriorityQueue&lt;int, int&gt; heap = new();\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        heap.Enqueue(nums[i], nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; nums.Length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.Peek()) {\n            heap.Dequeue();\n            heap.Enqueue(nums[i], nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.go<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums []int, k int) *minHeap {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    h := &amp;minHeap{}\n    heap.Init(h)\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for i := 0; i &lt; k; i++ {\n        heap.Push(h, nums[i])\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for i := k; i &lt; len(nums); i++ {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if nums[i] &gt; h.Top().(int) {\n            heap.Pop(h)\n            heap.Push(h, nums[i])\n        }\n    }\n    return h\n}\n</code></pre> top_k.swift<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums: [Int], k: Int) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5e76\u5c06\u524d k \u4e2a\u5143\u7d20\u5efa\u5806\n    var heap = Heap(nums.prefix(k))\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for i in nums.indices.dropFirst(k) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if nums[i] &gt; heap.min()! {\n            _ = heap.removeMin()\n            heap.insert(nums[i])\n        }\n    }\n    return heap.unordered\n}\n</code></pre> top_k.js<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap, val) {\n    // \u5143\u7d20\u53d6\u53cd\n    maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.pop();\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return maxHeap.getMaxHeap().map((num) =&gt; -num);\n}\n\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums, k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n    const maxHeap = new MaxHeap([]);\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (let i = k; i &lt; nums.length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.ts<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap: MaxHeap, val: number): void {\n    // \u5143\u7d20\u53d6\u53cd\n    maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap: MaxHeap): number {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.pop();\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap: MaxHeap): number {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap: MaxHeap): number[] {\n    // \u5143\u7d20\u53d6\u53cd\n    return maxHeap.getMaxHeap().map((num: number) =&gt; -num);\n}\n\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums: number[], k: number): number[] {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n    const maxHeap = new MaxHeap([]);\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (let i = k; i &lt; nums.length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.dart<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nMinHeap topKHeap(List&lt;int&gt; nums, int k) {\n  // \u521d\u59cb\u5316\u5c0f\u9876\u5806\uff0c\u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n  MinHeap heap = MinHeap(nums.sublist(0, k));\n  // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n  for (int i = k; i &lt; nums.length; i++) {\n    // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n    if (nums[i] &gt; heap.peek()) {\n      heap.pop();\n      heap.push(nums[i]);\n    }\n  }\n  return heap;\n}\n</code></pre> top_k.rs<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfn top_k_heap(nums: Vec&lt;i32&gt;, k: usize) -&gt; BinaryHeap&lt;Reverse&lt;i32&gt;&gt; {\n    // BinaryHeap \u662f\u5927\u9876\u5806\uff0c\u4f7f\u7528 Reverse \u5c06\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u5b9e\u73b0\u5c0f\u9876\u5806\n    let mut heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for &amp;num in nums.iter().take(k) {\n        heap.push(Reverse(num));\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for &amp;num in nums.iter().skip(k) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if num &gt; heap.peek().unwrap().0 {\n            heap.pop();\n            heap.push(Reverse(num));\n        }\n    }\n    heap\n}\n</code></pre> top_k.c<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid pushMinHeap(MaxHeap *maxHeap, int val) {\n    // \u5143\u7d20\u53d6\u53cd\n    push(maxHeap, -val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nint popMinHeap(MaxHeap *maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -pop(maxHeap);\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peekMinHeap(MaxHeap *maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -peek(maxHeap);\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n// \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\u7684\u51fd\u6570\nint *topKHeap(int *nums, int sizeNums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n    int *empty = (int *)malloc(0);\n    MaxHeap *maxHeap = newMaxHeap(empty, 0);\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; sizeNums; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    int *res = getMinHeap(maxHeap);\n    // \u91ca\u653e\u5185\u5b58\n    delMaxHeap(maxHeap);\n    return res;\n}\n</code></pre> top_k.kt<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfun topKHeap(nums: IntArray, k: Int): Queue&lt;Int&gt; {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    val heap = PriorityQueue&lt;Int&gt;()\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (i in 0..&lt;k) {\n        heap.offer(nums[i])\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (i in k..&lt;nums.size) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll()\n            heap.offer(nums[i])\n        }\n    }\n    return heap\n}\n</code></pre> top_k.rb<pre><code>[class]{}-[func]{top_k_heap}\n</code></pre> top_k.zig<pre><code>[class]{}-[func]{topKHeap}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u603b\u5171\u6267\u884c\u4e86 \\(n\\) \u8f6e\u5165\u5806\u548c\u51fa\u5806\uff0c\u5806\u7684\u6700\u5927\u957f\u5ea6\u4e3a \\(k\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002\u8be5\u65b9\u6cd5\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u5f53 \\(k\\) \u8f83\u5c0f\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \uff1b\u5f53 \\(k\\) \u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(O(n \\log n)\\) \u3002</p> <p>\u53e6\u5916\uff0c\u8be5\u65b9\u6cd5\u9002\u7528\u4e8e\u52a8\u6001\u6570\u636e\u6d41\u7684\u4f7f\u7528\u573a\u666f\u3002\u5728\u4e0d\u65ad\u52a0\u5165\u6570\u636e\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u6301\u7eed\u7ef4\u62a4\u5806\u5185\u7684\u5143\u7d20\uff0c\u4ece\u800c\u5b9e\u73b0\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u7684\u52a8\u6001\u66f4\u65b0\u3002</p>"},{"location":"chapter_introduction/","title":"\u7b2c 1 \u7ae0 \u00a0 \u521d\u8bc6\u7b97\u6cd5","text":"<p>Abstract</p> <p>\u4e00\u4f4d\u5c11\u5973\u7fe9\u7fe9\u8d77\u821e\uff0c\u4e0e\u6570\u636e\u4ea4\u7ec7\u5728\u4e00\u8d77\uff0c\u88d9\u6446\u4e0a\u98d8\u626c\u7740\u7b97\u6cd5\u7684\u65cb\u5f8b\u3002</p> <p>\u5979\u9080\u8bf7\u4f60\u5171\u821e\uff0c\u8bf7\u7d27\u8ddf\u5979\u7684\u6b65\u4f10\uff0c\u8e0f\u5165\u5145\u6ee1\u903b\u8f91\u4e0e\u7f8e\u611f\u7684\u7b97\u6cd5\u4e16\u754c\u3002</p> <p>\u8ba1\u7b97\u673a\u7684\u51fa\u73b0\u7ed9\u4e16\u754c\u5e26\u6765\u4e86\u5de8\u5927\u53d8\u9769\uff0c\u5b83\u51ed\u501f\u9ad8\u901f\u7684\u8ba1\u7b97\u80fd\u529b\u548c\u51fa\u8272\u7684\u53ef\u7f16\u7a0b\u6027\uff0c\u6210\u4e3a\u4e86\u6267\u884c\u7b97\u6cd5\u4e0e\u5904\u7406\u6570\u636e\u7684\u7406\u60f3\u5a92\u4ecb\u3002\u65e0\u8bba\u662f\u7535\u5b50\u6e38\u620f\u7684\u903c\u771f\u753b\u9762\u3001\u81ea\u52a8\u9a7e\u9a76\u7684\u667a\u80fd\u51b3\u7b56\uff0c\u8fd8\u662f AlphaGo \u7684\u7cbe\u5f69\u68cb\u5c40\u3001ChatGPT \u7684\u81ea\u7136\u4ea4\u4e92\uff0c\u8fd9\u4e9b\u5e94\u7528\u90fd\u662f\u7b97\u6cd5\u5728\u8ba1\u7b97\u673a\u4e0a\u7684\u7cbe\u5999\u6f14\u7ece\u3002</p> <p>\u4e8b\u5b9e\u4e0a\uff0c\u5728\u8ba1\u7b97\u673a\u95ee\u4e16\u4e4b\u524d\uff0c\u7b97\u6cd5\u548c\u6570\u636e\u7ed3\u6784\u5c31\u5df2\u7ecf\u5b58\u5728\u4e8e\u4e16\u754c\u7684\u5404\u4e2a\u89d2\u843d\u3002\u65e9\u671f\u7684\u7b97\u6cd5\u76f8\u5bf9\u7b80\u5355\uff0c\u4f8b\u5982\u53e4\u4ee3\u7684\u8ba1\u6570\u65b9\u6cd5\u548c\u5de5\u5177\u5236\u4f5c\u6b65\u9aa4\u7b49\u3002\u968f\u7740\u6587\u660e\u7684\u8fdb\u6b65\uff0c\u7b97\u6cd5\u9010\u6e10\u53d8\u5f97\u66f4\u52a0\u7cbe\u7ec6\u548c\u590d\u6742\u3002\u4ece\u5de7\u593a\u5929\u5de5\u7684\u5320\u4eba\u6280\u827a\u3001\u5230\u89e3\u653e\u751f\u4ea7\u529b\u7684\u5de5\u4e1a\u4ea7\u54c1\u3001\u518d\u5230\u5b87\u5b99\u8fd0\u884c\u7684\u79d1\u5b66\u89c4\u5f8b\uff0c\u51e0\u4e4e\u6bcf\u4e00\u4ef6\u5e73\u51e1\u6216\u4ee4\u4eba\u60ca\u53f9\u7684\u4e8b\u7269\u80cc\u540e\uff0c\u90fd\u9690\u85cf\u7740\u7cbe\u5999\u7684\u7b97\u6cd5\u601d\u60f3\u3002</p> <p>\u540c\u6837\uff0c\u6570\u636e\u7ed3\u6784\u65e0\u5904\u4e0d\u5728\uff1a\u5927\u5230\u793e\u4f1a\u7f51\u7edc\uff0c\u5c0f\u5230\u5730\u94c1\u7ebf\u8def\uff0c\u8bb8\u591a\u7cfb\u7edf\u90fd\u53ef\u4ee5\u5efa\u6a21\u4e3a\u201c\u56fe\u201d\uff1b\u5927\u5230\u4e00\u4e2a\u56fd\u5bb6\uff0c\u5c0f\u5230\u4e00\u4e2a\u5bb6\u5ead\uff0c\u793e\u4f1a\u7684\u4e3b\u8981\u7ec4\u7ec7\u5f62\u5f0f\u5448\u73b0\u51fa\u201c\u6811\u201d\u7684\u7279\u5f81\uff1b\u51ac\u5929\u7684\u8863\u670d\u5c31\u50cf\u201c\u6808\u201d\uff0c\u6700\u5148\u7a7f\u4e0a\u7684\u6700\u540e\u624d\u80fd\u8131\u4e0b\uff1b\u7fbd\u6bdb\u7403\u7b52\u5219\u5982\u540c\u201c\u961f\u5217\u201d\uff0c\u4e00\u7aef\u653e\u5165\u3001\u53e6\u4e00\u7aef\u53d6\u51fa\uff1b\u5b57\u5178\u5c31\u50cf\u4e00\u4e2a\u201c\u54c8\u5e0c\u8868\u201d\uff0c\u80fd\u591f\u5feb\u901f\u67e5\u627e\u76ee\u6807\u8bcd\u6761\u3002</p> <p>\u90a3\u4e48\uff0c\u4ec0\u4e48\u662f\u7b97\u6cd5\uff0c\u5b83\u5728\u6211\u4eec\u65e5\u5e38\u751f\u6d3b\u4e2d\u626e\u6f14\u7740\u600e\u6837\u7684\u89d2\u8272\uff1f\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u5c31\u8fd9\u4e9b\u95ee\u9898\u5c55\u5f00\u63a2\u8ba8\u3002</p>"},{"location":"chapter_introduction/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>1.1 \u00a0 \u7b97\u6cd5\u65e0\u5904\u4e0d\u5728</li> <li>1.2 \u00a0 \u7b97\u6cd5\u662f\u4ec0\u4e48</li> <li>1.3 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_introduction/algorithms_are_everywhere/","title":"1.1 \u00a0 \u7b97\u6cd5\u65e0\u5904\u4e0d\u5728","text":"<p>\u5f53\u6211\u4eec\u542c\u5230\u201c\u7b97\u6cd5\u201d\u8fd9\u4e2a\u8bcd\u65f6\uff0c\u5f88\u81ea\u7136\u5730\u4f1a\u60f3\u5230\u6570\u5b66\u3002\u7136\u800c\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7b97\u6cd5\u5e76\u4e0d\u6d89\u53ca\u590d\u6742\u6570\u5b66\uff0c\u800c\u662f\u66f4\u591a\u5730\u4f9d\u8d56\u57fa\u672c\u903b\u8f91\uff0c\u8fd9\u4e9b\u903b\u8f91\u5728\u6211\u4eec\u7684\u65e5\u5e38\u751f\u6d3b\u4e2d\u5904\u5904\u53ef\u89c1\u3002</p> <p>\u5728\u6b63\u5f0f\u63a2\u8ba8\u7b97\u6cd5\u4e4b\u524d\uff0c\u6709\u4e00\u4e2a\u6709\u8da3\u7684\u4e8b\u5b9e\u503c\u5f97\u5206\u4eab\uff1a\u4f60\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u5e76\u4e60\u60ef\u5c06\u5b83\u4eec\u5e94\u7528\u5230\u65e5\u5e38\u751f\u6d3b\u4e2d\u4e86\u3002\u4e0b\u9762\u6211\u5c06\u4e3e\u51e0\u4e2a\u5177\u4f53\u7684\u4f8b\u5b50\u6765\u8bc1\u5b9e\u8fd9\u4e00\u70b9\u3002</p> <p>\u4f8b\u4e00\uff1a\u67e5\u5b57\u5178\u3002\u5728\u5b57\u5178\u91cc\uff0c\u6bcf\u4e2a\u6c49\u5b57\u90fd\u5bf9\u5e94\u4e00\u4e2a\u62fc\u97f3\uff0c\u800c\u5b57\u5178\u662f\u6309\u7167\u62fc\u97f3\u5b57\u6bcd\u987a\u5e8f\u6392\u5217\u7684\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u67e5\u627e\u4e00\u4e2a\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u5b57\uff0c\u901a\u5e38\u4f1a\u6309\u7167\u56fe 1-1 \u6240\u793a\u7684\u65b9\u5f0f\u5b9e\u73b0\u3002</p> <ol> <li>\u7ffb\u5f00\u5b57\u5178\u7ea6\u4e00\u534a\u7684\u9875\u6570\uff0c\u67e5\u770b\u8be5\u9875\u7684\u9996\u5b57\u6bcd\u662f\u4ec0\u4e48\uff0c\u5047\u8bbe\u9996\u5b57\u6bcd\u4e3a \\(m\\) \u3002</li> <li>\u7531\u4e8e\u5728\u62fc\u97f3\u5b57\u6bcd\u8868\u4e2d \\(r\\) \u4f4d\u4e8e \\(m\\) \u4e4b\u540e\uff0c\u6240\u4ee5\u6392\u9664\u5b57\u5178\u524d\u534a\u90e8\u5206\uff0c\u67e5\u627e\u8303\u56f4\u7f29\u5c0f\u5230\u540e\u534a\u90e8\u5206\u3002</li> <li>\u4e0d\u65ad\u91cd\u590d\u6b65\u9aa4 <code>1.</code> \u548c \u6b65\u9aa4 <code>2.</code> \uff0c\u76f4\u81f3\u627e\u5230\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u9875\u7801\u4e3a\u6b62\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 1-1 \u00a0 \u67e5\u5b57\u5178\u6b65\u9aa4 </p> <p>\u67e5\u5b57\u5178\u8fd9\u4e2a\u5c0f\u5b66\u751f\u5fc5\u5907\u6280\u80fd\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u8457\u540d\u7684\u201c\u4e8c\u5206\u67e5\u627e\u201d\u7b97\u6cd5\u3002\u4ece\u6570\u636e\u7ed3\u6784\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b57\u5178\u89c6\u4e3a\u4e00\u4e2a\u5df2\u6392\u5e8f\u7684\u201c\u6570\u7ec4\u201d\uff1b\u4ece\u7b97\u6cd5\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4e0a\u8ff0\u67e5\u5b57\u5178\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u770b\u4f5c\u201c\u4e8c\u5206\u67e5\u627e\u201d\u3002</p> <p>\u4f8b\u4e8c\uff1a\u6574\u7406\u6251\u514b\u3002\u6211\u4eec\u5728\u6253\u724c\u65f6\uff0c\u6bcf\u5c40\u90fd\u9700\u8981\u6574\u7406\u624b\u4e2d\u7684\u6251\u514b\u724c\uff0c\u4f7f\u5176\u4ece\u5c0f\u5230\u5927\u6392\u5217\uff0c\u5b9e\u73b0\u6d41\u7a0b\u5982\u56fe 1-2 \u6240\u793a\u3002</p> <ol> <li>\u5c06\u6251\u514b\u724c\u5212\u5206\u4e3a\u201c\u6709\u5e8f\u201d\u548c\u201c\u65e0\u5e8f\u201d\u4e24\u90e8\u5206\uff0c\u5e76\u5047\u8bbe\u521d\u59cb\u72b6\u6001\u4e0b\u6700\u5de6 1 \u5f20\u6251\u514b\u724c\u5df2\u7ecf\u6709\u5e8f\u3002</li> <li>\u5728\u65e0\u5e8f\u90e8\u5206\u62bd\u51fa\u4e00\u5f20\u6251\u514b\u724c\uff0c\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\uff1b\u5b8c\u6210\u540e\u6700\u5de6 2 \u5f20\u6251\u514b\u5df2\u7ecf\u6709\u5e8f\u3002</li> <li>\u4e0d\u65ad\u5faa\u73af\u6b65\u9aa4 <code>2.</code> \uff0c\u6bcf\u4e00\u8f6e\u5c06\u4e00\u5f20\u6251\u514b\u724c\u4ece\u65e0\u5e8f\u90e8\u5206\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\uff0c\u76f4\u81f3\u6240\u6709\u6251\u514b\u724c\u90fd\u6709\u5e8f\u3002</li> </ol> <p></p> <p> \u56fe 1-2 \u00a0 \u6251\u514b\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u4e0a\u8ff0\u6574\u7406\u6251\u514b\u724c\u7684\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u201c\u63d2\u5165\u6392\u5e8f\u201d\u7b97\u6cd5\uff0c\u5b83\u5728\u5904\u7406\u5c0f\u578b\u6570\u636e\u96c6\u65f6\u975e\u5e38\u9ad8\u6548\u3002\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u7684\u6392\u5e8f\u5e93\u51fd\u6570\u4e2d\u90fd\u6709\u63d2\u5165\u6392\u5e8f\u7684\u8eab\u5f71\u3002</p> <p>\u4f8b\u4e09\uff1a\u8d27\u5e01\u627e\u96f6\u3002\u5047\u8bbe\u6211\u4eec\u5728\u8d85\u5e02\u8d2d\u4e70\u4e86 \\(69\\) \u5143\u7684\u5546\u54c1\uff0c\u7ed9\u4e86\u6536\u94f6\u5458 \\(100\\) \u5143\uff0c\u5219\u6536\u94f6\u5458\u9700\u8981\u627e\u6211\u4eec \\(31\\) \u5143\u3002\u4ed6\u4f1a\u5f88\u81ea\u7136\u5730\u5b8c\u6210\u5982\u56fe 1-3 \u6240\u793a\u7684\u601d\u8003\u3002</p> <ol> <li>\u53ef\u9009\u9879\u662f\u6bd4 \\(31\\) \u5143\u9762\u503c\u66f4\u5c0f\u7684\u8d27\u5e01\uff0c\u5305\u62ec \\(1\\) \u5143\u3001\\(5\\) \u5143\u3001\\(10\\) \u5143\u3001\\(20\\) \u5143\u3002</li> <li>\u4ece\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(20\\) \u5143\uff0c\u5269\u4f59 \\(31 - 20 = 11\\) \u5143\u3002</li> <li>\u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(10\\) \u5143\uff0c\u5269\u4f59 \\(11 - 10 = 1\\) \u5143\u3002</li> <li>\u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(1\\) \u5143\uff0c\u5269\u4f59 \\(1 - 1 = 0\\) \u5143\u3002</li> <li>\u5b8c\u6210\u627e\u96f6\uff0c\u65b9\u6848\u4e3a \\(20 + 10 + 1 = 31\\) \u5143\u3002</li> </ol> <p></p> <p> \u56fe 1-3 \u00a0 \u8d27\u5e01\u627e\u96f6\u8fc7\u7a0b </p> <p>\u5728\u4ee5\u4e0a\u6b65\u9aa4\u4e2d\uff0c\u6211\u4eec\u6bcf\u4e00\u6b65\u90fd\u91c7\u53d6\u5f53\u524d\u770b\u6765\u6700\u597d\u7684\u9009\u62e9\uff08\u5c3d\u53ef\u80fd\u7528\u5927\u9762\u989d\u7684\u8d27\u5e01\uff09\uff0c\u6700\u7ec8\u5f97\u5230\u4e86\u53ef\u884c\u7684\u627e\u96f6\u65b9\u6848\u3002\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u201c\u8d2a\u5fc3\u201d\u7b97\u6cd5\u3002</p> <p>\u5c0f\u5230\u70f9\u996a\u4e00\u9053\u83dc\uff0c\u5927\u5230\u661f\u9645\u822a\u884c\uff0c\u51e0\u4e4e\u6240\u6709\u95ee\u9898\u7684\u89e3\u51b3\u90fd\u79bb\u4e0d\u5f00\u7b97\u6cd5\u3002\u8ba1\u7b97\u673a\u7684\u51fa\u73b0\u4f7f\u5f97\u6211\u4eec\u80fd\u591f\u901a\u8fc7\u7f16\u7a0b\u5c06\u6570\u636e\u7ed3\u6784\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\uff0c\u540c\u65f6\u7f16\u5199\u4ee3\u7801\u8c03\u7528 CPU \u548c GPU \u6267\u884c\u7b97\u6cd5\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u6211\u4eec\u5c31\u80fd\u628a\u751f\u6d3b\u4e2d\u7684\u95ee\u9898\u8f6c\u79fb\u5230\u8ba1\u7b97\u673a\u4e0a\uff0c\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u89e3\u51b3\u5404\u79cd\u590d\u6742\u95ee\u9898\u3002</p> <p>Tip</p> <p>\u5982\u679c\u4f60\u5bf9\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u3001\u6570\u7ec4\u548c\u4e8c\u5206\u67e5\u627e\u7b49\u6982\u5ff5\u4ecd\u611f\u5230\u4e00\u77e5\u534a\u89e3\uff0c\u8bf7\u7ee7\u7eed\u5f80\u4e0b\u9605\u8bfb\uff0c\u672c\u4e66\u5c06\u5f15\u5bfc\u4f60\u8fc8\u5165\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u77e5\u8bc6\u6bbf\u5802\u3002</p>"},{"location":"chapter_introduction/summary/","title":"1.3 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u7b97\u6cd5\u5728\u65e5\u5e38\u751f\u6d3b\u4e2d\u65e0\u5904\u4e0d\u5728\uff0c\u5e76\u4e0d\u662f\u9065\u4e0d\u53ef\u53ca\u7684\u9ad8\u6df1\u77e5\u8bc6\u3002\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u7528\u4ee5\u89e3\u51b3\u751f\u6d3b\u4e2d\u7684\u5927\u5c0f\u95ee\u9898\u3002</li> <li>\u67e5\u5b57\u5178\u7684\u539f\u7406\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u76f8\u4e00\u81f4\u3002\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u4f53\u73b0\u4e86\u5206\u800c\u6cbb\u4e4b\u7684\u91cd\u8981\u7b97\u6cd5\u601d\u60f3\u3002</li> <li>\u6574\u7406\u6251\u514b\u7684\u8fc7\u7a0b\u4e0e\u63d2\u5165\u6392\u5e8f\u7b97\u6cd5\u975e\u5e38\u7c7b\u4f3c\u3002\u63d2\u5165\u6392\u5e8f\u7b97\u6cd5\u9002\u5408\u6392\u5e8f\u5c0f\u578b\u6570\u636e\u96c6\u3002</li> <li>\u8d27\u5e01\u627e\u96f6\u7684\u6b65\u9aa4\u672c\u8d28\u4e0a\u662f\u8d2a\u5fc3\u7b97\u6cd5\uff0c\u6bcf\u4e00\u6b65\u90fd\u91c7\u53d6\u5f53\u524d\u770b\u6765\u6700\u597d\u7684\u9009\u62e9\u3002</li> <li>\u7b97\u6cd5\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u800c\u6570\u636e\u7ed3\u6784\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002</li> <li>\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7d27\u5bc6\u76f8\u8fde\u3002\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\uff0c\u800c\u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u4f5c\u7528\u7684\u821e\u53f0\u3002</li> <li>\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7c7b\u6bd4\u4e3a\u62fc\u88c5\u79ef\u6728\uff0c\u79ef\u6728\u4ee3\u8868\u6570\u636e\uff0c\u79ef\u6728\u7684\u5f62\u72b6\u548c\u8fde\u63a5\u65b9\u5f0f\u7b49\u4ee3\u8868\u6570\u636e\u7ed3\u6784\uff0c\u62fc\u88c5\u79ef\u6728\u7684\u6b65\u9aa4\u5219\u5bf9\u5e94\u7b97\u6cd5\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/","title":"1.2 \u00a0 \u7b97\u6cd5\u662f\u4ec0\u4e48","text":""},{"location":"chapter_introduction/what_is_dsa/#121","title":"1.2.1 \u00a0 \u7b97\u6cd5\u5b9a\u4e49","text":"<p>\u300c\u7b97\u6cd5 algorithm\u300d\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u5b83\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u95ee\u9898\u662f\u660e\u786e\u7684\uff0c\u5305\u542b\u6e05\u6670\u7684\u8f93\u5165\u548c\u8f93\u51fa\u5b9a\u4e49\u3002</li> <li>\u5177\u6709\u53ef\u884c\u6027\uff0c\u80fd\u591f\u5728\u6709\u9650\u6b65\u9aa4\u3001\u65f6\u95f4\u548c\u5185\u5b58\u7a7a\u95f4\u4e0b\u5b8c\u6210\u3002</li> <li>\u5404\u6b65\u9aa4\u90fd\u6709\u786e\u5b9a\u7684\u542b\u4e49\uff0c\u5728\u76f8\u540c\u7684\u8f93\u5165\u548c\u8fd0\u884c\u6761\u4ef6\u4e0b\uff0c\u8f93\u51fa\u59cb\u7ec8\u76f8\u540c\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#122","title":"1.2.2 \u00a0 \u6570\u636e\u7ed3\u6784\u5b9a\u4e49","text":"<p>\u300c\u6570\u636e\u7ed3\u6784 data structure\u300d\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\uff0c\u5177\u6709\u4ee5\u4e0b\u8bbe\u8ba1\u76ee\u6807\u3002</p> <ul> <li>\u7a7a\u95f4\u5360\u7528\u5c3d\u91cf\u5c11\uff0c\u4ee5\u8282\u7701\u8ba1\u7b97\u673a\u5185\u5b58\u3002</li> <li>\u6570\u636e\u64cd\u4f5c\u5c3d\u53ef\u80fd\u5feb\u901f\uff0c\u6db5\u76d6\u6570\u636e\u8bbf\u95ee\u3001\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66f4\u65b0\u7b49\u3002</li> <li>\u63d0\u4f9b\u7b80\u6d01\u7684\u6570\u636e\u8868\u793a\u548c\u903b\u8f91\u4fe1\u606f\uff0c\u4ee5\u4fbf\u7b97\u6cd5\u9ad8\u6548\u8fd0\u884c\u3002</li> </ul> <p>\u6570\u636e\u7ed3\u6784\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5145\u6ee1\u6743\u8861\u7684\u8fc7\u7a0b\u3002\u5982\u679c\u60f3\u5728\u67d0\u65b9\u9762\u53d6\u5f97\u63d0\u5347\uff0c\u5f80\u5f80\u9700\u8981\u5728\u53e6\u4e00\u65b9\u9762\u4f5c\u51fa\u59a5\u534f\u3002\u4e0b\u9762\u4e3e\u4e24\u4e2a\u4f8b\u5b50\u3002</p> <ul> <li>\u94fe\u8868\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5728\u6570\u636e\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u4e0a\u66f4\u52a0\u4fbf\u6377\uff0c\u4f46\u727a\u7272\u4e86\u6570\u636e\u8bbf\u95ee\u901f\u5ea6\u3002</li> <li>\u56fe\u76f8\u8f83\u4e8e\u94fe\u8868\uff0c\u63d0\u4f9b\u4e86\u66f4\u4e30\u5bcc\u7684\u903b\u8f91\u4fe1\u606f\uff0c\u4f46\u9700\u8981\u5360\u7528\u66f4\u5927\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#123","title":"1.2.3 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb","text":"<p>\u5982\u56fe 1-4 \u6240\u793a\uff0c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u9ad8\u5ea6\u76f8\u5173\u3001\u7d27\u5bc6\u7ed3\u5408\uff0c\u5177\u4f53\u8868\u73b0\u5728\u4ee5\u4e0b\u4e09\u4e2a\u65b9\u9762\u3002</p> <ul> <li>\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\u3002\u6570\u636e\u7ed3\u6784\u4e3a\u7b97\u6cd5\u63d0\u4f9b\u4e86\u7ed3\u6784\u5316\u5b58\u50a8\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u64cd\u4f5c\u6570\u636e\u7684\u65b9\u6cd5\u3002</li> <li>\u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u4f5c\u7528\u7684\u821e\u53f0\u3002\u6570\u636e\u7ed3\u6784\u672c\u8eab\u4ec5\u5b58\u50a8\u6570\u636e\u4fe1\u606f\uff0c\u7ed3\u5408\u7b97\u6cd5\u624d\u80fd\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u3002</li> <li>\u7b97\u6cd5\u901a\u5e38\u53ef\u4ee5\u57fa\u4e8e\u4e0d\u540c\u7684\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff0c\u4f46\u6267\u884c\u6548\u7387\u53ef\u80fd\u76f8\u5dee\u5f88\u5927\uff0c\u9009\u62e9\u5408\u9002\u7684\u6570\u636e\u7ed3\u6784\u662f\u5173\u952e\u3002</li> </ul> <p></p> <p> \u56fe 1-4 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb </p> <p>\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u72b9\u5982\u56fe 1-5 \u6240\u793a\u7684\u62fc\u88c5\u79ef\u6728\u3002\u4e00\u5957\u79ef\u6728\uff0c\u9664\u4e86\u5305\u542b\u8bb8\u591a\u96f6\u4ef6\u4e4b\u5916\uff0c\u8fd8\u9644\u6709\u8be6\u7ec6\u7684\u7ec4\u88c5\u8bf4\u660e\u4e66\u3002\u6211\u4eec\u6309\u7167\u8bf4\u660e\u4e66\u4e00\u6b65\u6b65\u64cd\u4f5c\uff0c\u5c31\u80fd\u7ec4\u88c5\u51fa\u7cbe\u7f8e\u7684\u79ef\u6728\u6a21\u578b\u3002</p> <p></p> <p> \u56fe 1-5 \u00a0 \u62fc\u88c5\u79ef\u6728 </p> <p>\u4e24\u8005\u7684\u8be6\u7ec6\u5bf9\u5e94\u5173\u7cfb\u5982\u8868 1-1 \u6240\u793a\u3002</p> <p> \u8868 1-1 \u00a0 \u5c06\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7c7b\u6bd4\u4e3a\u62fc\u88c5\u79ef\u6728 </p> \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5 \u62fc\u88c5\u79ef\u6728 \u8f93\u5165\u6570\u636e \u672a\u62fc\u88c5\u7684\u79ef\u6728 \u6570\u636e\u7ed3\u6784 \u79ef\u6728\u7ec4\u7ec7\u5f62\u5f0f\uff0c\u5305\u62ec\u5f62\u72b6\u3001\u5927\u5c0f\u3001\u8fde\u63a5\u65b9\u5f0f\u7b49 \u7b97\u6cd5 \u628a\u79ef\u6728\u62fc\u6210\u76ee\u6807\u5f62\u6001\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u6b65\u9aa4 \u8f93\u51fa\u6570\u636e \u79ef\u6728\u6a21\u578b <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u662f\u72ec\u7acb\u4e8e\u7f16\u7a0b\u8bed\u8a00\u7684\u3002\u6b63\u56e0\u5982\u6b64\uff0c\u672c\u4e66\u5f97\u4ee5\u63d0\u4f9b\u57fa\u4e8e\u591a\u79cd\u7f16\u7a0b\u8bed\u8a00\u7684\u5b9e\u73b0\u3002</p> <p>\u7ea6\u5b9a\u4fd7\u6210\u7684\u7b80\u79f0</p> <p>\u5728\u5b9e\u9645\u8ba8\u8bba\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u201c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u201d\u7b80\u79f0\u4e3a\u201c\u7b97\u6cd5\u201d\u3002\u6bd4\u5982\u4f17\u6240\u5468\u77e5\u7684 LeetCode \u7b97\u6cd5\u9898\u76ee\uff0c\u5b9e\u9645\u4e0a\u540c\u65f6\u8003\u67e5\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e24\u65b9\u9762\u7684\u77e5\u8bc6\u3002</p>"},{"location":"chapter_paperbook/","title":"\u7eb8\u8d28\u4e66","text":"<p>\u7ecf\u8fc7\u957f\u65f6\u95f4\u7684\u6253\u78e8\uff0c\u300aHello \u7b97\u6cd5\u300b\u7eb8\u8d28\u4e66\u7ec8\u4e8e\u53d1\u5e03\u4e86\uff01\u6b64\u65f6\u7684\u5fc3\u60c5\u53ef\u4ee5\u7528\u4e00\u53e5\u8bd7\u6765\u5f62\u5bb9\uff1a</p> <p>\u8ffd\u98ce\u8d76\u6708\u83ab\u505c\u7559\uff0c\u5e73\u829c\u5c3d\u5904\u662f\u6625\u5c71\u3002</p> <p></p> <p>\u4ee5\u4e0b\u89c6\u9891\u5c55\u793a\u4e86\u7eb8\u8d28\u4e66\uff0c\u5e76\u4e14\u5305\u542b\u6211\u7684\u4e00\u4e9b\u601d\u8003\uff1a</p> <ul> <li>\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u91cd\u8981\u6027\u3002</li> <li>\u4e3a\u4ec0\u4e48\u5728\u7eb8\u8d28\u4e66\u4e2d\u9009\u62e9 Python\u3002</li> <li>\u5bf9\u77e5\u8bc6\u5206\u4eab\u7684\u7406\u89e3\u3002</li> </ul> <p>\u65b0\u4eba UP \u4e3b\uff0c\u8bf7\u591a\u591a\u5173\u7167\u3001\u4e00\u952e\u4e09\u8fde\uff5e\u8c22\u8c22\uff01</p> <p>\u9644\u7eb8\u8d28\u4e66\u5feb\u7167\uff1a</p> <p></p> <p></p>"},{"location":"chapter_paperbook/#_2","title":"\u4f18\u52bf\u4e0e\u4e0d\u8db3","text":"<p>\u603b\u7ed3\u4e00\u4e0b\u7eb8\u8d28\u4e66\u53ef\u80fd\u4f1a\u7ed9\u5927\u5bb6\u5e26\u6765\u60ca\u559c\u7684\u5730\u65b9\uff1a</p> <ul> <li>\u91c7\u7528\u5168\u5f69\u5370\u5237\uff0c\u80fd\u591f\u539f\u6c41\u539f\u5473\u5730\u53d1\u6325\u51fa\u672c\u4e66\u201c\u52a8\u753b\u56fe\u89e3\u201d\u7684\u4f18\u52bf\u3002</li> <li>\u8003\u7a76\u7eb8\u5f20\u6750\u8d28\uff0c\u65e2\u4fdd\u8bc1\u8272\u5f69\u9ad8\u5ea6\u8fd8\u539f\uff0c\u4e5f\u4fdd\u7559\u7eb8\u8d28\u4e66\u7279\u6709\u7684\u8d28\u611f\u3002</li> <li>\u5728\u4e0d\u63d0\u5347\u5b9a\u4ef7\u7684\u524d\u63d0\u4e0b\uff0c\u9644\u8d60\u601d\u7ef4\u5bfc\u56fe\u6298\u9875\u3001\u4e66\u7b7e\u3002</li> <li>\u7eb8\u8d28\u4e66\u3001\u7f51\u9875\u7248\u3001PDF \u7248\u5185\u5bb9\u540c\u6b65\uff0c\u968f\u610f\u5207\u6362\u9605\u8bfb\u3002</li> </ul> <p>Tip</p> <p>\u7531\u4e8e\u7eb8\u8d28\u4e66\u548c\u7f51\u9875\u7248\u7684\u540c\u6b65\u6210\u672c\u8f83\u5927\uff0c\u56e0\u6b64\u53ef\u80fd\u4f1a\u6709\u4e00\u4e9b\u7ec6\u8282\u4e0a\u7684\u4e0d\u540c\uff0c\u8bf7\u60a8\u89c1\u8c05\uff01</p> <p>\u5f53\u7136\uff0c\u7eb8\u8d28\u4e66\u4e5f\u6709\u4e00\u4e9b\u503c\u5f97\u5927\u5bb6\u5165\u624b\u524d\u8003\u8651\u7684\u5730\u65b9\uff1a</p> <ul> <li>\u4f7f\u7528 Python \u8bed\u8a00\uff0c\u53ef\u80fd\u4e0d\u5339\u914d\u4f60\u7684\u4e3b\u8bed\u8a00\uff08\u4e5f\u8bb8\u53ef\u4ee5\u8d81\u6b64\u673a\u4f1a\u7ec3\u4e60 Python\uff09\u3002</li> <li>\u5168\u5f69\u5370\u5237\u867d\u7136\u5927\u5e45\u63d0\u5347\u4e86\u9605\u8bfb\u4f53\u9a8c\uff0c\u4f46\u4ef7\u683c\u4f1a\u6bd4\u9ed1\u767d\u5370\u5237\u9ad8\u4e00\u4e9b\u3002</li> </ul> <p>Tip</p> <p>\u201c\u5370\u5237\u8d28\u91cf\u201d\u548c\u201c\u4ef7\u683c\u201d\u5c31\u50cf\u7b97\u6cd5\u4e2d\u7684\u201c\u65f6\u95f4\u6548\u7387\u201d\u548c\u201c\u7a7a\u95f4\u6548\u7387\u201d\uff0c\u96be\u4ee5\u4e24\u5168\u3002\u800c\u6211\u8ba4\u4e3a\uff0c\u201c\u5370\u5237\u8d28\u91cf\u201d\u5bf9\u5e94\u7684\u662f\u201c\u65f6\u95f4\u6548\u7387\u201d\uff0c\u66f4\u5e94\u8be5\u88ab\u6ce8\u91cd\u3002</p>"},{"location":"chapter_paperbook/#_3","title":"\u8d2d\u4e70\u94fe\u63a5","text":"<p>\u5982\u679c\u4f60\u5bf9\u7eb8\u8d28\u4e66\u611f\u5174\u8da3\uff0c\u53ef\u4ee5\u8003\u8651\u5165\u624b\u4e00\u672c\u3002\u6211\u4eec\u4e3a\u5927\u5bb6\u4e89\u53d6\u5230\u4e86\u65b0\u4e66 5 \u6298\u4f18\u60e0\uff0c\u8bf7\u89c1\u6b64\u94fe\u63a5\u6216\u626b\u63cf\u4ee5\u4e0b\u4e8c\u7ef4\u7801\uff1a</p> <p></p>"},{"location":"chapter_paperbook/#_4","title":"\u5c3e\u8bb0","text":"<p>\u8d77\u521d\uff0c\u6211\u4f4e\u4f30\u4e86\u7eb8\u8d28\u4e66\u51fa\u7248\u7684\u5de5\u4f5c\u91cf\uff0c\u4ee5\u4e3a\u53ea\u8981\u7ef4\u62a4\u597d\u4e86\u5f00\u6e90\u9879\u76ee\uff0c\u7eb8\u8d28\u7248\u5c31\u53ef\u4ee5\u901a\u8fc7\u67d0\u4e9b\u81ea\u52a8\u5316\u624b\u6bb5\u751f\u6210\u51fa\u6765\u3002\u5b9e\u8df5\u8bc1\u660e\uff0c\u7eb8\u8d28\u4e66\u7684\u751f\u4ea7\u6d41\u7a0b\u4e0e\u5f00\u6e90\u9879\u76ee\u7684\u66f4\u65b0\u673a\u5236\u5b58\u5728\u5f88\u5927\u7684\u4e0d\u540c\uff0c\u4e24\u8005\u4e4b\u95f4\u7684\u8f6c\u5316\u9700\u8981\u505a\u8bb8\u591a\u989d\u5916\u5de5\u4f5c\u3002</p> <p>\u4e00\u672c\u4e66\u7684\u521d\u7a3f\u4e0e\u8fbe\u5230\u51fa\u7248\u6807\u51c6\u7684\u5b9a\u7a3f\u4e4b\u95f4\u4ecd\u6709\u8f83\u957f\u8ddd\u79bb\uff0c\u9700\u8981\u51fa\u7248\u793e\uff08\u7b56\u5212\u3001\u7f16\u8f91\u3001\u8bbe\u8ba1\u3001\u5e02\u573a\u7b49\uff09\u4e0e\u4f5c\u8005\u7684\u901a\u529b\u5408\u4f5c\u3001\u957f\u671f\u96d5\u7422\u3002\u5728\u6b64\u611f\u8c22\u56fe\u7075\u7b56\u5212\u7f16\u8f91\u738b\u519b\u82b1\u3001\u4ee5\u53ca\u4eba\u6c11\u90ae\u7535\u51fa\u7248\u793e\u548c\u56fe\u7075\u793e\u533a\u6bcf\u4f4d\u53c2\u4e0e\u672c\u4e66\u51fa\u7248\u6d41\u7a0b\u7684\u5de5\u4f5c\u4eba\u5458\uff01</p> <p>\u5e0c\u671b\u8fd9\u672c\u4e66\u80fd\u591f\u5e2e\u52a9\u5230\u4f60\uff01</p>"},{"location":"chapter_preface/","title":"\u7b2c 0 \u7ae0 \u00a0 \u524d\u8a00","text":"<p>Abstract</p> <p>\u7b97\u6cd5\u72b9\u5982\u7f8e\u5999\u7684\u4ea4\u54cd\u4e50\uff0c\u6bcf\u4e00\u884c\u4ee3\u7801\u90fd\u50cf\u97f5\u5f8b\u822c\u6d41\u6dcc\u3002</p> <p>\u613f\u8fd9\u672c\u4e66\u5728\u4f60\u7684\u8111\u6d77\u4e2d\u8f7b\u8f7b\u54cd\u8d77\uff0c\u7559\u4e0b\u72ec\u7279\u800c\u6df1\u523b\u7684\u65cb\u5f8b\u3002</p> <p>\u4e24\u5e74\u524d\uff0c\u6211\u5728\u529b\u6263\u4e0a\u5206\u4eab\u4e86\u201c\u5251\u6307 Offer\u201d\u7cfb\u5217\u9898\u89e3\uff0c\u53d7\u5230\u4e86\u8bb8\u591a\u8bfb\u8005\u7684\u9f13\u52b1\u548c\u652f\u6301\u3002\u5728\u4e0e\u8bfb\u8005\u4ea4\u6d41\u671f\u95f4\uff0c\u6211\u6700\u5e38\u88ab\u95ee\u7684\u4e00\u4e2a\u95ee\u9898\u662f\u201c\u5982\u4f55\u5165\u95e8\u7b97\u6cd5\u201d\u3002\u9010\u6e10\u5730\uff0c\u6211\u5bf9\u8fd9\u4e2a\u95ee\u9898\u4ea7\u751f\u4e86\u6d53\u539a\u7684\u5174\u8da3\u3002</p> <p>\u4e24\u773c\u4e00\u62b9\u9ed1\u5730\u5237\u9898\u4f3c\u4e4e\u662f\u6700\u53d7\u6b22\u8fce\u7684\u65b9\u6cd5\uff0c\u7b80\u5355\u3001\u76f4\u63a5\u4e14\u6709\u6548\u3002\u7136\u800c\u5237\u9898\u5c31\u5982\u540c\u73a9\u201c\u626b\u96f7\u201d\u6e38\u620f\uff0c\u81ea\u5b66\u80fd\u529b\u5f3a\u7684\u4eba\u80fd\u591f\u987a\u5229\u5c06\u5730\u96f7\u9010\u4e2a\u6392\u6389\uff0c\u800c\u57fa\u7840\u4e0d\u8db3\u7684\u4eba\u5f88\u53ef\u80fd\u88ab\u70b8\u5f97\u6ee1\u5934\u662f\u5305\uff0c\u5e76\u5728\u632b\u6298\u4e2d\u6b65\u6b65\u9000\u7f29\u3002\u901a\u8bfb\u6559\u6750\u4e5f\u662f\u4e00\u79cd\u5e38\u89c1\u505a\u6cd5\uff0c\u4f46\u5bf9\u4e8e\u9762\u5411\u6c42\u804c\u7684\u4eba\u6765\u8bf4\uff0c\u6bd5\u4e1a\u8bba\u6587\u3001\u6295\u9012\u7b80\u5386\u3001\u51c6\u5907\u7b14\u8bd5\u548c\u9762\u8bd5\u5df2\u7ecf\u6d88\u8017\u4e86\u5927\u90e8\u5206\u7cbe\u529b\uff0c\u5543\u539a\u91cd\u7684\u4e66\u5f80\u5f80\u53d8\u6210\u4e86\u4e00\u9879\u8270\u5de8\u7684\u6311\u6218\u3002</p> <p>\u5982\u679c\u4f60\u4e5f\u9762\u4e34\u7c7b\u4f3c\u7684\u56f0\u6270\uff0c\u90a3\u4e48\u5f88\u5e78\u8fd0\u8fd9\u672c\u4e66\u201c\u627e\u201d\u5230\u4e86\u4f60\u3002\u672c\u4e66\u662f\u6211\u5bf9\u8fd9\u4e2a\u95ee\u9898\u7ed9\u51fa\u7684\u7b54\u6848\uff0c\u5373\u4f7f\u4e0d\u662f\u6700\u4f18\u89e3\uff0c\u4e5f\u81f3\u5c11\u662f\u4e00\u6b21\u79ef\u6781\u7684\u5c1d\u8bd5\u3002\u672c\u4e66\u867d\u7136\u4e0d\u8db3\u4ee5\u8ba9\u4f60\u76f4\u63a5\u62ff\u5230 Offer\uff0c\u4f46\u4f1a\u5f15\u5bfc\u4f60\u63a2\u7d22\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u201c\u77e5\u8bc6\u5730\u56fe\u201d\uff0c\u5e26\u4f60\u4e86\u89e3\u4e0d\u540c\u201c\u5730\u96f7\u201d\u7684\u5f62\u72b6\u3001\u5927\u5c0f\u548c\u5206\u5e03\u4f4d\u7f6e\uff0c\u8ba9\u4f60\u638c\u63e1\u5404\u79cd\u201c\u6392\u96f7\u65b9\u6cd5\u201d\u3002\u6709\u4e86\u8fd9\u4e9b\u672c\u9886\uff0c\u76f8\u4fe1\u4f60\u53ef\u4ee5\u66f4\u52a0\u81ea\u5982\u5730\u5237\u9898\u548c\u9605\u8bfb\u6587\u732e\uff0c\u9010\u6b65\u6784\u5efa\u8d77\u5b8c\u6574\u7684\u77e5\u8bc6\u4f53\u7cfb\u3002</p> <p>\u6211\u6df1\u6df1\u8d5e\u540c\u8d39\u66fc\u6559\u6388\u6240\u8a00\uff1a\u201cKnowledge isn't free. You have to pay attention.\u201d\u4ece\u8fd9\u4e2a\u610f\u4e49\u4e0a\u770b\uff0c\u8fd9\u672c\u4e66\u5e76\u975e\u5b8c\u5168\u201c\u514d\u8d39\u201d\u3002\u4e3a\u4e86\u4e0d\u8f9c\u8d1f\u4f60\u4e3a\u672c\u4e66\u6240\u4ed8\u51fa\u7684\u5b9d\u8d35\u201c\u6ce8\u610f\u529b\u201d\uff0c\u6211\u4f1a\u7aed\u5c3d\u6240\u80fd\uff0c\u6295\u5165\u6700\u5927\u7684\u201c\u6ce8\u610f\u529b\u201d\u6765\u5b8c\u6210\u672c\u4e66\u7684\u521b\u4f5c\u3002</p>"},{"location":"chapter_preface/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>0.1 \u00a0 \u5173\u4e8e\u672c\u4e66</li> <li>0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66</li> <li>0.3 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_preface/about_the_book/","title":"0.1 \u00a0 \u5173\u4e8e\u672c\u4e66","text":"<p>\u672c\u9879\u76ee\u65e8\u5728\u521b\u5efa\u4e00\u672c\u5f00\u6e90\u3001\u514d\u8d39\u3001\u5bf9\u65b0\u624b\u53cb\u597d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u6559\u7a0b\u3002</p> <ul> <li>\u5168\u4e66\u91c7\u7528\u52a8\u753b\u56fe\u89e3\uff0c\u7ed3\u6784\u5316\u5730\u8bb2\u89e3\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u5185\u5bb9\u6e05\u6670\u6613\u61c2\uff0c\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1\u3002</li> <li>\u7b97\u6cd5\u6e90\u4ee3\u7801\u7686\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u652f\u6301 Python\u3001C++\u3001Java\u3001C#\u3001Go\u3001Swift\u3001JavaScript\u3001TypeScript\u3001Dart\u3001Rust\u3001C \u548c Zig \u7b49\u8bed\u8a00\u3002</li> <li>\u9f13\u52b1\u8bfb\u8005\u5728\u7ebf\u4e0a\u7ae0\u8282\u8bc4\u8bba\u533a\u4e92\u5e2e\u4e92\u52a9\u3001\u5171\u540c\u8fdb\u6b65\uff0c\u63d0\u95ee\u4e0e\u8bc4\u8bba\u901a\u5e38\u53ef\u5728\u4e24\u65e5\u5185\u5f97\u5230\u56de\u590d\u3002</li> </ul>"},{"location":"chapter_preface/about_the_book/#011","title":"0.1.1 \u00a0 \u8bfb\u8005\u5bf9\u8c61","text":"<p>\u82e5\u4f60\u662f\u7b97\u6cd5\u521d\u5b66\u8005\uff0c\u4ece\u672a\u63a5\u89e6\u8fc7\u7b97\u6cd5\uff0c\u6216\u8005\u5df2\u7ecf\u6709\u4e00\u4e9b\u5237\u9898\u7ecf\u9a8c\uff0c\u5bf9\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6709\u6a21\u7cca\u7684\u8ba4\u8bc6\uff0c\u5728\u4f1a\u4e0e\u4e0d\u4f1a\u4e4b\u95f4\u53cd\u590d\u6a2a\u8df3\uff0c\u90a3\u4e48\u672c\u4e66\u6b63\u662f\u4e3a\u4f60\u91cf\u8eab\u5b9a\u5236\u7684\uff01</p> <p>\u5982\u679c\u4f60\u5df2\u7ecf\u79ef\u7d2f\u4e00\u5b9a\u7684\u5237\u9898\u91cf\uff0c\u719f\u6089\u5927\u90e8\u5206\u9898\u578b\uff0c\u90a3\u4e48\u672c\u4e66\u53ef\u52a9\u4f60\u56de\u987e\u4e0e\u68b3\u7406\u7b97\u6cd5\u77e5\u8bc6\u4f53\u7cfb\uff0c\u4ed3\u5e93\u6e90\u4ee3\u7801\u53ef\u4ee5\u5f53\u4f5c\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u6216\u201c\u7b97\u6cd5\u5b57\u5178\u201d\u6765\u4f7f\u7528\u3002</p> <p>\u82e5\u4f60\u662f\u7b97\u6cd5\u201c\u5927\u795e\u201d\uff0c\u6211\u4eec\u671f\u5f85\u6536\u5230\u4f60\u7684\u5b9d\u8d35\u5efa\u8bae\uff0c\u6216\u8005\u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c\u3002</p> <p>\u524d\u7f6e\u6761\u4ef6</p> <p>\u4f60\u9700\u8981\u81f3\u5c11\u5177\u5907\u4efb\u4e00\u8bed\u8a00\u7684\u7f16\u7a0b\u57fa\u7840\uff0c\u80fd\u591f\u9605\u8bfb\u548c\u7f16\u5199\u7b80\u5355\u4ee3\u7801\u3002</p>"},{"location":"chapter_preface/about_the_book/#012","title":"0.1.2 \u00a0 \u5185\u5bb9\u7ed3\u6784","text":"<p>\u672c\u4e66\u7684\u4e3b\u8981\u5185\u5bb9\u5982\u56fe 0-1 \u6240\u793a\u3002</p> <ul> <li>\u590d\u6742\u5ea6\u5206\u6790\uff1a\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u7684\u8bc4\u4ef7\u7ef4\u5ea6\u4e0e\u65b9\u6cd5\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u793a\u4f8b\u7b49\u3002</li> <li>\u6570\u636e\u7ed3\u6784\uff1a\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u6570\u636e\u7ed3\u6784\u7684\u5206\u7c7b\u65b9\u6cd5\u3002\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u7684\u5b9a\u4e49\u3001\u4f18\u7f3a\u70b9\u3001\u5e38\u7528\u64cd\u4f5c\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u5178\u578b\u5e94\u7528\u3001\u5b9e\u73b0\u65b9\u6cd5\u7b49\u3002</li> <li>\u7b97\u6cd5\uff1a\u641c\u7d22\u3001\u6392\u5e8f\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u8d2a\u5fc3\u7b49\u7b97\u6cd5\u7684\u5b9a\u4e49\u3001\u4f18\u7f3a\u70b9\u3001\u6548\u7387\u3001\u5e94\u7528\u573a\u666f\u3001\u89e3\u9898\u6b65\u9aa4\u548c\u793a\u4f8b\u95ee\u9898\u7b49\u3002</li> </ul> <p></p> <p> \u56fe 0-1 \u00a0 \u672c\u4e66\u4e3b\u8981\u5185\u5bb9 </p>"},{"location":"chapter_preface/about_the_book/#013","title":"0.1.3 \u00a0 \u81f4\u8c22","text":"<p>\u672c\u4e66\u5728\u5f00\u6e90\u793e\u533a\u4f17\u591a\u8d21\u732e\u8005\u7684\u5171\u540c\u52aa\u529b\u4e0b\u4e0d\u65ad\u5b8c\u5584\u3002\u611f\u8c22\u6bcf\u4e00\u4f4d\u6295\u5165\u65f6\u95f4\u4e0e\u7cbe\u529b\u7684\u64b0\u7a3f\u4eba\uff0c\u4ed6\u4eec\u662f\uff08\u6309\u7167 GitHub \u81ea\u52a8\u751f\u6210\u7684\u987a\u5e8f\uff09\uff1akrahets\u3001codingonion\u3001nuomi1\u3001Gonglja\u3001Reanon\u3001justin-tse\u3001danielsss\u3001hpstory\u3001S-N-O-R-L-A-X\u3001night-cruise\u3001msk397\u3001gvenusleo\u3001RiverTwilight\u3001gyt95\u3001zhuoqinyue\u3001Zuoxun\u3001Xia-Sang\u3001mingXta\u3001FangYuan33\u3001GN-Yu\u3001IsChristina\u3001xBLACKICEx\u3001guowei-gong\u3001Cathay-Chen\u3001mgisr\u3001JoseHung\u3001qualifier1024\u3001pengchzn\u3001Guanngxu\u3001longsizhuo\u3001L-Super\u3001what-is-me\u3001yuan0221\u3001lhxsm\u3001Slone123c\u3001WSL0809\u3001longranger2\u3001theNefelibatas\u3001xiongsp\u3001JeffersonHuang\u3001hongyun-robot\u3001K3v123\u3001yuelinxin\u3001a16su\u3001gaofer\u3001malone6\u3001Wonderdch\u3001xjr7670\u3001DullSword\u3001Horbin-Magician\u3001NI-SW\u3001reeswell\u3001XC-Zero\u3001XiaChuerwu\u3001yd-j\u3001iron-irax\u3001huawuque404\u3001MolDuM\u3001Nigh\u3001KorsChen\u3001foursevenlove\u300152coder\u3001bubble9um\u3001youshaoXG\u3001curly210102\u3001gltianwen\u3001fanchenggang\u3001Transmigration-zhou\u3001FloranceYeh\u3001FreddieLi\u3001ShiMaRing\u3001lipusheng\u3001Javesun99\u3001JackYang-hellobobo\u3001shanghai-Jerry\u30010130w\u3001Keynman\u3001psychelzh\u3001logan-qiu\u3001ZnYang2018\u3001MwumLi\u30011ch0\u3001Phoenix0415\u3001qingpeng9802\u3001Richard-Zhang1019\u3001QiLOL\u3001Suremotoo\u3001Turing-1024-Lee\u3001Evilrabbit520\u3001GaochaoZhu\u3001ZJKung\u3001linzeyan\u3001hezhizhen\u3001ZongYangL\u3001beintentional\u3001czruby\u3001coderlef\u3001dshlstarr\u3001szu17dmy\u3001fbigm\u3001gledfish\u3001hts0000\u3001boloboloda\u3001iStig\u3001jiaxianhua\u3001wenjianmin\u3001keshida\u3001kilikilikid\u3001lclc6\u3001lwbaptx\u3001liuxjerry\u3001lucaswangdev\u3001lyl625760\u3001chadyi\u3001noobcodemaker\u3001selear\u3001siqyka\u3001syd168\u30014yDX3906\u3001tao363\u3001wangwang105\u3001weibk\u3001yabo083\u3001yi427\u3001yishangzhang\u3001zhouLion\u3001baagod\u3001ElaBosak233\u3001xb534\u3001luluxia\u3001yanedie\u3001thomasq0\u3001YangXuanyi \u548c th1nk3r-ing \u3002</p> <p>\u672c\u4e66\u7684\u4ee3\u7801\u5ba1\u9605\u5de5\u4f5c\u7531 codingonion\u3001Gonglja\u3001gvenusleo\u3001hpstory\u3001justin-tse\u3001krahets\u3001night-cruise\u3001nuomi1 \u548c Reanon \u5b8c\u6210\uff08\u6309\u7167\u9996\u5b57\u6bcd\u987a\u5e8f\u6392\u5217\uff09\u3002\u611f\u8c22\u4ed6\u4eec\u4ed8\u51fa\u7684\u65f6\u95f4\u4e0e\u7cbe\u529b\uff0c\u6b63\u662f\u4ed6\u4eec\u786e\u4fdd\u4e86\u5404\u8bed\u8a00\u4ee3\u7801\u7684\u89c4\u8303\u4e0e\u7edf\u4e00\u3002</p> <p>\u5728\u672c\u4e66\u7684\u521b\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u5f97\u5230\u4e86\u8bb8\u591a\u4eba\u7684\u5e2e\u52a9\u3002</p> <ul> <li>\u611f\u8c22\u6211\u5728\u516c\u53f8\u7684\u5bfc\u5e08\u674e\u6c50\u535a\u58eb\uff0c\u5728\u4e00\u6b21\u7545\u8c08\u4e2d\u4f60\u9f13\u52b1\u6211\u201c\u5feb\u884c\u52a8\u8d77\u6765\u201d\uff0c\u575a\u5b9a\u4e86\u6211\u5199\u8fd9\u672c\u4e66\u7684\u51b3\u5fc3\uff1b</li> <li>\u611f\u8c22\u6211\u7684\u5973\u670b\u53cb\u6ce1\u6ce1\u4f5c\u4e3a\u672c\u4e66\u7684\u9996\u4f4d\u8bfb\u8005\uff0c\u4ece\u7b97\u6cd5\u5c0f\u767d\u7684\u89d2\u5ea6\u63d0\u51fa\u8bb8\u591a\u5b9d\u8d35\u5efa\u8bae\uff0c\u4f7f\u5f97\u672c\u4e66\u66f4\u9002\u5408\u65b0\u624b\u9605\u8bfb\uff1b</li> <li>\u611f\u8c22\u817e\u5b9d\u3001\u7426\u5b9d\u3001\u98de\u5b9d\u4e3a\u672c\u4e66\u8d77\u4e86\u4e00\u4e2a\u5bcc\u6709\u521b\u610f\u7684\u540d\u5b57\uff0c\u5524\u8d77\u5927\u5bb6\u5199\u4e0b\u7b2c\u4e00\u884c\u4ee3\u7801\u201cHello World!\u201d\u7684\u7f8e\u597d\u56de\u5fc6\uff1b</li> <li>\u611f\u8c22\u6821\u94e8\u5728\u77e5\u8bc6\u4ea7\u6743\u65b9\u9762\u63d0\u4f9b\u7684\u4e13\u4e1a\u5e2e\u52a9\uff0c\u8fd9\u5bf9\u672c\u5f00\u6e90\u4e66\u7684\u5b8c\u5584\u8d77\u5230\u4e86\u91cd\u8981\u4f5c\u7528\uff1b</li> <li>\u611f\u8c22\u82cf\u6f7c\u4e3a\u672c\u4e66\u8bbe\u8ba1\u4e86\u7cbe\u7f8e\u7684\u5c01\u9762\u548c logo \uff0c\u5e76\u5728\u6211\u7684\u5f3a\u8feb\u75c7\u7684\u9a71\u4f7f\u4e0b\u591a\u6b21\u8010\u5fc3\u4fee\u6539\uff1b</li> <li>\u611f\u8c22 @squidfunk \u63d0\u4f9b\u7684\u6392\u7248\u5efa\u8bae\uff0c\u4ee5\u53ca\u4ed6\u5f00\u53d1\u7684\u5f00\u6e90\u6587\u6863\u4e3b\u9898 Material-for-MkDocs \u3002</li> </ul> <p>\u5728\u5199\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u9605\u8bfb\u4e86\u8bb8\u591a\u5173\u4e8e\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u6559\u6750\u548c\u6587\u7ae0\u3002\u8fd9\u4e9b\u4f5c\u54c1\u4e3a\u672c\u4e66\u63d0\u4f9b\u4e86\u4f18\u79c0\u7684\u8303\u672c\uff0c\u786e\u4fdd\u4e86\u672c\u4e66\u5185\u5bb9\u7684\u51c6\u786e\u6027\u4e0e\u54c1\u8d28\u3002\u5728\u6b64\u611f\u8c22\u6240\u6709\u8001\u5e08\u548c\u524d\u8f88\u7684\u6770\u51fa\u8d21\u732e\uff01</p> <p>\u672c\u4e66\u5021\u5bfc\u624b\u8111\u5e76\u7528\u7684\u5b66\u4e60\u65b9\u5f0f\uff0c\u5728\u8fd9\u4e00\u70b9\u4e0a\u6211\u6df1\u53d7\u300a\u52a8\u624b\u5b66\u6df1\u5ea6\u5b66\u4e60\u300b\u7684\u542f\u53d1\u3002\u5728\u6b64\u5411\u5404\u4f4d\u8bfb\u8005\u5f3a\u70c8\u63a8\u8350\u8fd9\u672c\u4f18\u79c0\u7684\u8457\u4f5c\u3002</p> <p>\u8877\u5fc3\u611f\u8c22\u6211\u7684\u7236\u6bcd\uff0c\u6b63\u662f\u4f60\u4eec\u4e00\u76f4\u4ee5\u6765\u7684\u652f\u6301\u4e0e\u9f13\u52b1\uff0c\u8ba9\u6211\u6709\u673a\u4f1a\u505a\u8fd9\u4ef6\u5bcc\u6709\u8da3\u5473\u7684\u4e8b\u3002</p>"},{"location":"chapter_preface/suggestions/","title":"0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66","text":"<p>Tip</p> <p>\u4e3a\u4e86\u83b7\u5f97\u6700\u4f73\u7684\u9605\u8bfb\u4f53\u9a8c\uff0c\u5efa\u8bae\u4f60\u901a\u8bfb\u672c\u8282\u5185\u5bb9\u3002</p>"},{"location":"chapter_preface/suggestions/#021","title":"0.2.1 \u00a0 \u884c\u6587\u98ce\u683c\u7ea6\u5b9a","text":"<ul> <li>\u6807\u9898\u540e\u6807\u6ce8 <code>*</code> \u7684\u662f\u9009\u8bfb\u7ae0\u8282\uff0c\u5185\u5bb9\u76f8\u5bf9\u56f0\u96be\u3002\u5982\u679c\u4f60\u7684\u65f6\u95f4\u6709\u9650\uff0c\u53ef\u4ee5\u5148\u8df3\u8fc7\u3002</li> <li>\u91cd\u8981\u4e13\u6709\u540d\u8bcd\u53ca\u5176\u82f1\u6587\u7ffb\u8bd1\u4f1a\u7528 <code>\u300c \u300d</code> \u62ec\u53f7\u6807\u6ce8\uff0c\u4f8b\u5982 <code>\u300c\u6570\u7ec4 array\u300d</code> \u3002\u5efa\u8bae\u8bb0\u4f4f\u5b83\u4eec\uff0c\u4ee5\u4fbf\u9605\u8bfb\u6587\u732e\u3002</li> <li>\u4e13\u6709\u540d\u8bcd\u548c\u6709\u7279\u6307\u542b\u4e49\u7684\u8bcd\u53e5\u4f1a\u4f7f\u7528 <code>\u201c\u5f15\u53f7\u201d</code> \u6807\u6ce8\uff0c\u4ee5\u907f\u514d\u6b67\u4e49\u3002</li> <li>\u91cd\u8981\u540d\u8bcd\u3001\u91cd\u70b9\u5185\u5bb9\u548c\u603b\u7ed3\u6027\u8bed\u53e5\u4f1a \u52a0\u7c97 \uff0c\u8fd9\u7c7b\u6587\u5b57\u503c\u5f97\u7279\u522b\u5173\u6ce8\u3002</li> <li>\u5f53\u6d89\u53ca\u7f16\u7a0b\u8bed\u8a00\u4e4b\u95f4\u4e0d\u4e00\u81f4\u7684\u540d\u8bcd\u65f6\uff0c\u672c\u4e66\u5747\u4ee5 Python \u4e3a\u51c6\uff0c\u4f8b\u5982\u4f7f\u7528 <code>None</code> \u6765\u8868\u793a\u201c\u7a7a\u201d\u3002</li> <li>\u672c\u4e66\u90e8\u5206\u653e\u5f03\u4e86\u7f16\u7a0b\u8bed\u8a00\u7684\u6ce8\u91ca\u89c4\u8303\uff0c\u4ee5\u6362\u53d6\u66f4\u52a0\u7d27\u51d1\u7684\u5185\u5bb9\u6392\u7248\u3002\u6ce8\u91ca\u4e3b\u8981\u5206\u4e3a\u4e09\u79cd\u7c7b\u578b\uff1a\u6807\u9898\u6ce8\u91ca\u3001\u5185\u5bb9\u6ce8\u91ca\u3001\u591a\u884c\u6ce8\u91ca\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>\"\"\"\u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\"\"\"\n\n# \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n\"\"\"\n\u591a\u884c\n\u6ce8\u91ca\n\"\"\"\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>\n</code></pre> <pre><code>// \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n// \u591a\u884c\n// \u6ce8\u91ca\n</code></pre>"},{"location":"chapter_preface/suggestions/#022","title":"0.2.2 \u00a0 \u5728\u52a8\u753b\u56fe\u89e3\u4e2d\u9ad8\u6548\u5b66\u4e60","text":"<p>\u76f8\u8f83\u4e8e\u6587\u5b57\uff0c\u89c6\u9891\u548c\u56fe\u7247\u5177\u6709\u66f4\u9ad8\u7684\u4fe1\u606f\u5bc6\u5ea6\u548c\u7ed3\u6784\u5316\u7a0b\u5ea6\uff0c\u66f4\u6613\u4e8e\u7406\u89e3\u3002\u5728\u672c\u4e66\u4e2d\uff0c\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u5c06\u4e3b\u8981\u901a\u8fc7\u52a8\u753b\u4ee5\u56fe\u89e3\u5f62\u5f0f\u5c55\u793a\uff0c\u800c\u6587\u5b57\u5219\u4f5c\u4e3a\u89e3\u91ca\u4e0e\u8865\u5145\u3002</p> <p>\u5982\u679c\u4f60\u5728\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u53d1\u73b0\u67d0\u6bb5\u5185\u5bb9\u63d0\u4f9b\u4e86\u5982\u56fe 0-2 \u6240\u793a\u7684\u52a8\u753b\u56fe\u89e3\uff0c\u8bf7\u4ee5\u56fe\u4e3a\u4e3b\u3001\u4ee5\u6587\u5b57\u4e3a\u8f85\uff0c\u7efc\u5408\u4e24\u8005\u6765\u7406\u89e3\u5185\u5bb9\u3002</p> <p></p> <p> \u56fe 0-2 \u00a0 \u52a8\u753b\u56fe\u89e3\u793a\u4f8b </p>"},{"location":"chapter_preface/suggestions/#023","title":"0.2.3 \u00a0 \u5728\u4ee3\u7801\u5b9e\u8df5\u4e2d\u52a0\u6df1\u7406\u89e3","text":"<p>\u672c\u4e66\u7684\u914d\u5957\u4ee3\u7801\u6258\u7ba1\u5728 GitHub \u4ed3\u5e93\u3002\u5982\u56fe 0-3 \u6240\u793a\uff0c\u6e90\u4ee3\u7801\u9644\u6709\u6d4b\u8bd5\u6837\u4f8b\uff0c\u53ef\u4e00\u952e\u8fd0\u884c\u3002</p> <p>\u5982\u679c\u65f6\u95f4\u5141\u8bb8\uff0c\u5efa\u8bae\u4f60\u53c2\u7167\u4ee3\u7801\u81ea\u884c\u6572\u4e00\u904d\u3002\u5982\u679c\u5b66\u4e60\u65f6\u95f4\u6709\u9650\uff0c\u8bf7\u81f3\u5c11\u901a\u8bfb\u5e76\u8fd0\u884c\u6240\u6709\u4ee3\u7801\u3002</p> <p>\u4e0e\u9605\u8bfb\u4ee3\u7801\u76f8\u6bd4\uff0c\u7f16\u5199\u4ee3\u7801\u7684\u8fc7\u7a0b\u5f80\u5f80\u80fd\u5e26\u6765\u66f4\u591a\u6536\u83b7\u3002\u52a8\u624b\u5b66\uff0c\u624d\u662f\u771f\u7684\u5b66\u3002</p> <p></p> <p> \u56fe 0-3 \u00a0 \u8fd0\u884c\u4ee3\u7801\u793a\u4f8b </p> <p>\u8fd0\u884c\u4ee3\u7801\u7684\u524d\u7f6e\u5de5\u4f5c\u4e3b\u8981\u5206\u4e3a\u4e09\u6b65\u3002</p> <p>\u7b2c\u4e00\u6b65\uff1a\u5b89\u88c5\u672c\u5730\u7f16\u7a0b\u73af\u5883\u3002\u8bf7\u53c2\u7167\u9644\u5f55\u6240\u793a\u7684\u6559\u7a0b\u8fdb\u884c\u5b89\u88c5\uff0c\u5982\u679c\u5df2\u5b89\u88c5\uff0c\u5219\u53ef\u8df3\u8fc7\u6b64\u6b65\u9aa4\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u514b\u9686\u6216\u4e0b\u8f7d\u4ee3\u7801\u4ed3\u5e93\u3002\u524d\u5f80 GitHub \u4ed3\u5e93\u3002\u5982\u679c\u5df2\u7ecf\u5b89\u88c5 Git \uff0c\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u547d\u4ee4\u514b\u9686\u672c\u4ed3\u5e93\uff1a</p> <pre><code>git clone https://github.com/krahets/hello-algo.git\n</code></pre> <p>\u5f53\u7136\uff0c\u4f60\u4e5f\u53ef\u4ee5\u5728\u56fe 0-4 \u6240\u793a\u7684\u4f4d\u7f6e\uff0c\u70b9\u51fb\u201cDownload ZIP\u201d\u6309\u94ae\u76f4\u63a5\u4e0b\u8f7d\u4ee3\u7801\u538b\u7f29\u5305\uff0c\u7136\u540e\u5728\u672c\u5730\u89e3\u538b\u5373\u53ef\u3002</p> <p></p> <p> \u56fe 0-4 \u00a0 \u514b\u9686\u4ed3\u5e93\u4e0e\u4e0b\u8f7d\u4ee3\u7801 </p> <p>\u7b2c\u4e09\u6b65\uff1a\u8fd0\u884c\u6e90\u4ee3\u7801\u3002\u5982\u56fe 0-5 \u6240\u793a\uff0c\u5bf9\u4e8e\u9876\u90e8\u6807\u6709\u6587\u4ef6\u540d\u79f0\u7684\u4ee3\u7801\u5757\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u4ed3\u5e93\u7684 <code>codes</code> \u6587\u4ef6\u5939\u5185\u627e\u5230\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6\u3002\u6e90\u4ee3\u7801\u6587\u4ef6\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u5c06\u5e2e\u52a9\u4f60\u8282\u7701\u4e0d\u5fc5\u8981\u7684\u8c03\u8bd5\u65f6\u95f4\uff0c\u8ba9\u4f60\u80fd\u591f\u4e13\u6ce8\u4e8e\u5b66\u4e60\u5185\u5bb9\u3002</p> <p></p> <p> \u56fe 0-5 \u00a0 \u4ee3\u7801\u5757\u4e0e\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6 </p> <p>\u9664\u4e86\u672c\u5730\u8fd0\u884c\u4ee3\u7801\uff0c\u7f51\u9875\u7248\u8fd8\u652f\u6301 Python \u4ee3\u7801\u7684\u53ef\u89c6\u5316\u8fd0\u884c\uff08\u57fa\u4e8e pythontutor \u5b9e\u73b0\uff09\u3002\u5982\u56fe 0-6 \u6240\u793a\uff0c\u4f60\u53ef\u4ee5\u70b9\u51fb\u4ee3\u7801\u5757\u4e0b\u65b9\u7684\u201c\u53ef\u89c6\u5316\u8fd0\u884c\u201d\u6765\u5c55\u5f00\u89c6\u56fe\uff0c\u89c2\u5bdf\u7b97\u6cd5\u4ee3\u7801\u7684\u6267\u884c\u8fc7\u7a0b\uff1b\u4e5f\u53ef\u4ee5\u70b9\u51fb\u201c\u5168\u5c4f\u89c2\u770b\u201d\uff0c\u4ee5\u83b7\u5f97\u66f4\u597d\u7684\u9605\u89c8\u4f53\u9a8c\u3002</p> <p></p> <p> \u56fe 0-6 \u00a0 Python \u4ee3\u7801\u7684\u53ef\u89c6\u5316\u8fd0\u884c </p>"},{"location":"chapter_preface/suggestions/#024","title":"0.2.4 \u00a0 \u5728\u63d0\u95ee\u8ba8\u8bba\u4e2d\u5171\u540c\u6210\u957f","text":"<p>\u5728\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u8bf7\u4e0d\u8981\u8f7b\u6613\u8df3\u8fc7\u90a3\u4e9b\u6ca1\u5b66\u660e\u767d\u7684\u77e5\u8bc6\u70b9\u3002\u6b22\u8fce\u5728\u8bc4\u8bba\u533a\u63d0\u51fa\u4f60\u7684\u95ee\u9898\uff0c\u6211\u548c\u5c0f\u4f19\u4f34\u4eec\u5c06\u7aed\u8bda\u4e3a\u4f60\u89e3\u7b54\uff0c\u4e00\u822c\u60c5\u51b5\u4e0b\u53ef\u5728\u4e24\u5929\u5185\u56de\u590d\u3002</p> <p>\u5982\u56fe 0-7 \u6240\u793a\uff0c\u7f51\u9875\u7248\u6bcf\u4e2a\u7ae0\u8282\u7684\u5e95\u90e8\u90fd\u914d\u6709\u8bc4\u8bba\u533a\u3002\u5e0c\u671b\u4f60\u80fd\u591a\u5173\u6ce8\u8bc4\u8bba\u533a\u7684\u5185\u5bb9\u3002\u4e00\u65b9\u9762\uff0c\u4f60\u53ef\u4ee5\u4e86\u89e3\u5927\u5bb6\u9047\u5230\u7684\u95ee\u9898\uff0c\u4ece\u800c\u67e5\u6f0f\u8865\u7f3a\uff0c\u6fc0\u53d1\u66f4\u6df1\u5165\u7684\u601d\u8003\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u671f\u5f85\u4f60\u80fd\u6177\u6168\u5730\u56de\u7b54\u5176\u4ed6\u5c0f\u4f19\u4f34\u7684\u95ee\u9898\uff0c\u5206\u4eab\u4f60\u7684\u89c1\u89e3\uff0c\u5e2e\u52a9\u4ed6\u4eba\u8fdb\u6b65\u3002</p> <p></p> <p> \u56fe 0-7 \u00a0 \u8bc4\u8bba\u533a\u793a\u4f8b </p>"},{"location":"chapter_preface/suggestions/#025","title":"0.2.5 \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf","text":"<p>\u4ece\u603b\u4f53\u4e0a\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8fc7\u7a0b\u5212\u5206\u4e3a\u4e09\u4e2a\u9636\u6bb5\u3002</p> <ol> <li>\u9636\u6bb5\u4e00\uff1a\u7b97\u6cd5\u5165\u95e8\u3002\u6211\u4eec\u9700\u8981\u719f\u6089\u5404\u79cd\u6570\u636e\u7ed3\u6784\u7684\u7279\u70b9\u548c\u7528\u6cd5\uff0c\u5b66\u4e60\u4e0d\u540c\u7b97\u6cd5\u7684\u539f\u7406\u3001\u6d41\u7a0b\u3001\u7528\u9014\u548c\u6548\u7387\u7b49\u65b9\u9762\u7684\u5185\u5bb9\u3002</li> <li>\u9636\u6bb5\u4e8c\uff1a\u5237\u7b97\u6cd5\u9898\u3002\u5efa\u8bae\u4ece\u70ed\u95e8\u9898\u76ee\u5f00\u5237\uff0c\u5148\u79ef\u7d2f\u81f3\u5c11 100 \u9053\u9898\u76ee\uff0c\u719f\u6089\u4e3b\u6d41\u7684\u7b97\u6cd5\u95ee\u9898\u3002\u521d\u6b21\u5237\u9898\u65f6\uff0c\u201c\u77e5\u8bc6\u9057\u5fd8\u201d\u53ef\u80fd\u662f\u4e00\u4e2a\u6311\u6218\uff0c\u4f46\u8bf7\u653e\u5fc3\uff0c\u8fd9\u662f\u5f88\u6b63\u5e38\u7684\u3002\u6211\u4eec\u53ef\u4ee5\u6309\u7167\u201c\u827e\u5bbe\u6d69\u65af\u9057\u5fd8\u66f2\u7ebf\u201d\u6765\u590d\u4e60\u9898\u76ee\uff0c\u901a\u5e38\u5728\u8fdb\u884c 3\uff5e5 \u8f6e\u7684\u91cd\u590d\u540e\uff0c\u5c31\u80fd\u5c06\u5176\u7262\u8bb0\u5728\u5fc3\u3002\u63a8\u8350\u7684\u9898\u5355\u548c\u5237\u9898\u8ba1\u5212\u8bf7\u89c1\u6b64 GitHub \u4ed3\u5e93\u3002</li> <li>\u9636\u6bb5\u4e09\uff1a\u642d\u5efa\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5b66\u4e60\u65b9\u9762\uff0c\u6211\u4eec\u53ef\u4ee5\u9605\u8bfb\u7b97\u6cd5\u4e13\u680f\u6587\u7ae0\u3001\u89e3\u9898\u6846\u67b6\u548c\u7b97\u6cd5\u6559\u6750\uff0c\u4ee5\u4e0d\u65ad\u4e30\u5bcc\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5237\u9898\u65b9\u9762\uff0c\u53ef\u4ee5\u5c1d\u8bd5\u91c7\u7528\u8fdb\u9636\u5237\u9898\u7b56\u7565\uff0c\u5982\u6309\u4e13\u9898\u5206\u7c7b\u3001\u4e00\u9898\u591a\u89e3\u3001\u4e00\u89e3\u591a\u9898\u7b49\uff0c\u76f8\u5173\u7684\u5237\u9898\u5fc3\u5f97\u53ef\u4ee5\u5728\u5404\u4e2a\u793e\u533a\u627e\u5230\u3002</li> </ol> <p>\u5982\u56fe 0-8 \u6240\u793a\uff0c\u672c\u4e66\u5185\u5bb9\u4e3b\u8981\u6db5\u76d6\u201c\u9636\u6bb5\u4e00\u201d\uff0c\u65e8\u5728\u5e2e\u52a9\u4f60\u66f4\u9ad8\u6548\u5730\u5c55\u5f00\u9636\u6bb5\u4e8c\u548c\u9636\u6bb5\u4e09\u7684\u5b66\u4e60\u3002</p> <p></p> <p> \u56fe 0-8 \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf </p>"},{"location":"chapter_preface/summary/","title":"0.3 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u672c\u4e66\u7684\u4e3b\u8981\u53d7\u4f17\u662f\u7b97\u6cd5\u521d\u5b66\u8005\u3002\u5982\u679c\u4f60\u5df2\u6709\u4e00\u5b9a\u57fa\u7840\uff0c\u672c\u4e66\u80fd\u5e2e\u52a9\u4f60\u7cfb\u7edf\u56de\u987e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u4e66\u4e2d\u6e90\u4ee3\u7801\u4e5f\u53ef\u4f5c\u4e3a\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u4f7f\u7528\u3002</li> <li>\u4e66\u4e2d\u5185\u5bb9\u4e3b\u8981\u5305\u62ec\u590d\u6742\u5ea6\u5206\u6790\u3001\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e09\u90e8\u5206\uff0c\u6db5\u76d6\u4e86\u8be5\u9886\u57df\u7684\u5927\u90e8\u5206\u4e3b\u9898\u3002</li> <li>\u5bf9\u4e8e\u7b97\u6cd5\u65b0\u624b\uff0c\u5728\u521d\u5b66\u9636\u6bb5\u9605\u8bfb\u4e00\u672c\u5165\u95e8\u4e66\u81f3\u5173\u91cd\u8981\uff0c\u53ef\u4ee5\u5c11\u8d70\u8bb8\u591a\u5f2f\u8def\u3002</li> <li>\u4e66\u4e2d\u7684\u52a8\u753b\u56fe\u89e3\u901a\u5e38\u7528\u4e8e\u4ecb\u7ecd\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u3002\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u5e94\u7ed9\u4e88\u8fd9\u4e9b\u5185\u5bb9\u66f4\u591a\u5173\u6ce8\u3002</li> <li>\u5b9e\u8df5\u4e43\u5b66\u4e60\u7f16\u7a0b\u4e4b\u6700\u4f73\u9014\u5f84\u3002\u5f3a\u70c8\u5efa\u8bae\u8fd0\u884c\u6e90\u4ee3\u7801\u5e76\u4eb2\u81ea\u6572\u4ee3\u7801\u3002</li> <li>\u672c\u4e66\u7f51\u9875\u7248\u7684\u6bcf\u4e2a\u7ae0\u8282\u90fd\u8bbe\u6709\u8bc4\u8bba\u533a\uff0c\u6b22\u8fce\u968f\u65f6\u5206\u4eab\u4f60\u7684\u7591\u60d1\u4e0e\u89c1\u89e3\u3002</li> </ul>"},{"location":"chapter_reference/","title":"\u53c2\u8003\u6587\u732e","text":"<p>[1] Thomas H. Cormen, et al. Introduction to Algorithms (3<sup>rd</sup> Edition).</p> <p>[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1<sup>st</sup> Edition).</p> <p>[3] Robert Sedgewick, et al. Algorithms (4<sup>th</sup> Edition).</p> <p>[4] \u4e25\u851a\u654f. \u6570\u636e\u7ed3\u6784\uff08C \u8bed\u8a00\u7248\uff09.</p> <p>[5] \u9093\u4fca\u8f89. \u6570\u636e\u7ed3\u6784\uff08C++ \u8bed\u8a00\u7248\uff0c\u7b2c\u4e09\u7248\uff09.</p> <p>[6] \u9a6c\u514b \u827e\u4f26 \u7ef4\u65af\u8457\uff0c\u9648\u8d8a\u8bd1. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790\uff1aJava\u8bed\u8a00\u63cf\u8ff0\uff08\u7b2c\u4e09\u7248\uff09.</p> <p>[7] \u7a0b\u6770. \u5927\u8bdd\u6570\u636e\u7ed3\u6784.</p> <p>[8] \u738b\u4e89. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u7f8e.</p> <p>[9] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6<sup>th</sup> Edition).</p> <p>[10] Aston Zhang, et al. Dive into Deep Learning.</p>"},{"location":"chapter_searching/","title":"\u7b2c 10 \u7ae0 \u00a0 \u641c\u7d22","text":"<p>Abstract</p> <p>\u641c\u7d22\u662f\u4e00\u573a\u672a\u77e5\u7684\u5192\u9669\uff0c\u6211\u4eec\u6216\u8bb8\u9700\u8981\u8d70\u904d\u795e\u79d8\u7a7a\u95f4\u7684\u6bcf\u4e2a\u89d2\u843d\uff0c\u53c8\u6216\u8bb8\u53ef\u4ee5\u5feb\u901f\u9501\u5b9a\u76ee\u6807\u3002</p> <p>\u5728\u8fd9\u573a\u5bfb\u89c5\u4e4b\u65c5\u4e2d\uff0c\u6bcf\u4e00\u6b21\u63a2\u7d22\u90fd\u53ef\u80fd\u5f97\u5230\u4e00\u4e2a\u672a\u66fe\u6599\u60f3\u7684\u7b54\u6848\u3002</p>"},{"location":"chapter_searching/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>10.1 \u00a0 \u4e8c\u5206\u67e5\u627e</li> <li>10.2 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9</li> <li>10.3 \u00a0 \u4e8c\u5206\u67e5\u627e\u8fb9\u754c</li> <li>10.4 \u00a0 \u54c8\u5e0c\u4f18\u5316\u7b56\u7565</li> <li>10.5 \u00a0 \u91cd\u8bc6\u641c\u7d22\u7b97\u6cd5</li> <li>10.6 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_searching/binary_search/","title":"10.1 \u00a0 \u4e8c\u5206\u67e5\u627e","text":"<p>\u300c\u4e8c\u5206\u67e5\u627e binary search\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u9ad8\u6548\u641c\u7d22\u7b97\u6cd5\u3002\u5b83\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\u641c\u7d22\u8303\u56f4\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u4e3a\u6b62\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 <code>nums</code> \uff0c\u5143\u7d20\u6309\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\u4e14\u4e0d\u91cd\u590d\u3002\u8bf7\u67e5\u627e\u5e76\u8fd4\u56de\u5143\u7d20 <code>target</code> \u5728\u8be5\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u56fe 10-1 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 10-1 \u00a0 \u4e8c\u5206\u67e5\u627e\u793a\u4f8b\u6570\u636e </p> <p>\u5982\u56fe 10-2 \u6240\u793a\uff0c\u6211\u4eec\u5148\u521d\u59cb\u5316\u6307\u9488 \\(i = 0\\) \u548c \\(j = n - 1\\) \uff0c\u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u548c\u5c3e\u5143\u7d20\uff0c\u4ee3\u8868\u641c\u7d22\u533a\u95f4 \\([0, n - 1]\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u4e2d\u62ec\u53f7\u8868\u793a\u95ed\u533a\u95f4\uff0c\u5176\u5305\u542b\u8fb9\u754c\u503c\u672c\u8eab\u3002</p> <p>\u63a5\u4e0b\u6765\uff0c\u5faa\u73af\u6267\u884c\u4ee5\u4e0b\u4e24\u6b65\u3002</p> <ol> <li>\u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m = \\lfloor {(i + j) / 2} \\rfloor\\) \uff0c\u5176\u4e2d \\(\\lfloor \\: \\rfloor\\) \u8868\u793a\u5411\u4e0b\u53d6\u6574\u64cd\u4f5c\u3002</li> <li>\u5224\u65ad <code>nums[m]</code> \u548c <code>target</code> \u7684\u5927\u5c0f\u5173\u7cfb\uff0c\u5206\u4e3a\u4ee5\u4e0b\u4e09\u79cd\u60c5\u51b5\u3002<ol> <li>\u5f53 <code>nums[m] &lt; target</code> \u65f6\uff0c\u8bf4\u660e <code>target</code> \u5728\u533a\u95f4 \\([m + 1, j]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(i = m + 1\\) \u3002</li> <li>\u5f53 <code>nums[m] &gt; target</code> \u65f6\uff0c\u8bf4\u660e <code>target</code> \u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(j = m - 1\\) \u3002</li> <li>\u5f53 <code>nums[m] = target</code> \u65f6\uff0c\u8bf4\u660e\u627e\u5230 <code>target</code> \uff0c\u56e0\u6b64\u8fd4\u56de\u7d22\u5f15 \\(m\\) \u3002</li> </ol> </li> </ol> <p>\u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u76ee\u6807\u5143\u7d20\uff0c\u641c\u7d22\u533a\u95f4\u6700\u7ec8\u4f1a\u7f29\u5c0f\u4e3a\u7a7a\u3002\u6b64\u65f6\u8fd4\u56de \\(-1\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 10-2 \u00a0 \u4e8c\u5206\u67e5\u627e\u6d41\u7a0b </p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e \\(i\\) \u548c \\(j\\) \u90fd\u662f <code>int</code> \u7c7b\u578b\uff0c\u56e0\u6b64 \\(i + j\\) \u53ef\u80fd\u4f1a\u8d85\u51fa <code>int</code> \u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u3002\u4e3a\u4e86\u907f\u514d\u5927\u6570\u8d8a\u754c\uff0c\u6211\u4eec\u901a\u5e38\u91c7\u7528\u516c\u5f0f \\(m = \\lfloor {i + (j - i) / 2} \\rfloor\\) \u6765\u8ba1\u7b97\u4e2d\u70b9\u3002</p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search.py<pre><code>def binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\"\"\"\n    # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    i, j = 0, len(nums) - 1\n    # \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt;= j:\n        # \u7406\u8bba\u4e0a Python \u7684\u6570\u5b57\u53ef\u4ee5\u65e0\u9650\u5927\uff08\u53d6\u51b3\u4e8e\u5185\u5b58\u5927\u5c0f\uff09\uff0c\u65e0\u987b\u8003\u8651\u5927\u6570\u8d8a\u754c\u95ee\u9898\n        m = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        elif nums[m] &gt; target:\n            j = m - 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        else:\n            return m  # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n</code></pre> binary_search.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = nums.size() - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = nums.length - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint BinarySearch(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = nums.Length - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2;   // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        else                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    i, j := 0, len(nums)-1\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    for i &lt;= j {\n        m := i + (j-i)/2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums, target) {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    let i = 0,\n        j = nums.length - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        else return m; // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums: number[], target: number): number {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    let i = 0,\n        j = nums.length - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        } else {\n            // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    return -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n</code></pre> binary_search.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n  int i = 0, j = nums.length - 1;\n  // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n      j = m - 1;\n    } else {\n      // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n      return m;\n    }\n  }\n  // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    let mut i = 0;\n    let mut j = nums.len() as i32 - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        } else {\n            // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int *nums, int len, int target) {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = len - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    var i = 0\n    var j = nums.size - 1\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        else  // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>[class]{}-[func]{binary_search}\n</code></pre> binary_search.zig<pre><code>// \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\nfn binarySearch(comptime T: type, nums: std.ArrayList(T), target: T) T {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    var i: usize = 0;\n    var j: usize = nums.items.len - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        var m = i + (j - i) / 2;                // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums.items[m] &lt; target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        } else if (nums.items[m] &gt; target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        } else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return @intCast(m);\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \uff1a\u5728\u4e8c\u5206\u5faa\u73af\u4e2d\uff0c\u533a\u95f4\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\uff0c\u56e0\u6b64\u5faa\u73af\u6b21\u6570\u4e3a \\(\\log_2 n\\) \u3002</p> <p>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7a7a\u95f4\u3002</p>"},{"location":"chapter_searching/binary_search/#1011","title":"10.1.1 \u00a0 \u533a\u95f4\u8868\u793a\u65b9\u6cd5","text":"<p>\u9664\u4e86\u4e0a\u8ff0\u53cc\u95ed\u533a\u95f4\u5916\uff0c\u5e38\u89c1\u7684\u533a\u95f4\u8868\u793a\u8fd8\u6709\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u533a\u95f4\uff0c\u5b9a\u4e49\u4e3a \\([0, n)\\) \uff0c\u5373\u5de6\u8fb9\u754c\u5305\u542b\u81ea\u8eab\uff0c\u53f3\u8fb9\u754c\u4e0d\u5305\u542b\u81ea\u8eab\u3002\u5728\u8be5\u8868\u793a\u4e0b\uff0c\u533a\u95f4 \\([i, j)\\) \u5728 \\(i = j\\) \u65f6\u4e3a\u7a7a\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u8be5\u8868\u793a\u5b9e\u73b0\u5177\u6709\u76f8\u540c\u529f\u80fd\u7684\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search.py<pre><code>def binary_search_lcro(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09\"\"\"\n    # \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    i, j = 0, len(nums)\n    # \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt; j:\n        m = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n        elif nums[m] &gt; target:\n            j = m  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n        else:\n            return m  # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n</code></pre> binary_search.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(vector&lt;int&gt; &amp;nums, int target) {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = nums.size();\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = nums.length;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint BinarySearchLCRO(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = nums.Length;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2;   // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        else                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunc binarySearchLCRO(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    i, j := 0, len(nums)\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    for i &lt; j {\n        m := i + (j-i)/2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m\n        } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunc binarySearchLCRO(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    var i = nums.startIndex\n    var j = nums.endIndex\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt; j {\n        let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m\n        } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunction binarySearchLCRO(nums, target) {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    let i = 0,\n        j = nums.length;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        else return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunction binarySearchLCRO(nums: number[], target: number): number {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    let i = 0,\n        j = nums.length;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        } else {\n            // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    return -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n</code></pre> binary_search.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(List&lt;int&gt; nums, int target) {\n  // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n  int i = 0, j = nums.length;\n  // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n  while (i &lt; j) {\n    int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n      j = m;\n    } else {\n      // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n      return m;\n    }\n  }\n  // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfn binary_search_lcro(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    let mut i = 0;\n    let mut j = nums.len() as i32;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt; j {\n        let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        } else {\n            // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(int *nums, int len, int target) {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = len;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfun binarySearchLCRO(nums: IntArray, target: Int): Int {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    var i = 0\n    var j = nums.size\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        val m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m\n        else  // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>[class]{}-[func]{binary_search_lcro}\n</code></pre> binary_search.zig<pre><code>// \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09\nfn binarySearchLCRO(comptime T: type, nums: std.ArrayList(T), target: T) T {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    var i: usize = 0;\n    var j: usize = nums.items.len;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        var m = i + (j - i) / 2;                // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums.items[m] &lt; target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        } else if (nums.items[m] &gt; target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        } else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return @intCast(m);\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 10-3 \u6240\u793a\uff0c\u5728\u4e24\u79cd\u533a\u95f4\u8868\u793a\u4e0b\uff0c\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u521d\u59cb\u5316\u3001\u5faa\u73af\u6761\u4ef6\u548c\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\u7686\u6709\u6240\u4e0d\u540c\u3002</p> <p>\u7531\u4e8e\u201c\u53cc\u95ed\u533a\u95f4\u201d\u8868\u793a\u4e2d\u7684\u5de6\u53f3\u8fb9\u754c\u90fd\u88ab\u5b9a\u4e49\u4e3a\u95ed\u533a\u95f4\uff0c\u56e0\u6b64\u901a\u8fc7\u6307\u9488 \\(i\\) \u548c\u6307\u9488 \\(j\\) \u7f29\u5c0f\u533a\u95f4\u7684\u64cd\u4f5c\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u8fd9\u6837\u66f4\u4e0d\u5bb9\u6613\u51fa\u9519\uff0c\u56e0\u6b64\u4e00\u822c\u5efa\u8bae\u91c7\u7528\u201c\u53cc\u95ed\u533a\u95f4\u201d\u7684\u5199\u6cd5\u3002</p> <p></p> <p> \u56fe 10-3 \u00a0 \u4e24\u79cd\u533a\u95f4\u5b9a\u4e49 </p>"},{"location":"chapter_searching/binary_search/#1012","title":"10.1.2 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u4e8c\u5206\u67e5\u627e\u5728\u65f6\u95f4\u548c\u7a7a\u95f4\u65b9\u9762\u90fd\u6709\u8f83\u597d\u7684\u6027\u80fd\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\u7684\u65f6\u95f4\u6548\u7387\u9ad8\u3002\u5728\u5927\u6570\u636e\u91cf\u4e0b\uff0c\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5177\u6709\u663e\u8457\u4f18\u52bf\u3002\u4f8b\u5982\uff0c\u5f53\u6570\u636e\u5927\u5c0f \\(n = 2^{20}\\) \u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u9700\u8981 \\(2^{20} = 1048576\\) \u8f6e\u5faa\u73af\uff0c\u800c\u4e8c\u5206\u67e5\u627e\u4ec5\u9700 \\(\\log_2 2^{20} = 20\\) \u8f6e\u5faa\u73af\u3002</li> <li>\u4e8c\u5206\u67e5\u627e\u65e0\u987b\u989d\u5916\u7a7a\u95f4\u3002\u76f8\u8f83\u4e8e\u9700\u8981\u501f\u52a9\u989d\u5916\u7a7a\u95f4\u7684\u641c\u7d22\u7b97\u6cd5\uff08\u4f8b\u5982\u54c8\u5e0c\u67e5\u627e\uff09\uff0c\u4e8c\u5206\u67e5\u627e\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002</li> </ul> <p>\u7136\u800c\uff0c\u4e8c\u5206\u67e5\u627e\u5e76\u975e\u9002\u7528\u4e8e\u6240\u6709\u60c5\u51b5\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6709\u5e8f\u6570\u636e\u3002\u82e5\u8f93\u5165\u6570\u636e\u65e0\u5e8f\uff0c\u4e3a\u4e86\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u800c\u4e13\u95e8\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u4e0d\u507f\u5931\u3002\u56e0\u4e3a\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u6bd4\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u90fd\u66f4\u9ad8\u3002\u5bf9\u4e8e\u9891\u7e41\u63d2\u5165\u5143\u7d20\u7684\u573a\u666f\uff0c\u4e3a\u4fdd\u6301\u6570\u7ec4\u6709\u5e8f\u6027\uff0c\u9700\u8981\u5c06\u5143\u7d20\u63d2\u5165\u5230\u7279\u5b9a\u4f4d\u7f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u4e5f\u662f\u975e\u5e38\u6602\u8d35\u7684\u3002</li> <li>\u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002\u4e8c\u5206\u67e5\u627e\u9700\u8981\u8df3\u8dc3\u5f0f\uff08\u975e\u8fde\u7eed\u5730\uff09\u8bbf\u95ee\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u6267\u884c\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u7684\u6548\u7387\u8f83\u4f4e\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5e94\u7528\u5728\u94fe\u8868\u6216\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u5c0f\u6570\u636e\u91cf\u4e0b\uff0c\u7ebf\u6027\u67e5\u627e\u6027\u80fd\u66f4\u4f73\u3002\u5728\u7ebf\u6027\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u9700 1 \u6b21\u5224\u65ad\u64cd\u4f5c\uff1b\u800c\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u9700\u8981 1 \u6b21\u52a0\u6cd5\u30011 \u6b21\u9664\u6cd5\u30011 ~ 3 \u6b21\u5224\u65ad\u64cd\u4f5c\u30011 \u6b21\u52a0\u6cd5\uff08\u51cf\u6cd5\uff09\uff0c\u5171 4 ~ 6 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u56e0\u6b64\uff0c\u5f53\u6570\u636e\u91cf \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u53cd\u800c\u6bd4\u4e8c\u5206\u67e5\u627e\u66f4\u5feb\u3002</li> </ul>"},{"location":"chapter_searching/binary_search_edge/","title":"10.3 \u00a0 \u4e8c\u5206\u67e5\u627e\u8fb9\u754c","text":""},{"location":"chapter_searching/binary_search_edge/#1031","title":"10.3.1 \u00a0 \u67e5\u627e\u5de6\u8fb9\u754c","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\u3002\u8bf7\u8fd4\u56de\u6570\u7ec4\u4e2d\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 <code>target</code> \u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e2d\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002</p> <p>\u56de\u5fc6\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\u7684\u65b9\u6cd5\uff0c\u641c\u7d22\u5b8c\u6210\u540e \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u4e2a <code>target</code> \uff0c\u56e0\u6b64\u67e5\u627e\u63d2\u5165\u70b9\u672c\u8d28\u4e0a\u662f\u5728\u67e5\u627e\u6700\u5de6\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\u3002</p> <p>\u8003\u8651\u901a\u8fc7\u67e5\u627e\u63d2\u5165\u70b9\u7684\u51fd\u6570\u5b9e\u73b0\u67e5\u627e\u5de6\u8fb9\u754c\u3002\u8bf7\u6ce8\u610f\uff0c\u6570\u7ec4\u4e2d\u53ef\u80fd\u4e0d\u5305\u542b <code>target</code> \uff0c\u8fd9\u79cd\u60c5\u51b5\u53ef\u80fd\u5bfc\u81f4\u4ee5\u4e0b\u4e24\u79cd\u7ed3\u679c\u3002</p> <ul> <li>\u63d2\u5165\u70b9\u7684\u7d22\u5f15 \\(i\\) \u8d8a\u754c\u3002</li> <li>\u5143\u7d20 <code>nums[i]</code> \u4e0e <code>target</code> \u4e0d\u76f8\u7b49\u3002</li> </ul> <p>\u5f53\u9047\u5230\u4ee5\u4e0a\u4e24\u79cd\u60c5\u51b5\u65f6\uff0c\u76f4\u63a5\u8fd4\u56de \\(-1\\) \u5373\u53ef\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_edge.py<pre><code>def binary_search_left_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target\"\"\"\n    # \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    i = binary_search_insertion(nums, target)\n    # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == len(nums) or nums[i] != target:\n        return -1\n    # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n</code></pre> binary_search_edge.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    int i = binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.size() || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(int[] nums, int target) {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    int i = binary_search_insertion.binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.length || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint BinarySearchLeftEdge(int[] nums, int target) {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.Length || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunc binarySearchLeftEdge(nums []int, target int) int {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    i := binarySearchInsertion(nums, target)\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == len(nums) || nums[i] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n}\n</code></pre> binary_search_edge.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunc binarySearchLeftEdge(nums: [Int], target: Int) -&gt; Int {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    let i = binarySearchInsertion(nums: nums, target: target)\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == nums.endIndex || nums[i] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n}\n</code></pre> binary_search_edge.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunction binarySearchLeftEdge(nums, target) {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    const i = binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunction binarySearchLeftEdge(nums: Array&lt;number&gt;, target: number): number {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    const i = binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(List&lt;int&gt; nums, int target) {\n  // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n  int i = binarySearchInsertion(nums, target);\n  // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n  if (i == nums.length || nums[i] != target) {\n    return -1;\n  }\n  // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n  return i;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfn binary_search_left_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    let i = binary_search_insertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == nums.len() as i32 || nums[i as usize] != target {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    i\n}\n</code></pre> binary_search_edge.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(int *nums, int numSize, int target) {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    int i = binarySearchInsertion(nums, numSize, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == numSize || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfun binarySearchLeftEdge(nums: IntArray, target: Int): Int {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    val i = binarySearchInsertion(nums, target)\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.size || nums[i] != target) {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n}\n</code></pre> binary_search_edge.rb<pre><code>[class]{}-[func]{binary_search_left_edge}\n</code></pre> binary_search_edge.zig<pre><code>[class]{}-[func]{binarySearchLeftEdge}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_searching/binary_search_edge/#1032","title":"10.3.2 \u00a0 \u67e5\u627e\u53f3\u8fb9\u754c","text":"<p>\u90a3\u4e48\u5982\u4f55\u67e5\u627e\u6700\u53f3\u4e00\u4e2a <code>target</code> \u5462\uff1f\u6700\u76f4\u63a5\u7684\u65b9\u5f0f\u662f\u4fee\u6539\u4ee3\u7801\uff0c\u66ff\u6362\u5728 <code>nums[m] == target</code> \u60c5\u51b5\u4e0b\u7684\u6307\u9488\u6536\u7f29\u64cd\u4f5c\u3002\u4ee3\u7801\u5728\u6b64\u7701\u7565\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002</p> <p>\u4e0b\u9762\u6211\u4eec\u4ecb\u7ecd\u4e24\u79cd\u66f4\u52a0\u53d6\u5de7\u7684\u65b9\u6cd5\u3002</p>"},{"location":"chapter_searching/binary_search_edge/#1","title":"1. \u00a0 \u590d\u7528\u67e5\u627e\u5de6\u8fb9\u754c","text":"<p>\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u67e5\u627e\u6700\u5de6\u5143\u7d20\u7684\u51fd\u6570\u6765\u67e5\u627e\u6700\u53f3\u5143\u7d20\uff0c\u5177\u4f53\u65b9\u6cd5\u4e3a\uff1a\u5c06\u67e5\u627e\u6700\u53f3\u4e00\u4e2a <code>target</code> \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a <code>target + 1</code>\u3002</p> <p>\u5982\u56fe 10-7 \u6240\u793a\uff0c\u67e5\u627e\u5b8c\u6210\u540e\uff0c\u6307\u9488 \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u4e2a <code>target + 1</code>\uff08\u5982\u679c\u5b58\u5728\uff09\uff0c\u800c \\(j\\) \u6307\u5411\u6700\u53f3\u4e00\u4e2a <code>target</code> \uff0c\u56e0\u6b64\u8fd4\u56de \\(j\\) \u5373\u53ef\u3002</p> <p></p> <p> \u56fe 10-7 \u00a0 \u5c06\u67e5\u627e\u53f3\u8fb9\u754c\u8f6c\u5316\u4e3a\u67e5\u627e\u5de6\u8fb9\u754c </p> <p>\u8bf7\u6ce8\u610f\uff0c\u8fd4\u56de\u7684\u63d2\u5165\u70b9\u662f \\(i\\) \uff0c\u56e0\u6b64\u9700\u8981\u5c06\u5176\u51cf \\(1\\) \uff0c\u4ece\u800c\u83b7\u5f97 \\(j\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_edge.py<pre><code>def binary_search_right_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target\"\"\"\n    # \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    i = binary_search_insertion(nums, target + 1)\n    # j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    j = i - 1\n    # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 or nums[j] != target:\n        return -1\n    # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n</code></pre> binary_search_edge.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    int i = binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(int[] nums, int target) {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    int i = binary_search_insertion.binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint BinarySearchRightEdge(int[] nums, int target) {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunc binarySearchRightEdge(nums []int, target int) int {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    i := binarySearchInsertion(nums, target+1)\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    j := i - 1\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n}\n</code></pre> binary_search_edge.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunc binarySearchRightEdge(nums: [Int], target: Int) -&gt; Int {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    let i = binarySearchInsertion(nums: nums, target: target + 1)\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    let j = i - 1\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n}\n</code></pre> binary_search_edge.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunction binarySearchRightEdge(nums, target) {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    const i = binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    const j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunction binarySearchRightEdge(nums: Array&lt;number&gt;, target: number): number {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    const i = binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    const j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(List&lt;int&gt; nums, int target) {\n  // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n  int i = binarySearchInsertion(nums, target + 1);\n  // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n  int j = i - 1;\n  // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n  if (j == -1 || nums[j] != target) {\n    return -1;\n  }\n  // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n  return j;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfn binary_search_right_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    let i = binary_search_insertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    let j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 || nums[j as usize] != target {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    j\n}\n</code></pre> binary_search_edge.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(int *nums, int numSize, int target) {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    int i = binarySearchInsertion(nums, numSize, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfun binarySearchRightEdge(nums: IntArray, target: Int): Int {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    val i = binarySearchInsertion(nums, target + 1)\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    val j = i - 1\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n}\n</code></pre> binary_search_edge.rb<pre><code>[class]{}-[func]{binary_search_right_edge}\n</code></pre> binary_search_edge.zig<pre><code>[class]{}-[func]{binarySearchRightEdge}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_searching/binary_search_edge/#2","title":"2. \u00a0 \u8f6c\u5316\u4e3a\u67e5\u627e\u5143\u7d20","text":"<p>\u6211\u4eec\u77e5\u9053\uff0c\u5f53\u6570\u7ec4\u4e0d\u5305\u542b <code>target</code> \u65f6\uff0c\u6700\u7ec8 \\(i\\) \u548c \\(j\\) \u4f1a\u5206\u522b\u6307\u5411\u9996\u4e2a\u5927\u4e8e\u3001\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u3002</p> <p>\u56e0\u6b64\uff0c\u5982\u56fe 10-8 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u6784\u9020\u4e00\u4e2a\u6570\u7ec4\u4e2d\u4e0d\u5b58\u5728\u7684\u5143\u7d20\uff0c\u7528\u4e8e\u67e5\u627e\u5de6\u53f3\u8fb9\u754c\u3002</p> <ul> <li>\u67e5\u627e\u6700\u5de6\u4e00\u4e2a <code>target</code> \uff1a\u53ef\u4ee5\u8f6c\u5316\u4e3a\u67e5\u627e <code>target - 0.5</code> \uff0c\u5e76\u8fd4\u56de\u6307\u9488 \\(i\\) \u3002</li> <li>\u67e5\u627e\u6700\u53f3\u4e00\u4e2a <code>target</code> \uff1a\u53ef\u4ee5\u8f6c\u5316\u4e3a\u67e5\u627e <code>target + 0.5</code> \uff0c\u5e76\u8fd4\u56de\u6307\u9488 \\(j\\) \u3002</li> </ul> <p></p> <p> \u56fe 10-8 \u00a0 \u5c06\u67e5\u627e\u8fb9\u754c\u8f6c\u5316\u4e3a\u67e5\u627e\u5143\u7d20 </p> <p>\u4ee3\u7801\u5728\u6b64\u7701\u7565\uff0c\u4ee5\u4e0b\u4e24\u70b9\u503c\u5f97\u6ce8\u610f\u3002</p> <ul> <li>\u7ed9\u5b9a\u6570\u7ec4\u4e0d\u5305\u542b\u5c0f\u6570\uff0c\u8fd9\u610f\u5473\u7740\u6211\u4eec\u65e0\u987b\u5173\u5fc3\u5982\u4f55\u5904\u7406\u76f8\u7b49\u7684\u60c5\u51b5\u3002</li> <li>\u56e0\u4e3a\u8be5\u65b9\u6cd5\u5f15\u5165\u4e86\u5c0f\u6570\uff0c\u6240\u4ee5\u9700\u8981\u5c06\u51fd\u6570\u4e2d\u7684\u53d8\u91cf <code>target</code> \u6539\u4e3a\u6d6e\u70b9\u6570\u7c7b\u578b\uff08Python \u65e0\u987b\u6539\u52a8\uff09\u3002</li> </ul>"},{"location":"chapter_searching/binary_search_insertion/","title":"10.2 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9","text":"<p>\u4e8c\u5206\u67e5\u627e\u4e0d\u4ec5\u53ef\u7528\u4e8e\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff0c\u8fd8\u53ef\u7528\u4e8e\u89e3\u51b3\u8bb8\u591a\u53d8\u79cd\u95ee\u9898\uff0c\u6bd4\u5982\u641c\u7d22\u76ee\u6807\u5143\u7d20\u7684\u63d2\u5165\u4f4d\u7f6e\u3002</p>"},{"location":"chapter_searching/binary_search_insertion/#1021","title":"10.2.1 \u00a0 \u65e0\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u5143\u7d20 <code>target</code> \uff0c\u6570\u7ec4\u4e0d\u5b58\u5728\u91cd\u590d\u5143\u7d20\u3002\u73b0\u5c06 <code>target</code> \u63d2\u5165\u6570\u7ec4 <code>nums</code> \u4e2d\uff0c\u5e76\u4fdd\u6301\u5176\u6709\u5e8f\u6027\u3002\u82e5\u6570\u7ec4\u4e2d\u5df2\u5b58\u5728\u5143\u7d20 <code>target</code> \uff0c\u5219\u63d2\u5165\u5230\u5176\u5de6\u65b9\u3002\u8bf7\u8fd4\u56de\u63d2\u5165\u540e <code>target</code> \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15\u3002\u793a\u4f8b\u5982\u56fe 10-4 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 10-4 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\u793a\u4f8b\u6570\u636e </p> <p>\u5982\u679c\u60f3\u590d\u7528\u4e0a\u4e00\u8282\u7684\u4e8c\u5206\u67e5\u627e\u4ee3\u7801\uff0c\u5219\u9700\u8981\u56de\u7b54\u4ee5\u4e0b\u4e24\u4e2a\u95ee\u9898\u3002</p> <p>\u95ee\u9898\u4e00\uff1a\u5f53\u6570\u7ec4\u4e2d\u5305\u542b <code>target</code> \u65f6\uff0c\u63d2\u5165\u70b9\u7684\u7d22\u5f15\u662f\u5426\u662f\u8be5\u5143\u7d20\u7684\u7d22\u5f15\uff1f</p> <p>\u9898\u76ee\u8981\u6c42\u5c06 <code>target</code> \u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u5de6\u8fb9\uff0c\u8fd9\u610f\u5473\u7740\u65b0\u63d2\u5165\u7684 <code>target</code> \u66ff\u6362\u4e86\u539f\u6765 <code>target</code> \u7684\u4f4d\u7f6e\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5f53\u6570\u7ec4\u5305\u542b <code>target</code> \u65f6\uff0c\u63d2\u5165\u70b9\u7684\u7d22\u5f15\u5c31\u662f\u8be5 <code>target</code> \u7684\u7d22\u5f15\u3002</p> <p>\u95ee\u9898\u4e8c\uff1a\u5f53\u6570\u7ec4\u4e2d\u4e0d\u5b58\u5728 <code>target</code> \u65f6\uff0c\u63d2\u5165\u70b9\u662f\u54ea\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff1f</p> <p>\u8fdb\u4e00\u6b65\u601d\u8003\u4e8c\u5206\u67e5\u627e\u8fc7\u7a0b\uff1a\u5f53 <code>nums[m] &lt; target</code> \u65f6 \\(i\\) \u79fb\u52a8\uff0c\u8fd9\u610f\u5473\u7740\u6307\u9488 \\(i\\) \u5728\u5411\u5927\u4e8e\u7b49\u4e8e <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002\u540c\u7406\uff0c\u6307\u9488 \\(j\\) \u59cb\u7ec8\u5728\u5411\u5c0f\u4e8e\u7b49\u4e8e <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002</p> <p>\u56e0\u6b64\u4e8c\u5206\u7ed3\u675f\u65f6\u4e00\u5b9a\u6709\uff1a\\(i\\) \u6307\u5411\u9996\u4e2a\u5927\u4e8e <code>target</code> \u7684\u5143\u7d20\uff0c\\(j\\) \u6307\u5411\u9996\u4e2a\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u3002\u6613\u5f97\u5f53\u6570\u7ec4\u4e0d\u5305\u542b <code>target</code> \u65f6\uff0c\u63d2\u5165\u7d22\u5f15\u4e3a \\(i\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_insertion.py<pre><code>def binary_search_insertion_simple(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09\"\"\"\n    i, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while i &lt;= j:\n        m = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        elif nums[m] &gt; target:\n            j = m - 1  # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        else:\n            return m  # \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n    # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint BinarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        } else {\n            // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n            return m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target {\n            i = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if nums[m] &gt; target {\n            j = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(\n    nums: Array&lt;number&gt;,\n    target: number\n): number {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n    } else {\n      return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n    }\n  }\n  // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfn binary_search_insertion_simple(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m;\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfun binarySearchInsertionSimple(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>[class]{}-[func]{binary_search_insertion_simple}\n</code></pre> binary_search_insertion.zig<pre><code>[class]{}-[func]{binarySearchInsertionSimple}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_searching/binary_search_insertion/#1022","title":"10.2.2 \u00a0 \u5b58\u5728\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u5728\u4e0a\u4e00\u9898\u7684\u57fa\u7840\u4e0a\uff0c\u89c4\u5b9a\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u5176\u4f59\u4e0d\u53d8\u3002</p> <p>\u5047\u8bbe\u6570\u7ec4\u4e2d\u5b58\u5728\u591a\u4e2a <code>target</code> \uff0c\u5219\u666e\u901a\u4e8c\u5206\u67e5\u627e\u53ea\u80fd\u8fd4\u56de\u5176\u4e2d\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\uff0c\u800c\u65e0\u6cd5\u786e\u5b9a\u8be5\u5143\u7d20\u7684\u5de6\u8fb9\u548c\u53f3\u8fb9\u8fd8\u6709\u591a\u5c11 <code>target</code>\u3002</p> <p>\u9898\u76ee\u8981\u6c42\u5c06\u76ee\u6807\u5143\u7d20\u63d2\u5165\u5230\u6700\u5de6\u8fb9\uff0c\u6240\u4ee5\u6211\u4eec\u9700\u8981\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5de6\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\u3002\u521d\u6b65\u8003\u8651\u901a\u8fc7\u56fe 10-5 \u6240\u793a\u7684\u6b65\u9aa4\u5b9e\u73b0\u3002</p> <ol> <li>\u6267\u884c\u4e8c\u5206\u67e5\u627e\uff0c\u5f97\u5230\u4efb\u610f\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\uff0c\u8bb0\u4e3a \\(k\\) \u3002</li> <li>\u4ece\u7d22\u5f15 \\(k\\) \u5f00\u59cb\uff0c\u5411\u5de6\u8fdb\u884c\u7ebf\u6027\u904d\u5386\uff0c\u5f53\u627e\u5230\u6700\u5de6\u8fb9\u7684 <code>target</code> \u65f6\u8fd4\u56de\u3002</li> </ol> <p></p> <p> \u56fe 10-5 \u00a0 \u7ebf\u6027\u67e5\u627e\u91cd\u590d\u5143\u7d20\u7684\u63d2\u5165\u70b9 </p> <p>\u6b64\u65b9\u6cd5\u867d\u7136\u53ef\u7528\uff0c\u4f46\u5176\u5305\u542b\u7ebf\u6027\u67e5\u627e\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5f53\u6570\u7ec4\u4e2d\u5b58\u5728\u5f88\u591a\u91cd\u590d\u7684 <code>target</code> \u65f6\uff0c\u8be5\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\u3002</p> <p>\u73b0\u8003\u8651\u62d3\u5c55\u4e8c\u5206\u67e5\u627e\u4ee3\u7801\u3002\u5982\u56fe 10-6 \u6240\u793a\uff0c\u6574\u4f53\u6d41\u7a0b\u4fdd\u6301\u4e0d\u53d8\uff0c\u6bcf\u8f6e\u5148\u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m\\) \uff0c\u518d\u5224\u65ad <code>target</code> \u548c <code>nums[m]</code> \u7684\u5927\u5c0f\u5173\u7cfb\uff0c\u5206\u4e3a\u4ee5\u4e0b\u51e0\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u5f53 <code>nums[m] &lt; target</code> \u6216 <code>nums[m] &gt; target</code> \u65f6\uff0c\u8bf4\u660e\u8fd8\u6ca1\u6709\u627e\u5230 <code>target</code> \uff0c\u56e0\u6b64\u91c7\u7528\u666e\u901a\u4e8c\u5206\u67e5\u627e\u7684\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5411 <code>target</code> \u9760\u8fd1\u3002</li> <li>\u5f53 <code>nums[m] == target</code> \u65f6\uff0c\u8bf4\u660e\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u91c7\u7528 \\(j = m - 1\\) \u6765\u7f29\u5c0f\u533a\u95f4\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(j\\) \u5411\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002</li> </ul> <p>\u5faa\u73af\u5b8c\u6210\u540e\uff0c\\(i\\) \u6307\u5411\u6700\u5de6\u8fb9\u7684 <code>target</code> \uff0c\\(j\\) \u6307\u5411\u9996\u4e2a\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\uff0c\u56e0\u6b64\u7d22\u5f15 \\(i\\) \u5c31\u662f\u63d2\u5165\u70b9\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 10-6 \u00a0 \u4e8c\u5206\u67e5\u627e\u91cd\u590d\u5143\u7d20\u7684\u63d2\u5165\u70b9\u7684\u6b65\u9aa4 </p> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u5224\u65ad\u5206\u652f <code>nums[m] &gt; target</code> \u548c <code>nums[m] == target</code> \u7684\u64cd\u4f5c\u76f8\u540c\uff0c\u56e0\u6b64\u4e24\u8005\u53ef\u4ee5\u5408\u5e76\u3002</p> <p>\u5373\u4fbf\u5982\u6b64\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u5c06\u5224\u65ad\u6761\u4ef6\u4fdd\u6301\u5c55\u5f00\uff0c\u56e0\u4e3a\u5176\u903b\u8f91\u66f4\u52a0\u6e05\u6670\u3001\u53ef\u8bfb\u6027\u66f4\u597d\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_insertion.py<pre><code>def binary_search_insertion(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09\"\"\"\n    i, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while i &lt;= j:\n        m = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        elif nums[m] &gt; target:\n            j = m - 1  # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        else:\n            j = m - 1  # \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n    # \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint BinarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        } else {\n            // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target {\n            i = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if nums[m] &gt; target {\n            j = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1 // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums: Array&lt;number&gt;, target: number): number {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n    } else {\n      j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n    }\n  }\n  // \u8fd4\u56de\u63d2\u5165\u70b9 i\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\npub fn binary_search_insertion(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfun binarySearchInsertion(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1 // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>[class]{}-[func]{binary_search_insertion}\n</code></pre> binary_search_insertion.zig<pre><code>[class]{}-[func]{binarySearchInsertion}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>Tip</p> <p>\u672c\u8282\u7684\u4ee3\u7801\u90fd\u662f\u201c\u53cc\u95ed\u533a\u95f4\u201d\u5199\u6cd5\u3002\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u5199\u6cd5\u3002</p> <p>\u603b\u7684\u6765\u770b\uff0c\u4e8c\u5206\u67e5\u627e\u65e0\u975e\u5c31\u662f\u7ed9\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5206\u522b\u8bbe\u5b9a\u641c\u7d22\u76ee\u6807\uff0c\u76ee\u6807\u53ef\u80fd\u662f\u4e00\u4e2a\u5177\u4f53\u7684\u5143\u7d20\uff08\u4f8b\u5982 <code>target</code> \uff09\uff0c\u4e5f\u53ef\u80fd\u662f\u4e00\u4e2a\u5143\u7d20\u8303\u56f4\uff08\u4f8b\u5982\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\uff09\u3002</p> <p>\u5728\u4e0d\u65ad\u7684\u5faa\u73af\u4e8c\u5206\u4e2d\uff0c\u6307\u9488 \\(i\\) \u548c \\(j\\) \u90fd\u9010\u6e10\u903c\u8fd1\u9884\u5148\u8bbe\u5b9a\u7684\u76ee\u6807\u3002\u6700\u7ec8\uff0c\u5b83\u4eec\u6216\u662f\u6210\u529f\u627e\u5230\u7b54\u6848\uff0c\u6216\u662f\u8d8a\u8fc7\u8fb9\u754c\u540e\u505c\u6b62\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/","title":"10.4 \u00a0 \u54c8\u5e0c\u4f18\u5316\u7b56\u7565","text":"<p>\u5728\u7b97\u6cd5\u9898\u4e2d\uff0c\u6211\u4eec\u5e38\u901a\u8fc7\u5c06\u7ebf\u6027\u67e5\u627e\u66ff\u6362\u4e3a\u54c8\u5e0c\u67e5\u627e\u6765\u964d\u4f4e\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u7b97\u6cd5\u9898\u6765\u52a0\u6df1\u7406\u89e3\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6574\u6570\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u76ee\u6807\u5143\u7d20 <code>target</code> \uff0c\u8bf7\u5728\u6570\u7ec4\u4e2d\u641c\u7d22\u201c\u548c\u201d\u4e3a <code>target</code> \u7684\u4e24\u4e2a\u5143\u7d20\uff0c\u5e76\u8fd4\u56de\u5b83\u4eec\u7684\u6570\u7ec4\u7d22\u5f15\u3002\u8fd4\u56de\u4efb\u610f\u4e00\u4e2a\u89e3\u5373\u53ef\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/#1041","title":"10.4.1 \u00a0 \u7ebf\u6027\u67e5\u627e\uff1a\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4","text":"<p>\u8003\u8651\u76f4\u63a5\u904d\u5386\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\u3002\u5982\u56fe 10-9 \u6240\u793a\uff0c\u6211\u4eec\u5f00\u542f\u4e00\u4e2a\u4e24\u5c42\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u5224\u65ad\u4e24\u4e2a\u6574\u6570\u7684\u548c\u662f\u5426\u4e3a <code>target</code> \uff0c\u82e5\u662f\uff0c\u5219\u8fd4\u56de\u5b83\u4eec\u7684\u7d22\u5f15\u3002</p> <p></p> <p> \u56fe 10-9 \u00a0 \u7ebf\u6027\u67e5\u627e\u6c42\u89e3\u4e24\u6570\u4e4b\u548c </p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig two_sum.py<pre><code>def two_sum_brute_force(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"\u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\"\"\"\n    # \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for i in range(len(nums) - 1):\n        for j in range(i + 1, len(nums)):\n            if nums[i] + nums[j] == target:\n                return [i, j]\n    return []\n</code></pre> two_sum.cpp<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nvector&lt;int&gt; twoSumBruteForce(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return {i, j};\n        }\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] twoSumBruteForce(int[] nums, int target) {\n    int size = nums.length;\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return new int[] { i, j };\n        }\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] TwoSumBruteForce(int[] nums, int target) {\n    int size = nums.Length;\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return [i, j];\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums []int, target int) []int {\n    size := len(nums)\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for i := 0; i &lt; size-1; i++ {\n        for j := i + 1; i &lt; size; j++ {\n            if nums[i]+nums[j] == target {\n                return []int{i, j}\n            }\n        }\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums: [Int], target: Int) -&gt; [Int] {\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for i in nums.indices.dropLast() {\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[i] + nums[j] == target {\n                return [i, j]\n            }\n        }\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums, target) {\n    const n = nums.length;\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\n    const n = nums.length;\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* \u65b9\u6cd5\u4e00\uff1a \u66b4\u529b\u679a\u4e3e */\nList&lt;int&gt; twoSumBruteForce(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n  for (var i = 0; i &lt; size - 1; i++) {\n    for (var j = i + 1; j &lt; size; j++) {\n      if (nums[i] + nums[j] == target) return [i, j];\n    }\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\npub fn two_sum_brute_force(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    let size = nums.len();\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for i in 0..size - 1 {\n        for j in i + 1..size {\n            if nums[i] + nums[j] == target {\n                return Some(vec![i as i32, j as i32]);\n            }\n        }\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint *twoSumBruteForce(int *nums, int numsSize, int target, int *returnSize) {\n    for (int i = 0; i &lt; numsSize; ++i) {\n        for (int j = i + 1; j &lt; numsSize; ++j) {\n            if (nums[i] + nums[j] == target) {\n                int *res = malloc(sizeof(int) * 2);\n                res[0] = i, res[1] = j;\n                *returnSize = 2;\n                return res;\n            }\n        }\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfun twoSumBruteForce(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (i in 0..&lt;size - 1) {\n        for (j in i + 1..&lt;size) {\n            if (nums[i] + nums[j] == target) return intArrayOf(i, j)\n        }\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>[class]{}-[func]{two_sum_brute_force}\n</code></pre> two_sum.zig<pre><code>// \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\nfn twoSumBruteForce(nums: []i32, target: i32) ?[2]i32 {\n    var size: usize = nums.len;\n    var i: usize = 0;\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    while (i &lt; size - 1) : (i += 1) {\n        var j = i + 1;\n        while (j &lt; size) : (j += 1) {\n            if (nums[i] + nums[j] == target) {\n                return [_]i32{@intCast(i), @intCast(j)};\n            }\n        }\n    }\n    return null;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u6b64\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u975e\u5e38\u8017\u65f6\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/#1042","title":"10.4.2 \u00a0 \u54c8\u5e0c\u67e5\u627e\uff1a\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4","text":"<p>\u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u952e\u503c\u5bf9\u5206\u522b\u4e3a\u6570\u7ec4\u5143\u7d20\u548c\u5143\u7d20\u7d22\u5f15\u3002\u5faa\u73af\u904d\u5386\u6570\u7ec4\uff0c\u6bcf\u8f6e\u6267\u884c\u56fe 10-10 \u6240\u793a\u7684\u6b65\u9aa4\u3002</p> <ol> <li>\u5224\u65ad\u6570\u5b57 <code>target - nums[i]</code> \u662f\u5426\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u82e5\u662f\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u3002</li> <li>\u5c06\u952e\u503c\u5bf9 <code>nums[i]</code> \u548c\u7d22\u5f15 <code>i</code> \u6dfb\u52a0\u8fdb\u54c8\u5e0c\u8868\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> \u56fe 10-10 \u00a0 \u8f85\u52a9\u54c8\u5e0c\u8868\u6c42\u89e3\u4e24\u6570\u4e4b\u548c </p> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff0c\u4ec5\u9700\u5355\u5c42\u5faa\u73af\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig two_sum.py<pre><code>def two_sum_hash_table(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"\u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\"\"\"\n    # \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    dic = {}\n    # \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for i in range(len(nums)):\n        if target - nums[i] in dic:\n            return [dic[target - nums[i]], i]\n        dic[nums[i]] = i\n    return []\n</code></pre> two_sum.cpp<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nvector&lt;int&gt; twoSumHashTable(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    unordered_map&lt;int, int&gt; dic;\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.find(target - nums[i]) != dic.end()) {\n            return {dic[target - nums[i]], i};\n        }\n        dic.emplace(nums[i], i);\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] twoSumHashTable(int[] nums, int target) {\n    int size = nums.length;\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    Map&lt;Integer, Integer&gt; dic = new HashMap&lt;&gt;();\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.containsKey(target - nums[i])) {\n            return new int[] { dic.get(target - nums[i]), i };\n        }\n        dic.put(nums[i], i);\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] TwoSumHashTable(int[] nums, int target) {\n    int size = nums.Length;\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    Dictionary&lt;int, int&gt; dic = [];\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.ContainsKey(target - nums[i])) {\n            return [dic[target - nums[i]], i];\n        }\n        dic.Add(nums[i], i);\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums []int, target int) []int {\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    hashTable := map[int]int{}\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for idx, val := range nums {\n        if preIdx, ok := hashTable[target-val]; ok {\n            return []int{preIdx, idx}\n        }\n        hashTable[val] = idx\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums: [Int], target: Int) -&gt; [Int] {\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    var dic: [Int: Int] = [:]\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for i in nums.indices {\n        if let j = dic[target - nums[i]] {\n            return [j, i]\n        }\n        dic[nums[i]] = i\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums, target) {\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    let m = {};\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (m[target - nums[i]] !== undefined) {\n            return [m[target - nums[i]], i];\n        } else {\n            m[nums[i]] = i;\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    let m: Map&lt;number, number&gt; = new Map();\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (let i = 0; i &lt; nums.length; i++) {\n        let index = m.get(target - nums[i]);\n        if (index !== undefined) {\n            return [index, i];\n        } else {\n            m.set(nums[i], i);\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a \u8f85\u52a9\u54c8\u5e0c\u8868 */\nList&lt;int&gt; twoSumHashTable(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n  Map&lt;int, int&gt; dic = HashMap();\n  // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n  for (var i = 0; i &lt; size; i++) {\n    if (dic.containsKey(target - nums[i])) {\n      return [dic[target - nums[i]]!, i];\n    }\n    dic.putIfAbsent(nums[i], () =&gt; i);\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\npub fn two_sum_hash_table(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    let mut dic = HashMap::new();\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (i, num) in nums.iter().enumerate() {\n        match dic.get(&amp;(target - num)) {\n            Some(v) =&gt; return Some(vec![*v as i32, i as i32]),\n            None =&gt; dic.insert(num, i as i32),\n        };\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* \u54c8\u5e0c\u8868 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n\n/* \u54c8\u5e0c\u8868\u67e5\u8be2 */\nHashTable *find(HashTable *h, int key) {\n    HashTable *tmp;\n    HASH_FIND_INT(h, &amp;key, tmp);\n    return tmp;\n}\n\n/* \u54c8\u5e0c\u8868\u5143\u7d20\u63d2\u5165 */\nvoid insert(HashTable *h, int key, int val) {\n    HashTable *t = find(h, key);\n    if (t == NULL) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = key, tmp-&gt;val = val;\n        HASH_ADD_INT(h, key, tmp);\n    } else {\n        t-&gt;val = val;\n    }\n}\n\n/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint *twoSumHashTable(int *nums, int numsSize, int target, int *returnSize) {\n    HashTable *hashtable = NULL;\n    for (int i = 0; i &lt; numsSize; i++) {\n        HashTable *t = find(hashtable, target - nums[i]);\n        if (t != NULL) {\n            int *res = malloc(sizeof(int) * 2);\n            res[0] = t-&gt;val, res[1] = i;\n            *returnSize = 2;\n            return res;\n        }\n        insert(hashtable, nums[i], i);\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfun twoSumHashTable(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    val dic = HashMap&lt;Int, Int&gt;()\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (i in 0..&lt;size) {\n        if (dic.containsKey(target - nums[i])) {\n            return intArrayOf(dic[target - nums[i]]!!, i)\n        }\n        dic[nums[i]] = i\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>[class]{}-[func]{two_sum_hash_table}\n</code></pre> two_sum.zig<pre><code>// \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\nfn twoSumHashTable(nums: []i32, target: i32) !?[2]i32 {\n    var size: usize = nums.len;\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    var dic = std.AutoHashMap(i32, i32).init(std.heap.page_allocator);\n    defer dic.deinit();\n    var i: usize = 0;\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    while (i &lt; size) : (i += 1) {\n        if (dic.contains(target - nums[i])) {\n            return [_]i32{dic.get(target - nums[i]).?, @intCast(i)};\n        }\n        try dic.put(nums[i], @intCast(i));\n    }\n    return null;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u6b64\u65b9\u6cd5\u901a\u8fc7\u54c8\u5e0c\u67e5\u627e\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \uff0c\u5927\u5e45\u63d0\u5347\u8fd0\u884c\u6548\u7387\u3002</p> <p>\u7531\u4e8e\u9700\u8981\u7ef4\u62a4\u4e00\u4e2a\u989d\u5916\u7684\u54c8\u5e0c\u8868\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5c3d\u7ba1\u5982\u6b64\uff0c\u8be5\u65b9\u6cd5\u7684\u6574\u4f53\u65f6\u7a7a\u6548\u7387\u66f4\u4e3a\u5747\u8861\uff0c\u56e0\u6b64\u5b83\u662f\u672c\u9898\u7684\u6700\u4f18\u89e3\u6cd5\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/","title":"10.5 \u00a0 \u91cd\u8bc6\u641c\u7d22\u7b97\u6cd5","text":"<p>\u300c\u641c\u7d22\u7b97\u6cd5 searching algorithm\u300d\u7528\u4e8e\u5728\u6570\u636e\u7ed3\u6784\uff08\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u6216\u56fe\uff09\u4e2d\u641c\u7d22\u4e00\u4e2a\u6216\u4e00\u7ec4\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5143\u7d20\u3002</p> <p>\u641c\u7d22\u7b97\u6cd5\u53ef\u6839\u636e\u5b9e\u73b0\u601d\u8def\u5206\u4e3a\u4ee5\u4e0b\u4e24\u7c7b\u3002</p> <ul> <li>\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\uff0c\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u548c\u56fe\u7684\u904d\u5386\u7b49\u3002</li> <li>\u5229\u7528\u6570\u636e\u7ec4\u7ec7\u7ed3\u6784\u6216\u6570\u636e\u5305\u542b\u7684\u5148\u9a8c\u4fe1\u606f\uff0c\u5b9e\u73b0\u9ad8\u6548\u5143\u7d20\u67e5\u627e\uff0c\u4f8b\u5982\u4e8c\u5206\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u548c\u4e8c\u53c9\u641c\u7d22\u6811\u67e5\u627e\u7b49\u3002</li> </ul> <p>\u4e0d\u96be\u53d1\u73b0\uff0c\u8fd9\u4e9b\u77e5\u8bc6\u70b9\u90fd\u5df2\u5728\u524d\u9762\u7684\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u56e0\u6b64\u641c\u7d22\u7b97\u6cd5\u5bf9\u4e8e\u6211\u4eec\u6765\u8bf4\u5e76\u4e0d\u964c\u751f\u3002\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u4ece\u66f4\u52a0\u7cfb\u7edf\u7684\u89c6\u89d2\u5207\u5165\uff0c\u91cd\u65b0\u5ba1\u89c6\u641c\u7d22\u7b97\u6cd5\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1051","title":"10.5.1 \u00a0 \u66b4\u529b\u641c\u7d22","text":"<p>\u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u7684\u6bcf\u4e2a\u5143\u7d20\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002</p> <ul> <li>\u201c\u7ebf\u6027\u641c\u7d22\u201d\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\u7b49\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u5b83\u4ece\u6570\u636e\u7ed3\u6784\u7684\u4e00\u7aef\u5f00\u59cb\uff0c\u9010\u4e2a\u8bbf\u95ee\u5143\u7d20\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u5230\u8fbe\u53e6\u4e00\u7aef\u4ecd\u6ca1\u6709\u627e\u5230\u76ee\u6807\u5143\u7d20\u4e3a\u6b62\u3002</li> <li>\u201c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u548c\u201c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u662f\u56fe\u548c\u6811\u7684\u4e24\u79cd\u904d\u5386\u7b56\u7565\u3002\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\u9010\u5c42\u641c\u7d22\uff0c\u7531\u8fd1\u53ca\u8fdc\u5730\u8bbf\u95ee\u5404\u4e2a\u8282\u70b9\u3002\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\uff0c\u6cbf\u7740\u4e00\u6761\u8def\u5f84\u8d70\u5230\u5934\uff0c\u518d\u56de\u6eaf\u5e76\u5c1d\u8bd5\u5176\u4ed6\u8def\u5f84\uff0c\u76f4\u5230\u904d\u5386\u5b8c\u6574\u4e2a\u6570\u636e\u7ed3\u6784\u3002</li> </ul> <p>\u66b4\u529b\u641c\u7d22\u7684\u4f18\u70b9\u662f\u7b80\u5355\u4e14\u901a\u7528\u6027\u597d\uff0c\u65e0\u987b\u5bf9\u6570\u636e\u505a\u9884\u5904\u7406\u548c\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u7136\u800c\uff0c\u6b64\u7c7b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u5143\u7d20\u6570\u91cf\uff0c\u56e0\u6b64\u5728\u6570\u636e\u91cf\u8f83\u5927\u7684\u60c5\u51b5\u4e0b\u6027\u80fd\u8f83\u5dee\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1052","title":"10.5.2 \u00a0 \u81ea\u9002\u5e94\u641c\u7d22","text":"<p>\u81ea\u9002\u5e94\u641c\u7d22\u5229\u7528\u6570\u636e\u7684\u7279\u6709\u5c5e\u6027\uff08\u4f8b\u5982\u6709\u5e8f\u6027\uff09\u6765\u4f18\u5316\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ece\u800c\u66f4\u9ad8\u6548\u5730\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002</p> <ul> <li>\u201c\u4e8c\u5206\u67e5\u627e\u201d\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\u5b9e\u73b0\u9ad8\u6548\u67e5\u627e\uff0c\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002</li> <li>\u201c\u54c8\u5e0c\u67e5\u627e\u201d\u5229\u7528\u54c8\u5e0c\u8868\u5c06\u641c\u7d22\u6570\u636e\u548c\u76ee\u6807\u6570\u636e\u5efa\u7acb\u4e3a\u952e\u503c\u5bf9\u6620\u5c04\uff0c\u4ece\u800c\u5b9e\u73b0\u67e5\u8be2\u64cd\u4f5c\u3002</li> <li>\u201c\u6811\u67e5\u627e\u201d\u5728\u7279\u5b9a\u7684\u6811\u7ed3\u6784\uff08\u4f8b\u5982\u4e8c\u53c9\u641c\u7d22\u6811\uff09\u4e2d\uff0c\u57fa\u4e8e\u6bd4\u8f83\u8282\u70b9\u503c\u6765\u5feb\u901f\u6392\u9664\u8282\u70b9\uff0c\u4ece\u800c\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002</li> </ul> <p>\u6b64\u7c7b\u7b97\u6cd5\u7684\u4f18\u70b9\u662f\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002</p> <p>\u7136\u800c\uff0c\u4f7f\u7528\u8fd9\u4e9b\u7b97\u6cd5\u5f80\u5f80\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u9884\u5904\u7406\u3002\u4f8b\u5982\uff0c\u4e8c\u5206\u67e5\u627e\u9700\u8981\u9884\u5148\u5bf9\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\uff0c\u54c8\u5e0c\u67e5\u627e\u548c\u6811\u67e5\u627e\u90fd\u9700\u8981\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u7ef4\u62a4\u8fd9\u4e9b\u6570\u636e\u7ed3\u6784\u4e5f\u9700\u8981\u989d\u5916\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u5f00\u9500\u3002</p> <p>Tip</p> <p>\u81ea\u9002\u5e94\u641c\u7d22\u7b97\u6cd5\u5e38\u88ab\u79f0\u4e3a\u67e5\u627e\u7b97\u6cd5\uff0c\u4e3b\u8981\u7528\u4e8e\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u68c0\u7d22\u76ee\u6807\u5143\u7d20\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1053","title":"10.5.3 \u00a0 \u641c\u7d22\u65b9\u6cd5\u9009\u53d6","text":"<p>\u7ed9\u5b9a\u5927\u5c0f\u4e3a \\(n\\) \u7684\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u7ebf\u6027\u641c\u7d22\u3001\u4e8c\u5206\u67e5\u627e\u3001\u6811\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u7b49\u591a\u79cd\u65b9\u6cd5\u4ece\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\u3002\u5404\u4e2a\u65b9\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u5982\u56fe 10-11 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 10-11 \u00a0 \u591a\u79cd\u641c\u7d22\u7b56\u7565 </p> <p>\u4e0a\u8ff0\u51e0\u79cd\u65b9\u6cd5\u7684\u64cd\u4f5c\u6548\u7387\u4e0e\u7279\u6027\u5982\u8868 10-1 \u6240\u793a\u3002</p> <p> \u8868 10-1 \u00a0 \u67e5\u627e\u7b97\u6cd5\u6548\u7387\u5bf9\u6bd4 </p> \u7ebf\u6027\u641c\u7d22 \u4e8c\u5206\u67e5\u627e \u6811\u67e5\u627e \u54c8\u5e0c\u67e5\u627e \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u989d\u5916\u7a7a\u95f4 \\(O(1)\\) \\(O(1)\\) \\(O(n)\\) \\(O(n)\\) \u6570\u636e\u9884\u5904\u7406 / \u6392\u5e8f \\(O(n \\log n)\\) \u5efa\u6811 \\(O(n \\log n)\\) \u5efa\u54c8\u5e0c\u8868 \\(O(n)\\) \u6570\u636e\u662f\u5426\u6709\u5e8f \u65e0\u5e8f \u6709\u5e8f \u6709\u5e8f \u65e0\u5e8f <p>\u641c\u7d22\u7b97\u6cd5\u7684\u9009\u62e9\u8fd8\u53d6\u51b3\u4e8e\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u4e0e\u66f4\u65b0\u9891\u7387\u7b49\u3002</p> <p>\u7ebf\u6027\u641c\u7d22</p> <ul> <li>\u901a\u7528\u6027\u8f83\u597d\uff0c\u65e0\u987b\u4efb\u4f55\u6570\u636e\u9884\u5904\u7406\u64cd\u4f5c\u3002\u5047\u5982\u6211\u4eec\u4ec5\u9700\u67e5\u8be2\u4e00\u6b21\u6570\u636e\uff0c\u90a3\u4e48\u5176\u4ed6\u4e09\u79cd\u65b9\u6cd5\u7684\u6570\u636e\u9884\u5904\u7406\u7684\u65f6\u95f4\u6bd4\u7ebf\u6027\u641c\u7d22\u7684\u65f6\u95f4\u8fd8\u8981\u66f4\u957f\u3002</li> <li>\u9002\u7528\u4e8e\u4f53\u91cf\u8f83\u5c0f\u7684\u6570\u636e\uff0c\u6b64\u60c5\u51b5\u4e0b\u65f6\u95f4\u590d\u6742\u5ea6\u5bf9\u6548\u7387\u5f71\u54cd\u8f83\u5c0f\u3002</li> <li>\u9002\u7528\u4e8e\u6570\u636e\u66f4\u65b0\u9891\u7387\u8f83\u9ad8\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u8be5\u65b9\u6cd5\u4e0d\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u4efb\u4f55\u989d\u5916\u7ef4\u62a4\u3002</li> </ul> <p>\u4e8c\u5206\u67e5\u627e</p> <ul> <li>\u9002\u7528\u4e8e\u5927\u6570\u636e\u91cf\u7684\u60c5\u51b5\uff0c\u6548\u7387\u8868\u73b0\u7a33\u5b9a\uff0c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002</li> <li>\u6570\u636e\u91cf\u4e0d\u80fd\u8fc7\u5927\uff0c\u56e0\u4e3a\u5b58\u50a8\u6570\u7ec4\u9700\u8981\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u4e0d\u9002\u7528\u4e8e\u9ad8\u9891\u589e\u5220\u6570\u636e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u7ef4\u62a4\u6709\u5e8f\u6570\u7ec4\u7684\u5f00\u9500\u8f83\u5927\u3002</li> </ul> <p>\u54c8\u5e0c\u67e5\u627e</p> <ul> <li>\u9002\u5408\u5bf9\u67e5\u8be2\u6027\u80fd\u8981\u6c42\u5f88\u9ad8\u7684\u573a\u666f\uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</li> <li>\u4e0d\u9002\u5408\u9700\u8981\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u65e0\u6cd5\u7ef4\u62a4\u6570\u636e\u7684\u6709\u5e8f\u6027\u3002</li> <li>\u5bf9\u54c8\u5e0c\u51fd\u6570\u548c\u54c8\u5e0c\u51b2\u7a81\u5904\u7406\u7b56\u7565\u7684\u4f9d\u8d56\u6027\u8f83\u9ad8\uff0c\u5177\u6709\u8f83\u5927\u7684\u6027\u80fd\u52a3\u5316\u98ce\u9669\u3002</li> <li>\u4e0d\u9002\u5408\u6570\u636e\u91cf\u8fc7\u5927\u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u9700\u8981\u989d\u5916\u7a7a\u95f4\u6765\u6700\u5927\u7a0b\u5ea6\u5730\u51cf\u5c11\u51b2\u7a81\uff0c\u4ece\u800c\u63d0\u4f9b\u826f\u597d\u7684\u67e5\u8be2\u6027\u80fd\u3002</li> </ul> <p>\u6811\u67e5\u627e</p> <ul> <li>\u9002\u7528\u4e8e\u6d77\u91cf\u6570\u636e\uff0c\u56e0\u4e3a\u6811\u8282\u70b9\u5728\u5185\u5b58\u4e2d\u662f\u5206\u6563\u5b58\u50a8\u7684\u3002</li> <li>\u9002\u5408\u9700\u8981\u7ef4\u62a4\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\u3002</li> <li>\u5728\u6301\u7eed\u589e\u5220\u8282\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u4ea7\u751f\u503e\u659c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>\u82e5\u4f7f\u7528 AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u5219\u5404\u9879\u64cd\u4f5c\u53ef\u5728 \\(O(\\log n)\\) \u6548\u7387\u4e0b\u7a33\u5b9a\u8fd0\u884c\uff0c\u4f46\u7ef4\u62a4\u6811\u5e73\u8861\u7684\u64cd\u4f5c\u4f1a\u589e\u52a0\u989d\u5916\u7684\u5f00\u9500\u3002</li> </ul>"},{"location":"chapter_searching/summary/","title":"10.6 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u4e8c\u5206\u67e5\u627e\u4f9d\u8d56\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u901a\u8fc7\u5faa\u73af\u9010\u6b65\u7f29\u51cf\u4e00\u534a\u641c\u7d22\u533a\u95f4\u6765\u8fdb\u884c\u67e5\u627e\u3002\u5b83\u8981\u6c42\u8f93\u5165\u6570\u636e\u6709\u5e8f\uff0c\u4e14\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u6216\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u6570\u636e\u3002\u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\uff0c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u548c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u9002\u7528\u4e8e\u56fe\u548c\u6811\u3002\u6b64\u7c7b\u7b97\u6cd5\u901a\u7528\u6027\u597d\uff0c\u65e0\u987b\u5bf9\u6570\u636e\u8fdb\u884c\u9884\u5904\u7406\uff0c\u4f46\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u8f83\u9ad8\u3002</li> <li>\u54c8\u5e0c\u67e5\u627e\u3001\u6811\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u5c5e\u4e8e\u9ad8\u6548\u641c\u7d22\u65b9\u6cd5\uff0c\u53ef\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002\u6b64\u7c7b\u7b97\u6cd5\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \uff0c\u4f46\u901a\u5e38\u9700\u8981\u501f\u52a9\u989d\u5916\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u5bf9\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u548c\u66f4\u65b0\u9891\u7387\u7b49\u56e0\u7d20\u8fdb\u884c\u5177\u4f53\u5206\u6790\uff0c\u4ece\u800c\u9009\u62e9\u5408\u9002\u7684\u641c\u7d22\u65b9\u6cd5\u3002</li> <li>\u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u5c0f\u578b\u6216\u9891\u7e41\u66f4\u65b0\u7684\u6570\u636e\uff1b\u4e8c\u5206\u67e5\u627e\u9002\u7528\u4e8e\u5927\u578b\u3001\u6392\u5e8f\u7684\u6570\u636e\uff1b\u54c8\u5e0c\u67e5\u627e\u9002\u7528\u4e8e\u5bf9\u67e5\u8be2\u6548\u7387\u8981\u6c42\u8f83\u9ad8\u4e14\u65e0\u987b\u8303\u56f4\u67e5\u8be2\u7684\u6570\u636e\uff1b\u6811\u67e5\u627e\u9002\u7528\u4e8e\u9700\u8981\u7ef4\u62a4\u987a\u5e8f\u548c\u652f\u6301\u8303\u56f4\u67e5\u8be2\u7684\u5927\u578b\u52a8\u6001\u6570\u636e\u3002</li> <li>\u7528\u54c8\u5e0c\u67e5\u627e\u66ff\u6362\u7ebf\u6027\u67e5\u627e\u662f\u4e00\u79cd\u5e38\u7528\u7684\u4f18\u5316\u8fd0\u884c\u65f6\u95f4\u7684\u7b56\u7565\uff0c\u53ef\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002</li> </ul>"},{"location":"chapter_sorting/","title":"\u7b2c 11 \u7ae0 \u00a0 \u6392\u5e8f","text":"<p>Abstract</p> <p>\u6392\u5e8f\u72b9\u5982\u4e00\u628a\u5c06\u6df7\u4e71\u53d8\u4e3a\u79e9\u5e8f\u7684\u9b54\u6cd5\u94a5\u5319\uff0c\u4f7f\u6211\u4eec\u80fd\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u7406\u89e3\u4e0e\u5904\u7406\u6570\u636e\u3002</p> <p>\u65e0\u8bba\u662f\u7b80\u5355\u7684\u5347\u5e8f\uff0c\u8fd8\u662f\u590d\u6742\u7684\u5206\u7c7b\u6392\u5217\uff0c\u6392\u5e8f\u90fd\u5411\u6211\u4eec\u5c55\u793a\u4e86\u6570\u636e\u7684\u548c\u8c10\u7f8e\u611f\u3002</p>"},{"location":"chapter_sorting/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>11.1 \u00a0 \u6392\u5e8f\u7b97\u6cd5</li> <li>11.2 \u00a0 \u9009\u62e9\u6392\u5e8f</li> <li>11.3 \u00a0 \u5192\u6ce1\u6392\u5e8f</li> <li>11.4 \u00a0 \u63d2\u5165\u6392\u5e8f</li> <li>11.5 \u00a0 \u5feb\u901f\u6392\u5e8f</li> <li>11.6 \u00a0 \u5f52\u5e76\u6392\u5e8f</li> <li>11.7 \u00a0 \u5806\u6392\u5e8f</li> <li>11.8 \u00a0 \u6876\u6392\u5e8f</li> <li>11.9 \u00a0 \u8ba1\u6570\u6392\u5e8f</li> <li>11.10 \u00a0 \u57fa\u6570\u6392\u5e8f</li> <li>11.11 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_sorting/bubble_sort/","title":"11.3 \u00a0 \u5192\u6ce1\u6392\u5e8f","text":"<p>\u300c\u5192\u6ce1\u6392\u5e8f bubble sort\u300d\u901a\u8fc7\u8fde\u7eed\u5730\u6bd4\u8f83\u4e0e\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u5b9e\u73b0\u6392\u5e8f\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u5c31\u50cf\u6c14\u6ce1\u4ece\u5e95\u90e8\u5347\u5230\u9876\u90e8\u4e00\u6837\uff0c\u56e0\u6b64\u5f97\u540d\u5192\u6ce1\u6392\u5e8f\u3002</p> <p>\u5982\u56fe 11-4 \u6240\u793a\uff0c\u5192\u6ce1\u8fc7\u7a0b\u53ef\u4ee5\u5229\u7528\u5143\u7d20\u4ea4\u6362\u64cd\u4f5c\u6765\u6a21\u62df\uff1a\u4ece\u6570\u7ec4\u6700\u5de6\u7aef\u5f00\u59cb\u5411\u53f3\u904d\u5386\uff0c\u4f9d\u6b21\u6bd4\u8f83\u76f8\u90bb\u5143\u7d20\u5927\u5c0f\uff0c\u5982\u679c\u201c\u5de6\u5143\u7d20 &gt; \u53f3\u5143\u7d20\u201d\u5c31\u4ea4\u6362\u4e8c\u8005\u3002\u904d\u5386\u5b8c\u6210\u540e\uff0c\u6700\u5927\u7684\u5143\u7d20\u4f1a\u88ab\u79fb\u52a8\u5230\u6570\u7ec4\u7684\u6700\u53f3\u7aef\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-4 \u00a0 \u5229\u7528\u5143\u7d20\u4ea4\u6362\u64cd\u4f5c\u6a21\u62df\u5192\u6ce1 </p>"},{"location":"chapter_sorting/bubble_sort/#1131","title":"11.3.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6b65\u9aa4\u5982\u56fe 11-5 \u6240\u793a\u3002</p> <ol> <li>\u9996\u5148\uff0c\u5bf9 \\(n\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002</li> <li>\u63a5\u4e0b\u6765\uff0c\u5bf9\u5269\u4f59 \\(n - 1\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u7b2c\u4e8c\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002</li> <li>\u4ee5\u6b64\u7c7b\u63a8\uff0c\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u201c\u5192\u6ce1\u201d\u540e\uff0c\u524d \\(n - 1\\) \u5927\u7684\u5143\u7d20\u90fd\u88ab\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002</li> <li>\u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5c0f\u5143\u7d20\uff0c\u65e0\u987b\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002</li> </ol> <p></p> <p> \u56fe 11-5 \u00a0 \u5192\u6ce1\u6392\u5e8f\u6d41\u7a0b </p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig bubble_sort.py<pre><code>def bubble_sort(nums: list[int]):\n    \"\"\"\u5192\u6ce1\u6392\u5e8f\"\"\"\n    n = len(nums)\n    # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in range(n - 1, 0, -1):\n        # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n</code></pre> bubble_sort.cpp<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(vector&lt;int&gt; &amp;nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                // \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\n                swap(nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid BubbleSort(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums []int) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums: inout [Int]) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums.swapAt(j, j + 1)\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums: number[]): void {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(List&lt;int&gt; nums) {\n  // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n      }\n    }\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfn bubble_sort(nums: &amp;mut [i32]) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in (1..nums.len()).rev() {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int nums[], int size) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = size - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfun bubbleSort(nums: IntArray) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j] = nums[j+1].also { nums[j+1] = nums[j] }\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>[class]{}-[func]{bubble_sort}\n</code></pre> bubble_sort.zig<pre><code>// \u5192\u6ce1\u6392\u5e8f\nfn bubbleSort(nums: []i32) void {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    var i: usize = nums.len - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var j: usize = 0;\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/bubble_sort/#1132","title":"11.3.2 \u00a0 \u6548\u7387\u4f18\u5316","text":"<p>\u6211\u4eec\u53d1\u73b0\uff0c\u5982\u679c\u67d0\u8f6e\u201c\u5192\u6ce1\u201d\u4e2d\u6ca1\u6709\u6267\u884c\u4efb\u4f55\u4ea4\u6362\u64cd\u4f5c\uff0c\u8bf4\u660e\u6570\u7ec4\u5df2\u7ecf\u5b8c\u6210\u6392\u5e8f\uff0c\u53ef\u76f4\u63a5\u8fd4\u56de\u7ed3\u679c\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u589e\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d <code>flag</code> \u6765\u76d1\u6d4b\u8fd9\u79cd\u60c5\u51b5\uff0c\u4e00\u65e6\u51fa\u73b0\u5c31\u7acb\u5373\u8fd4\u56de\u3002</p> <p>\u7ecf\u8fc7\u4f18\u5316\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(n^2)\\) \uff1b\u4f46\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u53ef\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig bubble_sort.py<pre><code>def bubble_sort_with_flag(nums: list[int]):\n    \"\"\"\u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\"\"\"\n    n = len(nums)\n    # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in range(n - 1, 0, -1):\n        flag = False  # \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n                flag = True  # \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n        if not flag:\n            break  # \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n</code></pre> bubble_sort.cpp<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(vector&lt;int&gt; &amp;nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        bool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                // \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\n                swap(nums[j], nums[j + 1]);\n                flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag)\n            break; // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nvoid bubbleSortWithFlag(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        boolean flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag)\n            break; // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid BubbleSortWithFlag(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        bool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                flag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag) break;     // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums []int) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        flag := false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n                flag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if flag == false { // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        var flag = false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums.swapAt(j, j + 1)\n                flag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if !flag { // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag) break; // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums: number[]): void {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag) break; // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(List&lt;int&gt; nums) {\n  // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    bool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n    // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n      }\n    }\n    if (!flag) break; // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nfn bubble_sort_with_flag(nums: &amp;mut [i32]) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in (1..nums.len()).rev() {\n        let mut flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if !flag {\n            break; // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n        };\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(int nums[], int size) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = size - 1; i &gt; 0; i--) {\n        bool flag = false;\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n                flag = true;\n            }\n        }\n        if (!flag)\n            break;\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nfun bubbleSortWithFlag(nums: IntArray) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        var flag = false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j] = nums[j + 1].also { nums[j] = nums[j + 1] }\n                flag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag) break // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>[class]{}-[func]{bubble_sort_with_flag}\n</code></pre> bubble_sort.zig<pre><code>// \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\nfn bubbleSortWithFlag(nums: []i32) void {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    var i: usize = nums.len - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var flag = false;   // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        var j: usize = 0;\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true;\n            }\n        }\n        if (!flag) break;   // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/bubble_sort/#1133","title":"11.3.3 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5404\u8f6e\u201c\u5192\u6ce1\u201d\u904d\u5386\u7684\u6570\u7ec4\u957f\u5ea6\u4f9d\u6b21\u4e3a \\(n - 1\\)\u3001\\(n - 2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u603b\u548c\u4e3a \\((n - 1) n / 2\\) \u3002\u5728\u5f15\u5165 <code>flag</code> \u4f18\u5316\u540e\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5728\u201c\u5192\u6ce1\u201d\u4e2d\u9047\u5230\u76f8\u7b49\u5143\u7d20\u4e0d\u4ea4\u6362\u3002</li> </ul>"},{"location":"chapter_sorting/bucket_sort/","title":"11.8 \u00a0 \u6876\u6392\u5e8f","text":"<p>\u524d\u8ff0\u51e0\u79cd\u6392\u5e8f\u7b97\u6cd5\u90fd\u5c5e\u4e8e\u201c\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u901a\u8fc7\u6bd4\u8f83\u5143\u7d20\u95f4\u7684\u5927\u5c0f\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u6b64\u7c7b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u65e0\u6cd5\u8d85\u8d8a \\(O(n \\log n)\\) \u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u63a2\u8ba8\u51e0\u79cd\u201c\u975e\u6bd4\u8f83\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230\u7ebf\u6027\u9636\u3002</p> <p>\u300c\u6876\u6392\u5e8f bucket sort\u300d\u662f\u5206\u6cbb\u7b56\u7565\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u901a\u8fc7\u8bbe\u7f6e\u4e00\u4e9b\u5177\u6709\u5927\u5c0f\u987a\u5e8f\u7684\u6876\uff0c\u6bcf\u4e2a\u6876\u5bf9\u5e94\u4e00\u4e2a\u6570\u636e\u8303\u56f4\uff0c\u5c06\u6570\u636e\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff1b\u7136\u540e\uff0c\u5728\u6bcf\u4e2a\u6876\u5185\u90e8\u5206\u522b\u6267\u884c\u6392\u5e8f\uff1b\u6700\u7ec8\u6309\u7167\u6876\u7684\u987a\u5e8f\u5c06\u6240\u6709\u6570\u636e\u5408\u5e76\u3002</p>"},{"location":"chapter_sorting/bucket_sort/#1181","title":"11.8.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8003\u8651\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u5176\u5143\u7d20\u662f\u8303\u56f4 \\([0, 1)\\) \u5185\u7684\u6d6e\u70b9\u6570\u3002\u6876\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-13 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316 \\(k\\) \u4e2a\u6876\uff0c\u5c06 \\(n\\) \u4e2a\u5143\u7d20\u5206\u914d\u5230 \\(k\\) \u4e2a\u6876\u4e2d\u3002</li> <li>\u5bf9\u6bcf\u4e2a\u6876\u5206\u522b\u6267\u884c\u6392\u5e8f\uff08\u8fd9\u91cc\u91c7\u7528\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff09\u3002</li> <li>\u6309\u7167\u6876\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u5408\u5e76\u7ed3\u679c\u3002</li> </ol> <p></p> <p> \u56fe 11-13 \u00a0 \u6876\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b </p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig bucket_sort.py<pre><code>def bucket_sort(nums: list[float]):\n    \"\"\"\u6876\u6392\u5e8f\"\"\"\n    # \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    k = len(nums) // 2\n    buckets = [[] for _ in range(k)]\n    # 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for num in nums:\n        # \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        i = int(num * k)\n        # \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].append(num)\n    # 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for bucket in buckets:\n        # \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.sort()\n    # 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    i = 0\n    for bucket in buckets:\n        for num in bucket:\n            nums[i] = num\n            i += 1\n</code></pre> bucket_sort.cpp<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(vector&lt;float&gt; &amp;nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    int k = nums.size() / 2;\n    vector&lt;vector&lt;float&gt;&gt; buckets(k);\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (float num : nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        int i = num * k;\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\n        buckets[i].push_back(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        sort(bucket.begin(), bucket.end());\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    int i = 0;\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.java<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(float[] nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    int k = nums.length / 2;\n    List&lt;List&lt;Float&gt;&gt; buckets = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; k; i++) {\n        buckets.add(new ArrayList&lt;&gt;());\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (float num : nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        int i = (int) (num * k);\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets.get(i).add(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (List&lt;Float&gt; bucket : buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        Collections.sort(bucket);\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    int i = 0;\n    for (List&lt;Float&gt; bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.cs<pre><code>/* \u6876\u6392\u5e8f */\nvoid BucketSort(float[] nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    int k = nums.Length / 2;\n    List&lt;List&lt;float&gt;&gt; buckets = [];\n    for (int i = 0; i &lt; k; i++) {\n        buckets.Add([]);\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    foreach (float num in nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        int i = (int)(num * k);\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].Add(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    foreach (List&lt;float&gt; bucket in buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.Sort();\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    int j = 0;\n    foreach (List&lt;float&gt; bucket in buckets) {\n        foreach (float num in bucket) {\n            nums[j++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.go<pre><code>/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums []float64) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    k := len(nums) / 2\n    buckets := make([][]float64, k)\n    for i := 0; i &lt; k; i++ {\n        buckets[i] = make([]float64, 0)\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for _, num := range nums {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        i := int(num * float64(k))\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i] = append(buckets[i], num)\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for i := 0; i &lt; k; i++ {\n        // \u4f7f\u7528\u5185\u7f6e\u5207\u7247\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        sort.Float64s(buckets[i])\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    i := 0\n    for _, bucket := range buckets {\n        for _, num := range bucket {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> bucket_sort.swift<pre><code>/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums: inout [Double]) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    let k = nums.count / 2\n    var buckets = (0 ..&lt; k).map { _ in [Double]() }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for num in nums {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        let i = Int(num * Double(k))\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].append(num)\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for i in buckets.indices {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        buckets[i].sort()\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    var i = nums.startIndex\n    for bucket in buckets {\n        for num in bucket {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> bucket_sort.js<pre><code>/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    const k = nums.length / 2;\n    const buckets = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (const num of nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        const i = Math.floor(num * k);\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].push(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (const bucket of buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.ts<pre><code>/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums: number[]): void {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    const k = nums.length / 2;\n    const buckets: number[][] = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (const num of nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        const i = Math.floor(num * k);\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].push(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (const bucket of buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.dart<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(List&lt;double&gt; nums) {\n  // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n  int k = nums.length ~/ 2;\n  List&lt;List&lt;double&gt;&gt; buckets = List.generate(k, (index) =&gt; []);\n\n  // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n  for (double _num in nums) {\n    // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 _num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n    int i = (_num * k).toInt();\n    // \u5c06 _num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\n    buckets[i].add(_num);\n  }\n  // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n  for (List&lt;double&gt; bucket in buckets) {\n    bucket.sort();\n  }\n  // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n  int i = 0;\n  for (List&lt;double&gt; bucket in buckets) {\n    for (double _num in bucket) {\n      nums[i++] = _num;\n    }\n  }\n}\n</code></pre> bucket_sort.rs<pre><code>/* \u6876\u6392\u5e8f */\nfn bucket_sort(nums: &amp;mut [f64]) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    let k = nums.len() / 2;\n    let mut buckets = vec![vec![]; k];\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for &amp;mut num in &amp;mut *nums {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        let i = (num * k as f64) as usize;\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].push(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for bucket in &amp;mut buckets {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.sort_by(|a, b| a.partial_cmp(b).unwrap());\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    let mut i = 0;\n    for bucket in &amp;mut buckets {\n        for &amp;mut num in bucket {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> bucket_sort.c<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(float nums[], int size) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    int k = size / 2;\n    float **buckets = calloc(k, sizeof(float *));\n    for (int i = 0; i &lt; k; i++) {\n        // \u6bcf\u4e2a\u6876\u6700\u591a\u53ef\u4ee5\u5206\u914d size \u4e2a\u5143\u7d20\n        buckets[i] = calloc(size, sizeof(float));\n    }\n\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (int i = 0; i &lt; size; i++) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        int bucket_idx = nums[i] * k;\n        int j = 0;\n        // \u5982\u679c\u6876\u4e2d\u6709\u6570\u636e\u4e14\u6570\u636e\u5c0f\u4e8e\u5f53\u524d\u503c nums[i], \u8981\u5c06\u5176\u653e\u5230\u5f53\u524d\u6876\u7684\u540e\u9762\uff0c\u76f8\u5f53\u4e8e cpp \u4e2d\u7684 push_back\n        while (buckets[bucket_idx][j] &gt; 0 &amp;&amp; buckets[bucket_idx][j] &lt; nums[i]) {\n            j++;\n        }\n        float temp = nums[i];\n        while (j &lt; size &amp;&amp; buckets[bucket_idx][j] &gt; 0) {\n            swap(&amp;temp, &amp;buckets[bucket_idx][j]);\n            j++;\n        }\n        buckets[bucket_idx][j] = temp;\n    }\n\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (int i = 0; i &lt; k; i++) {\n        qsort(buckets[i], size, sizeof(float), compare_float);\n    }\n\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    for (int i = 0, j = 0; j &lt; k; j++) {\n        for (int l = 0; l &lt; size; l++) {\n            if (buckets[j][l] &gt; 0) {\n                nums[i++] = buckets[j][l];\n            }\n        }\n    }\n\n    // \u91ca\u653e\u4e0a\u8ff0\u5206\u914d\u7684\u5185\u5b58\n    for (int i = 0; i &lt; k; i++) {\n        free(buckets[i]);\n    }\n    free(buckets);\n}\n</code></pre> bucket_sort.kt<pre><code>/* \u6876\u6392\u5e8f */\nfun bucketSort(nums: FloatArray) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    val k = nums.size / 2\n    val buckets = ArrayList&lt;ArrayList&lt;Float&gt;&gt;()\n    for (i in 0..&lt;k) {\n        buckets.add(ArrayList())\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (num in nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        val i = (num * k).toInt()\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].add(num)\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (bucket in buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.sort()\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    var i = 0\n    for (bucket in buckets) {\n        for (num in bucket) {\n            nums[i++] = num\n        }\n    }\n}\n</code></pre> bucket_sort.rb<pre><code>[class]{}-[func]{bucket_sort}\n</code></pre> bucket_sort.zig<pre><code>[class]{}-[func]{bucketSort}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/bucket_sort/#1182","title":"11.8.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<p>\u6876\u6392\u5e8f\u9002\u7528\u4e8e\u5904\u7406\u4f53\u91cf\u5f88\u5927\u7684\u6570\u636e\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u6570\u636e\u5305\u542b 100 \u4e07\u4e2a\u5143\u7d20\uff0c\u7531\u4e8e\u7a7a\u95f4\u9650\u5236\uff0c\u7cfb\u7edf\u5185\u5b58\u65e0\u6cd5\u4e00\u6b21\u6027\u52a0\u8f7d\u6240\u6709\u6570\u636e\u3002\u6b64\u65f6\uff0c\u53ef\u4ee5\u5c06\u6570\u636e\u5206\u6210 1000 \u4e2a\u6876\uff0c\u7136\u540e\u5206\u522b\u5bf9\u6bcf\u4e2a\u6876\u8fdb\u884c\u6392\u5e8f\uff0c\u6700\u540e\u5c06\u7ed3\u679c\u5408\u5e76\u3002</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n + k)\\) \uff1a\u5047\u8bbe\u5143\u7d20\u5728\u5404\u4e2a\u6876\u5185\u5e73\u5747\u5206\u5e03\uff0c\u90a3\u4e48\u6bcf\u4e2a\u6876\u5185\u7684\u5143\u7d20\u6570\u91cf\u4e3a \\(\\frac{n}{k}\\) \u3002\u5047\u8bbe\u6392\u5e8f\u5355\u4e2a\u6876\u4f7f\u7528 \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) \u65f6\u95f4\uff0c\u5219\u6392\u5e8f\u6240\u6709\u6876\u4f7f\u7528 \\(O(n \\log\\frac{n}{k})\\) \u65f6\u95f4\u3002\u5f53\u6876\u6570\u91cf \\(k\\) \u6bd4\u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5219\u8d8b\u5411\u4e8e \\(O(n)\\) \u3002\u5408\u5e76\u7ed3\u679c\u65f6\u9700\u8981\u904d\u5386\u6240\u6709\u6876\u548c\u5143\u7d20\uff0c\u82b1\u8d39 \\(O(n + k)\\) \u65f6\u95f4\u3002</li> <li>\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u6570\u636e\u88ab\u5206\u914d\u5230\u4e00\u4e2a\u6876\u4e2d\uff0c\u4e14\u6392\u5e8f\u8be5\u6876\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n + k)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u9700\u8981\u501f\u52a9 \\(k\\) \u4e2a\u6876\u548c\u603b\u5171 \\(n\\) \u4e2a\u5143\u7d20\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u6876\u6392\u5e8f\u662f\u5426\u7a33\u5b9a\u53d6\u51b3\u4e8e\u6392\u5e8f\u6876\u5185\u5143\u7d20\u7684\u7b97\u6cd5\u662f\u5426\u7a33\u5b9a\u3002</li> </ul>"},{"location":"chapter_sorting/bucket_sort/#1183","title":"11.8.3 \u00a0 \u5982\u4f55\u5b9e\u73b0\u5e73\u5747\u5206\u914d","text":"<p>\u6876\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7406\u8bba\u4e0a\u53ef\u4ee5\u8fbe\u5230 \\(O(n)\\) \uff0c\u5173\u952e\u5728\u4e8e\u5c06\u5143\u7d20\u5747\u5300\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u56e0\u4e3a\u5b9e\u9645\u6570\u636e\u5f80\u5f80\u4e0d\u662f\u5747\u5300\u5206\u5e03\u7684\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u60f3\u8981\u5c06\u6dd8\u5b9d\u4e0a\u7684\u6240\u6709\u5546\u54c1\u6309\u4ef7\u683c\u8303\u56f4\u5e73\u5747\u5206\u914d\u5230 10 \u4e2a\u6876\u4e2d\uff0c\u4f46\u5546\u54c1\u4ef7\u683c\u5206\u5e03\u4e0d\u5747\uff0c\u4f4e\u4e8e 100 \u5143\u7684\u975e\u5e38\u591a\uff0c\u9ad8\u4e8e 1000 \u5143\u7684\u975e\u5e38\u5c11\u3002\u82e5\u5c06\u4ef7\u683c\u533a\u95f4\u5e73\u5747\u5212\u5206\u4e3a 10 \u4e2a\uff0c\u5404\u4e2a\u6876\u4e2d\u7684\u5546\u54c1\u6570\u91cf\u5dee\u8ddd\u4f1a\u975e\u5e38\u5927\u3002</p> <p>\u4e3a\u5b9e\u73b0\u5e73\u5747\u5206\u914d\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u8bbe\u5b9a\u4e00\u6761\u5927\u81f4\u7684\u5206\u754c\u7ebf\uff0c\u5c06\u6570\u636e\u7c97\u7565\u5730\u5206\u5230 3 \u4e2a\u6876\u4e2d\u3002\u5206\u914d\u5b8c\u6bd5\u540e\uff0c\u518d\u5c06\u5546\u54c1\u8f83\u591a\u7684\u6876\u7ee7\u7eed\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u76f4\u81f3\u6240\u6709\u6876\u4e2d\u7684\u5143\u7d20\u6570\u91cf\u5927\u81f4\u76f8\u7b49\u3002</p> <p>\u5982\u56fe 11-14 \u6240\u793a\uff0c\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u521b\u5efa\u4e00\u68f5\u9012\u5f52\u6811\uff0c\u76ee\u6807\u662f\u8ba9\u53f6\u8282\u70b9\u7684\u503c\u5c3d\u53ef\u80fd\u5e73\u5747\u3002\u5f53\u7136\uff0c\u4e0d\u4e00\u5b9a\u8981\u6bcf\u8f6e\u5c06\u6570\u636e\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u5177\u4f53\u5212\u5206\u65b9\u5f0f\u53ef\u6839\u636e\u6570\u636e\u7279\u70b9\u7075\u6d3b\u9009\u62e9\u3002</p> <p></p> <p> \u56fe 11-14 \u00a0 \u9012\u5f52\u5212\u5206\u6876 </p> <p>\u5982\u679c\u6211\u4eec\u63d0\u524d\u77e5\u9053\u5546\u54c1\u4ef7\u683c\u7684\u6982\u7387\u5206\u5e03\uff0c\u5219\u53ef\u4ee5\u6839\u636e\u6570\u636e\u6982\u7387\u5206\u5e03\u8bbe\u7f6e\u6bcf\u4e2a\u6876\u7684\u4ef7\u683c\u5206\u754c\u7ebf\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6570\u636e\u5206\u5e03\u5e76\u4e0d\u4e00\u5b9a\u9700\u8981\u7279\u610f\u7edf\u8ba1\uff0c\u4e5f\u53ef\u4ee5\u6839\u636e\u6570\u636e\u7279\u70b9\u91c7\u7528\u67d0\u79cd\u6982\u7387\u6a21\u578b\u8fdb\u884c\u8fd1\u4f3c\u3002</p> <p>\u5982\u56fe 11-15 \u6240\u793a\uff0c\u6211\u4eec\u5047\u8bbe\u5546\u54c1\u4ef7\u683c\u670d\u4ece\u6b63\u6001\u5206\u5e03\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5408\u7406\u5730\u8bbe\u5b9a\u4ef7\u683c\u533a\u95f4\uff0c\u4ece\u800c\u5c06\u5546\u54c1\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\u3002</p> <p></p> <p> \u56fe 11-15 \u00a0 \u6839\u636e\u6982\u7387\u5206\u5e03\u5212\u5206\u6876 </p>"},{"location":"chapter_sorting/counting_sort/","title":"11.9 \u00a0 \u8ba1\u6570\u6392\u5e8f","text":"<p>\u300c\u8ba1\u6570\u6392\u5e8f counting sort\u300d\u901a\u8fc7\u7edf\u8ba1\u5143\u7d20\u6570\u91cf\u6765\u5b9e\u73b0\u6392\u5e8f\uff0c\u901a\u5e38\u5e94\u7528\u4e8e\u6574\u6570\u6570\u7ec4\u3002</p>"},{"location":"chapter_sorting/counting_sort/#1191","title":"11.9.1 \u00a0 \u7b80\u5355\u5b9e\u73b0","text":"<p>\u5148\u6765\u770b\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\u3002\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d\u7684\u5143\u7d20\u90fd\u662f\u201c\u975e\u8d1f\u6574\u6570\u201d\uff0c\u8ba1\u6570\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-16 \u6240\u793a\u3002</p> <ol> <li>\u904d\u5386\u6570\u7ec4\uff0c\u627e\u51fa\u5176\u4e2d\u7684\u6700\u5927\u6570\u5b57\uff0c\u8bb0\u4e3a \\(m\\) \uff0c\u7136\u540e\u521b\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(m + 1\\) \u7684\u8f85\u52a9\u6570\u7ec4 <code>counter</code> \u3002</li> <li>\u501f\u52a9 <code>counter</code> \u7edf\u8ba1 <code>nums</code> \u4e2d\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\uff0c\u5176\u4e2d <code>counter[num]</code> \u5bf9\u5e94\u6570\u5b57 <code>num</code> \u7684\u51fa\u73b0\u6b21\u6570\u3002\u7edf\u8ba1\u65b9\u6cd5\u5f88\u7b80\u5355\uff0c\u53ea\u9700\u904d\u5386 <code>nums</code>\uff08\u8bbe\u5f53\u524d\u6570\u5b57\u4e3a <code>num</code>\uff09\uff0c\u6bcf\u8f6e\u5c06 <code>counter[num]</code> \u589e\u52a0 \\(1\\) \u5373\u53ef\u3002</li> <li>\u7531\u4e8e <code>counter</code> \u7684\u5404\u4e2a\u7d22\u5f15\u5929\u7136\u6709\u5e8f\uff0c\u56e0\u6b64\u76f8\u5f53\u4e8e\u6240\u6709\u6570\u5b57\u5df2\u7ecf\u6392\u5e8f\u597d\u4e86\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u904d\u5386 <code>counter</code> \uff0c\u6839\u636e\u5404\u6570\u5b57\u51fa\u73b0\u6b21\u6570\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u586b\u5165 <code>nums</code> \u5373\u53ef\u3002</li> </ol> <p></p> <p> \u56fe 11-16 \u00a0 \u8ba1\u6570\u6392\u5e8f\u6d41\u7a0b </p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig counting_sort.py<pre><code>def counting_sort_naive(nums: list[int]):\n    \"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n    # \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\n    # 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    m = 0\n    for num in nums:\n        m = max(m, num)\n    # 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    i = 0\n    for num in range(m + 1):\n        for _ in range(counter[num]):\n            nums[i] = num\n            i += 1\n</code></pre> counting_sort.cpp<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(vector&lt;int&gt; &amp;nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.java<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int[] nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid CountingSortNaive(int[] nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums []int) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    for i, num := 0, 0; num &lt; m+1; num++ {\n        for j := 0; j &lt; counter[num]; j++ {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.swift<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums: inout [Int]) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = nums.max()!\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    var i = 0\n    for num in 0 ..&lt; m + 1 {\n        for _ in 0 ..&lt; counter[num] {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums: number[]): void {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(List&lt;int&gt; nums) {\n  // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n  // counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73b0\u6b21\u6570\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n  int i = 0;\n  for (int _num = 0; _num &lt; m + 1; _num++) {\n    for (int j = 0; j &lt; counter[_num]; j++, i++) {\n      nums[i] = _num;\n    }\n  }\n}\n</code></pre> counting_sort.rs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfn counting_sort_naive(nums: &amp;mut [i32]) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = *nums.into_iter().max().unwrap();\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    let mut counter = vec![0; m as usize + 1];\n    for &amp;num in &amp;*nums {\n        counter[num as usize] += 1;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    let mut i = 0;\n    for num in 0..m + 1 {\n        for _ in 0..counter[num as usize] {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> counting_sort.c<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int nums[], int size) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int *counter = calloc(m + 1, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n    // 4. \u91ca\u653e\u5185\u5b58\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfun countingSortNaive(nums: IntArray) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    var m = 0\n    for (num in nums) {\n        m = max(m.toDouble(), num.toDouble()).toInt()\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    var i = 0\n    for (num in 0..&lt;m + 1) {\n        var j = 0\n        while (j &lt; counter[num]) {\n            nums[i] = num\n            j++\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.rb<pre><code>[class]{}-[func]{counting_sort_naive}\n</code></pre> counting_sort.zig<pre><code>[class]{}-[func]{countingSortNaive}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u8ba1\u6570\u6392\u5e8f\u4e0e\u6876\u6392\u5e8f\u7684\u8054\u7cfb</p> <p>\u4ece\u6876\u6392\u5e8f\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u8ba1\u6570\u6392\u5e8f\u4e2d\u7684\u8ba1\u6570\u6570\u7ec4 <code>counter</code> \u7684\u6bcf\u4e2a\u7d22\u5f15\u89c6\u4e3a\u4e00\u4e2a\u6876\uff0c\u5c06\u7edf\u8ba1\u6570\u91cf\u7684\u8fc7\u7a0b\u770b\u4f5c\u5c06\u5404\u4e2a\u5143\u7d20\u5206\u914d\u5230\u5bf9\u5e94\u7684\u6876\u4e2d\u3002\u672c\u8d28\u4e0a\uff0c\u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u5728\u6574\u578b\u6570\u636e\u4e0b\u7684\u4e00\u4e2a\u7279\u4f8b\u3002</p>"},{"location":"chapter_sorting/counting_sort/#1192","title":"11.9.2 \u00a0 \u5b8c\u6574\u5b9e\u73b0","text":"<p>\u7ec6\u5fc3\u7684\u8bfb\u8005\u53ef\u80fd\u53d1\u73b0\u4e86\uff0c\u5982\u679c\u8f93\u5165\u6570\u636e\u662f\u5bf9\u8c61\uff0c\u4e0a\u8ff0\u6b65\u9aa4 <code>3.</code> \u5c31\u5931\u6548\u4e86\u3002\u5047\u8bbe\u8f93\u5165\u6570\u636e\u662f\u5546\u54c1\u5bf9\u8c61\uff0c\u6211\u4eec\u60f3\u6309\u7167\u5546\u54c1\u4ef7\u683c\uff08\u7c7b\u7684\u6210\u5458\u53d8\u91cf\uff09\u5bf9\u5546\u54c1\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u4e0a\u8ff0\u7b97\u6cd5\u53ea\u80fd\u7ed9\u51fa\u4ef7\u683c\u7684\u6392\u5e8f\u7ed3\u679c\u3002</p> <p>\u90a3\u4e48\u5982\u4f55\u624d\u80fd\u5f97\u5230\u539f\u6570\u636e\u7684\u6392\u5e8f\u7ed3\u679c\u5462\uff1f\u6211\u4eec\u9996\u5148\u8ba1\u7b97 <code>counter</code> \u7684\u201c\u524d\u7f00\u548c\u201d\u3002\u987e\u540d\u601d\u4e49\uff0c\u7d22\u5f15 <code>i</code> \u5904\u7684\u524d\u7f00\u548c <code>prefix[i]</code> \u7b49\u4e8e\u6570\u7ec4\u524d <code>i</code> \u4e2a\u5143\u7d20\u4e4b\u548c\uff1a</p> \\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\] <p>\u524d\u7f00\u548c\u5177\u6709\u660e\u786e\u7684\u610f\u4e49\uff0c<code>prefix[num] - 1</code> \u4ee3\u8868\u5143\u7d20 <code>num</code> \u5728\u7ed3\u679c\u6570\u7ec4 <code>res</code> \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\u3002\u8fd9\u4e2a\u4fe1\u606f\u975e\u5e38\u5173\u952e\uff0c\u56e0\u4e3a\u5b83\u544a\u8bc9\u6211\u4eec\u5404\u4e2a\u5143\u7d20\u5e94\u8be5\u51fa\u73b0\u5728\u7ed3\u679c\u6570\u7ec4\u7684\u54ea\u4e2a\u4f4d\u7f6e\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5012\u5e8f\u904d\u5386\u539f\u6570\u7ec4 <code>nums</code> \u7684\u6bcf\u4e2a\u5143\u7d20 <code>num</code> \uff0c\u5728\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\u6267\u884c\u4ee5\u4e0b\u4e24\u6b65\u3002</p> <ol> <li>\u5c06 <code>num</code> \u586b\u5165\u6570\u7ec4 <code>res</code> \u7684\u7d22\u5f15 <code>prefix[num] - 1</code> \u5904\u3002</li> <li>\u4ee4\u524d\u7f00\u548c <code>prefix[num]</code> \u51cf\u5c0f \\(1\\) \uff0c\u4ece\u800c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e <code>num</code> \u7684\u7d22\u5f15\u3002</li> </ol> <p>\u904d\u5386\u5b8c\u6210\u540e\uff0c\u6570\u7ec4 <code>res</code> \u4e2d\u5c31\u662f\u6392\u5e8f\u597d\u7684\u7ed3\u679c\uff0c\u6700\u540e\u4f7f\u7528 <code>res</code> \u8986\u76d6\u539f\u6570\u7ec4 <code>nums</code> \u5373\u53ef\u3002\u56fe 11-17 \u5c55\u793a\u4e86\u5b8c\u6574\u7684\u8ba1\u6570\u6392\u5e8f\u6d41\u7a0b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-17 \u00a0 \u8ba1\u6570\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u8ba1\u6570\u6392\u5e8f\u7684\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig counting_sort.py<pre><code>def counting_sort(nums: list[int]):\n    \"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n    # \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\n    # 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    m = max(nums)\n    # 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    # \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for i in range(m):\n        counter[i + 1] += counter[i]\n    # 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    # \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    n = len(nums)\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        num = nums[i]\n        res[counter[num] - 1] = num  # \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num] -= 1  # \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    # \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in range(n):\n        nums[i] = res[i]\n</code></pre> counting_sort.cpp<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(vector&lt;int&gt; &amp;nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    int n = nums.size();\n    vector&lt;int&gt; res(n);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--;              // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    nums = res;\n}\n</code></pre> counting_sort.java<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int[] nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    int n = nums.length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid CountingSort(int[] nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    int n = nums.Length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums []int) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for i := 0; i &lt; m; i++ {\n        counter[i+1] += counter[i]\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    n := len(nums)\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        num := nums[i]\n        // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        res[counter[num]-1] = num\n        // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n        counter[num]--\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    copy(nums, res)\n}\n</code></pre> counting_sort.swift<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums: inout [Int]) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = nums.max()!\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for i in 0 ..&lt; m {\n        counter[i + 1] += counter[i]\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let num = nums[i]\n        res[counter[num] - 1] = num // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num] -= 1 // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    const n = nums.length;\n    const res = new Array(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums: number[]): void {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    const n = nums.length;\n    const res: number[] = new Array&lt;number&gt;(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(List&lt;int&gt; nums) {\n  // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n  // counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73b0\u6b21\u6570\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n  // \u5373 counter[_num]-1 \u662f _num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n  for (int i = 0; i &lt; m; i++) {\n    counter[i + 1] += counter[i];\n  }\n  // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n  // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n  int n = nums.length;\n  List&lt;int&gt; res = List.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int _num = nums[i];\n    res[counter[_num] - 1] = _num; // \u5c06 _num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n    counter[_num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e _num \u7684\u7d22\u5f15\n  }\n  // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n  nums.setAll(0, res);\n}\n</code></pre> counting_sort.rs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfn counting_sort(nums: &amp;mut [i32]) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = *nums.into_iter().max().unwrap();\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    let mut counter = vec![0; m as usize + 1];\n    for &amp;num in &amp;*nums {\n        counter[num as usize] += 1;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for i in 0..m as usize {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    let n = nums.len();\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let num = nums[i];\n        res[counter[num as usize] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num as usize] -= 1; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in 0..n {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.c<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int nums[], int size) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int *counter = calloc(m, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    int *res = malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--;              // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    memcpy(nums, res, size * sizeof(int));\n    // 5. \u91ca\u653e\u5185\u5b58\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfun countingSort(nums: IntArray) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    var m = 0\n    for (num in nums) {\n        m = max(m.toDouble(), num.toDouble()).toInt()\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (i in 0..&lt;m) {\n        counter[i + 1] += counter[i]\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    val n = nums.size\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val num = nums[i]\n        res[counter[num] - 1] = num // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]-- // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (i in 0..&lt;n) {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.rb<pre><code>[class]{}-[func]{counting_sort}\n</code></pre> counting_sort.zig<pre><code>[class]{}-[func]{countingSort}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/counting_sort/#1193","title":"11.9.3 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n + m)\\) \uff1a\u6d89\u53ca\u904d\u5386 <code>nums</code> \u548c\u904d\u5386 <code>counter</code> \uff0c\u90fd\u4f7f\u7528\u7ebf\u6027\u65f6\u95f4\u3002\u4e00\u822c\u60c5\u51b5\u4e0b \\(n \\gg m\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u4e8e \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n + m)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u501f\u52a9\u4e86\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) \u548c \\(m\\) \u7684\u6570\u7ec4 <code>res</code> \u548c <code>counter</code> \u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5411 <code>res</code> \u4e2d\u586b\u5145\u5143\u7d20\u7684\u987a\u5e8f\u662f\u201c\u4ece\u53f3\u5411\u5de6\u201d\u7684\uff0c\u56e0\u6b64\u5012\u5e8f\u904d\u5386 <code>nums</code> \u53ef\u4ee5\u907f\u514d\u6539\u53d8\u76f8\u7b49\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b9e\u73b0\u7a33\u5b9a\u6392\u5e8f\u3002\u5b9e\u9645\u4e0a\uff0c\u6b63\u5e8f\u904d\u5386 <code>nums</code> \u4e5f\u53ef\u4ee5\u5f97\u5230\u6b63\u786e\u7684\u6392\u5e8f\u7ed3\u679c\uff0c\u4f46\u7ed3\u679c\u662f\u975e\u7a33\u5b9a\u7684\u3002</li> </ul>"},{"location":"chapter_sorting/counting_sort/#1194","title":"11.9.4 \u00a0 \u5c40\u9650\u6027","text":"<p>\u770b\u5230\u8fd9\u91cc\uff0c\u4f60\u4e5f\u8bb8\u4f1a\u89c9\u5f97\u8ba1\u6570\u6392\u5e8f\u975e\u5e38\u5de7\u5999\uff0c\u4ec5\u901a\u8fc7\u7edf\u8ba1\u6570\u91cf\u5c31\u53ef\u4ee5\u5b9e\u73b0\u9ad8\u6548\u7684\u6392\u5e8f\u3002\u7136\u800c\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\u76f8\u5bf9\u8f83\u4e3a\u4e25\u683c\u3002</p> <p>\u8ba1\u6570\u6392\u5e8f\u53ea\u9002\u7528\u4e8e\u975e\u8d1f\u6574\u6570\u3002\u82e5\u60f3\u5c06\u5176\u7528\u4e8e\u5176\u4ed6\u7c7b\u578b\u7684\u6570\u636e\uff0c\u9700\u8981\u786e\u4fdd\u8fd9\u4e9b\u6570\u636e\u53ef\u4ee5\u8f6c\u6362\u4e3a\u975e\u8d1f\u6574\u6570\uff0c\u5e76\u4e14\u5728\u8f6c\u6362\u8fc7\u7a0b\u4e2d\u4e0d\u80fd\u6539\u53d8\u5404\u4e2a\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u5927\u5c0f\u5173\u7cfb\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u5305\u542b\u8d1f\u6570\u7684\u6574\u6570\u6570\u7ec4\uff0c\u53ef\u4ee5\u5148\u7ed9\u6240\u6709\u6570\u5b57\u52a0\u4e0a\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5168\u90e8\u6570\u5b57\u8f6c\u5316\u4e3a\u6b63\u6570\uff0c\u6392\u5e8f\u5b8c\u6210\u540e\u518d\u8f6c\u6362\u56de\u53bb\u3002</p> <p>\u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u6bd4\u5982\uff0c\u5728\u4e0a\u8ff0\u793a\u4f8b\u4e2d \\(m\\) \u4e0d\u80fd\u592a\u5927\uff0c\u5426\u5219\u4f1a\u5360\u7528\u8fc7\u591a\u7a7a\u95f4\u3002\u800c\u5f53 \\(n \\ll m\\) \u65f6\uff0c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\uff0c\u53ef\u80fd\u6bd4 \\(O(n \\log n)\\) \u7684\u6392\u5e8f\u7b97\u6cd5\u8fd8\u8981\u6162\u3002</p>"},{"location":"chapter_sorting/heap_sort/","title":"11.7 \u00a0 \u5806\u6392\u5e8f","text":"<p>Tip</p> <p>\u9605\u8bfb\u672c\u8282\u524d\uff0c\u8bf7\u786e\u4fdd\u5df2\u5b66\u5b8c\u201c\u5806\u201c\u7ae0\u8282\u3002</p> <p>\u300c\u5806\u6392\u5e8f heap sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5806\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\u7684\u9ad8\u6548\u6392\u5e8f\u7b97\u6cd5\u3002\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u5df2\u7ecf\u5b66\u8fc7\u7684\u201c\u5efa\u5806\u64cd\u4f5c\u201d\u548c\u201c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u201d\u5b9e\u73b0\u5806\u6392\u5e8f\u3002</p> <ol> <li>\u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5c0f\u9876\u5806\uff0c\u6b64\u65f6\u6700\u5c0f\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002</li> <li>\u4e0d\u65ad\u6267\u884c\u51fa\u5806\u64cd\u4f5c\uff0c\u4f9d\u6b21\u8bb0\u5f55\u51fa\u5806\u5143\u7d20\uff0c\u5373\u53ef\u5f97\u5230\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u7684\u5e8f\u5217\u3002</li> </ol> <p>\u4ee5\u4e0a\u65b9\u6cd5\u867d\u7136\u53ef\u884c\uff0c\u4f46\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u989d\u5916\u6570\u7ec4\u6765\u4fdd\u5b58\u5f39\u51fa\u7684\u5143\u7d20\uff0c\u6bd4\u8f83\u6d6a\u8d39\u7a7a\u95f4\u3002\u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u4e00\u79cd\u66f4\u52a0\u4f18\u96c5\u7684\u5b9e\u73b0\u65b9\u5f0f\u3002</p>"},{"location":"chapter_sorting/heap_sort/#1171","title":"11.7.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5806\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-12 \u6240\u793a\u3002</p> <ol> <li>\u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5927\u9876\u5806\u3002\u5b8c\u6210\u540e\uff0c\u6700\u5927\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002</li> <li>\u5c06\u5806\u9876\u5143\u7d20\uff08\u7b2c\u4e00\u4e2a\u5143\u7d20\uff09\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u6700\u540e\u4e00\u4e2a\u5143\u7d20\uff09\u4ea4\u6362\u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u5806\u7684\u957f\u5ea6\u51cf \\(1\\) \uff0c\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u52a0 \\(1\\) \u3002</li> <li>\u4ece\u5806\u9876\u5143\u7d20\u5f00\u59cb\uff0c\u4ece\u9876\u5230\u5e95\u6267\u884c\u5806\u5316\u64cd\u4f5c\uff08sift down\uff09\u3002\u5b8c\u6210\u5806\u5316\u540e\uff0c\u5806\u7684\u6027\u8d28\u5f97\u5230\u4fee\u590d\u3002</li> <li>\u5faa\u73af\u6267\u884c\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\u3002\u5faa\u73af \\(n - 1\\) \u8f6e\u540e\uff0c\u5373\u53ef\u5b8c\u6210\u6570\u7ec4\u6392\u5e8f\u3002</li> </ol> <p>Tip</p> <p>\u5b9e\u9645\u4e0a\uff0c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u4e2d\u4e5f\u5305\u542b\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\uff0c\u53ea\u662f\u591a\u4e86\u4e00\u4e2a\u5f39\u51fa\u5143\u7d20\u7684\u6b65\u9aa4\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-12 \u00a0 \u5806\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528\u4e86\u4e0e\u201c\u5806\u201d\u7ae0\u8282\u76f8\u540c\u7684\u4ece\u9876\u81f3\u5e95\u5806\u5316 <code>sift_down()</code> \u51fd\u6570\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e\u5806\u7684\u957f\u5ea6\u4f1a\u968f\u7740\u63d0\u53d6\u6700\u5927\u5143\u7d20\u800c\u51cf\u5c0f\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u7ed9 <code>sift_down()</code> \u51fd\u6570\u6dfb\u52a0\u4e00\u4e2a\u957f\u5ea6\u53c2\u6570 \\(n\\) \uff0c\u7528\u4e8e\u6307\u5b9a\u5806\u7684\u5f53\u524d\u6709\u6548\u957f\u5ea6\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig heap_sort.py<pre><code>def sift_down(nums: list[int], n: int, i: int):\n    \"\"\"\u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\n    while True:\n        # \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        l = 2 * i + 1\n        r = 2 * i + 2\n        ma = i\n        if l &lt; n and nums[l] &gt; nums[ma]:\n            ma = l\n        if r &lt; n and nums[r] &gt; nums[ma]:\n            ma = r\n        # \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i:\n            break\n        # \u4ea4\u6362\u4e24\u8282\u70b9\n        nums[i], nums[ma] = nums[ma], nums[i]\n        # \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n\ndef heap_sort(nums: list[int]):\n    \"\"\"\u5806\u6392\u5e8f\"\"\"\n    # \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in range(len(nums) // 2 - 1, -1, -1):\n        sift_down(nums, len(nums), i)\n    # \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for i in range(len(nums) - 1, 0, -1):\n        # \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        nums[0], nums[i] = nums[i], nums[0]\n        # \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        sift_down(nums, i, 0)\n</code></pre> heap_sort.cpp<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(vector&lt;int&gt; &amp;nums, int n, int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(nums[i], nums[ma]);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(vector&lt;int&gt; &amp;nums) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = nums.size() / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, nums.size(), i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (int i = nums.size() - 1; i &gt; 0; --i) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        swap(nums[0], nums[i]);\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.java<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int[] nums, int n, int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int[] nums) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = nums.length / 2 - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.cs<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid SiftDown(int[] nums, int n, int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        (nums[ma], nums[i]) = (nums[i], nums[ma]);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid HeapSort(int[] nums) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = nums.Length / 2 - 1; i &gt;= 0; i--) {\n        SiftDown(nums, nums.Length, i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        (nums[i], nums[0]) = (nums[0], nums[i]);\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        SiftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.go<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums *[]int, n, i int) {\n    for true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        l := 2*i + 1\n        r := 2*i + 2\n        ma := i\n        if l &lt; n &amp;&amp; (*nums)[l] &gt; (*nums)[ma] {\n            ma = l\n        }\n        if r &lt; n &amp;&amp; (*nums)[r] &gt; (*nums)[ma] {\n            ma = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        (*nums)[i], (*nums)[ma] = (*nums)[ma], (*nums)[i]\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums *[]int) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i := len(*nums)/2 - 1; i &gt;= 0; i-- {\n        siftDown(nums, len(*nums), i)\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for i := len(*nums) - 1; i &gt; 0; i-- {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        (*nums)[0], (*nums)[i] = (*nums)[i], (*nums)[0]\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.swift<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums: inout [Int], n: Int, i: Int) {\n    var i = i\n    while true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = 2 * i + 1\n        let r = 2 * i + 2\n        var ma = i\n        if l &lt; n, nums[l] &gt; nums[ma] {\n            ma = l\n        }\n        if r &lt; n, nums[r] &gt; nums[ma] {\n            ma = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        nums.swapAt(i, ma)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums: inout [Int]) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in stride(from: nums.count / 2 - 1, through: 0, by: -1) {\n        siftDown(nums: &amp;nums, n: nums.count, i: i)\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for i in nums.indices.dropFirst().reversed() {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        nums.swapAt(0, i)\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums: &amp;nums, n: i, i: 0)\n    }\n}\n</code></pre> heap_sort.js<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums, n, i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.ts<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums: number[], n: number, i: number): void {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums: number[]): void {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.dart<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(List&lt;int&gt; nums, int n, int i) {\n  while (true) {\n    // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n    int l = 2 * i + 1;\n    int r = 2 * i + 2;\n    int ma = i;\n    if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) ma = l;\n    if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) ma = r;\n    // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n    if (ma == i) break;\n    // \u4ea4\u6362\u4e24\u8282\u70b9\n    int temp = nums[i];\n    nums[i] = nums[ma];\n    nums[ma] = temp;\n    // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n    i = ma;\n  }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(List&lt;int&gt; nums) {\n  // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n  for (int i = nums.length ~/ 2 - 1; i &gt;= 0; i--) {\n    siftDown(nums, nums.length, i);\n  }\n  // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    int tmp = nums[0];\n    nums[0] = nums[i];\n    nums[i] = tmp;\n    // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n    siftDown(nums, i, 0);\n  }\n}\n</code></pre> heap_sort.rs<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfn sift_down(nums: &amp;mut [i32], n: usize, mut i: usize) {\n    loop {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let mut ma = i;\n        if l &lt; n &amp;&amp; nums[l] &gt; nums[ma] {\n            ma = l;\n        }\n        if r &lt; n &amp;&amp; nums[r] &gt; nums[ma] {\n            ma = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        let temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfn heap_sort(nums: &amp;mut [i32]) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in (0..=nums.len() / 2 - 1).rev() {\n        sift_down(nums, nums.len(), i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for i in (1..=nums.len() - 1).rev() {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        let tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        sift_down(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.c<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int nums[], int n, int i) {\n    while (1) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int nums[], int n) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = n / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, n, i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (int i = n - 1; i &gt; 0; --i) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.kt<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfun siftDown(nums: IntArray, n: Int, li: Int) {\n    var i = li\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        val l = 2 * i + 1\n        val r = 2 * i + 2\n        var ma = i\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) ma = l\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) ma = r\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) break\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        nums[i] = nums[ma].also { nums[ma] = nums[i] }\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfun heapSort(nums: IntArray) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (i in nums.size / 2 - 1 downTo 0) {\n        siftDown(nums, nums.size, i)\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (i in nums.size - 1 downTo 1) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        nums[0] = nums[i].also { nums[i] = nums[0] }\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.rb<pre><code>[class]{}-[func]{sift_down}\n\n[class]{}-[func]{heap_sort}\n</code></pre> heap_sort.zig<pre><code>[class]{}-[func]{siftDown}\n\n[class]{}-[func]{heapSort}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/heap_sort/#1172","title":"11.7.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5efa\u5806\u64cd\u4f5c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \uff0c\u5171\u5faa\u73af \\(n - 1\\) \u8f6e\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u51e0\u4e2a\u6307\u9488\u53d8\u91cf\u4f7f\u7528 \\(O(1)\\) \u7a7a\u95f4\u3002\u5143\u7d20\u4ea4\u6362\u548c\u5806\u5316\u64cd\u4f5c\u90fd\u662f\u5728\u539f\u6570\u7ec4\u4e0a\u8fdb\u884c\u7684\u3002</li> <li>\u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u4ea4\u6362\u5806\u9876\u5143\u7d20\u548c\u5806\u5e95\u5143\u7d20\u65f6\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002</li> </ul>"},{"location":"chapter_sorting/insertion_sort/","title":"11.4 \u00a0 \u63d2\u5165\u6392\u5e8f","text":"<p>\u300c\u63d2\u5165\u6392\u5e8f insertion sort\u300d\u662f\u4e00\u79cd\u7b80\u5355\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5b83\u7684\u5de5\u4f5c\u539f\u7406\u4e0e\u624b\u52a8\u6574\u7406\u4e00\u526f\u724c\u7684\u8fc7\u7a0b\u975e\u5e38\u76f8\u4f3c\u3002</p> <p>\u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u5728\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u4e00\u4e2a\u57fa\u51c6\u5143\u7d20\uff0c\u5c06\u8be5\u5143\u7d20\u4e0e\u5176\u5de6\u4fa7\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u5143\u7d20\u9010\u4e00\u6bd4\u8f83\u5927\u5c0f\uff0c\u5e76\u5c06\u8be5\u5143\u7d20\u63d2\u5165\u5230\u6b63\u786e\u7684\u4f4d\u7f6e\u3002</p> <p>\u56fe 11-6 \u5c55\u793a\u4e86\u6570\u7ec4\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\u6d41\u7a0b\u3002\u8bbe\u57fa\u51c6\u5143\u7d20\u4e3a <code>base</code> \uff0c\u6211\u4eec\u9700\u8981\u5c06\u4ece\u76ee\u6807\u7d22\u5f15\u5230 <code>base</code> \u4e4b\u95f4\u7684\u6240\u6709\u5143\u7d20\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\uff0c\u7136\u540e\u5c06 <code>base</code> \u8d4b\u503c\u7ed9\u76ee\u6807\u7d22\u5f15\u3002</p> <p></p> <p> \u56fe 11-6 \u00a0 \u5355\u6b21\u63d2\u5165\u64cd\u4f5c </p>"},{"location":"chapter_sorting/insertion_sort/#1141","title":"11.4.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u63d2\u5165\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-7 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6570\u7ec4\u7684\u7b2c 1 \u4e2a\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002</li> <li>\u9009\u53d6\u6570\u7ec4\u7684\u7b2c 2 \u4e2a\u5143\u7d20\u4f5c\u4e3a <code>base</code> \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u9009\u53d6\u7b2c 3 \u4e2a\u5143\u7d20\u4f5c\u4e3a <code>base</code> \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 3 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5728\u6700\u540e\u4e00\u8f6e\u4e2d\uff0c\u9009\u53d6\u6700\u540e\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a <code>base</code> \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6240\u6709\u5143\u7d20\u5747\u5df2\u6392\u5e8f\u3002</li> </ol> <p></p> <p> \u56fe 11-7 \u00a0 \u63d2\u5165\u6392\u5e8f\u6d41\u7a0b </p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig insertion_sort.py<pre><code>def insertion_sort(nums: list[int]):\n    \"\"\"\u63d2\u5165\u6392\u5e8f\"\"\"\n    # \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for i in range(1, len(nums)):\n        base = nums[i]\n        j = i - 1\n        # \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while j &gt;= 0 and nums[j] &gt; base:\n            nums[j + 1] = nums[j]  # \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j -= 1\n        nums[j + 1] = base  # \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n</code></pre> insertion_sort.cpp<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(vector&lt;int&gt; &amp;nums) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for (int i = 1; i &lt; nums.size(); i++) {\n        int base = nums[i], j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.java<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for (int i = 1; i &lt; nums.length; i++) {\n        int base = nums[i], j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base;        // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.cs<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid InsertionSort(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for (int i = 1; i &lt; nums.Length; i++) {\n        int bas = nums[i], j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; bas) {\n            nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = bas;         // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.go<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums []int) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for i := 1; i &lt; len(nums); i++ {\n        base := nums[i]\n        j := i - 1\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        for j &gt;= 0 &amp;&amp; nums[j] &gt; base {\n            nums[j+1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--\n        }\n        nums[j+1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.swift<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums: inout [Int]) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for i in nums.indices.dropFirst() {\n        let base = nums[i]\n        var j = i - 1\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while j &gt;= 0, nums[j] &gt; base {\n            nums[j + 1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j -= 1\n        }\n        nums[j + 1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.js<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for (let i = 1; i &lt; nums.length; i++) {\n        let base = nums[i],\n            j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.ts<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums: number[]): void {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for (let i = 1; i &lt; nums.length; i++) {\n        const base = nums[i];\n        let j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.dart<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(List&lt;int&gt; nums) {\n  // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n  for (int i = 1; i &lt; nums.length; i++) {\n    int base = nums[i], j = i - 1;\n    // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n    while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n      nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n      j--;\n    }\n    nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n  }\n}\n</code></pre> insertion_sort.rs<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfn insertion_sort(nums: &amp;mut [i32]) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for i in 1..nums.len() {\n        let (base, mut j) = (nums[i], (i - 1) as i32);\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while j &gt;= 0 &amp;&amp; nums[j as usize] &gt; base {\n            nums[(j + 1) as usize] = nums[j as usize]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j -= 1;\n        }\n        nums[(j + 1) as usize] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.c<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int nums[], int size) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for (int i = 1; i &lt; size; i++) {\n        int base = nums[i], j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            nums[j + 1] = nums[j];\n            j--;\n        }\n        // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n        nums[j + 1] = base;\n    }\n}\n</code></pre> insertion_sort.kt<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfun insertionSort(nums: IntArray) {\n    //\u5916\u5faa\u73af: \u5df2\u6392\u5e8f\u5143\u7d20\u4e3a 1, 2, ..., n\n    for (i in nums.indices) {\n        val base = nums[i]\n        var j = i - 1\n        // \u5185\u5faa\u73af: \u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--\n        }\n        nums[j + 1] = base        // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.rb<pre><code>[class]{}-[func]{insertion_sort}\n</code></pre> insertion_sort.zig<pre><code>// \u63d2\u5165\u6392\u5e8f\nfn insertionSort(nums: []i32) void {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    var i: usize = 1;\n    while (i &lt; nums.len) : (i += 1) {\n        var base = nums[i];\n        var j: usize = i;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 1 and nums[j - 1] &gt; base) : (j -= 1) {\n            nums[j] = nums[j - 1];  // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n        }\n        nums[j] = base;             // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/insertion_sort/#1142","title":"11.4.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u6b21\u63d2\u5165\u64cd\u4f5c\u5206\u522b\u9700\u8981\u5faa\u73af \\(n - 1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u6c42\u548c\u5f97\u5230 \\((n - 1) n / 2\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u5728\u9047\u5230\u6709\u5e8f\u6570\u636e\u65f6\uff0c\u63d2\u5165\u64cd\u4f5c\u4f1a\u63d0\u524d\u7ec8\u6b62\u3002\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u63d2\u5165\u6392\u5e8f\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u63d2\u5165\u64cd\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u4f1a\u5c06\u5143\u7d20\u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u4fa7\uff0c\u4e0d\u4f1a\u6539\u53d8\u5b83\u4eec\u7684\u987a\u5e8f\u3002</li> </ul>"},{"location":"chapter_sorting/insertion_sort/#1143","title":"11.4.3 \u00a0 \u63d2\u5165\u6392\u5e8f\u7684\u4f18\u52bf","text":"<p>\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u800c\u6211\u4eec\u5373\u5c06\u5b66\u4e60\u7684\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u5c3d\u7ba1\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u66f4\u9ad8\uff0c\u4f46\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u7684\u60c5\u51b5\u4e0b\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u66f4\u5feb\u3002</p> <p>\u8fd9\u4e2a\u7ed3\u8bba\u4e0e\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u7684\u9002\u7528\u60c5\u51b5\u7684\u7ed3\u8bba\u7c7b\u4f3c\u3002\u5feb\u901f\u6392\u5e8f\u8fd9\u7c7b \\(O(n \\log n)\\) \u7684\u7b97\u6cd5\u5c5e\u4e8e\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5f80\u5f80\u5305\u542b\u66f4\u591a\u5355\u5143\u8ba1\u7b97\u64cd\u4f5c\u3002\u800c\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\\(n^2\\) \u548c \\(n \\log n\\) \u7684\u6570\u503c\u6bd4\u8f83\u63a5\u8fd1\uff0c\u590d\u6742\u5ea6\u4e0d\u5360\u4e3b\u5bfc\u5730\u4f4d\uff0c\u6bcf\u8f6e\u4e2d\u7684\u5355\u5143\u64cd\u4f5c\u6570\u91cf\u8d77\u5230\u51b3\u5b9a\u6027\u4f5c\u7528\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java\uff09\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\u91c7\u7528\u4e86\u63d2\u5165\u6392\u5e8f\uff0c\u5927\u81f4\u601d\u8def\u4e3a\uff1a\u5bf9\u4e8e\u957f\u6570\u7ec4\uff0c\u91c7\u7528\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\uff1b\u5bf9\u4e8e\u77ed\u6570\u7ec4\uff0c\u76f4\u63a5\u4f7f\u7528\u63d2\u5165\u6392\u5e8f\u3002</p> <p>\u867d\u7136\u5192\u6ce1\u6392\u5e8f\u3001\u9009\u62e9\u6392\u5e8f\u548c\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u63d2\u5165\u6392\u5e8f\u7684\u4f7f\u7528\u9891\u7387\u663e\u8457\u9ad8\u4e8e\u5192\u6ce1\u6392\u5e8f\u548c\u9009\u62e9\u6392\u5e8f\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u5192\u6ce1\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u4ea4\u6362\u5b9e\u73b0\uff0c\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u4e34\u65f6\u53d8\u91cf\uff0c\u5171\u6d89\u53ca 3 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u63d2\u5165\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u8d4b\u503c\u5b9e\u73b0\uff0c\u4ec5\u9700 1 \u4e2a\u5355\u5143\u64cd\u4f5c\u3002\u56e0\u6b64\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u8ba1\u7b97\u5f00\u9500\u901a\u5e38\u6bd4\u63d2\u5165\u6392\u5e8f\u66f4\u9ad8\u3002</li> <li>\u9009\u62e9\u6392\u5e8f\u5728\u4efb\u4f55\u60c5\u51b5\u4e0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \u3002\u5982\u679c\u7ed9\u5b9a\u4e00\u7ec4\u90e8\u5206\u6709\u5e8f\u7684\u6570\u636e\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u6bd4\u9009\u62e9\u6392\u5e8f\u6548\u7387\u66f4\u9ad8\u3002</li> <li>\u9009\u62e9\u6392\u5e8f\u4e0d\u7a33\u5b9a\uff0c\u65e0\u6cd5\u5e94\u7528\u4e8e\u591a\u7ea7\u6392\u5e8f\u3002</li> </ul>"},{"location":"chapter_sorting/merge_sort/","title":"11.6 \u00a0 \u5f52\u5e76\u6392\u5e8f","text":"<p>\u300c\u5f52\u5e76\u6392\u5e8f merge sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5305\u542b\u56fe 11-10 \u6240\u793a\u7684\u201c\u5212\u5206\u201d\u548c\u201c\u5408\u5e76\u201d\u9636\u6bb5\u3002</p> <ol> <li>\u5212\u5206\u9636\u6bb5\uff1a\u901a\u8fc7\u9012\u5f52\u4e0d\u65ad\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5206\u5f00\uff0c\u5c06\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u8f6c\u6362\u4e3a\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002</li> <li>\u5408\u5e76\u9636\u6bb5\uff1a\u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u5212\u5206\uff0c\u5f00\u59cb\u5408\u5e76\uff0c\u6301\u7eed\u5730\u5c06\u5de6\u53f3\u4e24\u4e2a\u8f83\u77ed\u7684\u6709\u5e8f\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u8f83\u957f\u7684\u6709\u5e8f\u6570\u7ec4\uff0c\u76f4\u81f3\u7ed3\u675f\u3002</li> </ol> <p></p> <p> \u56fe 11-10 \u00a0 \u5f52\u5e76\u6392\u5e8f\u7684\u5212\u5206\u4e0e\u5408\u5e76\u9636\u6bb5 </p>"},{"location":"chapter_sorting/merge_sort/#1161","title":"11.6.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u5982\u56fe 11-11 \u6240\u793a\uff0c\u201c\u5212\u5206\u9636\u6bb5\u201d\u4ece\u9876\u81f3\u5e95\u9012\u5f52\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5207\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\u3002</p> <ol> <li>\u8ba1\u7b97\u6570\u7ec4\u4e2d\u70b9 <code>mid</code> \uff0c\u9012\u5f52\u5212\u5206\u5de6\u5b50\u6570\u7ec4\uff08\u533a\u95f4 <code>[left, mid]</code> \uff09\u548c\u53f3\u5b50\u6570\u7ec4\uff08\u533a\u95f4 <code>[mid + 1, right]</code> \uff09\u3002</li> <li>\u9012\u5f52\u6267\u884c\u6b65\u9aa4 <code>1.</code> \uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u533a\u95f4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u3002</li> </ol> <p>\u201c\u5408\u5e76\u9636\u6bb5\u201d\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ece\u957f\u5ea6\u4e3a 1 \u7684\u5b50\u6570\u7ec4\u5f00\u59cb\u5408\u5e76\uff0c\u5408\u5e76\u9636\u6bb5\u4e2d\u7684\u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u662f\u6709\u5e8f\u7684\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-11 \u00a0 \u5f52\u5e76\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u5f52\u5e76\u6392\u5e8f\u4e0e\u4e8c\u53c9\u6811\u540e\u5e8f\u904d\u5386\u7684\u9012\u5f52\u987a\u5e8f\u662f\u4e00\u81f4\u7684\u3002</p> <ul> <li>\u540e\u5e8f\u904d\u5386\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6811\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6811\uff0c\u6700\u540e\u5904\u7406\u6839\u8282\u70b9\u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6570\u7ec4\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6570\u7ec4\uff0c\u6700\u540e\u5904\u7406\u5408\u5e76\u3002</li> </ul> <p>\u5f52\u5e76\u6392\u5e8f\u7684\u5b9e\u73b0\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\u3002\u8bf7\u6ce8\u610f\uff0c<code>nums</code> \u7684\u5f85\u5408\u5e76\u533a\u95f4\u4e3a <code>[left, right]</code> \uff0c\u800c <code>tmp</code> \u7684\u5bf9\u5e94\u533a\u95f4\u4e3a <code>[0, right - left]</code> \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig merge_sort.py<pre><code>def merge(nums: list[int], left: int, mid: int, right: int):\n    \"\"\"\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\"\"\"\n    # \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    # \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    tmp = [0] * (right - left + 1)\n    # \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    i, j, k = left, mid + 1, 0\n    # \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid and j &lt;= right:\n        if nums[i] &lt;= nums[j]:\n            tmp[k] = nums[i]\n            i += 1\n        else:\n            tmp[k] = nums[j]\n            j += 1\n        k += 1\n    # \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid:\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    while j &lt;= right:\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    # \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for k in range(0, len(tmp)):\n        nums[left + k] = tmp[k]\n\ndef merge_sort(nums: list[int], left: int, right: int):\n    \"\"\"\u5f52\u5e76\u6392\u5e8f\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6\n    if left &gt;= right:\n        return  # \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    # \u5212\u5206\u9636\u6bb5\n    mid = (left + right) // 2  # \u8ba1\u7b97\u4e2d\u70b9\n    merge_sort(nums, left, mid)  # \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    merge_sort(nums, mid + 1, right)  # \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    # \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right)\n</code></pre> merge_sort.cpp<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    vector&lt;int&gt; tmp(right - left + 1);\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmp.size(); k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right)\n        return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    int mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.java<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(int[] nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    int[] tmp = new int[right - left + 1];\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right)\n        return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    int mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.cs<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid Merge(int[] nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    int[] tmp = new int[right - left + 1];\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmp.Length; ++k) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid MergeSort(int[] nums, int left, int right) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right) return;       // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    int mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\n    MergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    MergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    Merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.go<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunc merge(nums []int, left, mid, right int) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    tmp := make([]int, right-left+1)\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    i, j, k := left, mid+1, 0\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    for i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i++\n        } else {\n            tmp[k] = nums[j]\n            j++\n        }\n        k++\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    for i &lt;= mid {\n        tmp[k] = nums[i]\n        i++\n        k++\n    }\n    for j &lt;= right {\n        tmp[k] = nums[j]\n        j++\n        k++\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for k := 0; k &lt; len(tmp); k++ {\n        nums[left+k] = tmp[k]\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums []int, left, right int) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if left &gt;= right {\n        return\n    }\n    // \u5212\u5206\u9636\u6bb5\n    mid := (left + right) / 2\n    mergeSort(nums, left, mid)\n    mergeSort(nums, mid+1, right)\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.swift<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    var tmp = Array(repeating: 0, count: right - left + 1)\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    var i = left, j = mid + 1, k = 0\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid, j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i += 1\n        } else {\n            tmp[k] = nums[j]\n            j += 1\n        }\n        k += 1\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid {\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for k in tmp.indices {\n        nums[left + k] = tmp[k]\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if left &gt;= right { // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n        return\n    }\n    // \u5212\u5206\u9636\u6bb5\n    let mid = (left + right) / 2 // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums: &amp;nums, left: left, right: mid) // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums: &amp;nums, left: mid + 1, right: right) // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums: &amp;nums, left: left, mid: mid, right: right)\n}\n</code></pre> merge_sort.js<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunction merge(nums, left, mid, right) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    const tmp = new Array(right - left + 1);\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums, left, right) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    let mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.ts<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    const tmp = new Array(right - left + 1);\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    let mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.dart<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(List&lt;int&gt; nums, int left, int mid, int right) {\n  // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n  // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n  List&lt;int&gt; tmp = List.filled(right - left + 1, 0);\n  // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n  int i = left, j = mid + 1, k = 0;\n  // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n  while (i &lt;= mid &amp;&amp; j &lt;= right) {\n    if (nums[i] &lt;= nums[j])\n      tmp[k++] = nums[i++];\n    else\n      tmp[k++] = nums[j++];\n  }\n  // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n  while (i &lt;= mid) {\n    tmp[k++] = nums[i++];\n  }\n  while (j &lt;= right) {\n    tmp[k++] = nums[j++];\n  }\n  // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n  for (k = 0; k &lt; tmp.length; k++) {\n    nums[left + k] = tmp[k];\n  }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(List&lt;int&gt; nums, int left, int right) {\n  // \u7ec8\u6b62\u6761\u4ef6\n  if (left &gt;= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n  // \u5212\u5206\u9636\u6bb5\n  int mid = (left + right) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\n  mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n  mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n  // \u5408\u5e76\u9636\u6bb5\n  merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.rs<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfn merge(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    let tmp_size = right - left + 1;\n    let mut tmp = vec![0; tmp_size];\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    let (mut i, mut j, mut k) = (left, mid + 1, 0);\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i];\n            i += 1;\n        } else {\n            tmp[k] = nums[j];\n            j += 1;\n        }\n        k += 1;\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid {\n        tmp[k] = nums[i];\n        k += 1;\n        i += 1;\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j];\n        k += 1;\n        j += 1;\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for k in 0..tmp_size {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfn merge_sort(nums: &amp;mut [i32], left: usize, right: usize) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if left &gt;= right {\n        return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    }\n\n    // \u5212\u5206\u9636\u6bb5\n    let mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n    merge_sort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    merge_sort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.c<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(int *nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    int tmpSize = right - left + 1;\n    int *tmp = (int *)malloc(tmpSize * sizeof(int));\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmpSize; ++k) {\n        nums[left + k] = tmp[k];\n    }\n    // \u91ca\u653e\u5185\u5b58\n    free(tmp);\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int *nums, int left, int right) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right)\n        return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    int mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.kt<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfun merge(nums: IntArray, left: Int, mid: Int, right: Int) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    val tmp = IntArray(right - left + 1)\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    var i = left\n    var j = mid + 1\n    var k = 0\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) tmp[k++] = nums[i++]\n        else tmp[k++] = nums[j++]\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++]\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++]\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (l in tmp.indices) {\n        nums[left + l] = tmp[l]\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfun mergeSort(nums: IntArray, left: Int, right: Int) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right) return  // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    val mid = (left + right) / 2 // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid) // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right) // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.rb<pre><code>[class]{}-[func]{merge}\n\n[class]{}-[func]{merge_sort}\n</code></pre> merge_sort.zig<pre><code>// \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfn merge(nums: []i32, left: usize, mid: usize, right: usize) !void {\n    // \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\n    var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n    defer mem_arena.deinit();\n    const mem_allocator = mem_arena.allocator();\n    var tmp = try mem_allocator.alloc(i32, right + 1 - left);\n    std.mem.copy(i32, tmp, nums[left..right+1]);\n    // \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \n    var leftStart = left - left;\n    var leftEnd = mid - left;\n    // \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \n    var rightStart = mid + 1 - left;\n    var rightEnd = right - left;\n    // i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\n    var i = leftStart;\n    var j = rightStart;\n    // \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\n    var k = left;\n    while (k &lt;= right) : (k += 1) {\n        // \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n        if (i &gt; leftEnd) {\n            nums[k] = tmp[j];\n            j += 1;\n        // \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 &lt;= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n        } else if  (j &gt; rightEnd or tmp[i] &lt;= tmp[j]) {\n            nums[k] = tmp[i];\n            i += 1;\n        // \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 &gt; \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n        } else {\n            nums[k] = tmp[j];\n            j += 1;\n        }\n    }\n}\n\n// \u5f52\u5e76\u6392\u5e8f\nfn mergeSort(nums: []i32, left: usize, right: usize) !void {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right) return;              // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    var mid = (left + right) / 2;           // \u8ba1\u7b97\u4e2d\u70b9\n    try mergeSort(nums, left, mid);         // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    try mergeSort(nums, mid + 1, right);    // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    try merge(nums, left, mid, right);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/merge_sort/#1162","title":"11.6.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5212\u5206\u4ea7\u751f\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u5c42\u5408\u5e76\u7684\u603b\u64cd\u4f5c\u6570\u91cf\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u9012\u5f52\u6df1\u5ea6\u4e3a \\(\\log n\\) \uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002\u5408\u5e76\u64cd\u4f5c\u9700\u8981\u501f\u52a9\u8f85\u52a9\u6570\u7ec4\u5b9e\u73b0\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u5408\u5e76\u8fc7\u7a0b\u4e2d\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u6b21\u5e8f\u4fdd\u6301\u4e0d\u53d8\u3002</li> </ul>"},{"location":"chapter_sorting/merge_sort/#1163","title":"11.6.3 \u00a0 \u94fe\u8868\u6392\u5e8f","text":"<p>\u5bf9\u4e8e\u94fe\u8868\uff0c\u5f52\u5e76\u6392\u5e8f\u76f8\u8f83\u4e8e\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\u5177\u6709\u663e\u8457\u4f18\u52bf\uff0c\u53ef\u4ee5\u5c06\u94fe\u8868\u6392\u5e8f\u4efb\u52a1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(1)\\) \u3002</p> <ul> <li>\u5212\u5206\u9636\u6bb5\uff1a\u53ef\u4ee5\u4f7f\u7528\u201c\u8fed\u4ee3\u201d\u66ff\u4ee3\u201c\u9012\u5f52\u201d\u6765\u5b9e\u73b0\u94fe\u8868\u5212\u5206\u5de5\u4f5c\uff0c\u4ece\u800c\u7701\u53bb\u9012\u5f52\u4f7f\u7528\u7684\u6808\u5e27\u7a7a\u95f4\u3002</li> <li>\u5408\u5e76\u9636\u6bb5\uff1a\u5728\u94fe\u8868\u4e2d\uff0c\u8282\u70b9\u589e\u5220\u64cd\u4f5c\u4ec5\u9700\u6539\u53d8\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\u5b9e\u73b0\uff0c\u56e0\u6b64\u5408\u5e76\u9636\u6bb5\uff08\u5c06\u4e24\u4e2a\u77ed\u6709\u5e8f\u94fe\u8868\u5408\u5e76\u4e3a\u4e00\u4e2a\u957f\u6709\u5e8f\u94fe\u8868\uff09\u65e0\u987b\u521b\u5efa\u989d\u5916\u94fe\u8868\u3002</li> </ul> <p>\u5177\u4f53\u5b9e\u73b0\u7ec6\u8282\u6bd4\u8f83\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u76f8\u5173\u8d44\u6599\u8fdb\u884c\u5b66\u4e60\u3002</p>"},{"location":"chapter_sorting/quick_sort/","title":"11.5 \u00a0 \u5feb\u901f\u6392\u5e8f","text":"<p>\u300c\u5feb\u901f\u6392\u5e8f quick sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u8fd0\u884c\u9ad8\u6548\uff0c\u5e94\u7528\u5e7f\u6cdb\u3002</p> <p>\u5feb\u901f\u6392\u5e8f\u7684\u6838\u5fc3\u64cd\u4f5c\u662f\u201c\u54e8\u5175\u5212\u5206\u201d\uff0c\u5176\u76ee\u6807\u662f\uff1a\u9009\u62e9\u6570\u7ec4\u4e2d\u7684\u67d0\u4e2a\u5143\u7d20\u4f5c\u4e3a\u201c\u57fa\u51c6\u6570\u201d\uff0c\u5c06\u6240\u6709\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u5de6\u4fa7\uff0c\u800c\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u53f3\u4fa7\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u54e8\u5175\u5212\u5206\u7684\u6d41\u7a0b\u5982\u56fe 11-8 \u6240\u793a\u3002</p> <ol> <li>\u9009\u53d6\u6570\u7ec4\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u521d\u59cb\u5316\u4e24\u4e2a\u6307\u9488 <code>i</code> \u548c <code>j</code> \u5206\u522b\u6307\u5411\u6570\u7ec4\u7684\u4e24\u7aef\u3002</li> <li>\u8bbe\u7f6e\u4e00\u4e2a\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u4f7f\u7528 <code>i</code>\uff08<code>j</code>\uff09\u5206\u522b\u5bfb\u627e\u7b2c\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u5927\uff08\u5c0f\uff09\u7684\u5143\u7d20\uff0c\u7136\u540e\u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\u3002</li> <li>\u5faa\u73af\u6267\u884c\u6b65\u9aa4 <code>2.</code> \uff0c\u76f4\u5230 <code>i</code> \u548c <code>j</code> \u76f8\u9047\u65f6\u505c\u6b62\uff0c\u6700\u540e\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-8 \u00a0 \u54e8\u5175\u5212\u5206\u6b65\u9aa4 </p> <p>\u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u539f\u6570\u7ec4\u88ab\u5212\u5206\u6210\u4e09\u90e8\u5206\uff1a\u5de6\u5b50\u6570\u7ec4\u3001\u57fa\u51c6\u6570\u3001\u53f3\u5b50\u6570\u7ec4\uff0c\u4e14\u6ee1\u8db3\u201c\u5de6\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20 \\(\\leq\\) \u57fa\u51c6\u6570 \\(\\leq\\) \u53f3\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20\u201d\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u63a5\u4e0b\u6765\u53ea\u9700\u5bf9\u8fd9\u4e24\u4e2a\u5b50\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\u3002</p> <p>\u5feb\u901f\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565</p> <p>\u54e8\u5175\u5212\u5206\u7684\u5b9e\u8d28\u662f\u5c06\u4e00\u4e2a\u8f83\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u7b80\u5316\u4e3a\u4e24\u4e2a\u8f83\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"\u54e8\u5175\u5212\u5206\"\"\"\n    # \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        # \u5143\u7d20\u4ea4\u6362\n        nums[i], nums[j] = nums[j], nums[i]\n    # \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n</code></pre> quick_sort.cpp<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(vector&lt;int&gt; &amp;nums, int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.java<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int[] nums, int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint partition(int[] nums, int left, int right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid Swap(int[] nums, int i, int j) {\n    (nums[j], nums[i]) = (nums[i], nums[j]);\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint Partition(int[] nums, int left, int right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        Swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    Swap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.go<pre><code>/* \u54e8\u5175\u5212\u5206 */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        // \u5143\u7d20\u4ea4\u6362\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    nums[i], nums[left] = nums[left], nums[i]\n    return i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.swift<pre><code>/* \u54e8\u5175\u5212\u5206 */\nfunc partition(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    var i = left\n    var j = right\n    while i &lt; j {\n        while i &lt; j, nums[j] &gt;= nums[left] {\n            j -= 1 // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while i &lt; j, nums[i] &lt;= nums[left] {\n            i += 1 // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        nums.swapAt(i, j) // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    nums.swapAt(i, left) // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.js<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nswap(nums, i, j) {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums, left, right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        // \u5143\u7d20\u4ea4\u6362\n        this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nswap(nums: number[], i: number, j: number): void {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums: number[], left: number, right: number): number {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        // \u5143\u7d20\u4ea4\u6362\n        this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid _swap(List&lt;int&gt; nums, int i, int j) {\n  int tmp = nums[i];\n  nums[i] = nums[j];\n  nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n    _swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n  }\n  _swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n  return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rs<pre><code>/* \u54e8\u5175\u5212\u5206 */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        nums.swap(i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    nums.swap(i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.c<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int nums[], int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n            j--;\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n            i++;\n        }\n        // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n        swap(nums, i, j);\n    }\n    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    swap(nums, i, left);\n    // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n    return i;\n}\n</code></pre> quick_sort.kt<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nfun swap(nums: IntArray, i: Int, j: Int) {\n    nums[i] = nums[j].also { nums[j] = nums[i] }\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nfun partition(nums: IntArray, left: Int, right: Int): Int {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--           // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++           // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j)  // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left)   // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i              // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rb<pre><code>[class]{QuickSort}-[func]{partition}\n</code></pre> quick_sort.zig<pre><code>// \u5143\u7d20\u4ea4\u6362\nfn swap(nums: []i32, i: usize, j: usize) void {\n    var tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n// \u54e8\u5175\u5212\u5206\nfn partition(nums: []i32, left: usize, right: usize) usize {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    var i = left;\n    var j = right;\n    while (i &lt; j) {\n        while (i &lt; j and nums[j] &gt;= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j and nums[i] &lt;= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/quick_sort/#1151","title":"11.5.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u5feb\u901f\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-9 \u6240\u793a\u3002</p> <ol> <li>\u9996\u5148\uff0c\u5bf9\u539f\u6570\u7ec4\u6267\u884c\u4e00\u6b21\u201c\u54e8\u5175\u5212\u5206\u201d\uff0c\u5f97\u5230\u672a\u6392\u5e8f\u7684\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u3002</li> <li>\u7136\u540e\uff0c\u5bf9\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5206\u522b\u9012\u5f52\u6267\u884c\u201c\u54e8\u5175\u5212\u5206\u201d\u3002</li> <li>\u6301\u7eed\u9012\u5f52\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\uff0c\u4ece\u800c\u5b8c\u6210\u6574\u4e2a\u6570\u7ec4\u7684\u6392\u5e8f\u3002</li> </ol> <p></p> <p> \u56fe 11-9 \u00a0 \u5feb\u901f\u6392\u5e8f\u6d41\u7a0b </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"\u5feb\u901f\u6392\u5e8f\"\"\"\n    # \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if left &gt;= right:\n        return\n    # \u54e8\u5175\u5212\u5206\n    pivot = self.partition(nums, left, right)\n    # \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    self.quick_sort(nums, left, pivot - 1)\n    self.quick_sort(nums, pivot + 1, right)\n</code></pre> quick_sort.cpp<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right)\n        return;\n    // \u54e8\u5175\u5212\u5206\n    int pivot = partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.java<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right)\n        return;\n    // \u54e8\u5175\u5212\u5206\n    int pivot = partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid QuickSort(int[] nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right)\n        return;\n    // \u54e8\u5175\u5212\u5206\n    int pivot = Partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    QuickSort(nums, left, pivot - 1);\n    QuickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.go<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if left &gt;= right {\n        return\n    }\n    // \u54e8\u5175\u5212\u5206\n    pivot := q.partition(nums, left, right)\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    q.quickSort(nums, left, pivot-1)\n    q.quickSort(nums, pivot+1, right)\n}\n</code></pre> quick_sort.swift<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if left &gt;= right {\n        return\n    }\n    // \u54e8\u5175\u5212\u5206\n    let pivot = partition(nums: &amp;nums, left: left, right: right)\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums: &amp;nums, left: left, right: pivot - 1)\n    quickSort(nums: &amp;nums, left: pivot + 1, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums, left, right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) return;\n    // \u54e8\u5175\u5212\u5206\n    const pivot = this.partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums: number[], left: number, right: number): void {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) {\n        return;\n    }\n    // \u54e8\u5175\u5212\u5206\n    const pivot = this.partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n  if (left &gt;= right) return;\n  // \u54e8\u5175\u5212\u5206\n  int pivot = _partition(nums, left, right);\n  // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n  quickSort(nums, left, pivot - 1);\n  quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.rs<pre><code>/* \u5feb\u901f\u6392\u5e8f */\npub fn quick_sort(left: i32, right: i32, nums: &amp;mut [i32]) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if left &gt;= right {\n        return;\n    }\n    // \u54e8\u5175\u5212\u5206\n    let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    Self::quick_sort(left, pivot - 1, nums);\n    Self::quick_sort(pivot + 1, right, nums);\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n            j--;\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n            i++;\n        }\n        // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n        swap(nums, i, j);\n    }\n    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    swap(nums, i, left);\n    // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n    return i;\n}\n\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u5feb\u901f\u6392\u5e8f\nvoid quickSort(int nums[], int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) {\n        return;\n    }\n    // \u54e8\u5175\u5212\u5206\n    int pivot = partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.kt<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfun quickSort(nums: IntArray, left: Int, right: Int) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) return\n    // \u54e8\u5175\u5212\u5206\n    val pivot = partition(nums, left, right)\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1)\n    quickSort(nums, pivot + 1, right)\n}\n</code></pre> quick_sort.rb<pre><code>[class]{QuickSort}-[func]{quick_sort}\n</code></pre> quick_sort.zig<pre><code>// \u5feb\u901f\u6392\u5e8f\nfn quickSort(nums: []i32, left: usize, right: usize) void {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) return;\n    // \u54e8\u5175\u5212\u5206\n    var pivot = partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/quick_sort/#1152","title":"11.5.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u5e73\u5747\u60c5\u51b5\u4e0b\uff0c\u54e8\u5175\u5212\u5206\u7684\u9012\u5f52\u5c42\u6570\u4e3a \\(\\log n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u603b\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n \\log n)\\) \u65f6\u95f4\u3002\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u90fd\u5c06\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u5212\u5206\u4e3a\u957f\u5ea6\u4e3a \\(0\\) \u548c \\(n - 1\\) \u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u6b64\u65f6\u9012\u5f52\u5c42\u6570\u8fbe\u5230 \\(n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u5728\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u5012\u5e8f\u7684\u60c5\u51b5\u4e0b\uff0c\u8fbe\u5230\u6700\u5dee\u9012\u5f52\u6df1\u5ea6 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002\u6392\u5e8f\u64cd\u4f5c\u662f\u5728\u539f\u6570\u7ec4\u4e0a\u8fdb\u884c\u7684\uff0c\u672a\u501f\u52a9\u989d\u5916\u6570\u7ec4\u3002</li> <li>\u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u54e8\u5175\u5212\u5206\u7684\u6700\u540e\u4e00\u6b65\uff0c\u57fa\u51c6\u6570\u53ef\u80fd\u4f1a\u88ab\u4ea4\u6362\u81f3\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u4fa7\u3002</li> </ul>"},{"location":"chapter_sorting/quick_sort/#1153","title":"11.5.3 \u00a0 \u5feb\u901f\u6392\u5e8f\u4e3a\u4ec0\u4e48\u5feb","text":"<p>\u4ece\u540d\u79f0\u4e0a\u5c31\u80fd\u770b\u51fa\uff0c\u5feb\u901f\u6392\u5e8f\u5728\u6548\u7387\u65b9\u9762\u5e94\u8be5\u5177\u6709\u4e00\u5b9a\u7684\u4f18\u52bf\u3002\u5c3d\u7ba1\u5feb\u901f\u6392\u5e8f\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e0e\u201c\u5f52\u5e76\u6392\u5e8f\u201d\u548c\u201c\u5806\u6392\u5e8f\u201d\u76f8\u540c\uff0c\u4f46\u901a\u5e38\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u51fa\u73b0\u6700\u5dee\u60c5\u51b5\u7684\u6982\u7387\u5f88\u4f4e\uff1a\u867d\u7136\u5feb\u901f\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u6ca1\u6709\u5f52\u5e76\u6392\u5e8f\u7a33\u5b9a\uff0c\u4f46\u5728\u7edd\u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u80fd\u5728 \\(O(n \\log n)\\) \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u8fd0\u884c\u3002</li> <li>\u7f13\u5b58\u4f7f\u7528\u6548\u7387\u9ad8\uff1a\u5728\u6267\u884c\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u65f6\uff0c\u7cfb\u7edf\u53ef\u5c06\u6574\u4e2a\u5b50\u6570\u7ec4\u52a0\u8f7d\u5230\u7f13\u5b58\uff0c\u56e0\u6b64\u8bbf\u95ee\u5143\u7d20\u7684\u6548\u7387\u8f83\u9ad8\u3002\u800c\u50cf\u201c\u5806\u6392\u5e8f\u201d\u8fd9\u7c7b\u7b97\u6cd5\u9700\u8981\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u5143\u7d20\uff0c\u4ece\u800c\u7f3a\u4e4f\u8fd9\u4e00\u7279\u6027\u3002</li> <li>\u590d\u6742\u5ea6\u7684\u5e38\u6570\u7cfb\u6570\u5c0f\uff1a\u5728\u4e0a\u8ff0\u4e09\u79cd\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u6bd4\u8f83\u3001\u8d4b\u503c\u3001\u4ea4\u6362\u7b49\u64cd\u4f5c\u7684\u603b\u6570\u91cf\u6700\u5c11\u3002\u8fd9\u4e0e\u201c\u63d2\u5165\u6392\u5e8f\u201d\u6bd4\u201c\u5192\u6ce1\u6392\u5e8f\u201d\u66f4\u5feb\u7684\u539f\u56e0\u7c7b\u4f3c\u3002</li> </ul>"},{"location":"chapter_sorting/quick_sort/#1154","title":"11.5.4 \u00a0 \u57fa\u51c6\u6570\u4f18\u5316","text":"<p>\u5feb\u901f\u6392\u5e8f\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\u7684\u65f6\u95f4\u6548\u7387\u53ef\u80fd\u964d\u4f4e\u3002\u4e3e\u4e00\u4e2a\u6781\u7aef\u4f8b\u5b50\uff0c\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u662f\u5b8c\u5168\u5012\u5e8f\u7684\uff0c\u7531\u4e8e\u6211\u4eec\u9009\u62e9\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u5728\u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u57fa\u51c6\u6570\u88ab\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u53f3\u7aef\uff0c\u5bfc\u81f4\u5de6\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n - 1\\)\u3001\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(0\\) \u3002\u5982\u6b64\u9012\u5f52\u4e0b\u53bb\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u540e\u90fd\u6709\u4e00\u4e2a\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(0\\) \uff0c\u5206\u6cbb\u7b56\u7565\u5931\u6548\uff0c\u5feb\u901f\u6392\u5e8f\u9000\u5316\u4e3a\u201c\u5192\u6ce1\u6392\u5e8f\u201d\u7684\u8fd1\u4f3c\u5f62\u5f0f\u3002</p> <p>\u4e3a\u4e86\u5c3d\u91cf\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u53d1\u751f\uff0c\u6211\u4eec\u53ef\u4ee5\u4f18\u5316\u54e8\u5175\u5212\u5206\u4e2d\u7684\u57fa\u51c6\u6570\u7684\u9009\u53d6\u7b56\u7565\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u53ef\u4ee5\u968f\u673a\u9009\u53d6\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\u3002\u7136\u800c\uff0c\u5982\u679c\u8fd0\u6c14\u4e0d\u4f73\uff0c\u6bcf\u6b21\u90fd\u9009\u5230\u4e0d\u7406\u60f3\u7684\u57fa\u51c6\u6570\uff0c\u6548\u7387\u4ecd\u7136\u4e0d\u5c3d\u5982\u4eba\u610f\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u751f\u6210\u7684\u662f\u201c\u4f2a\u968f\u673a\u6570\u201d\u3002\u5982\u679c\u6211\u4eec\u9488\u5bf9\u4f2a\u968f\u673a\u6570\u5e8f\u5217\u6784\u5efa\u4e00\u4e2a\u7279\u5b9a\u7684\u6d4b\u8bd5\u6837\u4f8b\uff0c\u90a3\u4e48\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u4ecd\u7136\u53ef\u80fd\u52a3\u5316\u3002</p> <p>\u4e3a\u4e86\u8fdb\u4e00\u6b65\u6539\u8fdb\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6570\u7ec4\u4e2d\u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\uff08\u901a\u5e38\u4e3a\u6570\u7ec4\u7684\u9996\u3001\u5c3e\u3001\u4e2d\u70b9\u5143\u7d20\uff09\uff0c\u5e76\u5c06\u8fd9\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\u4f5c\u4e3a\u57fa\u51c6\u6570\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u57fa\u51c6\u6570\u201c\u65e2\u4e0d\u592a\u5c0f\u4e5f\u4e0d\u592a\u5927\u201d\u7684\u6982\u7387\u5c06\u5927\u5e45\u63d0\u5347\u3002\u5f53\u7136\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u9009\u53d6\u66f4\u591a\u5019\u9009\u5143\u7d20\uff0c\u4ee5\u8fdb\u4e00\u6b65\u63d0\u9ad8\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002\u91c7\u7528\u8fd9\u79cd\u65b9\u6cd5\u540e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u7684\u6982\u7387\u5927\u5927\u964d\u4f4e\u3002</p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def median_three(self, nums: list[int], left: int, mid: int, right: int) -&gt; int:\n    \"\"\"\u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\"\"\"\n    l, m, r = nums[left], nums[mid], nums[right]\n    if (l &lt;= m &lt;= r) or (r &lt;= m &lt;= l):\n        return mid  # m \u5728 l \u548c r \u4e4b\u95f4\n    if (m &lt;= l &lt;= r) or (r &lt;= l &lt;= m):\n        return left  # l \u5728 m \u548c r \u4e4b\u95f4\n    return right\n\ndef partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"\u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\"\"\"\n    # \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    med = self.median_three(nums, left, (left + right) // 2, right)\n    # \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    nums[left], nums[med] = nums[med], nums[left]\n    # \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        # \u5143\u7d20\u4ea4\u6362\n        nums[i], nums[j] = nums[j], nums[i]\n    # \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n</code></pre> quick_sort.cpp<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u95f4\n    return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.java<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u95f4\n    return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right) {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.cs<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint MedianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u95f4\n    return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint Partition(int[] nums, int left, int right) {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    int med = MedianThree(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    Swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        Swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    Swap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.go<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n    l, m, r := nums[left], nums[mid], nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m \u5728 l \u548c r \u4e4b\u95f4\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l \u5728 m \u548c r \u4e4b\u95f4\n    }\n    return right\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09*/\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    med := q.medianThree(nums, left, (left+right)/2, right)\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    nums[left], nums[med] = nums[med], nums[left]\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- //\u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ //\u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        //\u5143\u7d20\u4ea4\u6362\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    //\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    nums[i], nums[left] = nums[left], nums[i]\n    return i //\u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.swift<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -&gt; Int {\n    let l = nums[left]\n    let m = nums[mid]\n    let r = nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m \u5728 l \u548c r \u4e4b\u95f4\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l \u5728 m \u548c r \u4e4b\u95f4\n    }\n    return right\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    let med = medianThree(nums: nums, left: left, mid: (left + right) / 2, right: right)\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    nums.swapAt(left, med)\n    return partition(nums: &amp;nums, left: left, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(nums, left, mid, right) {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l \u5728 m \u548c r \u4e4b\u95f4\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums, left, right) {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    this.swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.ts<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(\n    nums: number[],\n    left: number,\n    mid: number,\n    right: number\n): number {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l \u5728 m \u548c r \u4e4b\u95f4\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums: number[], left: number, right: number): number {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    this.swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.dart<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint _medianThree(List&lt;int&gt; nums, int left, int mid, int right) {\n  int l = nums[left], m = nums[mid], r = nums[right];\n  if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n    return mid; // m \u5728 l \u548c r \u4e4b\u95f4\n  if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n    return left; // l \u5728 m \u548c r \u4e4b\u95f4\n  return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n  int med = _medianThree(nums, left, (left + right) ~/ 2, right);\n  // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n  _swap(nums, left, med);\n  // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n    _swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n  }\n  _swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n  return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rs<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfn median_three(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) -&gt; usize {\n    let (l, m, r) = (nums[left], nums[mid], nums[right]);\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid; // m \u5728 l \u548c r \u4e4b\u95f4\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left; // l \u5728 m \u548c r \u4e4b\u95f4\n    }\n    right\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    let med = Self::median_three(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    nums.swap(left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        nums.swap(i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    nums.swap(i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u4e2d\u4f4d\u57fa\u51c6\u6570\u4f18\u5316\uff09 */\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint medianThree(int nums[], int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u95f4\n    return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */ \nint partitionMedian(int nums[], int left, int right) {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.kt<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfun medianThree(nums: IntArray, left: Int, mid: Int, right: Int): Int {\n    val l = nums[left]\n    val m = nums[mid]\n    val r = nums[right]\n    if ((m in l..r) || (m in r..l))\n        return mid  // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((l in m..r) || (l in r..m))\n        return left // l \u5728 m \u548c r \u4e4b\u95f4\n    return right\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nfun partition(nums: IntArray, left: Int, right: Int): Int {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--           // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++           // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j)  // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left)   // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i              // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rb<pre><code>[class]{QuickSortMedian}-[func]{median_three}\n\n[class]{QuickSortMedian}-[func]{partition}\n</code></pre> quick_sort.zig<pre><code>// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nfn medianThree(nums: []i32, left: usize, mid: usize, right: usize) usize {\n    var l = nums[left];\n    var m = nums[mid];\n    var r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u95f4\n    return right;\n}\n\n// \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\nfn partition(nums: []i32, left: usize, right: usize) usize {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    var med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    var i = left;\n    var j = right;\n    while (i &lt; j) {\n        while (i &lt; j and nums[j] &gt;= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j and nums[i] &lt;= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/quick_sort/#1155","title":"11.5.5 \u00a0 \u5c3e\u9012\u5f52\u4f18\u5316","text":"<p>\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u53ef\u80fd\u5360\u7528\u7a7a\u95f4\u8f83\u591a\u3002\u4ee5\u5b8c\u5168\u6709\u5e8f\u7684\u8f93\u5165\u6570\u7ec4\u4e3a\u4f8b\uff0c\u8bbe\u9012\u5f52\u4e2d\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(m\\) \uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u90fd\u5c06\u4ea7\u751f\u957f\u5ea6\u4e3a \\(0\\) \u7684\u5de6\u5b50\u6570\u7ec4\u548c\u957f\u5ea6\u4e3a \\(m - 1\\) \u7684\u53f3\u5b50\u6570\u7ec4\uff0c\u8fd9\u610f\u5473\u7740\u6bcf\u4e00\u5c42\u9012\u5f52\u8c03\u7528\u51cf\u5c11\u7684\u95ee\u9898\u89c4\u6a21\u975e\u5e38\u5c0f\uff08\u53ea\u51cf\u5c11\u4e00\u4e2a\u5143\u7d20\uff09\uff0c\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\u4f1a\u8fbe\u5230 \\(n - 1\\) \uff0c\u6b64\u65f6\u9700\u8981\u5360\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002</p> <p>\u4e3a\u4e86\u9632\u6b62\u6808\u5e27\u7a7a\u95f4\u7684\u7d2f\u79ef\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6bcf\u8f6e\u54e8\u5175\u6392\u5e8f\u5b8c\u6210\u540e\uff0c\u6bd4\u8f83\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\uff0c\u4ec5\u5bf9\u8f83\u77ed\u7684\u5b50\u6570\u7ec4\u8fdb\u884c\u9012\u5f52\u3002\u7531\u4e8e\u8f83\u77ed\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(n / 2\\) \uff0c\u56e0\u6b64\u8fd9\u79cd\u65b9\u6cd5\u80fd\u786e\u4fdd\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff0c\u4ece\u800c\u5c06\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"\u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\"\"\"\n    # \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while left &lt; right:\n        # \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        pivot = self.partition(nums, left, right)\n        # \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if pivot - left &lt; right - pivot:\n            self.quick_sort(nums, left, pivot - 1)  # \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1  # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        else:\n            self.quick_sort(nums, pivot + 1, right)  # \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1  # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n</code></pre> quick_sort.cpp<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        int pivot = partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1;                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1;                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.java<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        int pivot = partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid QuickSort(int[] nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        int pivot = Partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            QuickSort(nums, left, pivot - 1);  // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1;  // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            QuickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.go<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09*/\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    for left &lt; right {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        pivot := q.partition(nums, left, right)\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if pivot-left &lt; right-pivot {\n            q.quickSort(nums, left, pivot-1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            q.quickSort(nums, pivot+1, right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.swift<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\n    var left = left\n    var right = right\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while left &lt; right {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        let pivot = partition(nums: &amp;nums, left: left, right: right)\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left) &lt; (right - pivot) {\n            quickSortTailCall(nums: &amp;nums, left: left, right: pivot - 1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            quickSortTailCall(nums: &amp;nums, left: pivot + 1, right: right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.js<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums, left, right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        let pivot = this.partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            this.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums: number[], left: number, right: number): void {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        let pivot = this.partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            this.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n  while (left &lt; right) {\n    // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n    int pivot = _partition(nums, left, right);\n    // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n    if (pivot - left &lt; right - pivot) {\n      quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n      left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n    } else {\n      quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n      right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n    }\n  }\n}\n</code></pre> quick_sort.rs<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\npub fn quick_sort(mut left: i32, mut right: i32, nums: &amp;mut [i32]) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while left &lt; right {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if pivot - left &lt; right - pivot {\n            Self::quick_sort(left, pivot - 1, nums); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            Self::quick_sort(pivot + 1, right, nums); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\n// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nvoid quickSortTailCall(int nums[], int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        int pivot = partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSortTailCall(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1;                         // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            quickSortTailCall(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1;                         // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.kt<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfun quickSort(nums: IntArray, left: Int, right: Int) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) return\n    // \u54e8\u5175\u5212\u5206\n    val pivot = partition(nums, left, right)\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1)\n    quickSort(nums, pivot + 1, right)\n}\n</code></pre> quick_sort.rb<pre><code>[class]{QuickSortTailCall}-[func]{quick_sort}\n</code></pre> quick_sort.zig<pre><code>// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nfn quickSort(nums: []i32, left_: usize, right_: usize) void {\n    var left = left_;\n    var right = right_;\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        var pivot = partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1);   // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1;                   // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right);  // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1;                  // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/radix_sort/","title":"11.10 \u00a0 \u57fa\u6570\u6392\u5e8f","text":"<p>\u4e0a\u4e00\u8282\u4ecb\u7ecd\u4e86\u8ba1\u6570\u6392\u5e8f\uff0c\u5b83\u9002\u7528\u4e8e\u6570\u636e\u91cf \\(n\\) \u8f83\u5927\u4f46\u6570\u636e\u8303\u56f4 \\(m\\) \u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u5bf9 \\(n = 10^6\\) \u4e2a\u5b66\u53f7\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u5b66\u53f7\u662f\u4e00\u4e2a \\(8\\) \u4f4d\u6570\u5b57\uff0c\u8fd9\u610f\u5473\u7740\u6570\u636e\u8303\u56f4 \\(m = 10^8\\) \u975e\u5e38\u5927\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u9700\u8981\u5206\u914d\u5927\u91cf\u5185\u5b58\u7a7a\u95f4\uff0c\u800c\u57fa\u6570\u6392\u5e8f\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u3002</p> <p>\u300c\u57fa\u6570\u6392\u5e8f radix sort\u300d\u7684\u6838\u5fc3\u601d\u60f3\u4e0e\u8ba1\u6570\u6392\u5e8f\u4e00\u81f4\uff0c\u4e5f\u901a\u8fc7\u7edf\u8ba1\u4e2a\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u57fa\u6570\u6392\u5e8f\u5229\u7528\u6570\u5b57\u5404\u4f4d\u4e4b\u95f4\u7684\u9012\u8fdb\u5173\u7cfb\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e00\u4f4d\u8fdb\u884c\u6392\u5e8f\uff0c\u4ece\u800c\u5f97\u5230\u6700\u7ec8\u7684\u6392\u5e8f\u7ed3\u679c\u3002</p>"},{"location":"chapter_sorting/radix_sort/#11101","title":"11.10.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u4ee5\u5b66\u53f7\u6570\u636e\u4e3a\u4f8b\uff0c\u5047\u8bbe\u6570\u5b57\u7684\u6700\u4f4e\u4f4d\u662f\u7b2c \\(1\\) \u4f4d\uff0c\u6700\u9ad8\u4f4d\u662f\u7b2c \\(8\\) \u4f4d\uff0c\u57fa\u6570\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-18 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316\u4f4d\u6570 \\(k = 1\\) \u3002</li> <li>\u5bf9\u5b66\u53f7\u7684\u7b2c \\(k\\) \u4f4d\u6267\u884c\u201c\u8ba1\u6570\u6392\u5e8f\u201d\u3002\u5b8c\u6210\u540e\uff0c\u6570\u636e\u4f1a\u6839\u636e\u7b2c \\(k\\) \u4f4d\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u3002</li> <li>\u5c06 \\(k\\) \u589e\u52a0 \\(1\\) \uff0c\u7136\u540e\u8fd4\u56de\u6b65\u9aa4 <code>2.</code> \u7ee7\u7eed\u8fed\u4ee3\uff0c\u76f4\u5230\u6240\u6709\u4f4d\u90fd\u6392\u5e8f\u5b8c\u6210\u540e\u7ed3\u675f\u3002</li> </ol> <p></p> <p> \u56fe 11-18 \u00a0 \u57fa\u6570\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b </p> <p>\u4e0b\u9762\u5256\u6790\u4ee3\u7801\u5b9e\u73b0\u3002\u5bf9\u4e8e\u4e00\u4e2a \\(d\\) \u8fdb\u5236\u7684\u6570\u5b57 \\(x\\) \uff0c\u8981\u83b7\u53d6\u5176\u7b2c \\(k\\) \u4f4d \\(x_k\\) \uff0c\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u8ba1\u7b97\u516c\u5f0f\uff1a</p> \\[ x_k = \\lfloor\\frac{x}{d^{k-1}}\\rfloor \\bmod d \\] <p>\u5176\u4e2d \\(\\lfloor a \\rfloor\\) \u8868\u793a\u5bf9\u6d6e\u70b9\u6570 \\(a\\) \u5411\u4e0b\u53d6\u6574\uff0c\u800c \\(\\bmod \\: d\\) \u8868\u793a\u5bf9 \\(d\\) \u53d6\u6a21\uff08\u53d6\u4f59\uff09\u3002\u5bf9\u4e8e\u5b66\u53f7\u6570\u636e\uff0c\\(d = 10\\) \u4e14 \\(k \\in [1, 8]\\) \u3002</p> <p>\u6b64\u5916\uff0c\u6211\u4eec\u9700\u8981\u5c0f\u5e45\u6539\u52a8\u8ba1\u6570\u6392\u5e8f\u4ee3\u7801\uff0c\u4f7f\u4e4b\u53ef\u4ee5\u6839\u636e\u6570\u5b57\u7684\u7b2c \\(k\\) \u4f4d\u8fdb\u884c\u6392\u5e8f\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig radix_sort.py<pre><code>def digit(num: int, exp: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\"\"\"\n    # \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num // exp) % 10\n\ndef counting_sort_digit(nums: list[int], exp: int):\n    \"\"\"\u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\"\"\"\n    # \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    counter = [0] * 10\n    n = len(nums)\n    # \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for i in range(n):\n        d = digit(nums[i], exp)  # \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d] += 1  # \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    # \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for i in range(1, 10):\n        counter[i] += counter[i - 1]\n    # \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        d = digit(nums[i], exp)\n        j = counter[d] - 1  # \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]  # \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1  # \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    # \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in range(n):\n        nums[i] = res[i]\n\ndef radix_sort(nums: list[int]):\n    \"\"\"\u57fa\u6570\u6392\u5e8f\"\"\"\n    # \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    m = max(nums)\n    # \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    exp = 1\n    while exp &lt;= m:\n        # \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        # k = 1 -&gt; exp = 1\n        # k = 2 -&gt; exp = 10\n        # \u5373 exp = 10^(k-1)\n        counting_sort_digit(nums, exp)\n        exp *= 10\n</code></pre> radix_sort.cpp<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(vector&lt;int&gt; &amp;nums, int exp) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    vector&lt;int&gt; counter(10, 0);\n    int n = nums.size();\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    vector&lt;int&gt; res(n, 0);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(vector&lt;int&gt; &amp;nums) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    int m = *max_element(nums.begin(), nums.end());\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (int exp = 1; exp &lt;= m; exp *= 10)\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.java<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int[] nums, int exp) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    int[] counter = new int[10];\n    int n = nums.length;\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int[] nums) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    int m = Integer.MIN_VALUE;\n    for (int num : nums)\n        if (num &gt; m)\n            m = num;\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (int exp = 1; exp &lt;= m; exp *= 10)\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.cs<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint Digit(int num, int exp) {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid CountingSortDigit(int[] nums, int exp) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    int[] counter = new int[10];\n    int n = nums.Length;\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (int i = 0; i &lt; n; i++) {\n        int d = Digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = Digit(nums[i], exp);\n        int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid RadixSort(int[] nums) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    int m = int.MinValue;\n    foreach (int num in nums) {\n        if (num &gt; m) m = num;\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (int exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        CountingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.go<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num, exp int) int {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums []int, exp int) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    counter := make([]int, 10)\n    n := len(nums)\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for i := 0; i &lt; n; i++ {\n        d := digit(nums[i], exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++             // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for i := 1; i &lt; 10; i++ {\n        counter[i] += counter[i-1]\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        d := digit(nums[i], exp)\n        j := counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]    // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--        // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i := 0; i &lt; n; i++ {\n        nums[i] = res[i]\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums []int) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    max := math.MinInt\n    for _, num := range nums {\n        if num &gt; max {\n            max = num\n        }\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for exp := 1; max &gt;= exp; exp *= 10 {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp)\n    }\n}\n</code></pre> radix_sort.swift<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num: Int, exp: Int) -&gt; Int {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    (num / exp) % 10\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums: inout [Int], exp: Int) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    var counter = Array(repeating: 0, count: 10)\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for i in nums.indices {\n        let d = digit(num: nums[i], exp: exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d] += 1 // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for i in 1 ..&lt; 10 {\n        counter[i] += counter[i - 1]\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let d = digit(num: nums[i], exp: exp)\n        let j = counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i] // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1 // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums: inout [Int]) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    var m = Int.min\n    for num in nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for exp in sequence(first: 1, next: { m &gt;= ($0 * 10) ? $0 * 10 : nil }) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums: &amp;nums, exp: exp)\n    }\n}\n</code></pre> radix_sort.js<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num, exp) {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return Math.floor(num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums, exp) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    let m = Number.MIN_VALUE;\n    for (const num of nums) {\n        if (num &gt; m) {\n            m = num;\n        }\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.ts<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num: number, exp: number): number {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return Math.floor(num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums: number[], exp: number): void {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums: number[]): void {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    let m = Number.MIN_VALUE;\n    for (const num of nums) {\n        if (num &gt; m) {\n            m = num;\n        }\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.dart<pre><code>/* \u83b7\u53d6\u5143\u7d20 _num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int _num, int exp) {\n  // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n  return (_num ~/ exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(List&lt;int&gt; nums, int exp) {\n  // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n  List&lt;int&gt; counter = List&lt;int&gt;.filled(10, 0);\n  int n = nums.length;\n  // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n  for (int i = 0; i &lt; n; i++) {\n    int d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n    counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n  }\n  // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n  for (int i = 1; i &lt; 10; i++) {\n    counter[i] += counter[i - 1];\n  }\n  // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n  List&lt;int&gt; res = List&lt;int&gt;.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int d = digit(nums[i], exp);\n    int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n    res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n    counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n  }\n  // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n  for (int i = 0; i &lt; n; i++) nums[i] = res[i];\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(List&lt;int&gt; nums) {\n  // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n  // dart \u4e2d int \u7684\u957f\u5ea6\u662f 64 \u4f4d\u7684\n  int m = -1 &lt;&lt; 63;\n  for (int _num in nums) if (_num &gt; m) m = _num;\n  // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n  for (int exp = 1; exp &lt;= m; exp *= 10)\n    // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n    // k = 1 -&gt; exp = 1\n    // k = 2 -&gt; exp = 10\n    // \u5373 exp = 10^(k-1)\n    countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.rs<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfn digit(num: i32, exp: i32) -&gt; usize {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return ((num / exp) % 10) as usize;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfn counting_sort_digit(nums: &amp;mut [i32], exp: i32) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    let mut counter = [0; 10];\n    let n = nums.len();\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for i in 0..n {\n        let d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d] += 1; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for i in 1..10 {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let d = digit(nums[i], exp);\n        let j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in 0..n {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfn radix_sort(nums: &amp;mut [i32]) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    let m = *nums.into_iter().max().unwrap();\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    let mut exp = 1;\n    while exp &lt;= m {\n        counting_sort_digit(nums, exp);\n        exp *= 10;\n    }\n}\n</code></pre> radix_sort.c<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int nums[], int size, int exp) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    int *counter = (int *)malloc((sizeof(int) * 10));\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (int i = 0; i &lt; size; i++) {\n        // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        int d = digit(nums[i], exp);\n        // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n        counter[d]++;\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    int *res = (int *)malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; size; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int nums[], int size) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    int max = INT32_MIN;\n    for (size_t i = 0; i &lt; size - 1; i++) {\n        if (nums[i] &gt; max) {\n            max = nums[i];\n        }\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (int exp = 1; max &gt;= exp; exp *= 10)\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, size, exp);\n}\n</code></pre> radix_sort.kt<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfun digit(num: Int, exp: Int): Int {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfun countingSortDigit(nums: IntArray, exp: Int) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    val counter = IntArray(10)\n    val n = nums.size\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (i in 0..&lt;n) {\n        val d = digit(nums[i], exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++ // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (i in 1..9) {\n        counter[i] += counter[i - 1]\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val d = digit(nums[i], exp)\n        val j = counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i] // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]-- // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (i in 0..&lt;n) nums[i] = res[i]\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfun radixSort(nums: IntArray) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    var m = Int.MIN_VALUE\n    for (num in nums) if (num &gt; m) m = num\n    var exp = 1\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    while (exp &lt;= m) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp)\n        exp *= 10\n    }\n}\n</code></pre> radix_sort.rb<pre><code>[class]{}-[func]{digit}\n\n[class]{}-[func]{counting_sort_digit}\n\n[class]{}-[func]{radix_sort}\n</code></pre> radix_sort.zig<pre><code>// \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\nfn digit(num: i32, exp: i32) i32 {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return @mod(@divFloor(num, exp), 10);\n}\n\n// \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\nfn countingSortDigit(nums: []i32, exp: i32) !void {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n    // defer mem_arena.deinit();\n    const mem_allocator = mem_arena.allocator();\n    var counter = try mem_allocator.alloc(usize, 10);\n    @memset(counter, 0);\n    var n = nums.len;\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (nums) |num| {\n        var d: u32 = @bitCast(digit(num, exp)); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d] += 1; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    var i: usize = 1;\n    while (i &lt; 10) : (i += 1) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    var res = try mem_allocator.alloc(i32, n);\n    i = n - 1;\n    while (i &gt;= 0) : (i -= 1) {\n        var d: u32 = @bitCast(digit(nums[i], exp));\n        var j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1;        // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n        if (i == 0) break;\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    i = 0;\n    while (i &lt; n) : (i += 1) {\n        nums[i] = res[i];\n    }\n}\n\n// \u57fa\u6570\u6392\u5e8f\nfn radixSort(nums: []i32) !void {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    var m: i32 = std.math.minInt(i32);\n    for (nums) |num| {\n        if (num &gt; m) m = num;\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    var exp: i32 = 1;\n    while (exp &lt;= m) : (exp *= 10) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        try countingSortDigit(nums, exp);    \n    }\n} \n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u4e3a\u4ec0\u4e48\u4ece\u6700\u4f4e\u4f4d\u5f00\u59cb\u6392\u5e8f\uff1f</p> <p>\u5728\u8fde\u7eed\u7684\u6392\u5e8f\u8f6e\u6b21\u4e2d\uff0c\u540e\u4e00\u8f6e\u6392\u5e8f\u4f1a\u8986\u76d6\u524d\u4e00\u8f6e\u6392\u5e8f\u7684\u7ed3\u679c\u3002\u4e3e\u4f8b\u6765\u8bf4\uff0c\u5982\u679c\u7b2c\u4e00\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a &lt; b\\) \uff0c\u800c\u7b2c\u4e8c\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a &gt; b\\) \uff0c\u90a3\u4e48\u7b2c\u4e8c\u8f6e\u7684\u7ed3\u679c\u5c06\u53d6\u4ee3\u7b2c\u4e00\u8f6e\u7684\u7ed3\u679c\u3002\u7531\u4e8e\u6570\u5b57\u7684\u9ad8\u4f4d\u4f18\u5148\u7ea7\u9ad8\u4e8e\u4f4e\u4f4d\uff0c\u56e0\u6b64\u5e94\u8be5\u5148\u6392\u5e8f\u4f4e\u4f4d\u518d\u6392\u5e8f\u9ad8\u4f4d\u3002</p>"},{"location":"chapter_sorting/radix_sort/#11102","title":"11.10.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<p>\u76f8\u8f83\u4e8e\u8ba1\u6570\u6392\u5e8f\uff0c\u57fa\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u503c\u8303\u56f4\u8f83\u5927\u7684\u60c5\u51b5\uff0c\u4f46\u524d\u63d0\u662f\u6570\u636e\u5fc5\u987b\u53ef\u4ee5\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u683c\u5f0f\uff0c\u4e14\u4f4d\u6570\u4e0d\u80fd\u8fc7\u5927\u3002\u4f8b\u5982\uff0c\u6d6e\u70b9\u6570\u4e0d\u9002\u5408\u4f7f\u7528\u57fa\u6570\u6392\u5e8f\uff0c\u56e0\u4e3a\u5176\u4f4d\u6570 \\(k\\) \u8fc7\u5927\uff0c\u53ef\u80fd\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(nk) \\gg O(n^2)\\) \u3002</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nk)\\)\uff1a\u8bbe\u6570\u636e\u91cf\u4e3a \\(n\\)\u3001\u6570\u636e\u4e3a \\(d\\) \u8fdb\u5236\u3001\u6700\u5927\u4f4d\u6570\u4e3a \\(k\\) \uff0c\u5219\u5bf9\u67d0\u4e00\u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(n + d)\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6240\u6709 \\(k\\) \u4f4d\u4f7f\u7528 \\(O((n + d)k)\\) \u65f6\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\\(d\\) \u548c \\(k\\) \u90fd\u76f8\u5bf9\u8f83\u5c0f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n + d)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u4e0e\u8ba1\u6570\u6392\u5e8f\u76f8\u540c\uff0c\u57fa\u6570\u6392\u5e8f\u9700\u8981\u501f\u52a9\u957f\u5ea6\u4e3a \\(n\\) \u548c \\(d\\) \u7684\u6570\u7ec4 <code>res</code> \u548c <code>counter</code> \u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u5f53\u8ba1\u6570\u6392\u5e8f\u7a33\u5b9a\u65f6\uff0c\u57fa\u6570\u6392\u5e8f\u4e5f\u7a33\u5b9a\uff1b\u5f53\u8ba1\u6570\u6392\u5e8f\u4e0d\u7a33\u5b9a\u65f6\uff0c\u57fa\u6570\u6392\u5e8f\u65e0\u6cd5\u4fdd\u8bc1\u5f97\u5230\u6b63\u786e\u7684\u6392\u5e8f\u7ed3\u679c\u3002</li> </ul>"},{"location":"chapter_sorting/selection_sort/","title":"11.2 \u00a0 \u9009\u62e9\u6392\u5e8f","text":"<p>\u300c\u9009\u62e9\u6392\u5e8f selection sort\u300d\u7684\u5de5\u4f5c\u539f\u7406\u975e\u5e38\u7b80\u5355\uff1a\u5f00\u542f\u4e00\u4e2a\u5faa\u73af\uff0c\u6bcf\u8f6e\u4ece\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u6700\u5c0f\u7684\u5143\u7d20\uff0c\u5c06\u5176\u653e\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u672b\u5c3e\u3002</p> <p>\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u9009\u62e9\u6392\u5e8f\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u56fe 11-2 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u672a\u6392\u5e8f\uff0c\u5373\u672a\u6392\u5e8f\uff08\u7d22\u5f15\uff09\u533a\u95f4\u4e3a \\([0, n-1]\\) \u3002</li> <li>\u9009\u53d6\u533a\u95f4 \\([0, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(0\\) \u5904\u7684\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 1 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u9009\u53d6\u533a\u95f4 \\([1, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(1\\) \u5904\u7684\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ee5\u6b64\u7c7b\u63a8\u3002\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u9009\u62e9\u4e0e\u4ea4\u6362\u540e\uff0c\u6570\u7ec4\u524d \\(n - 1\\) \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5927\u5143\u7d20\uff0c\u65e0\u987b\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-2 \u00a0 \u9009\u62e9\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u7528 \\(k\\) \u6765\u8bb0\u5f55\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig selection_sort.py<pre><code>def selection_sort(nums: list[int]):\n    \"\"\"\u9009\u62e9\u6392\u5e8f\"\"\"\n    n = len(nums)\n    # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for i in range(n - 1):\n        # \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        k = i\n        for j in range(i + 1, n):\n            if nums[j] &lt; nums[k]:\n                k = j  # \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        # \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        nums[i], nums[k] = nums[k], nums[i]\n</code></pre> selection_sort.cpp<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(vector&lt;int&gt; &amp;nums) {\n    int n = nums.size();\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        swap(nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.java<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int[] nums) {\n    int n = nums.length;\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.cs<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid SelectionSort(int[] nums) {\n    int n = nums.Length;\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        (nums[k], nums[i]) = (nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.go<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums []int) {\n    n := len(nums)\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for i := 0; i &lt; n-1; i++ {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        k := i\n        for j := i + 1; j &lt; n; j++ {\n            if nums[j] &lt; nums[k] {\n                // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n                k = j\n            }\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        nums[i], nums[k] = nums[k], nums[i]\n\n    }\n}\n</code></pre> selection_sort.swift<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums: inout [Int]) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for i in nums.indices.dropLast() {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        var k = i\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[j] &lt; nums[k] {\n                k = j // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        nums.swapAt(i, k)\n    }\n}\n</code></pre> selection_sort.js<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums) {\n    let n = nums.length;\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (let i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.ts<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums: number[]): void {\n    let n = nums.length;\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (let i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.dart<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(List&lt;int&gt; nums) {\n  int n = nums.length;\n  // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n  for (int i = 0; i &lt; n - 1; i++) {\n    // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n    int k = i;\n    for (int j = i + 1; j &lt; n; j++) {\n      if (nums[j] &lt; nums[k]) k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n    }\n    // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n    int temp = nums[i];\n    nums[i] = nums[k];\n    nums[k] = temp;\n  }\n}\n</code></pre> selection_sort.rs<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfn selection_sort(nums: &amp;mut [i32]) {\n    if nums.is_empty() {\n        return;\n    }\n    let n = nums.len();\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for i in 0..n - 1 {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        let mut k = i;\n        for j in i + 1..n {\n            if nums[j] &lt; nums[k] {\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        nums.swap(i, k);\n    }\n}\n</code></pre> selection_sort.c<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int nums[], int n) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.kt<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfun selectionSort(nums: IntArray) {\n    val n = nums.size\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (i in 0..&lt;n - 1) {\n        var k = i\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        for (j in i + 1..&lt;n) {\n            if (nums[j] &lt; nums[k]) k = j // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        nums[i] = nums[k].also { nums[k] = nums[i] }\n    }\n}\n</code></pre> selection_sort.rb<pre><code>[class]{}-[func]{selection_sort}\n</code></pre> selection_sort.zig<pre><code>[class]{}-[func]{selectionSort}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/selection_sort/#1121","title":"11.2.1 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5916\u5faa\u73af\u5171 \\(n - 1\\) \u8f6e\uff0c\u7b2c\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u6700\u540e\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(2\\) \uff0c\u5373\u5404\u8f6e\u5916\u5faa\u73af\u5206\u522b\u5305\u542b \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(3\\)\u3001\\(2\\) \u8f6e\u5185\u5faa\u73af\uff0c\u6c42\u548c\u4e3a \\(\\frac{(n - 1)(n + 2)}{2}\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5982\u56fe 11-3 \u6240\u793a\uff0c\u5143\u7d20 <code>nums[i]</code> \u6709\u53ef\u80fd\u88ab\u4ea4\u6362\u81f3\u4e0e\u5176\u76f8\u7b49\u7684\u5143\u7d20\u7684\u53f3\u8fb9\uff0c\u5bfc\u81f4\u4e24\u8005\u7684\u76f8\u5bf9\u987a\u5e8f\u53d1\u751f\u6539\u53d8\u3002</li> </ul> <p> \u56fe 11-3 \u00a0 \u9009\u62e9\u6392\u5e8f\u975e\u7a33\u5b9a\u793a\u4f8b </p>"},{"location":"chapter_sorting/sorting_algorithm/","title":"11.1 \u00a0 \u6392\u5e8f\u7b97\u6cd5","text":"<p>\u300c\u6392\u5e8f\u7b97\u6cd5 sorting algorithm\u300d\u7528\u4e8e\u5bf9\u4e00\u7ec4\u6570\u636e\u6309\u7167\u7279\u5b9a\u987a\u5e8f\u8fdb\u884c\u6392\u5217\u3002\u6392\u5e8f\u7b97\u6cd5\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u56e0\u4e3a\u6709\u5e8f\u6570\u636e\u901a\u5e38\u80fd\u591f\u88ab\u66f4\u9ad8\u6548\u5730\u67e5\u627e\u3001\u5206\u6790\u548c\u5904\u7406\u3002</p> <p>\u5982\u56fe 11-1 \u6240\u793a\uff0c\u6392\u5e8f\u7b97\u6cd5\u4e2d\u7684\u6570\u636e\u7c7b\u578b\u53ef\u4ee5\u662f\u6574\u6570\u3001\u6d6e\u70b9\u6570\u3001\u5b57\u7b26\u6216\u5b57\u7b26\u4e32\u7b49\u3002\u6392\u5e8f\u7684\u5224\u65ad\u89c4\u5219\u53ef\u6839\u636e\u9700\u6c42\u8bbe\u5b9a\uff0c\u5982\u6570\u5b57\u5927\u5c0f\u3001\u5b57\u7b26 ASCII \u7801\u987a\u5e8f\u6216\u81ea\u5b9a\u4e49\u89c4\u5219\u3002</p> <p></p> <p> \u56fe 11-1 \u00a0 \u6570\u636e\u7c7b\u578b\u548c\u5224\u65ad\u89c4\u5219\u793a\u4f8b </p>"},{"location":"chapter_sorting/sorting_algorithm/#1111","title":"11.1.1 \u00a0 \u8bc4\u4ef7\u7ef4\u5ea6","text":"<p>\u8fd0\u884c\u6548\u7387\uff1a\u6211\u4eec\u671f\u671b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c3d\u91cf\u4f4e\uff0c\u4e14\u603b\u4f53\u64cd\u4f5c\u6570\u91cf\u8f83\u5c11\uff08\u65f6\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\u53d8\u5c0f\uff09\u3002\u5bf9\u4e8e\u5927\u6570\u636e\u91cf\u7684\u60c5\u51b5\uff0c\u8fd0\u884c\u6548\u7387\u663e\u5f97\u5c24\u4e3a\u91cd\u8981\u3002</p> <p>\u5c31\u5730\u6027\uff1a\u987e\u540d\u601d\u4e49\uff0c\u300c\u539f\u5730\u6392\u5e8f\u300d\u901a\u8fc7\u5728\u539f\u6570\u7ec4\u4e0a\u76f4\u63a5\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\uff0c\u65e0\u987b\u501f\u52a9\u989d\u5916\u7684\u8f85\u52a9\u6570\u7ec4\uff0c\u4ece\u800c\u8282\u7701\u5185\u5b58\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u539f\u5730\u6392\u5e8f\u7684\u6570\u636e\u642c\u8fd0\u64cd\u4f5c\u8f83\u5c11\uff0c\u8fd0\u884c\u901f\u5ea6\u4e5f\u66f4\u5feb\u3002</p> <p>\u7a33\u5b9a\u6027\uff1a\u300c\u7a33\u5b9a\u6392\u5e8f\u300d\u5728\u5b8c\u6210\u6392\u5e8f\u540e\uff0c\u76f8\u7b49\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u76f8\u5bf9\u987a\u5e8f\u4e0d\u53d1\u751f\u6539\u53d8\u3002</p> <p>\u7a33\u5b9a\u6392\u5e8f\u662f\u591a\u7ea7\u6392\u5e8f\u573a\u666f\u7684\u5fc5\u8981\u6761\u4ef6\u3002\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u5b58\u50a8\u5b66\u751f\u4fe1\u606f\u7684\u8868\u683c\uff0c\u7b2c 1 \u5217\u548c\u7b2c 2 \u5217\u5206\u522b\u662f\u59d3\u540d\u548c\u5e74\u9f84\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u300c\u975e\u7a33\u5b9a\u6392\u5e8f\u300d\u53ef\u80fd\u5bfc\u81f4\u8f93\u5165\u6570\u636e\u7684\u6709\u5e8f\u6027\u4e27\u5931\uff1a</p> <pre><code># \u8f93\u5165\u6570\u636e\u662f\u6309\u7167\u59d3\u540d\u6392\u5e8f\u597d\u7684\n# (name, age)\n  ('A', 19)\n  ('B', 18)\n  ('C', 21)\n  ('D', 19)\n  ('E', 23)\n\n# \u5047\u8bbe\u4f7f\u7528\u975e\u7a33\u5b9a\u6392\u5e8f\u7b97\u6cd5\u6309\u5e74\u9f84\u6392\u5e8f\u5217\u8868\uff0c\n# \u7ed3\u679c\u4e2d ('D', 19) \u548c ('A', 19) \u7684\u76f8\u5bf9\u4f4d\u7f6e\u6539\u53d8\uff0c\n# \u8f93\u5165\u6570\u636e\u6309\u59d3\u540d\u6392\u5e8f\u7684\u6027\u8d28\u4e22\u5931\n  ('B', 18)\n  ('D', 19)\n  ('A', 19)\n  ('C', 21)\n  ('E', 23)\n</code></pre> <p>\u81ea\u9002\u5e94\u6027\uff1a\u300c\u81ea\u9002\u5e94\u6392\u5e8f\u300d\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u53d7\u8f93\u5165\u6570\u636e\u7684\u5f71\u54cd\uff0c\u5373\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5e76\u4e0d\u5b8c\u5168\u76f8\u7b49\u3002</p> <p>\u81ea\u9002\u5e94\u6027\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u6765\u8bc4\u4f30\u3002\u5982\u679c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u5dee\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u8bf4\u660e\u6392\u5e8f\u7b97\u6cd5\u5728\u67d0\u4e9b\u6570\u636e\u4e0b\u6027\u80fd\u53ef\u80fd\u52a3\u5316\uff0c\u56e0\u6b64\u88ab\u89c6\u4e3a\u8d1f\u9762\u5c5e\u6027\uff1b\u800c\u5982\u679c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u5219\u88ab\u89c6\u4e3a\u6b63\u9762\u5c5e\u6027\u3002</p> <p>\u662f\u5426\u57fa\u4e8e\u6bd4\u8f83\uff1a\u300c\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u300d\u4f9d\u8d56\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff08\\(&lt;\\)\u3001\\(=\\)\u3001\\(&gt;\\)\uff09\u6765\u5224\u65ad\u5143\u7d20\u7684\u76f8\u5bf9\u987a\u5e8f\uff0c\u4ece\u800c\u6392\u5e8f\u6574\u4e2a\u6570\u7ec4\uff0c\u7406\u8bba\u6700\u4f18\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u800c\u300c\u975e\u6bd4\u8f83\u6392\u5e8f\u300d\u4e0d\u4f7f\u7528\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(n)\\) \uff0c\u4f46\u5176\u901a\u7528\u6027\u76f8\u5bf9\u8f83\u5dee\u3002</p>"},{"location":"chapter_sorting/sorting_algorithm/#1112","title":"11.1.2 \u00a0 \u7406\u60f3\u6392\u5e8f\u7b97\u6cd5","text":"<p>\u8fd0\u884c\u5feb\u3001\u539f\u5730\u3001\u7a33\u5b9a\u3001\u6b63\u5411\u81ea\u9002\u5e94\u3001\u901a\u7528\u6027\u597d\u3002\u663e\u7136\uff0c\u8fc4\u4eca\u4e3a\u6b62\u5c1a\u672a\u53d1\u73b0\u517c\u5177\u4ee5\u4e0a\u6240\u6709\u7279\u6027\u7684\u6392\u5e8f\u7b97\u6cd5\u3002\u56e0\u6b64\uff0c\u5728\u9009\u62e9\u6392\u5e8f\u7b97\u6cd5\u65f6\uff0c\u9700\u8981\u6839\u636e\u5177\u4f53\u7684\u6570\u636e\u7279\u70b9\u548c\u95ee\u9898\u9700\u6c42\u6765\u51b3\u5b9a\u3002</p> <p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u5171\u540c\u5b66\u4e60\u5404\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5e76\u57fa\u4e8e\u4e0a\u8ff0\u8bc4\u4ef7\u7ef4\u5ea6\u5bf9\u5404\u4e2a\u6392\u5e8f\u7b97\u6cd5\u7684\u4f18\u7f3a\u70b9\u8fdb\u884c\u5206\u6790\u3002</p>"},{"location":"chapter_sorting/summary/","title":"11.11 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_sorting/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u5192\u6ce1\u6392\u5e8f\u901a\u8fc7\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u901a\u8fc7\u6dfb\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d\u6765\u5b9e\u73b0\u63d0\u524d\u8fd4\u56de\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(n)\\) \u3002</li> <li>\u63d2\u5165\u6392\u5e8f\u6bcf\u8f6e\u5c06\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u5143\u7d20\u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u6b63\u786e\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b8c\u6210\u6392\u5e8f\u3002\u867d\u7136\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u7531\u4e8e\u5355\u5143\u64cd\u4f5c\u76f8\u5bf9\u8f83\u5c11\uff0c\u56e0\u6b64\u5728\u5c0f\u6570\u636e\u91cf\u7684\u6392\u5e8f\u4efb\u52a1\u4e2d\u975e\u5e38\u53d7\u6b22\u8fce\u3002</li> <li>\u5feb\u901f\u6392\u5e8f\u57fa\u4e8e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u54e8\u5175\u5212\u5206\u4e2d\uff0c\u6709\u53ef\u80fd\u6bcf\u6b21\u90fd\u9009\u53d6\u5230\u6700\u5dee\u7684\u57fa\u51c6\u6570\uff0c\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u3002\u5f15\u5165\u4e2d\u4f4d\u6570\u57fa\u51c6\u6570\u6216\u968f\u673a\u57fa\u51c6\u6570\u53ef\u4ee5\u964d\u4f4e\u8fd9\u79cd\u52a3\u5316\u7684\u6982\u7387\u3002\u5c3e\u9012\u5f52\u65b9\u6cd5\u53ef\u4ee5\u6709\u6548\u5730\u51cf\u5c11\u9012\u5f52\u6df1\u5ea6\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(\\log n)\\) \u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\u5305\u62ec\u5212\u5206\u548c\u5408\u5e76\u4e24\u4e2a\u9636\u6bb5\uff0c\u5178\u578b\u5730\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\u3002\u5728\u5f52\u5e76\u6392\u5e8f\u4e2d\uff0c\u6392\u5e8f\u6570\u7ec4\u9700\u8981\u521b\u5efa\u8f85\u52a9\u6570\u7ec4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b\u7136\u800c\u6392\u5e8f\u94fe\u8868\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \u3002</li> <li>\u6876\u6392\u5e8f\u5305\u542b\u4e09\u4e2a\u6b65\u9aa4\uff1a\u6570\u636e\u5206\u6876\u3001\u6876\u5185\u6392\u5e8f\u548c\u5408\u5e76\u7ed3\u679c\u3002\u5b83\u540c\u6837\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u9002\u7528\u4e8e\u6570\u636e\u4f53\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u3002\u6876\u6392\u5e8f\u7684\u5173\u952e\u5728\u4e8e\u5bf9\u6570\u636e\u8fdb\u884c\u5e73\u5747\u5206\u914d\u3002</li> <li>\u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5b83\u901a\u8fc7\u7edf\u8ba1\u6570\u636e\u51fa\u73b0\u7684\u6b21\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u6709\u9650\u7684\u60c5\u51b5\uff0c\u5e76\u4e14\u8981\u6c42\u6570\u636e\u80fd\u591f\u8f6c\u6362\u4e3a\u6b63\u6574\u6570\u3002</li> <li>\u57fa\u6570\u6392\u5e8f\u901a\u8fc7\u9010\u4f4d\u6392\u5e8f\u6765\u5b9e\u73b0\u6570\u636e\u6392\u5e8f\uff0c\u8981\u6c42\u6570\u636e\u80fd\u591f\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u6570\u5b57\u3002</li> <li>\u603b\u7684\u6765\u8bf4\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5177\u6709\u9ad8\u6548\u7387\u3001\u7a33\u5b9a\u3001\u539f\u5730\u4ee5\u53ca\u6b63\u5411\u81ea\u9002\u5e94\u6027\u7b49\u4f18\u70b9\u3002\u7136\u800c\uff0c\u6b63\u5982\u5176\u4ed6\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e00\u6837\uff0c\u6ca1\u6709\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\u80fd\u591f\u540c\u65f6\u6ee1\u8db3\u6240\u6709\u8fd9\u4e9b\u6761\u4ef6\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u6839\u636e\u6570\u636e\u7684\u7279\u6027\u6765\u9009\u62e9\u5408\u9002\u7684\u6392\u5e8f\u7b97\u6cd5\u3002</li> <li>\u56fe 11-19 \u5bf9\u6bd4\u4e86\u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u6548\u7387\u3001\u7a33\u5b9a\u6027\u3001\u5c31\u5730\u6027\u548c\u81ea\u9002\u5e94\u6027\u7b49\u3002</li> </ul> <p> \u56fe 11-19 \u00a0 \u6392\u5e8f\u7b97\u6cd5\u5bf9\u6bd4 </p>"},{"location":"chapter_sorting/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u6392\u5e8f\u7b97\u6cd5\u7a33\u5b9a\u6027\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u662f\u5fc5\u9700\u7684\uff1f</p> <p>\u5728\u73b0\u5b9e\u4e2d\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u57fa\u4e8e\u5bf9\u8c61\u7684\u67d0\u4e2a\u5c5e\u6027\u8fdb\u884c\u6392\u5e8f\u3002\u4f8b\u5982\uff0c\u5b66\u751f\u6709\u59d3\u540d\u548c\u8eab\u9ad8\u4e24\u4e2a\u5c5e\u6027\uff0c\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u4e00\u4e2a\u591a\u7ea7\u6392\u5e8f\uff1a\u5148\u6309\u7167\u59d3\u540d\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u5230 <code>(A, 180) (B, 185) (C, 170) (D, 170)</code> \uff1b\u518d\u5bf9\u8eab\u9ad8\u8fdb\u884c\u6392\u5e8f\u3002\u7531\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e0d\u7a33\u5b9a\uff0c\u56e0\u6b64\u53ef\u80fd\u5f97\u5230 <code>(D, 170) (C, 170) (A, 180) (B, 185)</code> \u3002</p> <p>\u53ef\u4ee5\u53d1\u73b0\uff0c\u5b66\u751f D \u548c C \u7684\u4f4d\u7f6e\u53d1\u751f\u4e86\u4ea4\u6362\uff0c\u59d3\u540d\u7684\u6709\u5e8f\u6027\u88ab\u7834\u574f\u4e86\uff0c\u800c\u8fd9\u662f\u6211\u4eec\u4e0d\u5e0c\u671b\u770b\u5230\u7684\u3002</p> <p>Q\uff1a\u54e8\u5175\u5212\u5206\u4e2d\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u4e0e\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u7684\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\u5417\uff1f</p> <p>\u4e0d\u884c\uff0c\u5f53\u6211\u4eec\u4ee5\u6700\u5de6\u7aef\u5143\u7d20\u4e3a\u57fa\u51c6\u6570\u65f6\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u518d\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002\u8fd9\u4e2a\u7ed3\u8bba\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u6211\u4eec\u6765\u5256\u6790\u4e00\u4e0b\u539f\u56e0\u3002</p> <p>\u54e8\u5175\u5212\u5206 <code>partition()</code> \u7684\u6700\u540e\u4e00\u6b65\u662f\u4ea4\u6362 <code>nums[left]</code> \u548c <code>nums[i]</code> \u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u57fa\u51c6\u6570\u5de6\u8fb9\u7684\u5143\u7d20\u90fd <code>&lt;=</code> \u57fa\u51c6\u6570\uff0c\u8fd9\u5c31\u8981\u6c42\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u524d <code>nums[left] &gt;= nums[i]</code> \u5fc5\u987b\u6210\u7acb\u3002\u5047\u8bbe\u6211\u4eec\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\uff0c\u90a3\u4e48\u5982\u679c\u627e\u4e0d\u5230\u6bd4\u57fa\u51c6\u6570\u66f4\u5927\u7684\u5143\u7d20\uff0c\u5219\u4f1a\u5728 <code>i == j</code> \u65f6\u8df3\u51fa\u5faa\u73af\uff0c\u6b64\u65f6\u53ef\u80fd <code>nums[j] == nums[i] &gt; nums[left]</code>\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u6b64\u65f6\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u64cd\u4f5c\u4f1a\u628a\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u66f4\u5927\u7684\u5143\u7d20\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\uff0c\u5bfc\u81f4\u54e8\u5175\u5212\u5206\u5931\u8d25\u3002</p> <p>\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u6570\u7ec4 <code>[0, 0, 0, 0, 1]</code> \uff0c\u5982\u679c\u5148\u201c\u4ece\u5de6\u5411\u53f3\u67e5\u627e\u201d\uff0c\u54e8\u5175\u5212\u5206\u540e\u6570\u7ec4\u4e3a <code>[1, 0, 0, 0, 0]</code> \uff0c\u8fd9\u4e2a\u7ed3\u679c\u662f\u4e0d\u6b63\u786e\u7684\u3002</p> <p>\u518d\u6df1\u5165\u601d\u8003\u4e00\u4e0b\uff0c\u5982\u679c\u6211\u4eec\u9009\u62e9 <code>nums[right]</code> \u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u6b63\u597d\u53cd\u8fc7\u6765\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002</p> <p>Q\uff1a\u5173\u4e8e\u5c3e\u9012\u5f52\u4f18\u5316\uff0c\u4e3a\u4ec0\u4e48\u9009\u77ed\u7684\u6570\u7ec4\u80fd\u4fdd\u8bc1\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff1f</p> <p>\u9012\u5f52\u6df1\u5ea6\u5c31\u662f\u5f53\u524d\u672a\u8fd4\u56de\u7684\u9012\u5f52\u65b9\u6cd5\u7684\u6570\u91cf\u3002\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u6211\u4eec\u5c06\u539f\u6570\u7ec4\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\u3002\u5728\u5c3e\u9012\u5f52\u4f18\u5316\u540e\uff0c\u5411\u4e0b\u9012\u5f52\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u6700\u5927\u4e3a\u539f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a\u3002\u5047\u8bbe\u6700\u5dee\u60c5\u51b5\uff0c\u4e00\u76f4\u4e3a\u4e00\u534a\u957f\u5ea6\uff0c\u90a3\u4e48\u6700\u7ec8\u7684\u9012\u5f52\u6df1\u5ea6\u5c31\u662f \\(\\log n\\) \u3002</p> <p>\u56de\u987e\u539f\u59cb\u7684\u5feb\u901f\u6392\u5e8f\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u4f1a\u8fde\u7eed\u5730\u9012\u5f52\u957f\u5ea6\u8f83\u5927\u7684\u6570\u7ec4\uff0c\u6700\u5dee\u60c5\u51b5\u4e0b\u4e3a \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \u3002\u5c3e\u9012\u5f52\u4f18\u5316\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u51fa\u73b0\u3002</p> <p>Q\uff1a\u5f53\u6570\u7ec4\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u76f8\u7b49\u65f6\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n^2)\\) \u5417\uff1f\u8be5\u5982\u4f55\u5904\u7406\u8fd9\u79cd\u9000\u5316\u60c5\u51b5\uff1f</p> <p>\u662f\u7684\u3002\u5bf9\u4e8e\u8fd9\u79cd\u60c5\u51b5\uff0c\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u54e8\u5175\u5212\u5206\u5c06\u6570\u7ec4\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\uff1a\u5c0f\u4e8e\u3001\u7b49\u4e8e\u3001\u5927\u4e8e\u57fa\u51c6\u6570\u3002\u4ec5\u5411\u4e0b\u9012\u5f52\u5c0f\u4e8e\u548c\u5927\u4e8e\u7684\u4e24\u90e8\u5206\u3002\u5728\u8be5\u65b9\u6cd5\u4e0b\uff0c\u8f93\u5165\u5143\u7d20\u5168\u90e8\u76f8\u7b49\u7684\u6570\u7ec4\uff0c\u4ec5\u4e00\u8f6e\u54e8\u5175\u5212\u5206\u5373\u53ef\u5b8c\u6210\u6392\u5e8f\u3002</p> <p>Q\uff1a\u6876\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u4ec0\u4e48\u662f \\(O(n^2)\\) \uff1f</p> <p>\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u88ab\u5206\u81f3\u540c\u4e00\u4e2a\u6876\u4e2d\u3002\u5982\u679c\u6211\u4eec\u91c7\u7528\u4e00\u4e2a \\(O(n^2)\\) \u7b97\u6cd5\u6765\u6392\u5e8f\u8fd9\u4e9b\u5143\u7d20\uff0c\u5219\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_stack_and_queue/","title":"\u7b2c 5 \u7ae0 \u00a0 \u6808\u4e0e\u961f\u5217","text":"<p>Abstract</p> <p>\u6808\u5982\u540c\u53e0\u732b\u732b\uff0c\u800c\u961f\u5217\u5c31\u50cf\u732b\u732b\u6392\u961f\u3002</p> <p>\u4e24\u8005\u5206\u522b\u4ee3\u8868\u5148\u5165\u540e\u51fa\u548c\u5148\u5165\u5148\u51fa\u7684\u903b\u8f91\u5173\u7cfb\u3002</p>"},{"location":"chapter_stack_and_queue/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>5.1 \u00a0 \u6808</li> <li>5.2 \u00a0 \u961f\u5217</li> <li>5.3 \u00a0 \u53cc\u5411\u961f\u5217</li> <li>5.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_stack_and_queue/deque/","title":"5.3 \u00a0 \u53cc\u5411\u961f\u5217","text":"<p>\u5728\u961f\u5217\u4e2d\uff0c\u6211\u4eec\u4ec5\u80fd\u5220\u9664\u5934\u90e8\u5143\u7d20\u6216\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u3002\u5982\u56fe 5-7 \u6240\u793a\uff0c\u300c\u53cc\u5411\u961f\u5217 double-ended queue\u300d\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u7075\u6d3b\u6027\uff0c\u5141\u8bb8\u5728\u5934\u90e8\u548c\u5c3e\u90e8\u6267\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u6216\u5220\u9664\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 5-7 \u00a0 \u53cc\u5411\u961f\u5217\u7684\u64cd\u4f5c </p>"},{"location":"chapter_stack_and_queue/deque/#531","title":"5.3.1 \u00a0 \u53cc\u5411\u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"<p>\u53cc\u5411\u961f\u5217\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-3 \u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u79f0\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002</p> <p> \u8868 5-3 \u00a0 \u53cc\u5411\u961f\u5217\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 <code>push_first()</code> \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u9996 \\(O(1)\\) <code>push_last()</code> \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) <code>pop_first()</code> \u5220\u9664\u961f\u9996\u5143\u7d20 \\(O(1)\\) <code>pop_last()</code> \u5220\u9664\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) <code>peek_first()</code> \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) <code>peek_last()</code> \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) <p>\u540c\u6837\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u5df2\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig deque.py<pre><code>from collections import deque\n\n# \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217\ndeque: deque[int] = deque()\n\n# \u5143\u7d20\u5165\u961f\ndeque.append(2)      # \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.appendleft(3)  # \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.appendleft(1)\n\n# \u8bbf\u95ee\u5143\u7d20\nfront: int = deque[0]  # \u961f\u9996\u5143\u7d20\nrear: int = deque[-1]  # \u961f\u5c3e\u5143\u7d20\n\n# \u5143\u7d20\u51fa\u961f\npop_front: int = deque.popleft()  # \u961f\u9996\u5143\u7d20\u51fa\u961f\npop_rear: int = deque.pop()       # \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n# \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(deque)\n\n# \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(deque) == 0\n</code></pre> deque.cpp<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\ndeque&lt;int&gt; deque;\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint front = deque.front(); // \u961f\u9996\u5143\u7d20\nint back = deque.back();   // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\ndeque.pop_front();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.pop_back();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = deque.empty();\n</code></pre> deque.java<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nDeque&lt;Integer&gt; deque = new LinkedList&lt;&gt;();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.offerLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.offerFirst(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.peekFirst();  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.peekLast();    // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.pollFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.pollLast();    // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = deque.isEmpty();\n</code></pre> deque.cs<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 C# \u4e2d\uff0c\u5c06\u94fe\u8868 LinkedList \u770b\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nLinkedList&lt;int&gt; deque = new();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.AddLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.AddFirst(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.First.Value;  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.Last.Value;    // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\ndeque.RemoveFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.RemoveLast();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.Count;\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.Count == 0;\n</code></pre> deque_test.go<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u4f7f\u7528\ndeque := list.New()\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.PushBack(2)      // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.PushFront(1)\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nfront := deque.Front() // \u961f\u9996\u5143\u7d20\nrear := deque.Back()   // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\ndeque.Remove(front)    // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.Remove(rear)     // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize := deque.Len()\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := deque.Len() == 0\n</code></pre> deque.swift<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nvar deque: [Int] = []\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.append(2) // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.insert(1, at: 0)\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nlet peekFirst = deque.first! // \u961f\u9996\u5143\u7d20\nlet peekLast = deque.last! // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u4f7f\u7528 Array \u6a21\u62df\u65f6 popFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet popFirst = deque.removeFirst() // \u961f\u9996\u5143\u7d20\u51fa\u961f\nlet popLast = deque.removeLast() // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.count\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = deque.isEmpty\n</code></pre> deque.js<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque = [];\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst = deque[0];\nconst peekLast = deque[deque.length - 1];\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront = deque.shift();\nconst popBack = deque.pop();\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = deque.length;\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty = size === 0;\n</code></pre> deque.ts<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque: number[] = [];\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst: number = deque[0];\nconst peekLast: number = deque[deque.length - 1];\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront: number = deque.shift() as number;\nconst popBack: number = deque.pop() as number;\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size: number = deque.length;\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty: boolean = size === 0;\n</code></pre> deque.dart<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0cQueue \u88ab\u5b9a\u4e49\u4e3a\u53cc\u5411\u961f\u5217\nQueue&lt;int&gt; deque = Queue&lt;int&gt;();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.addLast(2);  // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.addFirst(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.first; // \u961f\u9996\u5143\u7d20\nint peekLast = deque.last;   // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.removeFirst(); // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.removeLast();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.length;\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.isEmpty;\n</code></pre> deque.rs<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2);  // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nif let Some(front) = deque.front() { // \u961f\u9996\u5143\u7d20\n}\nif let Some(rear) = deque.back() {   // \u961f\u5c3e\u5143\u7d20\n}\n\n/* \u5143\u7d20\u51fa\u961f */\nif let Some(pop_front) = deque.pop_front() { // \u961f\u9996\u5143\u7d20\u51fa\u961f\n}\nif let Some(pop_rear) = deque.pop_back() {   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.len();\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = deque.is_empty();\n</code></pre> deque.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u53cc\u5411\u961f\u5217\n</code></pre> deque.kt<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nval deque = LinkedList&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.offerLast(2)  // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.offerLast(5)\ndeque.offerLast(4)\ndeque.offerFirst(3) // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.offerFirst(1)\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nval peekFirst = deque.peekFirst() // \u961f\u9996\u5143\u7d20\nval peekLast = deque.peekLast()   // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\nval popFirst = deque.pollFirst() // \u961f\u9996\u5143\u7d20\u51fa\u961f\nval popLast = deque.pollLast()   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nval size = deque.size\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nval isEmpty = deque.isEmpty()\n</code></pre> deque.rb<pre><code>\n</code></pre> deque.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/deque/#532","title":"5.3.2 \u00a0 \u53cc\u5411\u961f\u5217\u5b9e\u73b0 *","text":"<p>\u53cc\u5411\u961f\u5217\u7684\u5b9e\u73b0\u4e0e\u961f\u5217\u7c7b\u4f3c\uff0c\u53ef\u4ee5\u9009\u62e9\u94fe\u8868\u6216\u6570\u7ec4\u4f5c\u4e3a\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002</p>"},{"location":"chapter_stack_and_queue/deque/#1","title":"1. \u00a0 \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u7684\u5b9e\u73b0","text":"<p>\u56de\u987e\u4e0a\u4e00\u8282\u5185\u5bb9\uff0c\u6211\u4eec\u4f7f\u7528\u666e\u901a\u5355\u5411\u94fe\u8868\u6765\u5b9e\u73b0\u961f\u5217\uff0c\u56e0\u4e3a\u5b83\u53ef\u4ee5\u65b9\u4fbf\u5730\u5220\u9664\u5934\u8282\u70b9\uff08\u5bf9\u5e94\u51fa\u961f\u64cd\u4f5c\uff09\u548c\u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0\u65b0\u8282\u70b9\uff08\u5bf9\u5e94\u5165\u961f\u64cd\u4f5c\uff09\u3002</p> <p>\u5bf9\u4e8e\u53cc\u5411\u961f\u5217\u800c\u8a00\uff0c\u5934\u90e8\u548c\u5c3e\u90e8\u90fd\u53ef\u4ee5\u6267\u884c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u53cc\u5411\u961f\u5217\u9700\u8981\u5b9e\u73b0\u53e6\u4e00\u4e2a\u5bf9\u79f0\u65b9\u5411\u7684\u64cd\u4f5c\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u91c7\u7528\u201c\u53cc\u5411\u94fe\u8868\u201d\u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u5982\u56fe 5-8 \u6240\u793a\uff0c\u6211\u4eec\u5c06\u53cc\u5411\u94fe\u8868\u7684\u5934\u8282\u70b9\u548c\u5c3e\u8282\u70b9\u89c6\u4e3a\u53cc\u5411\u961f\u5217\u7684\u961f\u9996\u548c\u961f\u5c3e\uff0c\u540c\u65f6\u5b9e\u73b0\u5728\u4e24\u7aef\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u7684\u529f\u80fd\u3002</p> LinkedListDequepush_last()push_first()pop_last()pop_first() <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 5-8 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_deque.py<pre><code>class ListNode:\n    \"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\"\"\"\n\n    def __init__(self, val: int):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self.val: int = val\n        self.next: ListNode | None = None  # \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n        self.prev: ListNode | None = None  # \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\nclass LinkedListDeque:\n    \"\"\"\u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._front: ListNode | None = None  # \u5934\u8282\u70b9 front\n        self._rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\n        self._size: int = 0  # \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self.size() == 0\n\n    def push(self, num: int, is_front: bool):\n        \"\"\"\u5165\u961f\u64cd\u4f5c\"\"\"\n        node = ListNode(num)\n        # \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if self.is_empty():\n            self._front = self._rear = node\n        # \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        elif is_front:\n            # \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            self._front.prev = node\n            node.next = self._front\n            self._front = node  # \u66f4\u65b0\u5934\u8282\u70b9\n        # \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else:\n            # \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            self._rear.next = node\n            node.prev = self._rear\n            self._rear = node  # \u66f4\u65b0\u5c3e\u8282\u70b9\n        self._size += 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n\n    def push_first(self, num: int):\n        \"\"\"\u961f\u9996\u5165\u961f\"\"\"\n        self.push(num, True)\n\n    def push_last(self, num: int):\n        \"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\n        self.push(num, False)\n\n    def pop(self, is_front: bool) -&gt; int:\n        \"\"\"\u51fa\u961f\u64cd\u4f5c\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        # \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if is_front:\n            val: int = self._front.val  # \u6682\u5b58\u5934\u8282\u70b9\u503c\n            # \u5220\u9664\u5934\u8282\u70b9\n            fnext: ListNode | None = self._front.next\n            if fnext != None:\n                fnext.prev = None\n                self._front.next = None\n            self._front = fnext  # \u66f4\u65b0\u5934\u8282\u70b9\n        # \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else:\n            val: int = self._rear.val  # \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            # \u5220\u9664\u5c3e\u8282\u70b9\n            rprev: ListNode | None = self._rear.prev\n            if rprev != None:\n                rprev.next = None\n                self._rear.prev = None\n            self._rear = rprev  # \u66f4\u65b0\u5c3e\u8282\u70b9\n        self._size -= 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val\n\n    def pop_first(self) -&gt; int:\n        \"\"\"\u961f\u9996\u51fa\u961f\"\"\"\n        return self.pop(True)\n\n    def pop_last(self) -&gt; int:\n        \"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\n        return self.pop(False)\n\n    def peek_first(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        return self._front.val\n\n    def peek_last(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        return self._rear.val\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n        node = self._front\n        res = [0] * self.size()\n        for i in range(self.size()):\n            res[i] = node.val\n            node = node.next\n        return res\n</code></pre> linkedlist_deque.cpp<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nstruct DoublyListNode {\n    int val;              // \u8282\u70b9\u503c\n    DoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n    DoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\u6307\u9488\n    DoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n    }\n};\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n  private:\n    DoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    int queSize = 0;              // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    LinkedListDeque() : front(nullptr), rear(nullptr) {\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~LinkedListDeque() {\n        // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n        DoublyListNode *pre, *cur = front;\n        while (cur != nullptr) {\n            pre = cur;\n            cur = cur-&gt;next;\n            delete pre;\n        }\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    void push(int num, bool isFront) {\n        DoublyListNode *node = new DoublyListNode(num);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty())\n            front = rear = node;\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front-&gt;prev = node;\n            node-&gt;next = front;\n            front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear-&gt;next = node;\n            node-&gt;prev = rear;\n            rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    int pop(bool isFront) {\n        if (isEmpty())\n            throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n        int val;\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            val = front-&gt;val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            DoublyListNode *fNext = front-&gt;next;\n            if (fNext != nullptr) {\n                fNext-&gt;prev = nullptr;\n                front-&gt;next = nullptr;\n            }\n            delete front;\n            front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        } else {\n            val = rear-&gt;val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            DoublyListNode *rPrev = rear-&gt;prev;\n            if (rPrev != nullptr) {\n                rPrev-&gt;next = nullptr;\n                rear-&gt;prev = nullptr;\n            }\n            delete rear;\n            rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    int popFirst() {\n        return pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    int popLast() {\n        return pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        return front-&gt;val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        return rear-&gt;val;\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    vector&lt;int&gt; toVector() {\n        DoublyListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_deque.java<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    int val; // \u8282\u70b9\u503c\n    ListNode next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    ListNode prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n    ListNode(int val) {\n        this.val = val;\n        prev = next = null;\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private int queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    public LinkedListDeque() {\n        front = rear = null;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    private void push(int num, boolean isFront) {\n        ListNode node = new ListNode(num);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty())\n            front = rear = node;\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front.prev = node;\n            node.next = front;\n            front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear.next = node;\n            node.prev = rear;\n            rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    private int pop(boolean isFront) {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        int val;\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            val = front.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            ListNode fNext = front.next;\n            if (fNext != null) {\n                fNext.prev = null;\n                front.next = null;\n            }\n            front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        } else {\n            val = rear.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            ListNode rPrev = rear.prev;\n            if (rPrev != null) {\n                rPrev.next = null;\n                rear.prev = null;\n            }\n            rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int popFirst() {\n        return pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int popLast() {\n        return pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return rear.val;\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.cs<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode(int val) {\n    public int val = val;       // \u8282\u70b9\u503c\n    public ListNode? next = null; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    public ListNode? prev = null; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    ListNode? front, rear; // \u5934\u8282\u70b9 front, \u5c3e\u8282\u70b9 rear\n    int queSize = 0;      // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    public LinkedListDeque() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    void Push(int num, bool isFront) {\n        ListNode node = new(num);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (IsEmpty()) {\n            front = node;\n            rear = node;\n        }\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front!.prev = node;\n            node.next = front;\n            front = node; // \u66f4\u65b0\u5934\u8282\u70b9                           \n        }\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear!.next = node;\n            node.prev = rear;\n            rear = node;  // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n\n        queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void PushFirst(int num) {\n        Push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void PushLast(int num) {\n        Push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    int? Pop(bool isFront) {\n        if (IsEmpty())\n            throw new Exception();\n        int? val;\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            val = front?.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            ListNode? fNext = front?.next;\n            if (fNext != null) {\n                fNext.prev = null;\n                front!.next = null;\n            }\n            front = fNext;   // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else {\n            val = rear?.val;  // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            ListNode? rPrev = rear?.prev;\n            if (rPrev != null) {\n                rPrev.next = null;\n                rear!.prev = null;\n            }\n            rear = rPrev;    // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n\n        queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int? PopFirst() {\n        return Pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int? PopLast() {\n        return Pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int? PeekFirst() {\n        if (IsEmpty())\n            throw new Exception();\n        return front?.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int? PeekLast() {\n        if (IsEmpty())\n            throw new Exception();\n        return rear?.val;\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int?[] ToArray() {\n        ListNode? node = front;\n        int?[] res = new int?[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node?.val;\n            node = node?.next;\n        }\n\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.go<pre><code>/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype linkedListDeque struct {\n    // \u4f7f\u7528\u5185\u7f6e\u5305 list\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u53cc\u7aef\u961f\u5217 */\nfunc newLinkedListDeque() *linkedListDeque {\n    return &amp;linkedListDeque{\n        data: list.New(),\n    }\n}\n\n/* \u961f\u9996\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushFirst(value any) {\n    s.data.PushFront(value)\n}\n\n/* \u961f\u5c3e\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushLast(value any) {\n    s.data.PushBack(value)\n}\n\n/* \u961f\u9996\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u961f\u5c3e\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListDeque) peekFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (s *linkedListDeque) peekLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListDeque) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListDeque) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListDeque) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_deque.swift<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    var val: Int // \u8282\u70b9\u503c\n    var next: ListNode? // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    weak var prev: ListNode? // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n    init(val: Int) {\n        self.val = val\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private var front: ListNode? // \u5934\u8282\u70b9 front\n    private var rear: ListNode? // \u5c3e\u8282\u70b9 rear\n    private var _size: Int // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    init() {\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    private func push(num: Int, isFront: Bool) {\n        let node = ListNode(val: num)\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if isEmpty() {\n            front = node\n            rear = node\n        }\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if isFront {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front?.prev = node\n            node.next = front\n            front = node // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear?.next = node\n            node.prev = rear\n            rear = node // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        _size += 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    func pushFirst(num: Int) {\n        push(num: num, isFront: true)\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    func pushLast(num: Int) {\n        push(num: num, isFront: false)\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    private func pop(isFront: Bool) -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        let val: Int\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if isFront {\n            val = front!.val // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            let fNext = front?.next\n            if fNext != nil {\n                fNext?.prev = nil\n                front?.next = nil\n            }\n            front = fNext // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else {\n            val = rear!.val // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            let rPrev = rear?.prev\n            if rPrev != nil {\n                rPrev?.next = nil\n                rear?.prev = nil\n            }\n            rear = rPrev // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        _size -= 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    func popFirst() -&gt; Int {\n        pop(isFront: true)\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    func popLast() -&gt; Int {\n        pop(isFront: false)\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return front!.val\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return rear!.val\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.js<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    val; // \u8282\u70b9\u503c\n\n    constructor(val) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    #front; // \u5934\u8282\u70b9 front\n    #rear; // \u5c3e\u8282\u70b9 rear\n    #queSize; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n        this.#queSize = 0;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\n    pushLast(val) {\n        const node = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            this.#rear.next = node;\n            node.prev = this.#rear;\n            this.#rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        this.#queSize++;\n    }\n\n    /* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\n    pushFirst(val) {\n        const node = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            this.#front.prev = node;\n            node.next = this.#front;\n            this.#front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        this.#queSize++;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\n    popLast() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5c3e\u8282\u70b9\n        let temp = this.#rear.prev;\n        if (temp !== null) {\n            temp.next = null;\n            this.#rear.prev = null;\n        }\n        this.#rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        this.#queSize--;\n        return value;\n    }\n\n    /* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\n    popFirst() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5934\u8282\u70b9\n        let temp = this.#front.next;\n        if (temp !== null) {\n            temp.prev = null;\n            this.#front.next = null;\n        }\n        this.#front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\n        this.#queSize--;\n        return value;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast() {\n        return this.#queSize === 0 ? null : this.#rear.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst() {\n        return this.#queSize === 0 ? null : this.#front.val;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u6253\u5370\u53cc\u5411\u961f\u5217 */\n    print() {\n        const arr = [];\n        let temp = this.#front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.ts<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    prev: ListNode; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    next: ListNode; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    val: number; // \u8282\u70b9\u503c\n\n    constructor(val: number) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private front: ListNode; // \u5934\u8282\u70b9 front\n    private rear: ListNode; // \u5c3e\u8282\u70b9 rear\n    private queSize: number; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n        this.queSize = 0;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\n    pushLast(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            this.rear.next = node;\n            node.prev = this.rear;\n            this.rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        this.queSize++;\n    }\n\n    /* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\n    pushFirst(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            this.front.prev = node;\n            node.next = this.front;\n            this.front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        this.queSize++;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\n    popLast(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5c3e\u8282\u70b9\n        let temp: ListNode = this.rear.prev;\n        if (temp !== null) {\n            temp.next = null;\n            this.rear.prev = null;\n        }\n        this.rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        this.queSize--;\n        return value;\n    }\n\n    /* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\n    popFirst(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5934\u8282\u70b9\n        let temp: ListNode = this.front.next;\n        if (temp !== null) {\n            temp.prev = null;\n            this.front.next = null;\n        }\n        this.front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\n        this.queSize--;\n        return value;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast(): number {\n        return this.queSize === 0 ? null : this.rear.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst(): number {\n        return this.queSize === 0 ? null : this.front.val;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u6253\u5370\u53cc\u5411\u961f\u5217 */\n    print(): void {\n        const arr: number[] = [];\n        let temp: ListNode = this.front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.dart<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n  int val; // \u8282\u70b9\u503c\n  ListNode? next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n  ListNode? prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n  ListNode(this.val, {this.next, this.prev});\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u5bf9\u5217 */\nclass LinkedListDeque {\n  late ListNode? _front; // \u5934\u8282\u70b9 _front\n  late ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\n  int _queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n  LinkedListDeque() {\n    this._front = null;\n    this._rear = null;\n  }\n\n  /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u957f\u5ea6 */\n  int size() {\n    return this._queSize;\n  }\n\n  /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return size() == 0;\n  }\n\n  /* \u5165\u961f\u64cd\u4f5c */\n  void push(int _num, bool isFront) {\n    final ListNode node = ListNode(_num);\n    if (isEmpty()) {\n      // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 _front \u548c _rear \u90fd\u6307\u5411 node\n      _front = _rear = node;\n    } else if (isFront) {\n      // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n      // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n      _front!.prev = node;\n      node.next = _front;\n      _front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n    } else {\n      // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n      // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n      _rear!.next = node;\n      node.prev = _rear;\n      _rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n    }\n    _queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n  }\n\n  /* \u961f\u9996\u5165\u961f */\n  void pushFirst(int _num) {\n    push(_num, true);\n  }\n\n  /* \u961f\u5c3e\u5165\u961f */\n  void pushLast(int _num) {\n    push(_num, false);\n  }\n\n  /* \u51fa\u961f\u64cd\u4f5c */\n  int? pop(bool isFront) {\n    // \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\n    if (isEmpty()) {\n      return null;\n    }\n    final int val;\n    if (isFront) {\n      // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n      val = _front!.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n      // \u5220\u9664\u5934\u8282\u70b9\n      ListNode? fNext = _front!.next;\n      if (fNext != null) {\n        fNext.prev = null;\n        _front!.next = null;\n      }\n      _front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n    } else {\n      // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n      val = _rear!.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n      // \u5220\u9664\u5c3e\u8282\u70b9\n      ListNode? rPrev = _rear!.prev;\n      if (rPrev != null) {\n        rPrev.next = null;\n        _rear!.prev = null;\n      }\n      _rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n    }\n    _queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    return val;\n  }\n\n  /* \u961f\u9996\u51fa\u961f */\n  int? popFirst() {\n    return pop(true);\n  }\n\n  /* \u961f\u5c3e\u51fa\u961f */\n  int? popLast() {\n    return pop(false);\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int? peekFirst() {\n    return _front?.val;\n  }\n\n  /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n  int? peekLast() {\n    return _rear?.val;\n  }\n\n  /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; _queSize; i++) {\n      res.add(node!.val);\n      node = node.next;\n    }\n    return res;\n  }\n}\n</code></pre> linkedlist_deque.rs<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\npub struct ListNode&lt;T&gt; {\n    pub val: T,                                 // \u8282\u70b9\u503c\n    pub next: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n    pub prev: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u524d\u9a71\u8282\u70b9\u6307\u9488\n}\n\nimpl&lt;T&gt; ListNode&lt;T&gt; {\n    pub fn new(val: T) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n        Rc::new(RefCell::new(ListNode {\n            val,\n            next: None,\n            prev: None,\n        }))\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListDeque&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5934\u8282\u70b9 front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // \u5c3e\u8282\u70b9 rear\n    que_size: usize,                         // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListDeque&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    pub fn push(&amp;mut self, num: T, is_front: bool) {\n        let node = ListNode::new(num);\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        if is_front {\n            match self.front.take() {\n                // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n                None =&gt; {\n                    self.rear = Some(node.clone());\n                    self.front = Some(node);\n                }\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n                Some(old_front) =&gt; {\n                    old_front.borrow_mut().prev = Some(node.clone());\n                    node.borrow_mut().next = Some(old_front);\n                    self.front = Some(node); // \u66f4\u65b0\u5934\u8282\u70b9\n                }\n            }\n        }\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else {\n            match self.rear.take() {\n                // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n                None =&gt; {\n                    self.front = Some(node.clone());\n                    self.rear = Some(node);\n                }\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n                Some(old_rear) =&gt; {\n                    old_rear.borrow_mut().next = Some(node.clone());\n                    node.borrow_mut().prev = Some(old_rear);\n                    self.rear = Some(node); // \u66f4\u65b0\u5c3e\u8282\u70b9\n                }\n            }\n        }\n        self.que_size += 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pub fn push_first(&amp;mut self, num: T) {\n        self.push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pub fn push_last(&amp;mut self, num: T) {\n        self.push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    pub fn pop(&amp;mut self, is_front: bool) -&gt; Option&lt;T&gt; {\n        // \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de None\n        if self.is_empty() {\n            return None;\n        };\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if is_front {\n            self.front.take().map(|old_front| {\n                match old_front.borrow_mut().next.take() {\n                    Some(new_front) =&gt; {\n                        new_front.borrow_mut().prev.take();\n                        self.front = Some(new_front); // \u66f4\u65b0\u5934\u8282\u70b9\n                    }\n                    None =&gt; {\n                        self.rear.take();\n                    }\n                }\n                self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n                Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n            })\n        }\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else {\n            self.rear.take().map(|old_rear| {\n                match old_rear.borrow_mut().prev.take() {\n                    Some(new_rear) =&gt; {\n                        new_rear.borrow_mut().next.take();\n                        self.rear = Some(new_rear); // \u66f4\u65b0\u5c3e\u8282\u70b9\n                    }\n                    None =&gt; {\n                        self.front.take();\n                    }\n                }\n                self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n                Rc::try_unwrap(old_rear).ok().unwrap().into_inner().val\n            })\n        }\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    pub fn pop_first(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    pub fn pop_last(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    pub fn peek_first(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    pub fn peek_last(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.rear.as_ref()\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.insert(0, node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_deque.c<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\ntypedef struct DoublyListNode {\n    int val;                     // \u8282\u70b9\u503c\n    struct DoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\n    struct DoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\n} DoublyListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nDoublyListNode *newDoublyListNode(int num) {\n    DoublyListNode *new = (DoublyListNode *)malloc(sizeof(DoublyListNode));\n    new-&gt;val = num;\n    new-&gt;next = NULL;\n    new-&gt;prev = NULL;\n    return new;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delDoublyListNode(DoublyListNode *node) {\n    free(node);\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\n    DoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    int queSize;                  // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n} LinkedListDeque;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListDeque *newLinkedListDeque() {\n    LinkedListDeque *deque = (LinkedListDeque *)malloc(sizeof(LinkedListDeque));\n    deque-&gt;front = NULL;\n    deque-&gt;rear = NULL;\n    deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListdeque(LinkedListDeque *deque) {\n    // \u91ca\u653e\u6240\u6709\u8282\u70b9\n    for (int i = 0; i &lt; deque-&gt;queSize &amp;&amp; deque-&gt;front != NULL; i++) {\n        DoublyListNode *tmp = deque-&gt;front;\n        deque-&gt;front = deque-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // \u91ca\u653e deque \u7ed3\u6784\u4f53\n    free(deque);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListDeque *deque) {\n    return (size(deque) == 0);\n}\n\n/* \u5165\u961f */\nvoid push(LinkedListDeque *deque, int num, bool isFront) {\n    DoublyListNode *node = newDoublyListNode(num);\n    // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411node\n    if (empty(deque)) {\n        deque-&gt;front = deque-&gt;rear = node;\n    }\n    // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n    else if (isFront) {\n        // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n        deque-&gt;front-&gt;prev = node;\n        node-&gt;next = deque-&gt;front;\n        deque-&gt;front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n    }\n    // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n    else {\n        // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n        deque-&gt;rear-&gt;next = node;\n        node-&gt;prev = deque-&gt;rear;\n        deque-&gt;rear = node;\n    }\n    deque-&gt;queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(LinkedListDeque *deque, int num) {\n    push(deque, num, true);\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(LinkedListDeque *deque, int num) {\n    push(deque, num, false);\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;front);\n    return deque-&gt;front-&gt;val;\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;rear);\n    return deque-&gt;rear-&gt;val;\n}\n\n/* \u51fa\u961f */\nint pop(LinkedListDeque *deque, bool isFront) {\n    if (empty(deque))\n        return -1;\n    int val;\n    // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n    if (isFront) {\n        val = peekFirst(deque); // \u6682\u5b58\u5934\u8282\u70b9\u503c\n        DoublyListNode *fNext = deque-&gt;front-&gt;next;\n        if (fNext) {\n            fNext-&gt;prev = NULL;\n            deque-&gt;front-&gt;next = NULL;\n            delDoublyListNode(deque-&gt;front);\n        }\n        deque-&gt;front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n    }\n    // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n    else {\n        val = peekLast(deque); // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n        DoublyListNode *rPrev = deque-&gt;rear-&gt;prev;\n        if (rPrev) {\n            rPrev-&gt;next = NULL;\n            deque-&gt;rear-&gt;prev = NULL;\n            delDoublyListNode(deque-&gt;rear);\n        }\n        deque-&gt;rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n    }\n    deque-&gt;queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    return val;\n}\n\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(LinkedListDeque *deque) {\n    return pop(deque, true);\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(LinkedListDeque *deque) {\n    return pop(deque, false);\n}\n\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListDeque(LinkedListDeque *deque) {\n    int *arr = malloc(sizeof(int) * deque-&gt;queSize);\n    // \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\n    int i;\n    DoublyListNode *node;\n    for (i = 0, node = deque-&gt;front; i &lt; deque-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, deque-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_deque.kt<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode(var value: Int) {\n    // \u8282\u70b9\u503c\n    var next: ListNode? = null // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    var prev: ListNode? = null // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private var front: ListNode? = null // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private var rear: ListNode? = null\n    private var queSize = 0 // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    fun push(num: Int, isFront: Boolean) {\n        val node = ListNode(num)\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty()) {\n            rear = node\n            front = rear\n            // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        } else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front?.prev = node\n            node.next = front\n            front = node // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear?.next = node\n            node.prev = rear\n            rear = node // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize++ // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    fun pushFirst(num: Int) {\n        push(num, true)\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    fun pushLast(num: Int) {\n        push(num, false)\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    fun pop(isFront: Boolean): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n\n        val value: Int\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            value = front!!.value // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            val fNext = front!!.next\n            if (fNext != null) {\n                fNext.prev = null\n                front!!.next = null\n            }\n            front = fNext // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        } else {\n            value = rear!!.value // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            val rPrev = rear!!.prev\n            if (rPrev != null) {\n                rPrev.next = null\n                rear!!.prev = null\n            }\n            rear = rPrev // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize-- // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return value\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    fun popFirst(): Int {\n        return pop(true)\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    fun popLast(): Int {\n        return pop(false)\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peekFirst(): Int {\n        if (isEmpty()) {\n            throw IndexOutOfBoundsException()\n\n        }\n        return front!!.value\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return rear!!.value\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!.value\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.rb<pre><code>[class]{ListNode}-[func]{}\n\n[class]{LinkedListDeque}-[func]{}\n</code></pre> linkedlist_deque.zig<pre><code>// \u53cc\u5411\u94fe\u8868\u8282\u70b9\nfn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = undefined,     // \u8282\u70b9\u503c\n        next: ?*Self = null,    // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n        prev: ?*Self = null,    // \u524d\u9a71\u8282\u70b9\u6307\u9488\n\n        // Initialize a list node with specific value\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n            self.prev = null;\n        }\n    };\n}\n\n// \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\nfn LinkedListDeque(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        front: ?*ListNode(T) = null,                    // \u5934\u8282\u70b9 front\n        rear: ?*ListNode(T) = null,                     // \u5c3e\u8282\u70b9 rear\n        que_size: usize = 0,                             // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.front = null;\n            self.rear = null;\n            self.que_size = 0;\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.que_size;\n        }\n\n        // \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u5165\u961f\u64cd\u4f5c\n        pub fn push(self: *Self, num: T, is_front: bool) !void {\n            var node = try self.mem_allocator.create(ListNode(T));\n            node.init(num);\n            // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n            if (self.isEmpty()) {\n                self.front = node;\n                self.rear = node;\n            // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n            } else if (is_front) {\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n                self.front.?.prev = node;\n                node.next = self.front;\n                self.front = node;  // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n            } else {\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n                self.rear.?.next = node;\n                node.prev = self.rear;\n                self.rear = node;   // \u66f4\u65b0\u5c3e\u8282\u70b9\n            }\n            self.que_size += 1;      // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        } \n\n        // \u961f\u9996\u5165\u961f\n        pub fn pushFirst(self: *Self, num: T) !void {\n            try self.push(num, true);\n        } \n\n        // \u961f\u5c3e\u5165\u961f\n        pub fn pushLast(self: *Self, num: T) !void {\n            try self.push(num, false);\n        } \n\n        // \u51fa\u961f\u64cd\u4f5c\n        pub fn pop(self: *Self, is_front: bool) T {\n            if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n            var val: T = undefined;\n            // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n            if (is_front) {\n                val = self.front.?.val;     // \u6682\u5b58\u5934\u8282\u70b9\u503c\n                // \u5220\u9664\u5934\u8282\u70b9\n                var fNext = self.front.?.next;\n                if (fNext != null) {\n                    fNext.?.prev = null;\n                    self.front.?.next = null;\n                }\n                self.front = fNext;         // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n            } else {\n                val = self.rear.?.val;      // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n                // \u5220\u9664\u5c3e\u8282\u70b9\n                var rPrev = self.rear.?.prev;\n                if (rPrev != null) {\n                    rPrev.?.next = null;\n                    self.rear.?.prev = null;\n                }\n                self.rear = rPrev;          // \u66f4\u65b0\u5c3e\u8282\u70b9\n            }\n            self.que_size -= 1;              // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n            return val;\n        } \n\n        // \u961f\u9996\u51fa\u961f\n        pub fn popFirst(self: *Self) T {\n            return self.pop(true);\n        } \n\n        // \u961f\u5c3e\u51fa\u961f\n        pub fn popLast(self: *Self) T {\n            return self.pop(false);\n        } \n\n        // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n        pub fn peekFirst(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n            return self.front.?.val;\n        }  \n\n        // \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\n        pub fn peekLast(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n            return self.rear.?.val;\n        }\n\n        // \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.front;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[i] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"<p>\u5982\u56fe 5-9 \u6240\u793a\uff0c\u4e0e\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7c7b\u4f3c\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4f7f\u7528\u73af\u5f62\u6570\u7ec4\u6765\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u3002</p> ArrayDequepush_last()push_first()pop_last()pop_first() <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 5-9 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u5728\u961f\u5217\u7684\u5b9e\u73b0\u57fa\u7840\u4e0a\uff0c\u4ec5\u9700\u589e\u52a0\u201c\u961f\u9996\u5165\u961f\u201d\u548c\u201c\u961f\u5c3e\u51fa\u961f\u201d\u7684\u65b9\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_deque.py<pre><code>class ArrayDeque:\n    \"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\n\n    def __init__(self, capacity: int):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._nums: list[int] = [0] * capacity\n        self._front: int = 0\n        self._size: int = 0\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self._size == 0\n\n    def index(self, i: int) -&gt; int:\n        \"\"\"\u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15\"\"\"\n        # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        # \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        # \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + self.capacity()) % self.capacity()\n\n    def push_first(self, num: int):\n        \"\"\"\u961f\u9996\u5165\u961f\"\"\"\n        if self._size == self.capacity():\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        # \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        self._front = self.index(self._front - 1)\n        # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        self._nums[self._front] = num\n        self._size += 1\n\n    def push_last(self, num: int):\n        \"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\n        if self._size == self.capacity():\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        # \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        rear = self.index(self._front + self._size)\n        # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop_first(self) -&gt; int:\n        \"\"\"\u961f\u9996\u51fa\u961f\"\"\"\n        num = self.peek_first()\n        # \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        self._front = self.index(self._front + 1)\n        self._size -= 1\n        return num\n\n    def pop_last(self) -&gt; int:\n        \"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\n        num = self.peek_last()\n        self._size -= 1\n        return num\n\n    def peek_first(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        return self._nums[self._front]\n\n    def peek_last(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        # \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        last = self.index(self._front + self._size - 1)\n        return self._nums[last]\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n        # \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        res = []\n        for i in range(self._size):\n            res.append(self._nums[self.index(self._front + i)])\n        return res\n</code></pre> array_deque.cpp<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n  private:\n    vector&lt;int&gt; nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;        // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;      // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    ArrayDeque(int capacity) {\n        nums.resize(capacity);\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    int capacity() {\n        return nums.size();\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    int index(int i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    void pushFirst(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    void pushLast(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        int rear = index(front + queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    int popFirst() {\n        int num = peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        return nums[front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    vector&lt;int&gt; toVector() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        vector&lt;int&gt; res(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n};\n</code></pre> array_deque.java<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    private int[] nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public ArrayDeque(int capacity) {\n        this.nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    private int index(int i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void pushFirst(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void pushLast(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        int rear = index(front + queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int popFirst() {\n        int num = peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int[] toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.cs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public ArrayDeque(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    int Index(int i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + Capacity()) % Capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void PushFirst(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = Index(front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void PushLast(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        int rear = Index(front + queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int PopFirst() {\n        int num = PeekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = Index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int PopLast() {\n        int num = PeekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int PeekFirst() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        return nums[front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int PeekLast() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = Index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int[] ToArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[Index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.go<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype arrayDeque struct {\n    nums        []int // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    queSize     int   // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n    queCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\n    return &amp;arrayDeque{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayDeque) size() int {\n    return q.queSize\n}\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayDeque) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nfunc (q *arrayDeque) index(i int) int {\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n    return (i + q.queCapacity) % q.queCapacity\n}\n\n/* \u961f\u9996\u5165\u961f */\nfunc (q *arrayDeque) pushFirst(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n        return\n    }\n    // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n    q.front = q.index(q.front - 1)\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n    q.nums[q.front] = num\n    q.queSize++\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nfunc (q *arrayDeque) pushLast(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n        return\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    rear := q.index(q.front + q.queSize)\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* \u961f\u9996\u51fa\u961f */\nfunc (q *arrayDeque) popFirst() any {\n    num := q.peekFirst()\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    q.front = q.index(q.front + 1)\n    q.queSize--\n    return num\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nfunc (q *arrayDeque) popLast() any {\n    num := q.peekLast()\n    q.queSize--\n    return num\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayDeque) peekFirst() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (q *arrayDeque) peekLast() any {\n    if q.isEmpty() {\n        return nil\n    }\n    // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    last := q.index(q.front + q.queSize - 1)\n    return q.nums[last]\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayDeque) toSlice() []int {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    res := make([]int, q.queSize)\n    for i, j := 0, q.front; i &lt; q.queSize; i++ {\n        res[i] = q.nums[q.index(j)]\n        j++\n    }\n    return res\n}\n</code></pre> array_deque.swift<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    private var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var _size: Int // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init(capacity: Int) {\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    private func index(i: Int) -&gt; Int {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        (i + capacity()) % capacity()\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    func pushFirst(num: Int) {\n        if size() == capacity() {\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(i: front - 1)\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num\n        _size += 1\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    func pushLast(num: Int) {\n        if size() == capacity() {\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        let rear = index(i: front + size())\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    func popFirst() -&gt; Int {\n        let num = peekFirst()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(i: front + 1)\n        _size -= 1\n        return num\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    func popLast() -&gt; Int {\n        let num = peekLast()\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return nums[front]\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        let last = index(i: front + size() - 1)\n        return nums[last]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    func toArray() -&gt; [Int] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        (front ..&lt; front + size()).map { nums[index(i: $0)] }\n    }\n}\n</code></pre> array_deque.js<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    #nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    #front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    #queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n        this.#front = 0;\n        this.#queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    capacity() {\n        return this.#nums.length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    index(i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pushFirst(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        this.#front = this.index(this.#front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        this.#nums[this.#front] = num;\n        this.#queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pushLast(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        const rear = this.index(this.#front + this.#queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    popFirst() {\n        const num = this.peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        this.#front = this.index(this.#front + 1);\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    popLast() {\n        const num = this.peekLast();\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.#nums[this.#front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        const last = this.index(this.#front + this.#queSize - 1);\n        return this.#nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const res = [];\n        for (let i = 0, j = this.#front; i &lt; this.#queSize; i++, j++) {\n            res[i] = this.#nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.ts<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    private nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private queSize: number; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = 0;\n        this.queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    capacity(): number {\n        return this.nums.length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    index(i: number): number {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pushFirst(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        this.front = this.index(this.front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        this.nums[this.front] = num;\n        this.queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pushLast(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        const rear: number = this.index(this.front + this.queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    popFirst(): number {\n        const num: number = this.peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        this.front = this.index(this.front + 1);\n        this.queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    popLast(): number {\n        const num: number = this.peekLast();\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.nums[this.front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        const last = this.index(this.front + this.queSize - 1);\n        return this.nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    toArray(): number[] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const res: number[] = [];\n        for (let i = 0, j = this.front; i &lt; this.queSize; i++, j++) {\n            res[i] = this.nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.dart<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n  late List&lt;int&gt; _nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n  late int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n  late int _queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  ArrayDeque(int capacity) {\n    this._nums = List.filled(capacity, 0);\n    this._front = this._queSize = 0;\n  }\n\n  /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n  int capacity() {\n    return _nums.length;\n  }\n\n  /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n  int index(int i) {\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n    return (i + capacity()) % capacity();\n  }\n\n  /* \u961f\u9996\u5165\u961f */\n  void pushFirst(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n    }\n    // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 _front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n    _front = index(_front - 1);\n    // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u9996\n    _nums[_front] = _num;\n    _queSize++;\n  }\n\n  /* \u961f\u5c3e\u5165\u961f */\n  void pushLast(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    int rear = index(_front + _queSize);\n    // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* \u961f\u9996\u51fa\u961f */\n  int popFirst() {\n    int _num = peekFirst();\n    // \u961f\u9996\u6307\u9488\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n    _front = index(_front + 1);\n    _queSize--;\n    return _num;\n  }\n\n  /* \u961f\u5c3e\u51fa\u961f */\n  int popLast() {\n    int _num = peekLast();\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int peekFirst() {\n    if (isEmpty()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n    }\n    return _nums[_front];\n  }\n\n  /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n  int peekLast() {\n    if (isEmpty()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n    }\n    // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    int last = index(_front + _queSize - 1);\n    return _nums[last];\n  }\n\n  /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n  List&lt;int&gt; toArray() {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[index(j)];\n    }\n    return res;\n  }\n}\n</code></pre> array_deque.rs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nstruct ArrayDeque {\n    nums: Vec&lt;i32&gt;,  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front: usize,    // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    que_size: usize, // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n}\n\nimpl ArrayDeque {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(capacity: usize) -&gt; Self {\n        Self {\n            nums: vec![0; capacity],\n            front: 0,\n            que_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    pub fn capacity(&amp;self) -&gt; usize {\n        self.nums.len()\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        self.que_size\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    fn index(&amp;self, i: i32) -&gt; usize {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return ((i + self.capacity() as i32) % self.capacity() as i32) as usize;\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pub fn push_first(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        self.front = self.index(self.front as i32 - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        self.nums[self.front] = num;\n        self.que_size += 1;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pub fn push_last(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        let rear = self.index(self.front as i32 + self.que_size as i32);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self.nums[rear] = num;\n        self.que_size += 1;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    fn pop_first(&amp;mut self) -&gt; i32 {\n        let num = self.peek_first();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        self.front = self.index(self.front as i32 + 1);\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    fn pop_last(&amp;mut self) -&gt; i32 {\n        let num = self.peek_last();\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fn peek_first(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        };\n        self.nums[self.front]\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fn peek_last(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        };\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        let last = self.index(self.front as i32 + self.que_size as i32 - 1);\n        self.nums[last]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    fn to_array(&amp;self) -&gt; Vec&lt;i32&gt; {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        let mut res = vec![0; self.que_size];\n        let mut j = self.front;\n        for i in 0..self.que_size {\n            res[i] = self.nums[self.index(j as i32)];\n            j += 1;\n        }\n        res\n    }\n}\n</code></pre> array_deque.c<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\n    int *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n    int queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayDeque;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayDeque *newArrayDeque(int capacity) {\n    ArrayDeque *deque = (ArrayDeque *)malloc(sizeof(ArrayDeque));\n    // \u521d\u59cb\u5316\u6570\u7ec4\n    deque-&gt;queCapacity = capacity;\n    deque-&gt;nums = (int *)malloc(sizeof(int) * deque-&gt;queCapacity);\n    deque-&gt;front = deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayDeque(ArrayDeque *deque) {\n    free(deque-&gt;nums);\n    free(deque);\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayDeque *deque) {\n    return deque-&gt;queCapacity;\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayDeque *deque) {\n    return deque-&gt;queSize == 0;\n}\n\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint dequeIndex(ArrayDeque *deque, int i) {\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n    return ((i + capacity(deque)) % capacity(deque));\n}\n\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\n        return;\n    }\n    // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u56de\u5230\u5c3e\u90e8\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front - 1);\n    // \u5c06 num \u6dfb\u52a0\u5230\u961f\u9996\n    deque-&gt;nums[deque-&gt;front] = num;\n    deque-&gt;queSize++;\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\n        return;\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    int rear = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize);\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    deque-&gt;nums[rear] = num;\n    deque-&gt;queSize++;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(ArrayDeque *deque) {\n    // \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\n    assert(empty(deque) == 0);\n    return deque-&gt;nums[deque-&gt;front];\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(ArrayDeque *deque) {\n    // \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\n    assert(empty(deque) == 0);\n    int last = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize - 1);\n    return deque-&gt;nums[last];\n}\n\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(ArrayDeque *deque) {\n    int num = peekFirst(deque);\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front + 1);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(ArrayDeque *deque) {\n    int num = peekLast(deque);\n    deque-&gt;queSize--;\n    return num;\n}\n</code></pre> array_deque.kt<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque(capacity: Int) {\n    private var nums = IntArray(capacity) // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var queSize = 0 // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    private fun index(i: Int): Int {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity()\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    fun pushFirst(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1)\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num\n        queSize++\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    fun pushLast(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        val rear = index(front + queSize)\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        queSize++\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    fun popFirst(): Int {\n        val num = peekFirst()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(front + 1)\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fun popLast(): Int {\n        val num = peekLast()\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        val last = index(front + queSize - 1)\n        return nums[last]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    fun toArray(): IntArray {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[index(j)]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_deque.rb<pre><code>[class]{ArrayDeque}-[func]{}\n</code></pre> array_deque.zig<pre><code>[class]{ArrayDeque}-[func]{}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#533","title":"5.3.3 \u00a0 \u53cc\u5411\u961f\u5217\u5e94\u7528","text":"<p>\u53cc\u5411\u961f\u5217\u517c\u5177\u6808\u4e0e\u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u5b83\u53ef\u4ee5\u5b9e\u73b0\u8fd9\u4e24\u8005\u7684\u6240\u6709\u5e94\u7528\u573a\u666f\uff0c\u540c\u65f6\u63d0\u4f9b\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\u3002</p> <p>\u6211\u4eec\u77e5\u9053\uff0c\u8f6f\u4ef6\u7684\u201c\u64a4\u9500\u201d\u529f\u80fd\u901a\u5e38\u4f7f\u7528\u6808\u6765\u5b9e\u73b0\uff1a\u7cfb\u7edf\u5c06\u6bcf\u6b21\u66f4\u6539\u64cd\u4f5c <code>push</code> \u5230\u6808\u4e2d\uff0c\u7136\u540e\u901a\u8fc7 <code>pop</code> \u5b9e\u73b0\u64a4\u9500\u3002\u7136\u800c\uff0c\u8003\u8651\u5230\u7cfb\u7edf\u8d44\u6e90\u7684\u9650\u5236\uff0c\u8f6f\u4ef6\u901a\u5e38\u4f1a\u9650\u5236\u64a4\u9500\u7684\u6b65\u6570\uff08\u4f8b\u5982\u4ec5\u5141\u8bb8\u4fdd\u5b58 \\(50\\) \u6b65\uff09\u3002\u5f53\u6808\u7684\u957f\u5ea6\u8d85\u8fc7 \\(50\\) \u65f6\uff0c\u8f6f\u4ef6\u9700\u8981\u5728\u6808\u5e95\uff08\u961f\u9996\uff09\u6267\u884c\u5220\u9664\u64cd\u4f5c\u3002\u4f46\u6808\u65e0\u6cd5\u5b9e\u73b0\u8be5\u529f\u80fd\uff0c\u6b64\u65f6\u5c31\u9700\u8981\u4f7f\u7528\u53cc\u5411\u961f\u5217\u6765\u66ff\u4ee3\u6808\u3002\u8bf7\u6ce8\u610f\uff0c\u201c\u64a4\u9500\u201d\u7684\u6838\u5fc3\u903b\u8f91\u4ecd\u7136\u9075\u5faa\u6808\u7684\u5148\u5165\u540e\u51fa\u539f\u5219\uff0c\u53ea\u662f\u53cc\u5411\u961f\u5217\u80fd\u591f\u66f4\u52a0\u7075\u6d3b\u5730\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u903b\u8f91\u3002</p>"},{"location":"chapter_stack_and_queue/queue/","title":"5.2 \u00a0 \u961f\u5217","text":"<p>\u300c\u961f\u5217 queue\u300d\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u89c4\u5219\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u987e\u540d\u601d\u4e49\uff0c\u961f\u5217\u6a21\u62df\u4e86\u6392\u961f\u73b0\u8c61\uff0c\u5373\u65b0\u6765\u7684\u4eba\u4e0d\u65ad\u52a0\u5165\u961f\u5217\u5c3e\u90e8\uff0c\u800c\u4f4d\u4e8e\u961f\u5217\u5934\u90e8\u7684\u4eba\u9010\u4e2a\u79bb\u5f00\u3002</p> <p>\u5982\u56fe 5-4 \u6240\u793a\uff0c\u6211\u4eec\u5c06\u961f\u5217\u5934\u90e8\u79f0\u4e3a\u201c\u961f\u9996\u201d\uff0c\u5c3e\u90e8\u79f0\u4e3a\u201c\u961f\u5c3e\u201d\uff0c\u5c06\u628a\u5143\u7d20\u52a0\u5165\u961f\u5c3e\u7684\u64cd\u4f5c\u79f0\u4e3a\u201c\u5165\u961f\u201d\uff0c\u5220\u9664\u961f\u9996\u5143\u7d20\u7684\u64cd\u4f5c\u79f0\u4e3a\u201c\u51fa\u961f\u201d\u3002</p> <p></p> <p> \u56fe 5-4 \u00a0 \u961f\u5217\u7684\u5148\u5165\u5148\u51fa\u89c4\u5219 </p>"},{"location":"chapter_stack_and_queue/queue/#521","title":"5.2.1 \u00a0 \u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"<p>\u961f\u5217\u7684\u5e38\u89c1\u64cd\u4f5c\u5982\u8868 5-2 \u6240\u793a\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u65b9\u6cd5\u540d\u79f0\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002\u6211\u4eec\u5728\u6b64\u91c7\u7528\u4e0e\u6808\u76f8\u540c\u7684\u65b9\u6cd5\u547d\u540d\u3002</p> <p> \u8868 5-2 \u00a0 \u961f\u5217\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 <code>push()</code> \u5143\u7d20\u5165\u961f\uff0c\u5373\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) <code>pop()</code> \u961f\u9996\u5143\u7d20\u51fa\u961f \\(O(1)\\) <code>peek()</code> \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) <p>\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u73b0\u6210\u7684\u961f\u5217\u7c7b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig queue.py<pre><code>from collections import deque\n\n# \u521d\u59cb\u5316\u961f\u5217\n# \u5728 Python \u4e2d\uff0c\u6211\u4eec\u4e00\u822c\u5c06\u53cc\u5411\u961f\u5217\u7c7b deque \u5f53\u4f5c\u961f\u5217\u4f7f\u7528\n# \u867d\u7136 queue.Queue() \u662f\u7eaf\u6b63\u7684\u961f\u5217\u7c7b\uff0c\u4f46\u4e0d\u592a\u597d\u7528\uff0c\u56e0\u6b64\u4e0d\u63a8\u8350\nque: deque[int] = deque()\n\n# \u5143\u7d20\u5165\u961f\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n\n# \u8bbf\u95ee\u961f\u9996\u5143\u7d20\nfront: int = que[0]\n\n# \u5143\u7d20\u51fa\u961f\npop: int = que.popleft()\n\n# \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(que)\n\n# \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(que) == 0\n</code></pre> queue.cpp<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\nqueue&lt;int&gt; queue;\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint front = queue.front();\n\n/* \u5143\u7d20\u51fa\u961f */\nqueue.pop();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = queue.empty();\n</code></pre> queue.java<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\nQueue&lt;Integer&gt; queue = new LinkedList&lt;&gt;();\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.peek();\n\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.poll();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = queue.isEmpty();\n</code></pre> queue.cs<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\nQueue&lt;int&gt; queue = new();\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.Peek();\n\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.Dequeue();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.Count;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.Count == 0;\n</code></pre> queue_test.go<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u961f\u5217\u6765\u4f7f\u7528\nqueue := list.New()\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek := queue.Front()\n\n/* \u5143\u7d20\u51fa\u961f */\npop := queue.Front()\nqueue.Remove(pop)\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nsize := queue.Len()\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := queue.Len() == 0\n</code></pre> queue.swift<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nvar queue: [Int] = []\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nlet peek = queue.first!\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u7531\u4e8e\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 removeFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet pool = queue.removeFirst()\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = queue.count\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = queue.isEmpty\n</code></pre> queue.js<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nconst queue = [];\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n</code></pre> queue.ts<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528 \nconst queue: number[] = [];\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n</code></pre> queue.dart<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0c\u961f\u5217\u7c7b Qeque \u662f\u53cc\u5411\u961f\u5217\uff0c\u4e5f\u53ef\u4f5c\u4e3a\u961f\u5217\u4f7f\u7528\nQueue&lt;int&gt; queue = Queue();\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.first;\n\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.removeFirst();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.length;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.isEmpty;\n</code></pre> queue.rs<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Rust \u4e2d\u4f7f\u7528\u53cc\u5411\u961f\u5217\u4f5c\u4e3a\u666e\u901a\u961f\u5217\u6765\u4f7f\u7528\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(1);\ndeque.push_back(3);\ndeque.push_back(2);\ndeque.push_back(5);\ndeque.push_back(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nif let Some(front) = deque.front() {\n}\n\n/* \u5143\u7d20\u51fa\u961f */\nif let Some(pop) = deque.pop_front() {\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.len();\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = deque.is_empty();\n</code></pre> queue.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u961f\u5217\n</code></pre> queue.kt<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\nval queue = LinkedList&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.offer(1)\nqueue.offer(3)\nqueue.offer(2)\nqueue.offer(5)\nqueue.offer(4)\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nval peek = queue.peek()\n\n/* \u5143\u7d20\u51fa\u961f */\nval pop = queue.poll()\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nval size = queue.size\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nval isEmpty = queue.isEmpty()\n</code></pre> queue.rb<pre><code>\n</code></pre> queue.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/queue/#522","title":"5.2.2 \u00a0 \u961f\u5217\u5b9e\u73b0","text":"<p>\u4e3a\u4e86\u5b9e\u73b0\u961f\u5217\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u4ee5\u5728\u4e00\u7aef\u6dfb\u52a0\u5143\u7d20\uff0c\u5e76\u5728\u53e6\u4e00\u7aef\u5220\u9664\u5143\u7d20\uff0c\u94fe\u8868\u548c\u6570\u7ec4\u90fd\u7b26\u5408\u8981\u6c42\u3002</p>"},{"location":"chapter_stack_and_queue/queue/#1","title":"1. \u00a0 \u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"<p>\u5982\u56fe 5-5 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u201c\u5934\u8282\u70b9\u201d\u548c\u201c\u5c3e\u8282\u70b9\u201d\u5206\u522b\u89c6\u4e3a\u201c\u961f\u9996\u201d\u548c\u201c\u961f\u5c3e\u201d\uff0c\u89c4\u5b9a\u961f\u5c3e\u4ec5\u53ef\u6dfb\u52a0\u8282\u70b9\uff0c\u961f\u9996\u4ec5\u53ef\u5220\u9664\u8282\u70b9\u3002</p> LinkedListQueuepush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-5 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u4ee5\u4e0b\u662f\u7528\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_queue.py<pre><code>class LinkedListQueue:\n    \"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._front: ListNode | None = None  # \u5934\u8282\u70b9 front\n        self._rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return not self._front\n\n    def push(self, num: int):\n        \"\"\"\u5165\u961f\"\"\"\n        # \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        node = ListNode(num)\n        # \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if self._front is None:\n            self._front = node\n            self._rear = node\n        # \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        else:\n            self._rear.next = node\n            self._rear = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u961f\"\"\"\n        num = self.peek()\n        # \u5220\u9664\u5934\u8282\u70b9\n        self._front = self._front.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\n        return self._front.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        queue = []\n        temp = self._front\n        while temp:\n            queue.append(temp.val)\n            temp = temp.next\n        return queue\n</code></pre> linkedlist_queue.cpp<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n  private:\n    ListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    int queSize;\n\n  public:\n    LinkedListQueue() {\n        front = nullptr;\n        rear = nullptr;\n        queSize = 0;\n    }\n\n    ~LinkedListQueue() {\n        // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n        freeMemoryLinkedList(front);\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u961f */\n    void push(int num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        ListNode *node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == nullptr) {\n            front = node;\n            rear = node;\n        }\n        // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        else {\n            rear-&gt;next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    int pop() {\n        int num = peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        ListNode *tmp = front;\n        front = front-&gt;next;\n        // \u91ca\u653e\u5185\u5b58\n        delete tmp;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peek() {\n        if (size() == 0)\n            throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n        return front-&gt;val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_queue.java<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    private ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f */\n    public void push(int num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        ListNode node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == null) {\n            front = node;\n            rear = node;\n        // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int pop() {\n        int num = peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front.next;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.cs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    ListNode? front, rear;  // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear \n    int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u961f */\n    public void Push(int num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        ListNode node = new(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == null) {\n            front = node;\n            rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else if (rear != null) {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int Pop() {\n        int num = Peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front?.next;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return front!.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] ToArray() {\n        if (front == null)\n            return [];\n\n        ListNode? node = front;\n        int[] res = new int[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.go<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntype linkedListQueue struct {\n    // \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u961f\u5217\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newLinkedListQueue() *linkedListQueue {\n    return &amp;linkedListQueue{\n        data: list.New(),\n    }\n}\n\n/* \u5165\u961f */\nfunc (s *linkedListQueue) push(value any) {\n    s.data.PushBack(value)\n}\n\n/* \u51fa\u961f */\nfunc (s *linkedListQueue) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListQueue) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListQueue) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListQueue) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListQueue) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_queue.swift<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    private var front: ListNode? // \u5934\u8282\u70b9\n    private var rear: ListNode? // \u5c3e\u8282\u70b9\n    private var _size: Int\n\n    init() {\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u961f */\n    func push(num: Int) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        let node = ListNode(x: num)\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if front == nil {\n            front = node\n            rear = node\n        }\n        // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        else {\n            rear?.next = node\n            rear = node\n        }\n        _size += 1\n    }\n\n    /* \u51fa\u961f */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front?.next\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return front!.val\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.js<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    #front; // \u5934\u8282\u70b9 #front\n    #rear; // \u5c3e\u8282\u70b9 #rear\n    #queSize = 0;\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        const node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (!this.#front) {\n            this.#front = node;\n            this.#rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            this.#rear.next = node;\n            this.#rear = node;\n        }\n        this.#queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop() {\n        const num = this.peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        this.#front = this.#front.next;\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek() {\n        if (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.#front.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray() {\n        let node = this.#front;\n        const res = new Array(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.ts<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    private front: ListNode | null; // \u5934\u8282\u70b9 front\n    private rear: ListNode | null; // \u5c3e\u8282\u70b9 rear\n    private queSize: number = 0;\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num: number): void {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        const node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (!this.front) {\n            this.front = node;\n            this.rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            this.rear!.next = node;\n            this.rear = node;\n        }\n        this.queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop(): number {\n        const num = this.peek();\n        if (!this.front) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        // \u5220\u9664\u5934\u8282\u70b9\n        this.front = this.front.next;\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek(): number {\n        if (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.front!.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray(): number[] {\n        let node = this.front;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.dart<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n  ListNode? _front; // \u5934\u8282\u70b9 _front\n  ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\n  int _queSize = 0; // \u961f\u5217\u957f\u5ea6\n\n  LinkedListQueue() {\n    _front = null;\n    _rear = null;\n  }\n\n  /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u5165\u961f */\n  void push(int _num) {\n    // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 _num\n    final node = ListNode(_num);\n    // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n    if (_front == null) {\n      _front = node;\n      _rear = node;\n    } else {\n      // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n      _rear!.next = node;\n      _rear = node;\n    }\n    _queSize++;\n  }\n\n  /* \u51fa\u961f */\n  int pop() {\n    final int _num = peek();\n    // \u5220\u9664\u5934\u8282\u70b9\n    _front = _front!.next;\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int peek() {\n    if (_queSize == 0) {\n      throw Exception('\u961f\u5217\u4e3a\u7a7a');\n    }\n    return _front!.val;\n  }\n\n  /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; queue = [];\n    while (node != null) {\n      queue.add(node.val);\n      node = node.next;\n    }\n    return queue;\n  }\n}\n</code></pre> linkedlist_queue.rs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListQueue&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5934\u8282\u70b9 front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // \u5c3e\u8282\u70b9 rear\n    que_size: usize,                         // \u961f\u5217\u7684\u957f\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListQueue&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u961f */\n    pub fn push(&amp;mut self, num: T) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        let new_rear = ListNode::new(num);\n        match self.rear.take() {\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n            Some(old_rear) =&gt; {\n                old_rear.borrow_mut().next = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n            // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n            None =&gt; {\n                self.front = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n        }\n        self.que_size += 1;\n    }\n\n    /* \u51fa\u961f */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.front.take().map(|old_front| {\n            match old_front.borrow_mut().next.take() {\n                Some(new_front) =&gt; {\n                    self.front = Some(new_front);\n                }\n                None =&gt; {\n                    self.rear.take();\n                }\n            }\n            self.que_size -= 1;\n            Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n        })\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.insert(0, node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_queue.c<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\n    ListNode *front, *rear;\n    int queSize;\n} LinkedListQueue;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListQueue *newLinkedListQueue() {\n    LinkedListQueue *queue = (LinkedListQueue *)malloc(sizeof(LinkedListQueue));\n    queue-&gt;front = NULL;\n    queue-&gt;rear = NULL;\n    queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListQueue(LinkedListQueue *queue) {\n    // \u91ca\u653e\u6240\u6709\u8282\u70b9\n    while (queue-&gt;front != NULL) {\n        ListNode *tmp = queue-&gt;front;\n        queue-&gt;front = queue-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // \u91ca\u653e queue \u7ed3\u6784\u4f53\n    free(queue);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListQueue *queue) {\n    return (size(queue) == 0);\n}\n\n/* \u5165\u961f */\nvoid push(LinkedListQueue *queue, int num) {\n    // \u5c3e\u8282\u70b9\u5904\u6dfb\u52a0 node\n    ListNode *node = newListNode(num);\n    // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n    if (queue-&gt;front == NULL) {\n        queue-&gt;front = node;\n        queue-&gt;rear = node;\n    }\n    // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n    else {\n        queue-&gt;rear-&gt;next = node;\n        queue-&gt;rear = node;\n    }\n    queue-&gt;queSize++;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(LinkedListQueue *queue) {\n    assert(size(queue) &amp;&amp; queue-&gt;front);\n    return queue-&gt;front-&gt;val;\n}\n\n/* \u51fa\u961f */\nint pop(LinkedListQueue *queue) {\n    int num = peek(queue);\n    ListNode *tmp = queue-&gt;front;\n    queue-&gt;front = queue-&gt;front-&gt;next;\n    free(tmp);\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListQueue(LinkedListQueue *queue) {\n    int *arr = malloc(sizeof(int) * queue-&gt;queSize);\n    // \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\n    int i;\n    ListNode *node;\n    for (i = 0, node = queue-&gt;front; i &lt; queue-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, queue-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_queue.kt<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue(\n    // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private var front: ListNode? = null,\n    private var rear: ListNode? = null,\n    private var queSize: Int = 0\n) {\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u961f */\n    fun push(num: Int) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        val node = ListNode(num)\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == null) {\n            front = node\n            rear = node\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            rear?.next = node\n            rear = node\n        }\n        queSize++\n    }\n\n    /* \u51fa\u961f */\n    fun pop(): Int {\n        val num = peek()\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front?.next\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!.value\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!.value\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.rb<pre><code>[class]{LinkedListQueue}-[func]{}\n</code></pre> linkedlist_queue.zig<pre><code>// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\nfn LinkedListQueue(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        front: ?*inc.ListNode(T) = null,                // \u5934\u8282\u70b9 front\n        rear: ?*inc.ListNode(T) = null,                 // \u5c3e\u8282\u70b9 rear\n        que_size: usize = 0,                            // \u961f\u5217\u7684\u957f\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.front = null;\n            self.rear = null;\n            self.que_size = 0;\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.que_size;\n        }\n\n        // \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.size() == 0) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\n            return self.front.?.val;\n        }  \n\n        // \u5165\u961f\n        pub fn push(self: *Self, num: T) !void {\n            // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n            var node = try self.mem_allocator.create(inc.ListNode(T));\n            node.init(num);\n            // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n            if (self.front == null) {\n                self.front = node;\n                self.rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n            } else {\n                self.rear.?.next = node;\n                self.rear = node;\n            }\n            self.que_size += 1;\n        } \n\n        // \u51fa\u961f\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            // \u5220\u9664\u5934\u8282\u70b9\n            self.front = self.front.?.next;\n            self.que_size -= 1;\n            return num;\n        } \n\n        // \u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.front;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[i] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/queue/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"<p>\u5728\u6570\u7ec4\u4e2d\u5220\u9664\u9996\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4f1a\u5bfc\u81f4\u51fa\u961f\u64cd\u4f5c\u6548\u7387\u8f83\u4f4e\u3002\u7136\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u5de7\u5999\u65b9\u6cd5\u6765\u907f\u514d\u8fd9\u4e2a\u95ee\u9898\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u53d8\u91cf <code>front</code> \u6307\u5411\u961f\u9996\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u5e76\u7ef4\u62a4\u4e00\u4e2a\u53d8\u91cf <code>size</code> \u7528\u4e8e\u8bb0\u5f55\u961f\u5217\u957f\u5ea6\u3002\u5b9a\u4e49 <code>rear = front + size</code> \uff0c\u8fd9\u4e2a\u516c\u5f0f\u8ba1\u7b97\u51fa\u7684 <code>rear</code> \u6307\u5411\u961f\u5c3e\u5143\u7d20\u4e4b\u540e\u7684\u4e0b\u4e00\u4e2a\u4f4d\u7f6e\u3002</p> <p>\u57fa\u4e8e\u6b64\u8bbe\u8ba1\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u5143\u7d20\u7684\u6709\u6548\u533a\u95f4\u4e3a <code>[front, rear - 1]</code>\uff0c\u5404\u79cd\u64cd\u4f5c\u7684\u5b9e\u73b0\u65b9\u6cd5\u5982\u56fe 5-6 \u6240\u793a\u3002</p> <ul> <li>\u5165\u961f\u64cd\u4f5c\uff1a\u5c06\u8f93\u5165\u5143\u7d20\u8d4b\u503c\u7ed9 <code>rear</code> \u7d22\u5f15\u5904\uff0c\u5e76\u5c06 <code>size</code> \u589e\u52a0 1 \u3002</li> <li>\u51fa\u961f\u64cd\u4f5c\uff1a\u53ea\u9700\u5c06 <code>front</code> \u589e\u52a0 1 \uff0c\u5e76\u5c06 <code>size</code> \u51cf\u5c11 1 \u3002</li> </ul> <p>\u53ef\u4ee5\u770b\u5230\uff0c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u90fd\u53ea\u9700\u8fdb\u884c\u4e00\u6b21\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002</p> ArrayQueuepush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-6 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\u4e00\u4e2a\u95ee\u9898\uff1a\u5728\u4e0d\u65ad\u8fdb\u884c\u5165\u961f\u548c\u51fa\u961f\u7684\u8fc7\u7a0b\u4e2d\uff0c<code>front</code> \u548c <code>rear</code> \u90fd\u5728\u5411\u53f3\u79fb\u52a8\uff0c\u5f53\u5b83\u4eec\u5230\u8fbe\u6570\u7ec4\u5c3e\u90e8\u65f6\u5c31\u65e0\u6cd5\u7ee7\u7eed\u79fb\u52a8\u4e86\u3002\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u89c6\u4e3a\u9996\u5c3e\u76f8\u63a5\u7684\u201c\u73af\u5f62\u6570\u7ec4\u201d\u3002</p> <p>\u5bf9\u4e8e\u73af\u5f62\u6570\u7ec4\uff0c\u6211\u4eec\u9700\u8981\u8ba9 <code>front</code> \u6216 <code>rear</code> \u5728\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u76f4\u63a5\u56de\u5230\u6570\u7ec4\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002\u8fd9\u79cd\u5468\u671f\u6027\u89c4\u5f8b\u53ef\u4ee5\u901a\u8fc7\u201c\u53d6\u4f59\u64cd\u4f5c\u201d\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_queue.py<pre><code>class ArrayQueue:\n    \"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\n\n    def __init__(self, size: int):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._nums: list[int] = [0] * size  # \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n        self._front: int = 0  # \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n        self._size: int = 0  # \u961f\u5217\u957f\u5ea6\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"\u5165\u961f\"\"\"\n        if self._size == self.capacity():\n            raise IndexError(\"\u961f\u5217\u5df2\u6ee1\")\n        # \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        rear: int = (self._front + self._size) % self.capacity()\n        # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u961f\"\"\"\n        num: int = self.peek()\n        # \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        self._front = (self._front + 1) % self.capacity()\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\n        return self._nums[self._front]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        res = [0] * self.size()\n        j: int = self._front\n        for i in range(self.size()):\n            res[i] = self._nums[(j % self.capacity())]\n            j += 1\n        return res\n</code></pre> array_queue.cpp<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n  private:\n    int *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;     // \u961f\u5217\u957f\u5ea6\n    int queCapacity; // \u961f\u5217\u5bb9\u91cf\n\n  public:\n    ArrayQueue(int capacity) {\n        // \u521d\u59cb\u5316\u6570\u7ec4\n        nums = new int[capacity];\n        queCapacity = capacity;\n        front = queSize = 0;\n    }\n\n    ~ArrayQueue() {\n        delete[] nums;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    int capacity() {\n        return queCapacity;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f */\n    void push(int num) {\n        if (queSize == queCapacity) {\n            cout &lt;&lt; \"\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        int rear = (front + queSize) % queCapacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    int pop() {\n        int num = peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % queCapacity;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peek() {\n        if (isEmpty())\n            throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n        return nums[front];\n    }\n\n    /* \u5c06\u6570\u7ec4\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        vector&lt;int&gt; arr(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            arr[i] = nums[j % queCapacity];\n        }\n        return arr;\n    }\n};\n</code></pre> array_queue.java<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    private int[] nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private int queSize; // \u961f\u5217\u957f\u5ea6\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u961f */\n    public void push(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        int rear = (front + queSize) % capacity();\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int pop() {\n        int num = peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % capacity();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    public int[] toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.cs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize; // \u961f\u5217\u957f\u5ea6\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u961f */\n    public void Push(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        int rear = (front + queSize) % Capacity();\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int Pop() {\n        int num = Peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % Capacity();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return nums[front];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    public int[] ToArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % this.Capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.go<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntype arrayQueue struct {\n    nums        []int // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    queSize     int   // \u961f\u5217\u957f\u5ea6\n    queCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\n    return &amp;arrayQueue{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayQueue) size() int {\n    return q.queSize\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayQueue) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* \u5165\u961f */\nfunc (q *arrayQueue) push(num int) {\n    // \u5f53 rear == queCapacity \u8868\u793a\u961f\u5217\u5df2\u6ee1\n    if q.queSize == q.queCapacity {\n        return\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n    rear := (q.front + q.queSize) % q.queCapacity\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* \u51fa\u961f */\nfunc (q *arrayQueue) pop() any {\n    num := q.peek()\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n    q.front = (q.front + 1) % q.queCapacity\n    q.queSize--\n    return num\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayQueue) peek() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayQueue) toSlice() []int {\n    rear := (q.front + q.queSize)\n    if rear &gt;= q.queCapacity {\n        rear %= q.queCapacity\n        return append(q.nums[q.front:], q.nums[:rear]...)\n    }\n    return q.nums[q.front:rear]\n}\n</code></pre> array_queue.swift<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    private var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var _size: Int // \u961f\u5217\u957f\u5ea6\n\n    init(capacity: Int) {\n        // \u521d\u59cb\u5316\u6570\u7ec4\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u961f */\n    func push(num: Int) {\n        if size() == capacity() {\n            print(\"\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        let rear = (front + size()) % capacity()\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* \u51fa\u961f */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % capacity()\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return nums[front]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    func toArray() -&gt; [Int] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        (front ..&lt; front + size()).map { nums[$0 % capacity()] }\n    }\n}\n</code></pre> array_queue.js<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    #nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    #front = 0; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    #queSize = 0; // \u961f\u5217\u957f\u5ea6\n\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    get capacity() {\n        return this.#nums.length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num) {\n        if (this.size === this.capacity) {\n            console.log('\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        const rear = (this.#front + this.size) % this.capacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop() {\n        const num = this.peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        this.#front = (this.#front + 1) % this.capacity;\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek() {\n        if (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.#nums[this.#front];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.#front; i &lt; this.size; i++, j++) {\n            arr[i] = this.#nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.ts<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    private nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private queSize: number; // \u961f\u5217\u957f\u5ea6\n\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = this.queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    get capacity(): number {\n        return this.nums.length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num: number): void {\n        if (this.size === this.capacity) {\n            console.log('\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        const rear = (this.front + this.queSize) % this.capacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop(): number {\n        const num = this.peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        this.front = (this.front + 1) % this.capacity;\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek(): number {\n        if (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.nums[this.front];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray(): number[] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.front; i &lt; this.size; i++, j++) {\n            arr[i] = this.nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.dart<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n  late List&lt;int&gt; _nums; // \u7528\u4e8e\u50a8\u5b58\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n  late int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n  late int _queSize; // \u961f\u5217\u957f\u5ea6\n\n  ArrayQueue(int capacity) {\n    _nums = List.filled(capacity, 0);\n    _front = _queSize = 0;\n  }\n\n  /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n  int capaCity() {\n    return _nums.length;\n  }\n\n  /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u5165\u961f */\n  void push(int _num) {\n    if (_queSize == capaCity()) {\n      throw Exception(\"\u961f\u5217\u5df2\u6ee1\");\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n    int rear = (_front + _queSize) % capaCity();\n    // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* \u51fa\u961f */\n  int pop() {\n    int _num = peek();\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n    _front = (_front + 1) % capaCity();\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"\u961f\u5217\u4e3a\u7a7a\");\n    }\n    return _nums[_front];\n  }\n\n  /* \u8fd4\u56de Array */\n  List&lt;int&gt; toArray() {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    final List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[j % capaCity()];\n    }\n    return res;\n  }\n}\n</code></pre> array_queue.rs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nstruct ArrayQueue {\n    nums: Vec&lt;i32&gt;,    // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front: i32,        // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    que_size: i32,     // \u961f\u5217\u957f\u5ea6\n    que_capacity: i32, // \u961f\u5217\u5bb9\u91cf\n}\n\nimpl ArrayQueue {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(capacity: i32) -&gt; ArrayQueue {\n        ArrayQueue {\n            nums: vec![0; capacity as usize],\n            front: 0,\n            que_size: 0,\n            que_capacity: capacity,\n        }\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    fn capacity(&amp;self) -&gt; i32 {\n        self.que_capacity\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    fn size(&amp;self) -&gt; i32 {\n        self.que_size\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* \u5165\u961f */\n    fn push(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        let rear = (self.front + self.que_size) % self.que_capacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self.nums[rear as usize] = num;\n        self.que_size += 1;\n    }\n\n    /* \u51fa\u961f */\n    fn pop(&amp;mut self) -&gt; i32 {\n        let num = self.peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        self.front = (self.front + 1) % self.que_capacity;\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fn peek(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"index out of bounds\");\n        }\n        self.nums[self.front as usize]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    fn to_vector(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let cap = self.que_capacity;\n        let mut j = self.front;\n        let mut arr = vec![0; self.que_size as usize];\n        for i in 0..self.que_size {\n            arr[i as usize] = self.nums[(j % cap) as usize];\n            j += 1;\n        }\n        arr\n    }\n}\n</code></pre> array_queue.c<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\n    int *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n    int queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayQueue;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayQueue *newArrayQueue(int capacity) {\n    ArrayQueue *queue = (ArrayQueue *)malloc(sizeof(ArrayQueue));\n    // \u521d\u59cb\u5316\u6570\u7ec4\n    queue-&gt;queCapacity = capacity;\n    queue-&gt;nums = (int *)malloc(sizeof(int) * queue-&gt;queCapacity);\n    queue-&gt;front = queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayQueue(ArrayQueue *queue) {\n    free(queue-&gt;nums);\n    free(queue);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayQueue *queue) {\n    return queue-&gt;queCapacity;\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayQueue *queue) {\n    return queue-&gt;queSize == 0;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(ArrayQueue *queue) {\n    assert(size(queue) != 0);\n    return queue-&gt;nums[queue-&gt;front];\n}\n\n/* \u5165\u961f */\nvoid push(ArrayQueue *queue, int num) {\n    if (size(queue) == capacity(queue)) {\n        printf(\"\u961f\u5217\u5df2\u6ee1\\r\\n\");\n        return;\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n    int rear = (queue-&gt;front + queue-&gt;queSize) % queue-&gt;queCapacity;\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    queue-&gt;nums[rear] = num;\n    queue-&gt;queSize++;\n}\n\n/* \u51fa\u961f */\nint pop(ArrayQueue *queue) {\n    int num = peek(queue);\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n    queue-&gt;front = (queue-&gt;front + 1) % queue-&gt;queCapacity;\n    queue-&gt;queSize--;\n    return num;\n}\n</code></pre> array_queue.kt<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue(capacity: Int) {\n    private val nums = IntArray(capacity) // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var queSize = 0 // \u961f\u5217\u957f\u5ea6\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* \u5165\u961f */\n    fun push(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        val rear = (front + queSize) % capacity()\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        queSize++\n    }\n\n    /* \u51fa\u961f */\n    fun pop(): Int {\n        val num = peek()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % capacity()\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    fun toArray(): IntArray {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[j % capacity()]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_queue.rb<pre><code>[class]{ArrayQueue}-[func]{}\n</code></pre> array_queue.zig<pre><code>// \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\nfn ArrayQueue(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        nums: []T = undefined,                          // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4     \n        cap: usize = 0,                                 // \u961f\u5217\u5bb9\u91cf\n        front: usize = 0,                               // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n        queSize: usize = 0,                             // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6570\u7ec4\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator, cap: usize) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.cap = cap;\n            self.nums = try self.mem_allocator.alloc(T, self.cap);\n            @memset(self.nums, @as(T, 0));\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\n        pub fn capacity(self: *Self) usize {\n            return self.cap;\n        }\n\n        // \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.queSize;\n        }\n\n        // \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.queSize == 0;\n        }\n\n        // \u5165\u961f\n        pub fn push(self: *Self, num: T) !void {\n            if (self.size() == self.capacity()) {\n                std.debug.print(\"\u961f\u5217\u5df2\u6ee1\\n\", .{});\n                return;\n            }\n            // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n            // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n            var rear = (self.front + self.queSize) % self.capacity();\n            // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n            self.nums[rear] = num;\n            self.queSize += 1;\n        } \n\n        // \u51fa\u961f\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n            self.front = (self.front + 1) % self.capacity();\n            self.queSize -= 1;\n            return num;\n        } \n\n        // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\n            return self.nums[self.front];\n        } \n\n        // \u8fd4\u56de\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            var j: usize = self.front;\n            while (i &lt; self.size()) : ({ i += 1; j += 1; }) {\n                res[i] = self.nums[j % self.capacity()];\n            }\n            return res;\n        }\n    };\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u4ee5\u4e0a\u5b9e\u73b0\u7684\u961f\u5217\u4ecd\u7136\u5177\u6709\u5c40\u9650\u6027\uff1a\u5176\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u7136\u800c\uff0c\u8fd9\u4e2a\u95ee\u9898\u4e0d\u96be\u89e3\u51b3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u66ff\u6362\u4e3a\u52a8\u6001\u6570\u7ec4\uff0c\u4ece\u800c\u5f15\u5165\u6269\u5bb9\u673a\u5236\u3002\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u5c1d\u8bd5\u81ea\u884c\u5b9e\u73b0\u3002</p> <p>\u4e24\u79cd\u5b9e\u73b0\u7684\u5bf9\u6bd4\u7ed3\u8bba\u4e0e\u6808\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002</p>"},{"location":"chapter_stack_and_queue/queue/#523","title":"5.2.3 \u00a0 \u961f\u5217\u5178\u578b\u5e94\u7528","text":"<ul> <li>\u6dd8\u5b9d\u8ba2\u5355\u3002\u8d2d\u7269\u8005\u4e0b\u5355\u540e\uff0c\u8ba2\u5355\u5c06\u52a0\u5165\u961f\u5217\u4e2d\uff0c\u7cfb\u7edf\u968f\u540e\u4f1a\u6839\u636e\u987a\u5e8f\u5904\u7406\u961f\u5217\u4e2d\u7684\u8ba2\u5355\u3002\u5728\u53cc\u5341\u4e00\u671f\u95f4\uff0c\u77ed\u65f6\u95f4\u5185\u4f1a\u4ea7\u751f\u6d77\u91cf\u8ba2\u5355\uff0c\u9ad8\u5e76\u53d1\u6210\u4e3a\u5de5\u7a0b\u5e08\u4eec\u9700\u8981\u91cd\u70b9\u653b\u514b\u7684\u95ee\u9898\u3002</li> <li>\u5404\u7c7b\u5f85\u529e\u4e8b\u9879\u3002\u4efb\u4f55\u9700\u8981\u5b9e\u73b0\u201c\u5148\u6765\u540e\u5230\u201d\u529f\u80fd\u7684\u573a\u666f\uff0c\u4f8b\u5982\u6253\u5370\u673a\u7684\u4efb\u52a1\u961f\u5217\u3001\u9910\u5385\u7684\u51fa\u9910\u961f\u5217\u7b49\uff0c\u961f\u5217\u5728\u8fd9\u4e9b\u573a\u666f\u4e2d\u53ef\u4ee5\u6709\u6548\u5730\u7ef4\u62a4\u5904\u7406\u987a\u5e8f\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/stack/","title":"5.1 \u00a0 \u6808","text":"<p>\u300c\u6808 stack\u300d\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u903b\u8f91\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u5c06\u6808\u7c7b\u6bd4\u4e3a\u684c\u9762\u4e0a\u7684\u4e00\u645e\u76d8\u5b50\uff0c\u5982\u679c\u60f3\u53d6\u51fa\u5e95\u90e8\u7684\u76d8\u5b50\uff0c\u5219\u9700\u8981\u5148\u5c06\u4e0a\u9762\u7684\u76d8\u5b50\u4f9d\u6b21\u79fb\u8d70\u3002\u6211\u4eec\u5c06\u76d8\u5b50\u66ff\u6362\u4e3a\u5404\u79cd\u7c7b\u578b\u7684\u5143\u7d20\uff08\u5982\u6574\u6570\u3001\u5b57\u7b26\u3001\u5bf9\u8c61\u7b49\uff09\uff0c\u5c31\u5f97\u5230\u4e86\u6808\u8fd9\u79cd\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u5982\u56fe 5-1 \u6240\u793a\uff0c\u6211\u4eec\u628a\u5806\u53e0\u5143\u7d20\u7684\u9876\u90e8\u79f0\u4e3a\u201c\u6808\u9876\u201d\uff0c\u5e95\u90e8\u79f0\u4e3a\u201c\u6808\u5e95\u201d\u3002\u5c06\u628a\u5143\u7d20\u6dfb\u52a0\u5230\u6808\u9876\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u5165\u6808\u201d\uff0c\u5220\u9664\u6808\u9876\u5143\u7d20\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u51fa\u6808\u201d\u3002</p> <p></p> <p> \u56fe 5-1 \u00a0 \u6808\u7684\u5148\u5165\u540e\u51fa\u89c4\u5219 </p>"},{"location":"chapter_stack_and_queue/stack/#511","title":"5.1.1 \u00a0 \u6808\u7684\u5e38\u7528\u64cd\u4f5c","text":"<p>\u6808\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-1 \u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002\u5728\u6b64\uff0c\u6211\u4eec\u4ee5\u5e38\u89c1\u7684 <code>push()</code>\u3001<code>pop()</code>\u3001<code>peek()</code> \u547d\u540d\u4e3a\u4f8b\u3002</p> <p> \u8868 5-1 \u00a0 \u6808\u7684\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5 \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 <code>push()</code> \u5143\u7d20\u5165\u6808\uff08\u6dfb\u52a0\u81f3\u6808\u9876\uff09 \\(O(1)\\) <code>pop()</code> \u6808\u9876\u5143\u7d20\u51fa\u6808 \\(O(1)\\) <code>peek()</code> \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \\(O(1)\\) <p>\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u6808\u7c7b\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u8bed\u8a00\u53ef\u80fd\u6ca1\u6709\u4e13\u95e8\u63d0\u4f9b\u6808\u7c7b\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u5c06\u8be5\u8bed\u8a00\u7684\u201c\u6570\u7ec4\u201d\u6216\u201c\u94fe\u8868\u201d\u5f53\u4f5c\u6808\u6765\u4f7f\u7528\uff0c\u5e76\u5728\u7a0b\u5e8f\u903b\u8f91\u4e0a\u5ffd\u7565\u4e0e\u6808\u65e0\u5173\u7684\u64cd\u4f5c\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig stack.py<pre><code># \u521d\u59cb\u5316\u6808\n# Python \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a list \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nstack: list[int] = []\n\n# \u5143\u7d20\u5165\u6808\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n# \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npeek: int = stack[-1]\n\n# \u5143\u7d20\u51fa\u6808\npop: int = stack.pop()\n\n# \u83b7\u53d6\u6808\u7684\u957f\u5ea6\nsize: int = len(stack)\n\n# \u5224\u65ad\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(stack) == 0\n</code></pre> stack.cpp<pre><code>/* \u521d\u59cb\u5316\u6808 */\nstack&lt;int&gt; stack;\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top = stack.top();\n\n/* \u5143\u7d20\u51fa\u6808 */\nstack.pop(); // \u65e0\u8fd4\u56de\u503c\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool empty = stack.empty();\n</code></pre> stack.java<pre><code>/* \u521d\u59cb\u5316\u6808 */\nStack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.peek();\n\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = stack.isEmpty();\n</code></pre> stack.cs<pre><code>/* \u521d\u59cb\u5316\u6808 */\nStack&lt;int&gt; stack = new();\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.Peek();\n\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.Pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.Count;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.Count == 0;\n</code></pre> stack_test.go<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// \u5728 Go \u4e2d\uff0c\u63a8\u8350\u5c06 Slice \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack []int\n\n/* \u5143\u7d20\u5165\u6808 */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek := stack[len(stack)-1]\n\n/* \u5143\u7d20\u51fa\u6808 */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nsize := len(stack)\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nisEmpty := len(stack) == 0\n</code></pre> stack.swift<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack: [Int] = []\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet peek = stack.last!\n\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.removeLast()\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.count\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = stack.isEmpty\n</code></pre> stack.js<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack = [];\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length-1];\n\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n</code></pre> stack.ts<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack: number[] = [];\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length - 1];\n\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n</code></pre> stack.dart<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// Dart \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nList&lt;int&gt; stack = [];\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.last;\n\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.removeLast();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.length;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.isEmpty;\n</code></pre> stack.rs<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// \u628a Vec \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nlet mut stack: Vec&lt;i32&gt; = Vec::new();\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet top = stack.last().unwrap();\n\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.pop().unwrap();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.len();\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = stack.is_empty();\n</code></pre> stack.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u6808\n</code></pre> stack.kt<pre><code>/* \u521d\u59cb\u5316\u6808 */\nval stack = Stack&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1)\nstack.push(3)\nstack.push(2)\nstack.push(5)\nstack.push(4)\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nval peek = stack.peek()\n\n/* \u5143\u7d20\u51fa\u6808 */\nval pop = stack.pop()\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nval size = stack.size\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nval isEmpty = stack.isEmpty()\n</code></pre> stack.rb<pre><code>\n</code></pre> stack.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#512","title":"5.1.2 \u00a0 \u6808\u7684\u5b9e\u73b0","text":"<p>\u4e3a\u4e86\u6df1\u5165\u4e86\u89e3\u6808\u7684\u8fd0\u884c\u673a\u5236\uff0c\u6211\u4eec\u6765\u5c1d\u8bd5\u81ea\u5df1\u5b9e\u73b0\u4e00\u4e2a\u6808\u7c7b\u3002</p> <p>\u6808\u9075\u5faa\u5148\u5165\u540e\u51fa\u7684\u539f\u5219\uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u80fd\u5728\u6808\u9876\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\u3002\u7136\u800c\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u90fd\u53ef\u4ee5\u5728\u4efb\u610f\u4f4d\u7f6e\u6dfb\u52a0\u548c\u5220\u9664\u5143\u7d20\uff0c\u56e0\u6b64\u6808\u53ef\u4ee5\u89c6\u4e3a\u4e00\u79cd\u53d7\u9650\u5236\u7684\u6570\u7ec4\u6216\u94fe\u8868\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u201c\u5c4f\u853d\u201d\u6570\u7ec4\u6216\u94fe\u8868\u7684\u90e8\u5206\u65e0\u5173\u64cd\u4f5c\uff0c\u4f7f\u5176\u5bf9\u5916\u8868\u73b0\u7684\u903b\u8f91\u7b26\u5408\u6808\u7684\u7279\u6027\u3002</p>"},{"location":"chapter_stack_and_queue/stack/#1","title":"1. \u00a0 \u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"<p>\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u5934\u8282\u70b9\u89c6\u4e3a\u6808\u9876\uff0c\u5c3e\u8282\u70b9\u89c6\u4e3a\u6808\u5e95\u3002</p> <p>\u5982\u56fe 5-2 \u6240\u793a\uff0c\u5bf9\u4e8e\u5165\u6808\u64cd\u4f5c\uff0c\u6211\u4eec\u53ea\u9700\u5c06\u5143\u7d20\u63d2\u5165\u94fe\u8868\u5934\u90e8\uff0c\u8fd9\u79cd\u8282\u70b9\u63d2\u5165\u65b9\u6cd5\u88ab\u79f0\u4e3a\u201c\u5934\u63d2\u6cd5\u201d\u3002\u800c\u5bf9\u4e8e\u51fa\u6808\u64cd\u4f5c\uff0c\u53ea\u9700\u5c06\u5934\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u5373\u53ef\u3002</p> LinkedListStackpush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-2 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u5165\u6808\u51fa\u6808\u64cd\u4f5c </p> <p>\u4ee5\u4e0b\u662f\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u793a\u4f8b\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_stack.py<pre><code>class LinkedListStack:\n    \"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._peek: ListNode | None = None\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return not self._peek\n\n    def push(self, val: int):\n        \"\"\"\u5165\u6808\"\"\"\n        node = ListNode(val)\n        node.next = self._peek\n        self._peek = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u6808\"\"\"\n        num = self.peek()\n        self._peek = self._peek.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u6808\u4e3a\u7a7a\")\n        return self._peek.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        arr = []\n        node = self._peek\n        while node:\n            arr.append(node.val)\n            node = node.next\n        arr.reverse()\n        return arr\n</code></pre> linkedlist_stack.cpp<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n  private:\n    ListNode *stackTop; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    int stkSize;        // \u6808\u7684\u957f\u5ea6\n\n  public:\n    LinkedListStack() {\n        stackTop = nullptr;\n        stkSize = 0;\n    }\n\n    ~LinkedListStack() {\n        // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n        freeMemoryLinkedList(stackTop);\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    int size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    void push(int num) {\n        ListNode *node = new ListNode(num);\n        node-&gt;next = stackTop;\n        stackTop = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    int pop() {\n        int num = top();\n        ListNode *tmp = stackTop;\n        stackTop = stackTop-&gt;next;\n        // \u91ca\u653e\u5185\u5b58\n        delete tmp;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"\u6808\u4e3a\u7a7a\");\n        return stackTop-&gt;val;\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = stackTop;\n        vector&lt;int&gt; res(size());\n        for (int i = res.size() - 1; i &gt;= 0; i--) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_stack.java<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    private ListNode stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private int stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void push(int num) {\n        ListNode node = new ListNode(num);\n        node.next = stackPeek;\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    public int pop() {\n        int num = peek();\n        stackPeek = stackPeek.next;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stackPeek.val;\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] toArray() {\n        ListNode node = stackPeek;\n        int[] res = new int[size()];\n        for (int i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.cs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    ListNode? stackPeek;  // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    int stkSize = 0;   // \u6808\u7684\u957f\u5ea6\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int Size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void Push(int num) {\n        ListNode node = new(num) {\n            next = stackPeek\n        };\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    public int Pop() {\n        int num = Peek();\n        stackPeek = stackPeek!.next;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stackPeek!.val;\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] ToArray() {\n        if (stackPeek == null)\n            return [];\n\n        ListNode? node = stackPeek;\n        int[] res = new int[Size()];\n        for (int i = res.Length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.go<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntype linkedListStack struct {\n    // \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u6808\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u6808 */\nfunc newLinkedListStack() *linkedListStack {\n    return &amp;linkedListStack{\n        data: list.New(),\n    }\n}\n\n/* \u5165\u6808 */\nfunc (s *linkedListStack) push(value int) {\n    s.data.PushBack(value)\n}\n\n/* \u51fa\u6808 */\nfunc (s *linkedListStack) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc (s *linkedListStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc (s *linkedListStack) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListStack) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListStack) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_stack.swift<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    private var _peek: ListNode? // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private var _size: Int // \u6808\u7684\u957f\u5ea6\n\n    init() {\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u6808 */\n    func push(num: Int) {\n        let node = ListNode(x: num)\n        node.next = _peek\n        _peek = node\n        _size += 1\n    }\n\n    /* \u51fa\u6808 */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        _peek = _peek?.next\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u6808\u4e3a\u7a7a\")\n        }\n        return _peek!.val\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        var node = _peek\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices.reversed() {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.js<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    #stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    #stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n    constructor() {\n        this.#stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num) {\n        const node = new ListNode(num);\n        node.next = this.#stackPeek;\n        this.#stackPeek = node;\n        this.#stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    pop() {\n        const num = this.peek();\n        this.#stackPeek = this.#stackPeek.next;\n        this.#stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    peek() {\n        if (!this.#stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.#stackPeek.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray() {\n        let node = this.#stackPeek;\n        const res = new Array(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.ts<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    private stackPeek: ListNode | null; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private stkSize: number = 0; // \u6808\u7684\u957f\u5ea6\n\n    constructor() {\n        this.stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num: number): void {\n        const node = new ListNode(num);\n        node.next = this.stackPeek;\n        this.stackPeek = node;\n        this.stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    pop(): number {\n        const num = this.peek();\n        if (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n        this.stackPeek = this.stackPeek.next;\n        this.stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    peek(): number {\n        if (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.stackPeek.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray(): number[] {\n        let node = this.stackPeek;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.dart<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u7c7b\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n  ListNode? _stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n  int _stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n  LinkedListStack() {\n    _stackPeek = null;\n  }\n\n  /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n  int size() {\n    return _stkSize;\n  }\n\n  /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _stkSize == 0;\n  }\n\n  /* \u5165\u6808 */\n  void push(int _num) {\n    final ListNode node = ListNode(_num);\n    node.next = _stackPeek;\n    _stackPeek = node;\n    _stkSize++;\n  }\n\n  /* \u51fa\u6808 */\n  int pop() {\n    final int _num = peek();\n    _stackPeek = _stackPeek!.next;\n    _stkSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n  int peek() {\n    if (_stackPeek == null) {\n      throw Exception(\"\u6808\u4e3a\u7a7a\");\n    }\n    return _stackPeek!.val;\n  }\n\n  /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a List \u5e76\u8fd4\u56de */\n  List&lt;int&gt; toList() {\n    ListNode? node = _stackPeek;\n    List&lt;int&gt; list = [];\n    while (node != null) {\n      list.add(node.val);\n      node = node.next;\n    }\n    list = list.reversed.toList();\n    return list;\n  }\n}\n</code></pre> linkedlist_stack.rs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\n#[allow(dead_code)]\npub struct LinkedListStack&lt;T&gt; {\n    stack_peek: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    stk_size: usize,                              // \u6808\u7684\u957f\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListStack&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            stack_peek: None,\n            stk_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.stk_size;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    pub fn push(&amp;mut self, num: T) {\n        let node = ListNode::new(num);\n        node.borrow_mut().next = self.stack_peek.take();\n        self.stack_peek = Some(node);\n        self.stk_size += 1;\n    }\n\n    /* \u51fa\u6808 */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack_peek.take().map(|old_head| {\n            match old_head.borrow_mut().next.take() {\n                Some(new_head) =&gt; {\n                    self.stack_peek = Some(new_head);\n                }\n                None =&gt; {\n                    self.stack_peek = None;\n                }\n            }\n            self.stk_size -= 1;\n            Rc::try_unwrap(old_head).ok().unwrap().into_inner().val\n        })\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.stack_peek.as_ref()\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.push(node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_stack.c<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\n    ListNode *top; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    int size;      // \u6808\u7684\u957f\u5ea6\n} LinkedListStack;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListStack *newLinkedListStack() {\n    LinkedListStack *s = malloc(sizeof(LinkedListStack));\n    s-&gt;top = NULL;\n    s-&gt;size = 0;\n    return s;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListStack(LinkedListStack *s) {\n    while (s-&gt;top) {\n        ListNode *n = s-&gt;top-&gt;next;\n        free(s-&gt;top);\n        s-&gt;top = n;\n    }\n    free(s);\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(LinkedListStack *s) {\n    return s-&gt;size;\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(LinkedListStack *s) {\n    return size(s) == 0;\n}\n\n/* \u5165\u6808 */\nvoid push(LinkedListStack *s, int num) {\n    ListNode *node = (ListNode *)malloc(sizeof(ListNode));\n    node-&gt;next = s-&gt;top; // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6307\u9488\u57df\n    node-&gt;val = num;     // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6570\u636e\u57df\n    s-&gt;top = node;       // \u66f4\u65b0\u6808\u9876\n    s-&gt;size++;           // \u66f4\u65b0\u6808\u5927\u5c0f\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(LinkedListStack *s) {\n    if (s-&gt;size == 0) {\n        printf(\"\u6808\u4e3a\u7a7a\\n\");\n        return INT_MAX;\n    }\n    return s-&gt;top-&gt;val;\n}\n\n/* \u51fa\u6808 */\nint pop(LinkedListStack *s) {\n    int val = peek(s);\n    ListNode *tmp = s-&gt;top;\n    s-&gt;top = s-&gt;top-&gt;next;\n    // \u91ca\u653e\u5185\u5b58\n    free(tmp);\n    s-&gt;size--;\n    return val;\n}\n</code></pre> linkedlist_stack.kt<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack(\n    private var stackPeek: ListNode? = null, // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private var stkSize: Int = 0 // \u6808\u7684\u957f\u5ea6\n) {\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return stkSize\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u6808 */\n    fun push(num: Int) {\n        val node = ListNode(num)\n        node.next = stackPeek\n        stackPeek = node\n        stkSize++\n    }\n\n    /* \u51fa\u6808 */\n    fun pop(): Int? {\n        val num = peek()\n        stackPeek = stackPeek?.next\n        stkSize--;\n        return num\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    fun peek(): Int? {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stackPeek?.value\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    fun toArray(): IntArray {\n        var node = stackPeek\n        val res = IntArray(size())\n        for (i in res.size - 1 downTo 0) {\n            res[i] = node?.value!!\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.rb<pre><code>[class]{LinkedListStack}-[func]{}\n</code></pre> linkedlist_stack.zig<pre><code>// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\nfn LinkedListStack(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        stack_top: ?*inc.ListNode(T) = null,             // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n        stk_size: usize = 0,                             // \u6808\u7684\u957f\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,    // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.stack_top = null;\n            self.stk_size = 0;\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u6808\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.stk_size;\n        }\n\n        // \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8bbf\u95ee\u6808\u9876\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.size() == 0) @panic(\"\u6808\u4e3a\u7a7a\");\n            return self.stack_top.?.val;\n        }  \n\n        // \u5165\u6808\n        pub fn push(self: *Self, num: T) !void {\n            var node = try self.mem_allocator.create(inc.ListNode(T));\n            node.init(num);\n            node.next = self.stack_top;\n            self.stack_top = node;\n            self.stk_size += 1;\n        } \n\n        // \u51fa\u6808\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            self.stack_top = self.stack_top.?.next;\n            self.stk_size -= 1;\n            return num;\n        } \n\n        // \u5c06\u6808\u8f6c\u6362\u4e3a\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.stack_top;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[res.len - i - 1] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"<p>\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u7684\u5c3e\u90e8\u4f5c\u4e3a\u6808\u9876\u3002\u5982\u56fe 5-3 \u6240\u793a\uff0c\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u5206\u522b\u5bf9\u5e94\u5728\u6570\u7ec4\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u4e0e\u5220\u9664\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(1)\\) \u3002</p> ArrayStackpush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-3 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u6808\u7684\u5165\u6808\u51fa\u6808\u64cd\u4f5c </p> <p>\u7531\u4e8e\u5165\u6808\u7684\u5143\u7d20\u53ef\u80fd\u4f1a\u6e90\u6e90\u4e0d\u65ad\u5730\u589e\u52a0\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u52a8\u6001\u6570\u7ec4\uff0c\u8fd9\u6837\u5c31\u65e0\u987b\u81ea\u884c\u5904\u7406\u6570\u7ec4\u6269\u5bb9\u95ee\u9898\u3002\u4ee5\u4e0b\u4e3a\u793a\u4f8b\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_stack.py<pre><code>class ArrayStack:\n    \"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._stack: list[int] = []\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\n        return len(self._stack)\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self._stack == []\n\n    def push(self, item: int):\n        \"\"\"\u5165\u6808\"\"\"\n        self._stack.append(item)\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u6808\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u6808\u4e3a\u7a7a\")\n        return self._stack.pop()\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u6808\u4e3a\u7a7a\")\n        return self._stack[-1]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        return self._stack\n</code></pre> array_stack.cpp<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n  private:\n    vector&lt;int&gt; stack;\n\n  public:\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    int size() {\n        return stack.size();\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return stack.size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    void push(int num) {\n        stack.push_back(num);\n    }\n\n    /* \u51fa\u6808 */\n    int pop() {\n        int num = top();\n        stack.pop_back();\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"\u6808\u4e3a\u7a7a\");\n        return stack.back();\n    }\n\n    /* \u8fd4\u56de Vector */\n    vector&lt;int&gt; toVector() {\n        return stack;\n    }\n};\n</code></pre> array_stack.java<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    private ArrayList&lt;Integer&gt; stack;\n\n    public ArrayStack() {\n        // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n        stack = new ArrayList&lt;&gt;();\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int size() {\n        return stack.size();\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void push(int num) {\n        stack.add(num);\n    }\n\n    /* \u51fa\u6808 */\n    public int pop() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.remove(size() - 1);\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.get(size() - 1);\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public Object[] toArray() {\n        return stack.toArray();\n    }\n}\n</code></pre> array_stack.cs<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    List&lt;int&gt; stack;\n    public ArrayStack() {\n        // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n        stack = [];\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int Size() {\n        return stack.Count;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void Push(int num) {\n        stack.Add(num);\n    }\n\n    /* \u51fa\u6808 */\n    public int Pop() {\n        if (IsEmpty())\n            throw new Exception();\n        var val = Peek();\n        stack.RemoveAt(Size() - 1);\n        return val;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stack[Size() - 1];\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] ToArray() {\n        return [.. stack];\n    }\n}\n</code></pre> array_stack.go<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntype arrayStack struct {\n    data []int // \u6570\u636e\n}\n\n/* \u521d\u59cb\u5316\u6808 */\nfunc newArrayStack() *arrayStack {\n    return &amp;arrayStack{\n        // \u8bbe\u7f6e\u6808\u7684\u957f\u5ea6\u4e3a 0\uff0c\u5bb9\u91cf\u4e3a 16\n        data: make([]int, 0, 16),\n    }\n}\n\n/* \u6808\u7684\u957f\u5ea6 */\nfunc (s *arrayStack) size() int {\n    return len(s.data)\n}\n\n/* \u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *arrayStack) isEmpty() bool {\n    return s.size() == 0\n}\n\n/* \u5165\u6808 */\nfunc (s *arrayStack) push(v int) {\n    // \u5207\u7247\u4f1a\u81ea\u52a8\u6269\u5bb9\n    s.data = append(s.data, v)\n}\n\n/* \u51fa\u6808 */\nfunc (s *arrayStack) pop() any {\n    val := s.peek()\n    s.data = s.data[:len(s.data)-1]\n    return val\n}\n\n/* \u83b7\u53d6\u6808\u9876\u5143\u7d20 */\nfunc (s *arrayStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    val := s.data[len(s.data)-1]\n    return val\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (s *arrayStack) toSlice() []int {\n    return s.data\n}\n</code></pre> array_stack.swift<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    private var stack: [Int]\n\n    init() {\n        // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n        stack = []\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        stack.count\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        stack.isEmpty\n    }\n\n    /* \u5165\u6808 */\n    func push(num: Int) {\n        stack.append(num)\n    }\n\n    /* \u51fa\u6808 */\n    @discardableResult\n    func pop() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u6808\u4e3a\u7a7a\")\n        }\n        return stack.removeLast()\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u6808\u4e3a\u7a7a\")\n        }\n        return stack.last!\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        stack\n    }\n}\n</code></pre> array_stack.js<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    #stack;\n    constructor() {\n        this.#stack = [];\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#stack.length;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#stack.length === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num) {\n        this.#stack.push(num);\n    }\n\n    /* \u51fa\u6808 */\n    pop() {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.#stack.pop();\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    top() {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.#stack[this.#stack.length - 1];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        return this.#stack;\n    }\n}\n</code></pre> array_stack.ts<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    private stack: number[];\n    constructor() {\n        this.stack = [];\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.stack.length;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.stack.length === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num: number): void {\n        this.stack.push(num);\n    }\n\n    /* \u51fa\u6808 */\n    pop(): number | undefined {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.stack.pop();\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    top(): number | undefined {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.stack[this.stack.length - 1];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        return this.stack;\n    }\n}\n</code></pre> array_stack.dart<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n  late List&lt;int&gt; _stack;\n  ArrayStack() {\n    _stack = [];\n  }\n\n  /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n  int size() {\n    return _stack.length;\n  }\n\n  /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _stack.isEmpty;\n  }\n\n  /* \u5165\u6808 */\n  void push(int _num) {\n    _stack.add(_num);\n  }\n\n  /* \u51fa\u6808 */\n  int pop() {\n    if (isEmpty()) {\n      throw Exception(\"\u6808\u4e3a\u7a7a\");\n    }\n    return _stack.removeLast();\n  }\n\n  /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"\u6808\u4e3a\u7a7a\");\n    }\n    return _stack.last;\n  }\n\n  /* \u5c06\u6808\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n  List&lt;int&gt; toArray() =&gt; _stack;\n}\n</code></pre> array_stack.rs<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nstruct ArrayStack&lt;T&gt; {\n    stack: Vec&lt;T&gt;,\n}\n\nimpl&lt;T&gt; ArrayStack&lt;T&gt; {\n    /* \u521d\u59cb\u5316\u6808 */\n    fn new() -&gt; ArrayStack&lt;T&gt; {\n        ArrayStack::&lt;T&gt; {\n            stack: Vec::&lt;T&gt;::new(),\n        }\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    fn size(&amp;self) -&gt; usize {\n        self.stack.len()\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.size() == 0\n    }\n\n    /* \u5165\u6808 */\n    fn push(&amp;mut self, num: T) {\n        self.stack.push(num);\n    }\n\n    /* \u51fa\u6808 */\n    fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack.pop()\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    fn peek(&amp;self) -&gt; Option&lt;&amp;T&gt; {\n        if self.is_empty() {\n            panic!(\"\u6808\u4e3a\u7a7a\")\n        };\n        self.stack.last()\n    }\n\n    /* \u8fd4\u56de &amp;Vec */\n    fn to_array(&amp;self) -&gt; &amp;Vec&lt;T&gt; {\n        &amp;self.stack\n    }\n}\n</code></pre> array_stack.c<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\n    int *data;\n    int size;\n} ArrayStack;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayStack *newArrayStack() {\n    ArrayStack *stack = malloc(sizeof(ArrayStack));\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5927\u5bb9\u91cf\uff0c\u907f\u514d\u6269\u5bb9\n    stack-&gt;data = malloc(sizeof(int) * MAX_SIZE);\n    stack-&gt;size = 0;\n    return stack;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayStack(ArrayStack *stack) {\n    free(stack-&gt;data);\n    free(stack);\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(ArrayStack *stack) {\n    return stack-&gt;size;\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(ArrayStack *stack) {\n    return stack-&gt;size == 0;\n}\n\n/* \u5165\u6808 */\nvoid push(ArrayStack *stack, int num) {\n    if (stack-&gt;size == MAX_SIZE) {\n        printf(\"\u6808\u5df2\u6ee1\\n\");\n        return;\n    }\n    stack-&gt;data[stack-&gt;size] = num;\n    stack-&gt;size++;\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(ArrayStack *stack) {\n    if (stack-&gt;size == 0) {\n        printf(\"\u6808\u4e3a\u7a7a\\n\");\n        return INT_MAX;\n    }\n    return stack-&gt;data[stack-&gt;size - 1];\n}\n\n/* \u51fa\u6808 */\nint pop(ArrayStack *stack) {\n    int val = peek(stack);\n    stack-&gt;size--;\n    return val;\n}\n</code></pre> array_stack.kt<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n    private val stack = ArrayList&lt;Int&gt;()\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return stack.size\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u6808 */\n    fun push(num: Int) {\n        stack.add(num)\n    }\n\n    /* \u51fa\u6808 */\n    fun pop(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack.removeAt(size() - 1)\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack[size() - 1]\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    fun toArray(): Array&lt;Any&gt; {\n        return stack.toArray()\n    }\n}\n</code></pre> array_stack.rb<pre><code>[class]{ArrayStack}-[func]{}\n</code></pre> array_stack.zig<pre><code>// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\nfn ArrayStack(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        stack: ?std.ArrayList(T) = null,     \n\n        // \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) void {\n            if (self.stack == null) {\n                self.stack = std.ArrayList(T).init(allocator);\n            }\n        }\n\n        // \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.stack == null) return;\n            self.stack.?.deinit();\n        }\n\n        // \u83b7\u53d6\u6808\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.stack.?.items.len;\n        }\n\n        // \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8bbf\u95ee\u6808\u9876\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u6808\u4e3a\u7a7a\");\n            return self.stack.?.items[self.size() - 1];\n        }  \n\n        // \u5165\u6808\n        pub fn push(self: *Self, num: T) !void {\n            try self.stack.?.append(num);\n        } \n\n        // \u51fa\u6808\n        pub fn pop(self: *Self) T {\n            var num = self.stack.?.pop();\n            return num;\n        } \n\n        // \u8fd4\u56de ArrayList\n        pub fn toList(self: *Self) std.ArrayList(T) {\n            return self.stack.?;\n        }\n    };\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#513","title":"5.1.3 \u00a0 \u4e24\u79cd\u5b9e\u73b0\u5bf9\u6bd4","text":"<p>\u652f\u6301\u64cd\u4f5c</p> <p>\u4e24\u79cd\u5b9e\u73b0\u90fd\u652f\u6301\u6808\u5b9a\u4e49\u4e2d\u7684\u5404\u9879\u64cd\u4f5c\u3002\u6570\u7ec4\u5b9e\u73b0\u989d\u5916\u652f\u6301\u968f\u673a\u8bbf\u95ee\uff0c\u4f46\u8fd9\u5df2\u8d85\u51fa\u4e86\u6808\u7684\u5b9a\u4e49\u8303\u7574\uff0c\u56e0\u6b64\u4e00\u822c\u4e0d\u4f1a\u7528\u5230\u3002</p> <p>\u65f6\u95f4\u6548\u7387</p> <p>\u5728\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0\u4e2d\uff0c\u5165\u6808\u548c\u51fa\u6808\u64cd\u4f5c\u90fd\u5728\u9884\u5148\u5206\u914d\u597d\u7684\u8fde\u7eed\u5185\u5b58\u4e2d\u8fdb\u884c\uff0c\u5177\u6709\u5f88\u597d\u7684\u7f13\u5b58\u672c\u5730\u6027\uff0c\u56e0\u6b64\u6548\u7387\u8f83\u9ad8\u3002\u7136\u800c\uff0c\u5982\u679c\u5165\u6808\u65f6\u8d85\u51fa\u6570\u7ec4\u5bb9\u91cf\uff0c\u4f1a\u89e6\u53d1\u6269\u5bb9\u673a\u5236\uff0c\u5bfc\u81f4\u8be5\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53d8\u4e3a \\(O(n)\\) \u3002</p> <p>\u5728\u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0\u4e2d\uff0c\u94fe\u8868\u7684\u6269\u5bb9\u975e\u5e38\u7075\u6d3b\uff0c\u4e0d\u5b58\u5728\u4e0a\u8ff0\u6570\u7ec4\u6269\u5bb9\u65f6\u6548\u7387\u964d\u4f4e\u7684\u95ee\u9898\u3002\u4f46\u662f\uff0c\u5165\u6808\u64cd\u4f5c\u9700\u8981\u521d\u59cb\u5316\u8282\u70b9\u5bf9\u8c61\u5e76\u4fee\u6539\u6307\u9488\uff0c\u56e0\u6b64\u6548\u7387\u76f8\u5bf9\u8f83\u4f4e\u3002\u4e0d\u8fc7\uff0c\u5982\u679c\u5165\u6808\u5143\u7d20\u672c\u8eab\u5c31\u662f\u8282\u70b9\u5bf9\u8c61\uff0c\u90a3\u4e48\u53ef\u4ee5\u7701\u53bb\u521d\u59cb\u5316\u6b65\u9aa4\uff0c\u4ece\u800c\u63d0\u9ad8\u6548\u7387\u3002</p> <p>\u7efc\u4e0a\u6240\u8ff0\uff0c\u5f53\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u7684\u5143\u7d20\u662f\u57fa\u672c\u6570\u636e\u7c7b\u578b\u65f6\uff0c\u4f8b\u5982 <code>int</code> \u6216 <code>double</code> \uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\u3002</p> <ul> <li>\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u5728\u89e6\u53d1\u6269\u5bb9\u65f6\u6548\u7387\u4f1a\u964d\u4f4e\uff0c\u4f46\u7531\u4e8e\u6269\u5bb9\u662f\u4f4e\u9891\u64cd\u4f5c\uff0c\u56e0\u6b64\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002</li> <li>\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u53ef\u4ee5\u63d0\u4f9b\u66f4\u52a0\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002</li> </ul> <p>\u7a7a\u95f4\u6548\u7387</p> <p>\u5728\u521d\u59cb\u5316\u5217\u8868\u65f6\uff0c\u7cfb\u7edf\u4f1a\u4e3a\u5217\u8868\u5206\u914d\u201c\u521d\u59cb\u5bb9\u91cf\u201d\uff0c\u8be5\u5bb9\u91cf\u53ef\u80fd\u8d85\u51fa\u5b9e\u9645\u9700\u6c42\uff1b\u5e76\u4e14\uff0c\u6269\u5bb9\u673a\u5236\u901a\u5e38\u662f\u6309\u7167\u7279\u5b9a\u500d\u7387\uff08\u4f8b\u5982 2 \u500d\uff09\u8fdb\u884c\u6269\u5bb9\u7684\uff0c\u6269\u5bb9\u540e\u7684\u5bb9\u91cf\u4e5f\u53ef\u80fd\u8d85\u51fa\u5b9e\u9645\u9700\u6c42\u3002\u56e0\u6b64\uff0c\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u53ef\u80fd\u9020\u6210\u4e00\u5b9a\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002</p> <p>\u7136\u800c\uff0c\u7531\u4e8e\u94fe\u8868\u8282\u70b9\u9700\u8981\u989d\u5916\u5b58\u50a8\u6307\u9488\uff0c\u56e0\u6b64\u94fe\u8868\u8282\u70b9\u5360\u7528\u7684\u7a7a\u95f4\u76f8\u5bf9\u8f83\u5927\u3002</p> <p>\u7efc\u4e0a\uff0c\u6211\u4eec\u4e0d\u80fd\u7b80\u5355\u5730\u786e\u5b9a\u54ea\u79cd\u5b9e\u73b0\u66f4\u52a0\u8282\u7701\u5185\u5b58\uff0c\u9700\u8981\u9488\u5bf9\u5177\u4f53\u60c5\u51b5\u8fdb\u884c\u5206\u6790\u3002</p>"},{"location":"chapter_stack_and_queue/stack/#514","title":"5.1.4 \u00a0 \u6808\u7684\u5178\u578b\u5e94\u7528","text":"<ul> <li>\u6d4f\u89c8\u5668\u4e2d\u7684\u540e\u9000\u4e0e\u524d\u8fdb\u3001\u8f6f\u4ef6\u4e2d\u7684\u64a4\u9500\u4e0e\u53cd\u64a4\u9500\u3002\u6bcf\u5f53\u6211\u4eec\u6253\u5f00\u65b0\u7684\u7f51\u9875\uff0c\u6d4f\u89c8\u5668\u5c31\u4f1a\u5bf9\u4e0a\u4e00\u4e2a\u7f51\u9875\u6267\u884c\u5165\u6808\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u540e\u9000\u64cd\u4f5c\u56de\u5230\u4e0a\u4e00\u4e2a\u7f51\u9875\u3002\u540e\u9000\u64cd\u4f5c\u5b9e\u9645\u4e0a\u662f\u5728\u6267\u884c\u51fa\u6808\u3002\u5982\u679c\u8981\u540c\u65f6\u652f\u6301\u540e\u9000\u548c\u524d\u8fdb\uff0c\u90a3\u4e48\u9700\u8981\u4e24\u4e2a\u6808\u6765\u914d\u5408\u5b9e\u73b0\u3002</li> <li>\u7a0b\u5e8f\u5185\u5b58\u7ba1\u7406\u3002\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\uff0c\u7cfb\u7edf\u90fd\u4f1a\u5728\u6808\u9876\u6dfb\u52a0\u4e00\u4e2a\u6808\u5e27\uff0c\u7528\u4e8e\u8bb0\u5f55\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u4fe1\u606f\u3002\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u5411\u4e0b\u9012\u63a8\u9636\u6bb5\u4f1a\u4e0d\u65ad\u6267\u884c\u5165\u6808\u64cd\u4f5c\uff0c\u800c\u5411\u4e0a\u56de\u6eaf\u9636\u6bb5\u5219\u4f1a\u4e0d\u65ad\u6267\u884c\u51fa\u6808\u64cd\u4f5c\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/summary/","title":"5.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_stack_and_queue/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u6808\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002</li> <li>\u5728\u65f6\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u5177\u6709\u8f83\u9ad8\u7684\u5e73\u5747\u6548\u7387\uff0c\u4f46\u5728\u6269\u5bb9\u8fc7\u7a0b\u4e2d\uff0c\u5355\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u52a3\u5316\u81f3 \\(O(n)\\) \u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u6808\u7684\u94fe\u8868\u5b9e\u73b0\u5177\u6709\u66f4\u4e3a\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002</li> <li>\u5728\u7a7a\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u53ef\u80fd\u5bfc\u81f4\u4e00\u5b9a\u7a0b\u5ea6\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002\u4f46\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u94fe\u8868\u8282\u70b9\u6240\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u6bd4\u6570\u7ec4\u5143\u7d20\u66f4\u5927\u3002</li> <li>\u961f\u5217\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u540c\u6837\u53ef\u4ee5\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u7684\u5bf9\u6bd4\u4e0a\uff0c\u961f\u5217\u7684\u7ed3\u8bba\u4e0e\u524d\u8ff0\u6808\u7684\u7ed3\u8bba\u76f8\u4f3c\u3002</li> <li>\u53cc\u5411\u961f\u5217\u662f\u4e00\u79cd\u5177\u6709\u66f4\u9ad8\u81ea\u7531\u5ea6\u7684\u961f\u5217\uff0c\u5b83\u5141\u8bb8\u5728\u4e24\u7aef\u8fdb\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u662f\u5426\u662f\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\uff1f</p> <p>\u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u529f\u80fd\u672c\u8d28\u4e0a\u662f\u201c\u6808\u201d\u7684\u4f53\u73b0\u3002\u5f53\u7528\u6237\u8bbf\u95ee\u4e00\u4e2a\u65b0\u9875\u9762\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u88ab\u6dfb\u52a0\u5230\u6808\u9876\uff1b\u5f53\u7528\u6237\u70b9\u51fb\u540e\u9000\u6309\u94ae\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u4ece\u6808\u9876\u5f39\u51fa\u3002\u4f7f\u7528\u53cc\u5411\u961f\u5217\u53ef\u4ee5\u65b9\u4fbf\u5730\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u64cd\u4f5c\uff0c\u8fd9\u4e2a\u5728\u201c\u53cc\u5411\u961f\u5217\u201d\u7ae0\u8282\u6709\u63d0\u5230\u3002</p> <p>Q\uff1a\u5728\u51fa\u6808\u540e\uff0c\u662f\u5426\u9700\u8981\u91ca\u653e\u51fa\u6808\u8282\u70b9\u7684\u5185\u5b58\uff1f</p> <p>\u5982\u679c\u540e\u7eed\u4ecd\u9700\u8981\u4f7f\u7528\u5f39\u51fa\u8282\u70b9\uff0c\u5219\u4e0d\u9700\u8981\u91ca\u653e\u5185\u5b58\u3002\u82e5\u4e4b\u540e\u4e0d\u9700\u8981\u7528\u5230\uff0c<code>Java</code> \u548c <code>Python</code> \u7b49\u8bed\u8a00\u62e5\u6709\u81ea\u52a8\u5783\u573e\u56de\u6536\u673a\u5236\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\uff1b\u5728 <code>C</code> \u548c <code>C++</code> \u4e2d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\u3002</p> <p>Q\uff1a\u53cc\u5411\u961f\u5217\u50cf\u662f\u4e24\u4e2a\u6808\u62fc\u63a5\u5728\u4e86\u4e00\u8d77\uff0c\u5b83\u7684\u7528\u9014\u662f\u4ec0\u4e48\uff1f</p> <p>\u53cc\u5411\u961f\u5217\u5c31\u50cf\u662f\u6808\u548c\u961f\u5217\u7684\u7ec4\u5408\u6216\u4e24\u4e2a\u6808\u62fc\u5728\u4e86\u4e00\u8d77\u3002\u5b83\u8868\u73b0\u7684\u662f\u6808 + \u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u53ef\u4ee5\u5b9e\u73b0\u6808\u4e0e\u961f\u5217\u7684\u6240\u6709\u5e94\u7528\uff0c\u5e76\u4e14\u66f4\u52a0\u7075\u6d3b\u3002</p> <p>Q\uff1a\u64a4\u9500\uff08undo\uff09\u548c\u53cd\u64a4\u9500\uff08redo\uff09\u5177\u4f53\u662f\u5982\u4f55\u5b9e\u73b0\u7684\uff1f</p> <p>\u4f7f\u7528\u4e24\u4e2a\u6808\uff0c\u6808 <code>A</code> \u7528\u4e8e\u64a4\u9500\uff0c\u6808 <code>B</code> \u7528\u4e8e\u53cd\u64a4\u9500\u3002</p> <ol> <li>\u6bcf\u5f53\u7528\u6237\u6267\u884c\u4e00\u4e2a\u64cd\u4f5c\uff0c\u5c06\u8fd9\u4e2a\u64cd\u4f5c\u538b\u5165\u6808 <code>A</code> \uff0c\u5e76\u6e05\u7a7a\u6808 <code>B</code> \u3002</li> <li>\u5f53\u7528\u6237\u6267\u884c\u201c\u64a4\u9500\u201d\u65f6\uff0c\u4ece\u6808 <code>A</code> \u4e2d\u5f39\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u5e76\u5c06\u5176\u538b\u5165\u6808 <code>B</code> \u3002</li> <li>\u5f53\u7528\u6237\u6267\u884c\u201c\u53cd\u64a4\u9500\u201d\u65f6\uff0c\u4ece\u6808 <code>B</code> \u4e2d\u5f39\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u5e76\u5c06\u5176\u538b\u5165\u6808 <code>A</code> \u3002</li> </ol>"},{"location":"chapter_tree/","title":"\u7b2c 7 \u7ae0 \u00a0 \u6811","text":"<p>Abstract</p> <p>\u53c2\u5929\u5927\u6811\u5145\u6ee1\u751f\u547d\u529b\uff0c\u6839\u6df1\u53f6\u8302\uff0c\u5206\u679d\u6276\u758f\u3002</p> <p>\u5b83\u4e3a\u6211\u4eec\u5c55\u73b0\u4e86\u6570\u636e\u5206\u6cbb\u7684\u751f\u52a8\u5f62\u6001\u3002</p>"},{"location":"chapter_tree/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>7.1 \u00a0 \u4e8c\u53c9\u6811</li> <li>7.2 \u00a0 \u4e8c\u53c9\u6811\u904d\u5386</li> <li>7.3 \u00a0 \u4e8c\u53c9\u6811\u6570\u7ec4\u8868\u793a</li> <li>7.4 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811</li> <li>7.5 \u00a0 AVL \u6811 *</li> <li>7.6 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_tree/array_representation_of_tree/","title":"7.3 \u00a0 \u4e8c\u53c9\u6811\u6570\u7ec4\u8868\u793a","text":"<p>\u5728\u94fe\u8868\u8868\u793a\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u5b58\u50a8\u5355\u5143\u4e3a\u8282\u70b9 <code>TreeNode</code> \uff0c\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u6307\u9488\u76f8\u8fde\u63a5\u3002\u4e0a\u4e00\u8282\u4ecb\u7ecd\u4e86\u94fe\u8868\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7684\u5404\u9879\u57fa\u672c\u64cd\u4f5c\u3002</p> <p>\u90a3\u4e48\uff0c\u6211\u4eec\u80fd\u5426\u7528\u6570\u7ec4\u6765\u8868\u793a\u4e8c\u53c9\u6811\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\u3002</p>"},{"location":"chapter_tree/array_representation_of_tree/#731","title":"7.3.1 \u00a0 \u8868\u793a\u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"<p>\u5148\u5206\u6790\u4e00\u4e2a\u7b80\u5355\u6848\u4f8b\u3002\u7ed9\u5b9a\u4e00\u68f5\u5b8c\u7f8e\u4e8c\u53c9\u6811\uff0c\u6211\u4eec\u5c06\u6240\u6709\u8282\u70b9\u6309\u7167\u5c42\u5e8f\u904d\u5386\u7684\u987a\u5e8f\u5b58\u50a8\u5728\u4e00\u4e2a\u6570\u7ec4\u4e2d\uff0c\u5219\u6bcf\u4e2a\u8282\u70b9\u90fd\u5bf9\u5e94\u552f\u4e00\u7684\u6570\u7ec4\u7d22\u5f15\u3002</p> <p>\u6839\u636e\u5c42\u5e8f\u904d\u5386\u7684\u7279\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa\u7236\u8282\u70b9\u7d22\u5f15\u4e0e\u5b50\u8282\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u201c\u6620\u5c04\u516c\u5f0f\u201d\uff1a\u82e5\u67d0\u8282\u70b9\u7684\u7d22\u5f15\u4e3a \\(i\\) \uff0c\u5219\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \u3002\u56fe 7-12 \u5c55\u793a\u4e86\u5404\u4e2a\u8282\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u6620\u5c04\u5173\u7cfb\u3002</p> <p></p> <p> \u56fe 7-12 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a </p> <p>\u6620\u5c04\u516c\u5f0f\u7684\u89d2\u8272\u76f8\u5f53\u4e8e\u94fe\u8868\u4e2d\u7684\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\u3002\u7ed9\u5b9a\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\u8282\u70b9\uff0c\u6211\u4eec\u90fd\u53ef\u4ee5\u901a\u8fc7\u6620\u5c04\u516c\u5f0f\u6765\u8bbf\u95ee\u5b83\u7684\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u3002</p>"},{"location":"chapter_tree/array_representation_of_tree/#732","title":"7.3.2 \u00a0 \u8868\u793a\u4efb\u610f\u4e8c\u53c9\u6811","text":"<p>\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u4e00\u4e2a\u7279\u4f8b\uff0c\u5728\u4e8c\u53c9\u6811\u7684\u4e2d\u95f4\u5c42\u901a\u5e38\u5b58\u5728\u8bb8\u591a <code>None</code> \u3002\u7531\u4e8e\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5e76\u4e0d\u5305\u542b\u8fd9\u4e9b <code>None</code> \uff0c\u56e0\u6b64\u6211\u4eec\u65e0\u6cd5\u4ec5\u51ed\u8be5\u5e8f\u5217\u6765\u63a8\u6d4b <code>None</code> \u7684\u6570\u91cf\u548c\u5206\u5e03\u4f4d\u7f6e\u3002\u8fd9\u610f\u5473\u7740\u5b58\u5728\u591a\u79cd\u4e8c\u53c9\u6811\u7ed3\u6784\u90fd\u7b26\u5408\u8be5\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002</p> <p>\u5982\u56fe 7-13 \u6240\u793a\uff0c\u7ed9\u5b9a\u4e00\u68f5\u975e\u5b8c\u7f8e\u4e8c\u53c9\u6811\uff0c\u4e0a\u8ff0\u6570\u7ec4\u8868\u793a\u65b9\u6cd5\u5df2\u7ecf\u5931\u6548\u3002</p> <p></p> <p> \u56fe 7-13 \u00a0 \u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5bf9\u5e94\u591a\u79cd\u4e8c\u53c9\u6811\u53ef\u80fd\u6027 </p> <p>\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u4e2d\u663e\u5f0f\u5730\u5199\u51fa\u6240\u6709 <code>None</code> \u3002\u5982\u56fe 7-14 \u6240\u793a\uff0c\u8fd9\u6837\u5904\u7406\u540e\uff0c\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5c31\u53ef\u4ee5\u552f\u4e00\u8868\u793a\u4e8c\u53c9\u6811\u4e86\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\n# \u4f7f\u7528 None \u6765\u8868\u793a\u7a7a\u4f4d\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c INT_MAX \u6807\u8bb0\u7a7a\u4f4d\nvector&lt;int&gt; tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u7684\u5305\u88c5\u7c7b Integer \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nint?[] tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 any \u7c7b\u578b\u7684\u5207\u7247, \u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 Int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nList&lt;int?&gt; tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 None \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c\u6807\u8bb0\u7a7a\u4f4d\uff0c\u56e0\u6b64\u8981\u6c42\u8282\u70b9\u503c\u4e0d\u80fd\u4e3a INT_MAX\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nval tree = mutableListOf( 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 )\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p></p> <p> \u56fe 7-14 \u00a0 \u4efb\u610f\u7c7b\u578b\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a </p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u4f7f\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u56de\u987e\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u5b9a\u4e49\uff0c<code>None</code> \u53ea\u51fa\u73b0\u5728\u6700\u5e95\u5c42\u4e14\u9760\u53f3\u7684\u4f4d\u7f6e\uff0c\u56e0\u6b64\u6240\u6709 <code>None</code> \u4e00\u5b9a\u51fa\u73b0\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u672b\u5c3e\u3002</p> <p>\u8fd9\u610f\u5473\u7740\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u5b8c\u5168\u4e8c\u53c9\u6811\u65f6\uff0c\u53ef\u4ee5\u7701\u7565\u5b58\u50a8\u6240\u6709 <code>None</code> \uff0c\u975e\u5e38\u65b9\u4fbf\u3002\u56fe 7-15 \u7ed9\u51fa\u4e86\u4e00\u4e2a\u4f8b\u5b50\u3002</p> <p></p> <p> \u56fe 7-15 \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a </p> <p>\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u68f5\u57fa\u4e8e\u6570\u7ec4\u8868\u793a\u7684\u4e8c\u53c9\u6811\uff0c\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u64cd\u4f5c\u3002</p> <ul> <li>\u7ed9\u5b9a\u67d0\u8282\u70b9\uff0c\u83b7\u53d6\u5b83\u7684\u503c\u3001\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u3001\u7236\u8282\u70b9\u3002</li> <li>\u83b7\u53d6\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u3001\u540e\u5e8f\u904d\u5386\u3001\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_binary_tree.py<pre><code>class ArrayBinaryTree:\n    \"\"\"\u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b\"\"\"\n\n    def __init__(self, arr: list[int | None]):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._tree = list(arr)\n\n    def size(self):\n        \"\"\"\u5217\u8868\u5bb9\u91cf\"\"\"\n        return len(self._tree)\n\n    def val(self, i: int) -&gt; int:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c\"\"\"\n        # \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 or i &gt;= self.size():\n            return None\n        return self._tree[i]\n\n    def left(self, i: int) -&gt; int | None:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n        return 2 * i + 1\n\n    def right(self, i: int) -&gt; int | None:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n        return 2 * i + 2\n\n    def parent(self, i: int) -&gt; int | None:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n        return (i - 1) // 2\n\n    def level_order(self) -&gt; list[int]:\n        \"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for i in range(self.size()):\n            if self.val(i) is not None:\n                self.res.append(self.val(i))\n        return self.res\n\n    def dfs(self, i: int, order: str):\n        \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\n        if self.val(i) is None:\n            return\n        # \u524d\u5e8f\u904d\u5386\n        if order == \"pre\":\n            self.res.append(self.val(i))\n        self.dfs(self.left(i), order)\n        # \u4e2d\u5e8f\u904d\u5386\n        if order == \"in\":\n            self.res.append(self.val(i))\n        self.dfs(self.right(i), order)\n        # \u540e\u5e8f\u904d\u5386\n        if order == \"post\":\n            self.res.append(self.val(i))\n\n    def pre_order(self) -&gt; list[int]:\n        \"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        self.dfs(0, order=\"pre\")\n        return self.res\n\n    def in_order(self) -&gt; list[int]:\n        \"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        self.dfs(0, order=\"in\")\n        return self.res\n\n    def post_order(self) -&gt; list[int]:\n        \"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        self.dfs(0, order=\"post\")\n        return self.res\n</code></pre> array_binary_tree.cpp<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    ArrayBinaryTree(vector&lt;int&gt; arr) {\n        tree = arr;\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    int size() {\n        return tree.size();\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    int val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size())\n            return INT_MAX;\n        return tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    int left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    int right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    int parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; levelOrder() {\n        vector&lt;int&gt; res;\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != INT_MAX)\n                res.push_back(val(i));\n        }\n        return res;\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; preOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; inOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; postOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"post\", res);\n        return res;\n    }\n\n  private:\n    vector&lt;int&gt; tree;\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    void dfs(int i, string order, vector&lt;int&gt; &amp;res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (val(i) == INT_MAX)\n            return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order == \"pre\")\n            res.push_back(val(i));\n        dfs(left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order == \"in\")\n            res.push_back(val(i));\n        dfs(right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order == \"post\")\n            res.push_back(val(i));\n    }\n};\n</code></pre> array_binary_tree.java<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    private List&lt;Integer&gt; tree;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public ArrayBinaryTree(List&lt;Integer&gt; arr) {\n        tree = new ArrayList&lt;&gt;(arr);\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    public int size() {\n        return tree.size();\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    public Integer val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size())\n            return null;\n        return tree.get(i);\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public Integer left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public Integer right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    public Integer parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; levelOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != null)\n                res.add(val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    private void dfs(Integer i, String order, List&lt;Integer&gt; res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (val(i) == null)\n            return;\n        // \u524d\u5e8f\u904d\u5386\n        if (\"pre\".equals(order))\n            res.add(val(i));\n        dfs(left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (\"in\".equals(order))\n            res.add(val(i));\n        dfs(right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (\"post\".equals(order))\n            res.add(val(i));\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; preOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; inOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; postOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.cs<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree(List&lt;int?&gt; arr) {\n    List&lt;int?&gt; tree = new(arr);\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    public int Size() {\n        return tree.Count;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    public int? Val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= Size())\n            return null;\n        return tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public int Left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public int Right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    public int Parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; LevelOrder() {\n        List&lt;int&gt; res = [];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (int i = 0; i &lt; Size(); i++) {\n            if (Val(i).HasValue)\n                res.Add(Val(i)!.Value);\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    void DFS(int i, string order, List&lt;int&gt; res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (!Val(i).HasValue)\n            return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order == \"pre\")\n            res.Add(Val(i)!.Value);\n        DFS(Left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order == \"in\")\n            res.Add(Val(i)!.Value);\n        DFS(Right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order == \"post\")\n            res.Add(Val(i)!.Value);\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; PreOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; InOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"in\", res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; PostOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.go<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\ntype arrayBinaryTree struct {\n    tree []any\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newArrayBinaryTree(arr []any) *arrayBinaryTree {\n    return &amp;arrayBinaryTree{\n        tree: arr,\n    }\n}\n\n/* \u5217\u8868\u5bb9\u91cf */\nfunc (abt *arrayBinaryTree) size() int {\n    return len(abt.tree)\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nfunc (abt *arrayBinaryTree) val(i int) any {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if i &lt; 0 || i &gt;= abt.size() {\n        return nil\n    }\n    return abt.tree[i]\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) left(i int) int {\n    return 2*i + 1\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) right(i int) int {\n    return 2*i + 2\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) parent(i int) int {\n    return (i - 1) / 2\n}\n\n/* \u5c42\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) levelOrder() []any {\n    var res []any\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n    for i := 0; i &lt; abt.size(); i++ {\n        if abt.val(i) != nil {\n            res = append(res, abt.val(i))\n        }\n    }\n    return res\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nfunc (abt *arrayBinaryTree) dfs(i int, order string, res *[]any) {\n    // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n    if abt.val(i) == nil {\n        return\n    }\n    // \u524d\u5e8f\u904d\u5386\n    if order == \"pre\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.left(i), order, res)\n    // \u4e2d\u5e8f\u904d\u5386\n    if order == \"in\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.right(i), order, res)\n    // \u540e\u5e8f\u904d\u5386\n    if order == \"post\" {\n        *res = append(*res, abt.val(i))\n    }\n}\n\n/* \u524d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) preOrder() []any {\n    var res []any\n    abt.dfs(0, \"pre\", &amp;res)\n    return res\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) inOrder() []any {\n    var res []any\n    abt.dfs(0, \"in\", &amp;res)\n    return res\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) postOrder() []any {\n    var res []any\n    abt.dfs(0, \"post\", &amp;res)\n    return res\n}\n</code></pre> array_binary_tree.swift<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    private var tree: [Int?]\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init(arr: [Int?]) {\n        tree = arr\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    func size() -&gt; Int {\n        tree.count\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    func val(i: Int) -&gt; Int? {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 || i &gt;= size() {\n            return nil\n        }\n        return tree[i]\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    func left(i: Int) -&gt; Int {\n        2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    func right(i: Int) -&gt; Int {\n        2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    func parent(i: Int) -&gt; Int {\n        (i - 1) / 2\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    func levelOrder() -&gt; [Int] {\n        var res: [Int] = []\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for i in 0 ..&lt; size() {\n            if let val = val(i: i) {\n                res.append(val)\n            }\n        }\n        return res\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    private func dfs(i: Int, order: String, res: inout [Int]) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        guard let val = val(i: i) else {\n            return\n        }\n        // \u524d\u5e8f\u904d\u5386\n        if order == \"pre\" {\n            res.append(val)\n        }\n        dfs(i: left(i: i), order: order, res: &amp;res)\n        // \u4e2d\u5e8f\u904d\u5386\n        if order == \"in\" {\n            res.append(val)\n        }\n        dfs(i: right(i: i), order: order, res: &amp;res)\n        // \u540e\u5e8f\u904d\u5386\n        if order == \"post\" {\n            res.append(val)\n        }\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    func preOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"pre\", res: &amp;res)\n        return res\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    func inOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"in\", res: &amp;res)\n        return res\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    func postOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"post\", res: &amp;res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.js<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    #tree;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(arr) {\n        this.#tree = arr;\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    size() {\n        return this.#tree.length;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    val(i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    left(i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    right(i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    parent(i) {\n        return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    levelOrder() {\n        let res = [];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    #dfs(i, order, res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (this.val(i) === null) return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    preOrder() {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    inOrder() {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    postOrder() {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.ts<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    #tree: (number | null)[];\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(arr: (number | null)[]) {\n        this.#tree = arr;\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    size(): number {\n        return this.#tree.length;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    val(i: number): number | null {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    left(i: number): number {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    right(i: number): number {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    parent(i: number): number {\n        return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    levelOrder(): number[] {\n        let res = [];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    #dfs(i: number, order: Order, res: (number | null)[]): void {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (this.val(i) === null) return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    preOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    inOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    postOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.dart<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n  late List&lt;int?&gt; _tree;\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  ArrayBinaryTree(this._tree);\n\n  /* \u5217\u8868\u5bb9\u91cf */\n  int size() {\n    return _tree.length;\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n  int? val(int i) {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if (i &lt; 0 || i &gt;= size()) {\n      return null;\n    }\n    return _tree[i];\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n  int? left(int i) {\n    return 2 * i + 1;\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n  int? right(int i) {\n    return 2 * i + 2;\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n  int? parent(int i) {\n    return (i - 1) ~/ 2;\n  }\n\n  /* \u5c42\u5e8f\u904d\u5386 */\n  List&lt;int&gt; levelOrder() {\n    List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; size(); i++) {\n      if (val(i) != null) {\n        res.add(val(i)!);\n      }\n    }\n    return res;\n  }\n\n  /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n  void dfs(int i, String order, List&lt;int?&gt; res) {\n    // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n    if (val(i) == null) {\n      return;\n    }\n    // \u524d\u5e8f\u904d\u5386\n    if (order == 'pre') {\n      res.add(val(i));\n    }\n    dfs(left(i)!, order, res);\n    // \u4e2d\u5e8f\u904d\u5386\n    if (order == 'in') {\n      res.add(val(i));\n    }\n    dfs(right(i)!, order, res);\n    // \u540e\u5e8f\u904d\u5386\n    if (order == 'post') {\n      res.add(val(i));\n    }\n  }\n\n  /* \u524d\u5e8f\u904d\u5386 */\n  List&lt;int?&gt; preOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'pre', res);\n    return res;\n  }\n\n  /* \u4e2d\u5e8f\u904d\u5386 */\n  List&lt;int?&gt; inOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'in', res);\n    return res;\n  }\n\n  /* \u540e\u5e8f\u904d\u5386 */\n  List&lt;int?&gt; postOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'post', res);\n    return res;\n  }\n}\n</code></pre> array_binary_tree.rs<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nstruct ArrayBinaryTree {\n    tree: Vec&lt;Option&lt;i32&gt;&gt;,\n}\n\nimpl ArrayBinaryTree {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(arr: Vec&lt;Option&lt;i32&gt;&gt;) -&gt; Self {\n        Self { tree: arr }\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    fn size(&amp;self) -&gt; i32 {\n        self.tree.len() as i32\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    fn val(&amp;self, i: i32) -&gt; Option&lt;i32&gt; {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 || i &gt;= self.size() {\n            None\n        } else {\n            self.tree[i as usize]\n        }\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fn left(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fn right(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    fn parent(&amp;self, i: i32) -&gt; i32 {\n        (i - 1) / 2\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    fn level_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for i in 0..self.size() {\n            if let Some(val) = self.val(i) {\n                res.push(val)\n            }\n        }\n        res\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    fn dfs(&amp;self, i: i32, order: &amp;str, res: &amp;mut Vec&lt;i32&gt;) {\n        if self.val(i).is_none() {\n            return;\n        }\n        let val = self.val(i).unwrap();\n        // \u524d\u5e8f\u904d\u5386\n        if order == \"pre\" {\n            res.push(val);\n        }\n        self.dfs(self.left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if order == \"in\" {\n            res.push(val);\n        }\n        self.dfs(self.right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if order == \"post\" {\n            res.push(val);\n        }\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    fn pre_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"pre\", &amp;mut res);\n        res\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    fn in_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"in\", &amp;mut res);\n        res\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    fn post_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"post\", &amp;mut res);\n        res\n    }\n}\n</code></pre> array_binary_tree.c<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7ed3\u6784\u4f53 */\ntypedef struct {\n    int *tree;\n    int size;\n} ArrayBinaryTree;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayBinaryTree *newArrayBinaryTree(int *arr, int arrSize) {\n    ArrayBinaryTree *abt = (ArrayBinaryTree *)malloc(sizeof(ArrayBinaryTree));\n    abt-&gt;tree = malloc(sizeof(int) * arrSize);\n    memcpy(abt-&gt;tree, arr, sizeof(int) * arrSize);\n    abt-&gt;size = arrSize;\n    return abt;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayBinaryTree(ArrayBinaryTree *abt) {\n    free(abt-&gt;tree);\n    free(abt);\n}\n\n/* \u5217\u8868\u5bb9\u91cf */\nint size(ArrayBinaryTree *abt) {\n    return abt-&gt;size;\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nint val(ArrayBinaryTree *abt, int i) {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if (i &lt; 0 || i &gt;= size(abt))\n        return INT_MAX;\n    return abt-&gt;tree[i];\n}\n\n/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; size(abt); i++) {\n        if (val(abt, i) != INT_MAX)\n            res[index++] = val(abt, i);\n    }\n    *returnSize = index;\n    return res;\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nvoid dfs(ArrayBinaryTree *abt, int i, char *order, int *res, int *index) {\n    // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n    if (val(abt, i) == INT_MAX)\n        return;\n    // \u524d\u5e8f\u904d\u5386\n    if (strcmp(order, \"pre\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, left(i), order, res, index);\n    // \u4e2d\u5e8f\u904d\u5386\n    if (strcmp(order, \"in\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, right(i), order, res, index);\n    // \u540e\u5e8f\u904d\u5386\n    if (strcmp(order, \"post\") == 0)\n        res[(*index)++] = val(abt, i);\n}\n\n/* \u524d\u5e8f\u904d\u5386 */\nint *preOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"pre\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nint *inOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"in\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nint *postOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"post\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n</code></pre> array_binary_tree.kt<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree(val tree: List&lt;Int?&gt;) {\n    /* \u5217\u8868\u5bb9\u91cf */\n    fun size(): Int {\n        return tree.size\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    fun value(i: Int): Int? {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size()) return null\n        return tree[i]\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    fun parent(i: Int): Int {\n        return (i - 1) / 2\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    fun levelOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (i in 0..&lt;size()) {\n            if (value(i) != null) res.add(value(i))\n        }\n        return res\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    fun dfs(i: Int, order: String, res: MutableList&lt;Int?&gt;) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (value(i) == null) return\n        // \u524d\u5e8f\u904d\u5386\n        if (\"pre\" == order) res.add(value(i))\n        dfs(left(i), order, res)\n        // \u4e2d\u5e8f\u904d\u5386\n        if (\"in\" == order) res.add(value(i))\n        dfs(right(i), order, res)\n        // \u540e\u5e8f\u904d\u5386\n        if (\"post\" == order) res.add(value(i))\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    fun preOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        dfs(0, \"pre\", res)\n        return res\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    fun inOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        dfs(0, \"in\", res)\n        return res\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    fun postOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        dfs(0, \"post\", res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.rb<pre><code>[class]{ArrayBinaryTree}-[func]{}\n</code></pre> array_binary_tree.zig<pre><code>[class]{ArrayBinaryTree}-[func]{}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_tree/array_representation_of_tree/#733","title":"7.3.3 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\u4e3b\u8981\u6709\u4ee5\u4e0b\u4f18\u70b9\u3002</p> <ul> <li>\u6570\u7ec4\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u5bf9\u7f13\u5b58\u53cb\u597d\uff0c\u8bbf\u95ee\u4e0e\u904d\u5386\u901f\u5ea6\u8f83\u5feb\u3002</li> <li>\u4e0d\u9700\u8981\u5b58\u50a8\u6307\u9488\uff0c\u6bd4\u8f83\u8282\u7701\u7a7a\u95f4\u3002</li> <li>\u5141\u8bb8\u968f\u673a\u8bbf\u95ee\u8282\u70b9\u3002</li> </ul> <p>\u7136\u800c\uff0c\u6570\u7ec4\u8868\u793a\u4e5f\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\u3002</p> <ul> <li>\u6570\u7ec4\u5b58\u50a8\u9700\u8981\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5b58\u50a8\u6570\u636e\u91cf\u8fc7\u5927\u7684\u6811\u3002</li> <li>\u589e\u5220\u8282\u70b9\u9700\u8981\u901a\u8fc7\u6570\u7ec4\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u5b9e\u73b0\uff0c\u6548\u7387\u8f83\u4f4e\u3002</li> <li>\u5f53\u4e8c\u53c9\u6811\u4e2d\u5b58\u5728\u5927\u91cf <code>None</code> \u65f6\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u7684\u8282\u70b9\u6570\u636e\u6bd4\u91cd\u8f83\u4f4e\uff0c\u7a7a\u95f4\u5229\u7528\u7387\u8f83\u4f4e\u3002</li> </ul>"},{"location":"chapter_tree/avl_tree/","title":"7.5 \u00a0 AVL \u6811 *","text":"<p>\u5728\u201c\u4e8c\u53c9\u641c\u7d22\u6811\u201d\u7ae0\u8282\u4e2d\u6211\u4eec\u63d0\u5230\uff0c\u5728\u591a\u6b21\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u9000\u5316\u4e3a\u94fe\u8868\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c06\u4ece \\(O(\\log n)\\) \u52a3\u5316\u4e3a \\(O(n)\\) \u3002</p> <p>\u5982\u56fe 7-24 \u6240\u793a\uff0c\u7ecf\u8fc7\u4e24\u6b21\u5220\u9664\u8282\u70b9\u64cd\u4f5c\uff0c\u8fd9\u68f5\u4e8c\u53c9\u641c\u7d22\u6811\u4fbf\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\u3002</p> <p></p> <p> \u56fe 7-24 \u00a0 AVL \u6811\u5728\u5220\u9664\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316 </p> <p>\u518d\u4f8b\u5982\uff0c\u5728\u56fe 7-25 \u6240\u793a\u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e24\u4e2a\u8282\u70b9\u540e\uff0c\u6811\u5c06\u4e25\u91cd\u5411\u5de6\u503e\u659c\uff0c\u67e5\u627e\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u968f\u4e4b\u52a3\u5316\u3002</p> <p></p> <p> \u56fe 7-25 \u00a0 AVL \u6811\u5728\u63d2\u5165\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316 </p> <p>1962 \u5e74 G. M. Adelson-Velsky \u548c E. M. Landis \u5728\u8bba\u6587\u201cAn algorithm for the organization of information\u201d\u4e2d\u63d0\u51fa\u4e86\u300cAVL \u6811\u300d\u3002\u8bba\u6587\u4e2d\u8be6\u7ec6\u63cf\u8ff0\u4e86\u4e00\u7cfb\u5217\u64cd\u4f5c\uff0c\u786e\u4fdd\u5728\u6301\u7eed\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4e0d\u4f1a\u9000\u5316\uff0c\u4ece\u800c\u4f7f\u5f97\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4fdd\u6301\u5728 \\(O(\\log n)\\) \u7ea7\u522b\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5728\u9700\u8981\u9891\u7e41\u8fdb\u884c\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u7684\u573a\u666f\u4e2d\uff0cAVL \u6811\u80fd\u59cb\u7ec8\u4fdd\u6301\u9ad8\u6548\u7684\u6570\u636e\u64cd\u4f5c\u6027\u80fd\uff0c\u5177\u6709\u5f88\u597d\u7684\u5e94\u7528\u4ef7\u503c\u3002</p>"},{"location":"chapter_tree/avl_tree/#751-avl","title":"7.5.1 \u00a0 AVL \u6811\u5e38\u89c1\u672f\u8bed","text":"<p>AVL \u6811\u65e2\u662f\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u4e5f\u662f\u5e73\u8861\u4e8c\u53c9\u6811\uff0c\u540c\u65f6\u6ee1\u8db3\u8fd9\u4e24\u7c7b\u4e8c\u53c9\u6811\u7684\u6240\u6709\u6027\u8d28\uff0c\u56e0\u6b64\u662f\u4e00\u79cd\u300c\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811 balanced binary search tree\u300d\u3002</p>"},{"location":"chapter_tree/avl_tree/#1","title":"1. \u00a0 \u8282\u70b9\u9ad8\u5ea6","text":"<p>\u7531\u4e8e AVL \u6811\u7684\u76f8\u5173\u64cd\u4f5c\u9700\u8981\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u4e3a\u8282\u70b9\u7c7b\u6dfb\u52a0 <code>height</code> \u53d8\u91cf\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class TreeNode:\n    \"\"\"AVL \u6811\u8282\u70b9\u7c7b\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                 # \u8282\u70b9\u503c\n        self.height: int = 0                # \u8282\u70b9\u9ad8\u5ea6\n        self.left: TreeNode | None = None   # \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n        self.right: TreeNode | None = None  # \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nstruct TreeNode {\n    int val{};          // \u8282\u70b9\u503c\n    int height = 0;     // \u8282\u70b9\u9ad8\u5ea6\n    TreeNode *left{};   // \u5de6\u5b50\u8282\u70b9\n    TreeNode *right{};  // \u53f3\u5b50\u8282\u70b9\n    TreeNode() = default;\n    explicit TreeNode(int x) : val(x){}\n};\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    public int val;        // \u8282\u70b9\u503c\n    public int height;     // \u8282\u70b9\u9ad8\u5ea6\n    public TreeNode left;  // \u5de6\u5b50\u8282\u70b9\n    public TreeNode right; // \u53f3\u5b50\u8282\u70b9\n    public TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode(int? x) {\n    public int? val = x;    // \u8282\u70b9\u503c\n    public int height;      // \u8282\u70b9\u9ad8\u5ea6\n    public TreeNode? left;  // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    public TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\n    Val    int       // \u8282\u70b9\u503c\n    Height int       // \u8282\u70b9\u9ad8\u5ea6\n    Left   *TreeNode // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    Right  *TreeNode // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    var val: Int // \u8282\u70b9\u503c\n    var height: Int // \u8282\u70b9\u9ad8\u5ea6\n    var left: TreeNode? // \u5de6\u5b50\u8282\u70b9\n    var right: TreeNode? // \u53f3\u5b50\u8282\u70b9\n\n    init(x: Int) {\n        val = x\n        height = 0\n    }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    val; // \u8282\u70b9\u503c\n    height; //\u8282\u70b9\u9ad8\u5ea6\n    left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n    right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n    constructor(val, left, right, height) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    val: number;            // \u8282\u70b9\u503c\n    height: number;         // \u8282\u70b9\u9ad8\u5ea6\n    left: TreeNode | null;  // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n    right: TreeNode | null; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n    constructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height; \n        this.left = left === undefined ? null : left; \n        this.right = right === undefined ? null : right; \n    }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n  int val;         // \u8282\u70b9\u503c\n  int height;      // \u8282\u70b9\u9ad8\u5ea6\n  TreeNode? left;  // \u5de6\u5b50\u8282\u70b9\n  TreeNode? right; // \u53f3\u5b50\u8282\u70b9\n  TreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\n    val: i32,                               // \u8282\u70b9\u503c\n    height: i32,                            // \u8282\u70b9\u9ad8\u5ea6\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // \u5de6\u5b50\u8282\u70b9\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // \u53f3\u5b50\u8282\u70b9\n}\n\nimpl TreeNode {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            height: 0,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nTreeNode struct TreeNode {\n    int val;\n    int height;\n    struct TreeNode *left;\n    struct TreeNode *right;\n} TreeNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode(val _val: Int) {  // \u8282\u70b9\u503c\n    val height: Int = 0          // \u8282\u70b9\u9ad8\u5ea6\n    val left: TreeNode? = null   // \u5de6\u5b50\u8282\u70b9\n    val right: TreeNode? = null  // \u53f3\u5b50\u8282\u70b9\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>\u201c\u8282\u70b9\u9ad8\u5ea6\u201d\u662f\u6307\u4ece\u8be5\u8282\u70b9\u5230\u5b83\u7684\u6700\u8fdc\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u5373\u6240\u7ecf\u8fc7\u7684\u201c\u8fb9\u201d\u7684\u6570\u91cf\u3002\u9700\u8981\u7279\u522b\u6ce8\u610f\u7684\u662f\uff0c\u53f6\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a \\(0\\) \uff0c\u800c\u7a7a\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a \\(-1\\) \u3002\u6211\u4eec\u5c06\u521b\u5efa\u4e24\u4e2a\u5de5\u5177\u51fd\u6570\uff0c\u5206\u522b\u7528\u4e8e\u83b7\u53d6\u548c\u66f4\u65b0\u8282\u70b9\u7684\u9ad8\u5ea6\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def height(self, node: TreeNode | None) -&gt; int:\n    \"\"\"\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\"\"\"\n    # \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    if node is not None:\n        return node.height\n    return -1\n\ndef update_height(self, node: TreeNode | None):\n    \"\"\"\u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\"\"\"\n    # \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height = max([self.height(node.left), self.height(node.right)]) + 1\n</code></pre> avl_tree.cpp<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node == nullptr ? -1 : node-&gt;height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node-&gt;height = max(height(node-&gt;left), height(node-&gt;right)) + 1;\n}\n</code></pre> avl_tree.java<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint Height(TreeNode? node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid UpdateHeight(TreeNode node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height = Math.Max(Height(node.left), Height(node.right)) + 1;\n}\n</code></pre> avl_tree.go<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    if node != nil {\n        return node.Height\n    }\n    return -1\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\n    lh := t.height(node.Left)\n    rh := t.height(node.Right)\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    if lh &gt; rh {\n        node.Height = lh + 1\n    } else {\n        node.Height = rh + 1\n    }\n}\n</code></pre> avl_tree.swift<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc height(node: TreeNode?) -&gt; Int {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    node?.height ?? -1\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc updateHeight(node: TreeNode?) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n</code></pre> avl_tree.js<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\n#updateHeight(node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node: TreeNode): number {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nupdateHeight(node: TreeNode): void {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode? node) {\n  // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n  return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode? node) {\n  // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n  node!.height = max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfn height(node: OptionTreeNodeRc) -&gt; i32 {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    match node {\n        Some(node) =&gt; node.borrow().height,\n        None =&gt; -1,\n    }\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfn update_height(node: OptionTreeNodeRc) {\n    if let Some(node) = node {\n        let left = node.borrow().left.clone();\n        let right = node.borrow().right.clone();\n        // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n        node.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    if (node != NULL) {\n        return node-&gt;height;\n    }\n    return -1;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n    int lh = height(node-&gt;left);\n    int rh = height(node-&gt;right);\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    if (lh &gt; rh) {\n        node-&gt;height = lh + 1;\n    } else {\n        node-&gt;height = rh + 1;\n    }\n}\n</code></pre> avl_tree.kt<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfun height(node: TreeNode?): Int {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node?.height ?: -1\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfun updateHeight(node: TreeNode?) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node?.height = (max(height(node?.left).toDouble(), height(node?.right).toDouble()) + 1).toInt()\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{height}\n\n[class]{AVLTree}-[func]{update_height}\n</code></pre> avl_tree.zig<pre><code>// \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\nfn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n    _ = self;\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return if (node == null) -1 else node.?.height;\n}\n\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nfn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.?.height = @max(self.height(node.?.left), self.height(node.?.right)) + 1;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2","title":"2. \u00a0 \u8282\u70b9\u5e73\u8861\u56e0\u5b50","text":"<p>\u8282\u70b9\u7684\u300c\u5e73\u8861\u56e0\u5b50 balance factor\u300d\u5b9a\u4e49\u4e3a\u8282\u70b9\u5de6\u5b50\u6811\u7684\u9ad8\u5ea6\u51cf\u53bb\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\uff0c\u540c\u65f6\u89c4\u5b9a\u7a7a\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4e3a \\(0\\) \u3002\u6211\u4eec\u540c\u6837\u5c06\u83b7\u53d6\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u7684\u529f\u80fd\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def balance_factor(self, node: TreeNode | None) -&gt; int:\n    \"\"\"\u83b7\u53d6\u5e73\u8861\u56e0\u5b50\"\"\"\n    # \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if node is None:\n        return 0\n    # \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return self.height(node.left) - self.height(node.right)\n</code></pre> avl_tree.cpp<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == nullptr)\n        return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.java<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null)\n        return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.cs<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint BalanceFactor(TreeNode? node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return Height(node.left) - Height(node.right);\n}\n</code></pre> avl_tree.go<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if node == nil {\n        return 0\n    }\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return t.height(node.Left) - t.height(node.Right)\n}\n</code></pre> avl_tree.swift<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc balanceFactor(node: TreeNode?) -&gt; Int {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    guard let node = node else { return 0 }\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node: node.left) - height(node: node.right)\n}\n</code></pre> avl_tree.js<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node === null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.ts<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node: TreeNode): number {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node === null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.dart<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node) {\n  // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n  if (node == null) return 0;\n  // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n  return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.rs<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfn balance_factor(node: OptionTreeNodeRc) -&gt; i32 {\n    match node {\n        // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n        None =&gt; 0,\n        // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n        Some(node) =&gt; {\n            Self::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())\n        }\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == NULL) {\n        return 0;\n    }\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.kt<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfun balanceFactor(node: TreeNode?): Int {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null) return 0\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node.left) - height(node.right)\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{balance_factor}\n</code></pre> avl_tree.zig<pre><code>// \u83b7\u53d6\u5e73\u8861\u56e0\u5b50\nfn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return self.height(node.?.left) - self.height(node.?.right);\n}\n</code></pre> <p>Note</p> <p>\u8bbe\u5e73\u8861\u56e0\u5b50\u4e3a \\(f\\) \uff0c\u5219\u4e00\u68f5 AVL \u6811\u7684\u4efb\u610f\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7686\u6ee1\u8db3 \\(-1 \\le f \\le 1\\) \u3002</p>"},{"location":"chapter_tree/avl_tree/#752-avl","title":"7.5.2 \u00a0 AVL \u6811\u65cb\u8f6c","text":"<p>AVL \u6811\u7684\u7279\u70b9\u5728\u4e8e\u201c\u65cb\u8f6c\u201d\u64cd\u4f5c\uff0c\u5b83\u80fd\u591f\u5728\u4e0d\u5f71\u54cd\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u524d\u63d0\u4e0b\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u65cb\u8f6c\u64cd\u4f5c\u65e2\u80fd\u4fdd\u6301\u201c\u4e8c\u53c9\u641c\u7d22\u6811\u201d\u7684\u6027\u8d28\uff0c\u4e5f\u80fd\u4f7f\u6811\u91cd\u65b0\u53d8\u4e3a\u201c\u5e73\u8861\u4e8c\u53c9\u6811\u201d\u3002</p> <p>\u6211\u4eec\u5c06\u5e73\u8861\u56e0\u5b50\u7edd\u5bf9\u503c \\(&gt; 1\\) \u7684\u8282\u70b9\u79f0\u4e3a\u201c\u5931\u8861\u8282\u70b9\u201d\u3002\u6839\u636e\u8282\u70b9\u5931\u8861\u60c5\u51b5\u7684\u4e0d\u540c\uff0c\u65cb\u8f6c\u64cd\u4f5c\u5206\u4e3a\u56db\u79cd\uff1a\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3002\u4e0b\u9762\u8be6\u7ec6\u4ecb\u7ecd\u8fd9\u4e9b\u65cb\u8f6c\u64cd\u4f5c\u3002</p>"},{"location":"chapter_tree/avl_tree/#1_1","title":"1. \u00a0 \u53f3\u65cb","text":"<p>\u5982\u56fe 7-26 \u6240\u793a\uff0c\u8282\u70b9\u4e0b\u65b9\u4e3a\u5e73\u8861\u56e0\u5b50\u3002\u4ece\u5e95\u81f3\u9876\u770b\uff0c\u4e8c\u53c9\u6811\u4e2d\u9996\u4e2a\u5931\u8861\u8282\u70b9\u662f\u201c\u8282\u70b9 3\u201d\u3002\u6211\u4eec\u5173\u6ce8\u4ee5\u8be5\u5931\u8861\u8282\u70b9\u4e3a\u6839\u8282\u70b9\u7684\u5b50\u6811\uff0c\u5c06\u8be5\u8282\u70b9\u8bb0\u4e3a <code>node</code> \uff0c\u5176\u5de6\u5b50\u8282\u70b9\u8bb0\u4e3a <code>child</code> \uff0c\u6267\u884c\u201c\u53f3\u65cb\u201d\u64cd\u4f5c\u3002\u5b8c\u6210\u53f3\u65cb\u540e\uff0c\u5b50\u6811\u6062\u590d\u5e73\u8861\uff0c\u5e76\u4e14\u4ecd\u7136\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-26 \u00a0 \u53f3\u65cb\u64cd\u4f5c\u6b65\u9aa4 </p> <p>\u5982\u56fe 7-27 \u6240\u793a\uff0c\u5f53\u8282\u70b9 <code>child</code> \u6709\u53f3\u5b50\u8282\u70b9\uff08\u8bb0\u4e3a <code>grand_child</code> \uff09\u65f6\uff0c\u9700\u8981\u5728\u53f3\u65cb\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 <code>grand_child</code> \u4f5c\u4e3a <code>node</code> \u7684\u5de6\u5b50\u8282\u70b9\u3002</p> <p></p> <p> \u56fe 7-27 \u00a0 \u6709 grand_child \u7684\u53f3\u65cb\u64cd\u4f5c </p> <p>\u201c\u5411\u53f3\u65cb\u8f6c\u201d\u662f\u4e00\u79cd\u5f62\u8c61\u5316\u7684\u8bf4\u6cd5\uff0c\u5b9e\u9645\u4e0a\u9700\u8981\u901a\u8fc7\u4fee\u6539\u8282\u70b9\u6307\u9488\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def right_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u53f3\u65cb\u64cd\u4f5c\"\"\"\n    child = node.left\n    grand_child = child.right\n    # \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node\n    node.left = grand_child\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    self.update_height(child)\n    # \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;left;\n    TreeNode *grandChild = child-&gt;right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode rightRotate(TreeNode node) {\n    TreeNode child = node.left;\n    TreeNode grandChild = child.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? RightRotate(TreeNode? node) {\n    TreeNode? child = node?.left;\n    TreeNode? grandChild = child?.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\n    child := node.Left\n    grandChild := child.Right\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.Right = node\n    node.Left = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc rightRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.left\n    let grandChild = child?.right\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child?.right = node\n    node?.left = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\n#rightRotate(node) {\n    const child = node.left;\n    const grandChild = child.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nrightRotate(node: TreeNode): TreeNode {\n    const child = node.left;\n    const grandChild = child.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node) {\n  TreeNode? child = node!.left;\n  TreeNode? grandChild = child!.right;\n  // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n  child.right = node;\n  node.left = grandChild;\n  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  updateHeight(node);\n  updateHeight(child);\n  // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfn right_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().left.clone().unwrap();\n            let grand_child = child.borrow().right.clone();\n            // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n            child.borrow_mut().right = Some(node.clone());\n            node.borrow_mut().left = grand_child;\n            // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;left;\n    grandChild = child-&gt;right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfun rightRotate(node: TreeNode?): TreeNode {\n    val child = node!!.left\n    val grandChild = child!!.right\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node\n    node.left = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node)\n    updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{right_rotate}\n</code></pre> avl_tree.zig<pre><code>// \u53f3\u65cb\u64cd\u4f5c\nfn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    var child = node.?.left;\n    var grandChild = child.?.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.?.right = node;\n    node.?.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.updateHeight(node);\n    self.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2_1","title":"2. \u00a0 \u5de6\u65cb","text":"<p>\u76f8\u5e94\u5730\uff0c\u5982\u679c\u8003\u8651\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u201c\u955c\u50cf\u201d\uff0c\u5219\u9700\u8981\u6267\u884c\u56fe 7-28 \u6240\u793a\u7684\u201c\u5de6\u65cb\u201d\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 7-28 \u00a0 \u5de6\u65cb\u64cd\u4f5c </p> <p>\u540c\u7406\uff0c\u5982\u56fe 7-29 \u6240\u793a\uff0c\u5f53\u8282\u70b9 <code>child</code> \u6709\u5de6\u5b50\u8282\u70b9\uff08\u8bb0\u4e3a <code>grand_child</code> \uff09\u65f6\uff0c\u9700\u8981\u5728\u5de6\u65cb\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 <code>grand_child</code> \u4f5c\u4e3a <code>node</code> \u7684\u53f3\u5b50\u8282\u70b9\u3002</p> <p></p> <p> \u56fe 7-29 \u00a0 \u6709 grand_child \u7684\u5de6\u65cb\u64cd\u4f5c </p> <p>\u53ef\u4ee5\u89c2\u5bdf\u5230\uff0c\u53f3\u65cb\u548c\u5de6\u65cb\u64cd\u4f5c\u5728\u903b\u8f91\u4e0a\u662f\u955c\u50cf\u5bf9\u79f0\u7684\uff0c\u5b83\u4eec\u5206\u522b\u89e3\u51b3\u7684\u4e24\u79cd\u5931\u8861\u60c5\u51b5\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u57fa\u4e8e\u5bf9\u79f0\u6027\uff0c\u6211\u4eec\u53ea\u9700\u5c06\u53f3\u65cb\u7684\u5b9e\u73b0\u4ee3\u7801\u4e2d\u7684\u6240\u6709\u7684 <code>left</code> \u66ff\u6362\u4e3a <code>right</code> \uff0c\u5c06\u6240\u6709\u7684 <code>right</code> \u66ff\u6362\u4e3a <code>left</code> \uff0c\u5373\u53ef\u5f97\u5230\u5de6\u65cb\u7684\u5b9e\u73b0\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def left_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u5de6\u65cb\u64cd\u4f5c\"\"\"\n    child = node.right\n    grand_child = child.left\n    # \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node\n    node.right = grand_child\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    self.update_height(child)\n    # \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;right;\n    TreeNode *grandChild = child-&gt;left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode leftRotate(TreeNode node) {\n    TreeNode child = node.right;\n    TreeNode grandChild = child.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? LeftRotate(TreeNode? node) {\n    TreeNode? child = node?.right;\n    TreeNode? grandChild = child?.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\n    child := node.Right\n    grandChild := child.Left\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.Left = node\n    node.Right = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc leftRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.right\n    let grandChild = child?.left\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child?.left = node\n    node?.right = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\n#leftRotate(node) {\n    const child = node.right;\n    const grandChild = child.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nleftRotate(node: TreeNode): TreeNode {\n    const child = node.right;\n    const grandChild = child.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node) {\n  TreeNode? child = node!.right;\n  TreeNode? grandChild = child!.left;\n  // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n  child.left = node;\n  node.right = grandChild;\n  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  updateHeight(node);\n  updateHeight(child);\n  // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfn left_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().right.clone().unwrap();\n            let grand_child = child.borrow().left.clone();\n            // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n            child.borrow_mut().left = Some(node.clone());\n            node.borrow_mut().right = grand_child;\n            // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;right;\n    grandChild = child-&gt;left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfun leftRotate(node: TreeNode?): TreeNode {\n    val child = node!!.right\n    val grandChild = child!!.left\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node\n    node.right = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node)\n    updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{left_rotate}\n</code></pre> avl_tree.zig<pre><code>// \u5de6\u65cb\u64cd\u4f5c\nfn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    var child = node.?.right;\n    var grandChild = child.?.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.?.left = node;\n    node.?.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.updateHeight(node);\n    self.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3","title":"3. \u00a0 \u5148\u5de6\u65cb\u540e\u53f3\u65cb","text":"<p>\u5bf9\u4e8e\u56fe 7-30 \u4e2d\u7684\u5931\u8861\u8282\u70b9 3 \uff0c\u4ec5\u4f7f\u7528\u5de6\u65cb\u6216\u53f3\u65cb\u90fd\u65e0\u6cd5\u4f7f\u5b50\u6811\u6062\u590d\u5e73\u8861\u3002\u6b64\u65f6\u9700\u8981\u5148\u5bf9 <code>child</code> \u6267\u884c\u201c\u5de6\u65cb\u201d\uff0c\u518d\u5bf9 <code>node</code> \u6267\u884c\u201c\u53f3\u65cb\u201d\u3002</p> <p></p> <p> \u56fe 7-30 \u00a0 \u5148\u5de6\u65cb\u540e\u53f3\u65cb </p>"},{"location":"chapter_tree/avl_tree/#4","title":"4. \u00a0 \u5148\u53f3\u65cb\u540e\u5de6\u65cb","text":"<p>\u5982\u56fe 7-31 \u6240\u793a\uff0c\u5bf9\u4e8e\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u955c\u50cf\u60c5\u51b5\uff0c\u9700\u8981\u5148\u5bf9 <code>child</code> \u6267\u884c\u201c\u53f3\u65cb\u201d\uff0c\u518d\u5bf9 <code>node</code> \u6267\u884c\u201c\u5de6\u65cb\u201d\u3002</p> <p></p> <p> \u56fe 7-31 \u00a0 \u5148\u53f3\u65cb\u540e\u5de6\u65cb </p>"},{"location":"chapter_tree/avl_tree/#5","title":"5. \u00a0 \u65cb\u8f6c\u7684\u9009\u62e9","text":"<p>\u56fe 7-32 \u5c55\u793a\u7684\u56db\u79cd\u5931\u8861\u60c5\u51b5\u4e0e\u4e0a\u8ff0\u6848\u4f8b\u9010\u4e2a\u5bf9\u5e94\uff0c\u5206\u522b\u9700\u8981\u91c7\u7528\u53f3\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5de6\u65cb\u7684\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 7-32 \u00a0 AVL \u6811\u7684\u56db\u79cd\u65cb\u8f6c\u60c5\u51b5 </p> <p>\u5982\u4e0b\u8868\u6240\u793a\uff0c\u6211\u4eec\u901a\u8fc7\u5224\u65ad\u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4ee5\u53ca\u8f83\u9ad8\u4e00\u4fa7\u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7684\u6b63\u8d1f\u53f7\uff0c\u6765\u786e\u5b9a\u5931\u8861\u8282\u70b9\u5c5e\u4e8e\u56fe 7-32 \u4e2d\u7684\u54ea\u79cd\u60c5\u51b5\u3002</p> <p> \u8868 7-3 \u00a0 \u56db\u79cd\u65cb\u8f6c\u60c5\u51b5\u7684\u9009\u62e9\u6761\u4ef6 </p> \u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5e94\u91c7\u7528\u7684\u65cb\u8f6c\u65b9\u6cd5 \\(&gt; 1\\) \uff08\u5de6\u504f\u6811\uff09 \\(\\geq 0\\) \u53f3\u65cb \\(&gt; 1\\) \uff08\u5de6\u504f\u6811\uff09 \\(&lt;0\\) \u5148\u5de6\u65cb\u540e\u53f3\u65cb \\(&lt; -1\\) \uff08\u53f3\u504f\u6811\uff09 \\(\\leq 0\\) \u5de6\u65cb \\(&lt; -1\\) \uff08\u53f3\u504f\u6811\uff09 \\(&gt;0\\) \u5148\u53f3\u65cb\u540e\u5de6\u65cb <p>\u4e3a\u4e86\u4fbf\u4e8e\u4f7f\u7528\uff0c\u6211\u4eec\u5c06\u65cb\u8f6c\u64cd\u4f5c\u5c01\u88c5\u6210\u4e00\u4e2a\u51fd\u6570\u3002\u6709\u4e86\u8fd9\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u5c31\u80fd\u5bf9\u5404\u79cd\u5931\u8861\u60c5\u51b5\u8fdb\u884c\u65cb\u8f6c\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\"\"\"\n    # \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    balance_factor = self.balance_factor(node)\n    # \u5de6\u504f\u6811\n    if balance_factor &gt; 1:\n        if self.balance_factor(node.left) &gt;= 0:\n            # \u53f3\u65cb\n            return self.right_rotate(node)\n        else:\n            # \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = self.left_rotate(node.left)\n            return self.right_rotate(node)\n    # \u53f3\u504f\u6811\n    elif balance_factor &lt; -1:\n        if self.balance_factor(node.right) &lt;= 0:\n            # \u5de6\u65cb\n            return self.left_rotate(node)\n        else:\n            # \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = self.right_rotate(node.right)\n            return self.left_rotate(node)\n    # \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n</code></pre> avl_tree.cpp<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int _balanceFactor = balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (_balanceFactor &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (_balanceFactor &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode rotate(TreeNode node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int balanceFactor = balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = leftRotate(node.left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = rightRotate(node.right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? Rotate(TreeNode? node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int balanceFactorInt = BalanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactorInt &gt; 1) {\n        if (BalanceFactor(node?.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return RightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node!.left = LeftRotate(node!.left);\n            return RightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactorInt &lt; -1) {\n        if (BalanceFactor(node?.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return LeftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node!.right = RightRotate(node!.right);\n            return LeftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    // Go \u63a8\u8350\u77ed\u53d8\u91cf\uff0c\u8fd9\u91cc bf \u6307\u4ee3 t.balanceFactor\n    bf := t.balanceFactor(node)\n    // \u5de6\u504f\u6811\n    if bf &gt; 1 {\n        if t.balanceFactor(node.Left) &gt;= 0 {\n            // \u53f3\u65cb\n            return t.rightRotate(node)\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.Left = t.leftRotate(node.Left)\n            return t.rightRotate(node)\n        }\n    }\n    // \u53f3\u504f\u6811\n    if bf &lt; -1 {\n        if t.balanceFactor(node.Right) &lt;= 0 {\n            // \u5de6\u65cb\n            return t.leftRotate(node)\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.Right = t.rightRotate(node.Right)\n            return t.leftRotate(node)\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc rotate(node: TreeNode?) -&gt; TreeNode? {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    let balanceFactor = balanceFactor(node: node)\n    // \u5de6\u504f\u6811\n    if balanceFactor &gt; 1 {\n        if self.balanceFactor(node: node?.left) &gt;= 0 {\n            // \u53f3\u65cb\n            return rightRotate(node: node)\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node?.left = leftRotate(node: node?.left)\n            return rightRotate(node: node)\n        }\n    }\n    // \u53f3\u504f\u6811\n    if balanceFactor &lt; -1 {\n        if self.balanceFactor(node: node?.right) &lt;= 0 {\n            // \u5de6\u65cb\n            return leftRotate(node: node)\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node?.right = rightRotate(node: node?.right)\n            return leftRotate(node: node)\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n#rotate(node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    const balanceFactor = this.balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return this.#rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = this.#leftRotate(node.left);\n            return this.#rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return this.#leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = this.#rightRotate(node.right);\n            return this.#leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nrotate(node: TreeNode): TreeNode {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    const balanceFactor = this.balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return this.rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = this.leftRotate(node.left);\n            return this.rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return this.leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = this.rightRotate(node.right);\n            return this.leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node) {\n  // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n  int factor = balanceFactor(node);\n  // \u5de6\u504f\u6811\n  if (factor &gt; 1) {\n    if (balanceFactor(node!.left) &gt;= 0) {\n      // \u53f3\u65cb\n      return rightRotate(node);\n    } else {\n      // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n      node.left = leftRotate(node.left);\n      return rightRotate(node);\n    }\n  }\n  // \u53f3\u504f\u6811\n  if (factor &lt; -1) {\n    if (balanceFactor(node!.right) &lt;= 0) {\n      // \u5de6\u65cb\n      return leftRotate(node);\n    } else {\n      // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n      node.right = rightRotate(node.right);\n      return leftRotate(node);\n    }\n  }\n  // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfn rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    let balance_factor = Self::balance_factor(node.clone());\n    // \u5de6\u504f\u6811\n    if balance_factor &gt; 1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().left.clone()) &gt;= 0 {\n            // \u53f3\u65cb\n            Self::right_rotate(Some(node))\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            let left = node.borrow().left.clone();\n            node.borrow_mut().left = Self::left_rotate(left);\n            Self::right_rotate(Some(node))\n        }\n    }\n    // \u53f3\u504f\u6811\n    else if balance_factor &lt; -1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().right.clone()) &lt;= 0 {\n            // \u5de6\u65cb\n            Self::left_rotate(Some(node))\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            let right = node.borrow().right.clone();\n            node.borrow_mut().right = Self::right_rotate(right);\n            Self::left_rotate(Some(node))\n        }\n    } else {\n        // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n        node\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int bf = balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (bf &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (bf &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfun rotate(node: TreeNode): TreeNode {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    val balanceFactor = balanceFactor(node)\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node)\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = leftRotate(node.left)\n            return rightRotate(node)\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node)\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = rightRotate(node.right)\n            return leftRotate(node)\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{rotate}\n</code></pre> avl_tree.zig<pre><code>// \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nfn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    var balance_factor = self.balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balance_factor &gt; 1) {\n        if (self.balanceFactor(node.?.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return self.rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.?.left = self.leftRotate(node.?.left);\n            return self.rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balance_factor &lt; -1) {\n        if (self.balanceFactor(node.?.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return self.leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.?.right = self.rightRotate(node.?.right);\n            return self.leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#753-avl","title":"7.5.3 \u00a0 AVL \u6811\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_tree/avl_tree/#1_2","title":"1. \u00a0 \u63d2\u5165\u8282\u70b9","text":"<p>AVL \u6811\u7684\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u4e0e\u4e8c\u53c9\u641c\u7d22\u6811\u5728\u4e3b\u4f53\u4e0a\u7c7b\u4f3c\u3002\u552f\u4e00\u7684\u533a\u522b\u5728\u4e8e\uff0c\u5728 AVL \u6811\u4e2d\u63d2\u5165\u8282\u70b9\u540e\uff0c\u4ece\u8be5\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u53ef\u80fd\u4f1a\u51fa\u73b0\u4e00\u7cfb\u5217\u5931\u8861\u8282\u70b9\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4ece\u8fd9\u4e2a\u8282\u70b9\u5f00\u59cb\uff0c\u81ea\u5e95\u5411\u4e0a\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def insert(self, val):\n    \"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n    self._root = self.insert_helper(self._root, val)\n\ndef insert_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode:\n    \"\"\"\u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\n    if node is None:\n        return TreeNode(val)\n    # 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9\n    if val &lt; node.val:\n        node.left = self.insert_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.insert_helper(node.right, val)\n    else:\n        # \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    # 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    else\n        return node;    // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode insertHelper(TreeNode node, int val) {\n    if (node == null)\n        return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val)\n        node.left = insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = insertHelper(node.right, val);\n    else\n        return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int val) {\n    root = InsertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? InsertHelper(TreeNode? node, int val) {\n    if (node == null) return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val)\n        node.left = InsertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = InsertHelper(node.right, val);\n    else\n        return node;     // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    UpdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = Rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc (t *aVLTree) insert(val int) {\n    t.root = t.insertHelper(t.root, val)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return NewTreeNode(val)\n    }\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if val &lt; node.Val.(int) {\n        node.Left = t.insertHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.insertHelper(node.Right, val)\n    } else {\n        // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = t.rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(val: Int) {\n    root = insertHelper(node: root, val: val)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc insertHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return TreeNode(x: val)\n    }\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if val &lt; node!.val {\n        node?.left = insertHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = insertHelper(node: node?.right, val: val)\n    } else {\n        return node // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    updateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node: node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(val) {\n    this.root = this.#insertHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#insertHelper(node, val) {\n    if (node === null) return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val) node.left = this.#insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#insertHelper(node.right, val);\n    else return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    this.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.#rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(val: number): void {\n    this.root = this.insertHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\ninsertHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val) {\n        node.left = this.insertHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.insertHelper(node.right, val);\n    } else {\n        return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    this.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n  root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val) {\n  if (node == null) return TreeNode(val);\n  /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n  if (val &lt; node.val)\n    node.left = insertHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = insertHelper(node.right, val);\n  else\n    return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n  updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n  node = rotate(node);\n  // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfn insert(&amp;mut self, val: i32) {\n    self.root = Self::insert_helper(self.root.clone(), val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn insert_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n            match {\n                let node_val = node.borrow().val;\n                node_val\n            }\n            .cmp(&amp;val)\n            {\n                Ordering::Greater =&gt; {\n                    let left = node.borrow().left.clone();\n                    node.borrow_mut().left = Self::insert_helper(left, val);\n                }\n                Ordering::Less =&gt; {\n                    let right = node.borrow().right.clone();\n                    node.borrow_mut().right = Self::insert_helper(right, val);\n                }\n                Ordering::Equal =&gt; {\n                    return Some(node); // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n                }\n            }\n            Self::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n\n            /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n            node = Self::rotate(Some(node)).unwrap();\n            // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(node)\n        }\n        None =&gt; Some(TreeNode::new(val)),\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(AVLTree *tree, int val) {\n    tree-&gt;root = insertHelper(tree-&gt;root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == NULL) {\n        return newTreeNode(val);\n    }\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    } else {\n        // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node;\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfun insert(value: Int) {\n    root = insertHelper(root, value)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfun insertHelper(n: TreeNode?, value: Int): TreeNode {\n    if (n == null)\n        return TreeNode(value)\n    var node = n\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (value &lt; node.value) node.left = insertHelper(node.left, value)\n    else if (value &gt; node.value) node.right = insertHelper(node.right, value)\n    else return node // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n\n    updateHeight(node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{insert}\n\n[class]{AVLTree}-[func]{insert_helper}\n</code></pre> avl_tree.zig<pre><code>// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, val: T) !void {\n    self.root = (try self.insertHelper(self.root, val)).?;\n}\n\n// \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {\n    var node = node_;\n    if (node == null) {\n        var tmp_node = try self.mem_allocator.create(inc.TreeNode(T));\n        tmp_node.init(val);\n        return tmp_node;\n    }\n    // 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9\n    if (val &lt; node.?.val) {\n        node.?.left = try self.insertHelper(node.?.left, val);\n    } else if (val &gt; node.?.val) {\n        node.?.right = try self.insertHelper(node.?.right, val);\n    } else {\n        return node;            // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    self.updateHeight(node);    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    // 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    node = self.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2_2","title":"2. \u00a0 \u5220\u9664\u8282\u70b9","text":"<p>\u7c7b\u4f3c\u5730\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5220\u9664\u8282\u70b9\u65b9\u6cd5\u7684\u57fa\u7840\u4e0a\uff0c\u9700\u8981\u4ece\u5e95\u81f3\u9876\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def remove(self, val: int):\n    \"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n    self._root = self.remove_helper(self._root, val)\n\ndef remove_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode | None:\n    \"\"\"\u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\n    if node is None:\n        return None\n    # 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664\n    if val &lt; node.val:\n        node.left = self.remove_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.remove_helper(node.right, val)\n    else:\n        if node.left is None or node.right is None:\n            child = node.left or node.right\n            # \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if child is None:\n                return None\n            # \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else:\n                node = child\n        else:\n            # \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            temp = node.right\n            while temp.left is not None:\n                temp = temp.left\n            node.right = self.remove_helper(node.right, temp.val)\n            node.val = temp.val\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    # 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return nullptr;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    else {\n        if (node-&gt;left == nullptr || node-&gt;right == nullptr) {\n            TreeNode *child = node-&gt;left != nullptr ? node-&gt;left : node-&gt;right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == nullptr) {\n                delete node;\n                return nullptr;\n            }\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else {\n                delete node;\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != nullptr) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode removeHelper(TreeNode node, int val) {\n    if (node == null)\n        return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val)\n        node.left = removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = removeHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode child = node.left != null ? node.left : node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null)\n                return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else\n                node = child;\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int val) {\n    root = RemoveHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? RemoveHelper(TreeNode? node, int val) {\n    if (node == null) return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val)\n        node.left = RemoveHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = RemoveHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode? child = node.left ?? node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null)\n                return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else\n                node = child;\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode? temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = RemoveHelper(node.right, temp.val!.Value);\n            node.val = temp.val;\n        }\n    }\n    UpdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = Rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc (t *aVLTree) remove(val int) {\n    t.root = t.removeHelper(t.root, val)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return nil\n    }\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if val &lt; node.Val.(int) {\n        node.Left = t.removeHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.removeHelper(node.Right, val)\n    } else {\n        if node.Left == nil || node.Right == nil {\n            child := node.Left\n            if node.Right != nil {\n                child = node.Right\n            }\n            if child == nil {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n                return nil\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                node = child\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            temp := node.Right\n            for temp.Left != nil {\n                temp = temp.Left\n            }\n            node.Right = t.removeHelper(node.Right, temp.Val.(int))\n            node.Val = temp.Val\n        }\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = t.rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc remove(val: Int) {\n    root = removeHelper(node: root, val: val)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc removeHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return nil\n    }\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if val &lt; node!.val {\n        node?.left = removeHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = removeHelper(node: node?.right, val: val)\n    } else {\n        if node?.left == nil || node?.right == nil {\n            let child = node?.left ?? node?.right\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if child == nil {\n                return nil\n            }\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else {\n                node = child\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            var temp = node?.right\n            while temp?.left != nil {\n                temp = temp?.left\n            }\n            node?.right = removeHelper(node: node?.right, val: temp!.val)\n            node?.val = temp!.val\n        }\n    }\n    updateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node: node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(val) {\n    this.root = this.#removeHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#removeHelper(node, val) {\n    if (node === null) return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val) node.left = this.#removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#removeHelper(node.right, val);\n    else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child === null) return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else node = child;\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.#removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.#rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(val: number): void {\n    this.root = this.removeHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nremoveHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val) {\n        node.left = this.removeHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.removeHelper(node.right, val);\n    } else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child === null) {\n                return null;\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n  root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val) {\n  if (node == null) return null;\n  /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n  if (val &lt; node.val)\n    node.left = removeHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = removeHelper(node.right, val);\n  else {\n    if (node.left == null || node.right == null) {\n      TreeNode? child = node.left ?? node.right;\n      // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n      if (child == null)\n        return null;\n      // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n      else\n        node = child;\n    } else {\n      // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n      TreeNode? temp = node.right;\n      while (temp!.left != null) {\n        temp = temp.left;\n      }\n      node.right = removeHelper(node.right, temp.val);\n      node.val = temp.val;\n    }\n  }\n  updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n  node = rotate(node);\n  // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfn remove(&amp;self, val: i32) {\n    Self::remove_helper(self.root.clone(), val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn remove_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n            if val &lt; node.borrow().val {\n                let left = node.borrow().left.clone();\n                node.borrow_mut().left = Self::remove_helper(left, val);\n            } else if val &gt; node.borrow().val {\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, val);\n            } else if node.borrow().left.is_none() || node.borrow().right.is_none() {\n                let child = if node.borrow().left.is_some() {\n                    node.borrow().left.clone()\n                } else {\n                    node.borrow().right.clone()\n                };\n                match child {\n                    // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n                    None =&gt; {\n                        return None;\n                    }\n                    // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                    Some(child) =&gt; node = child,\n                }\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n                let mut temp = node.borrow().right.clone().unwrap();\n                loop {\n                    let temp_left = temp.borrow().left.clone();\n                    if temp_left.is_none() {\n                        break;\n                    }\n                    temp = temp_left.unwrap();\n                }\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);\n                node.borrow_mut().val = temp.borrow().val;\n            }\n            Self::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n\n            /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n            node = Self::rotate(Some(node)).unwrap();\n            // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(node)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(AVLTree *tree, int val) {\n    TreeNode *root = removeHelper(tree-&gt;root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    TreeNode *child, *grandChild;\n    if (node == NULL) {\n        return NULL;\n    }\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    } else {\n        if (node-&gt;left == NULL || node-&gt;right == NULL) {\n            child = node-&gt;left;\n            if (node-&gt;right != NULL) {\n                child = node-&gt;right;\n            }\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == NULL) {\n                return NULL;\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != NULL) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfun remove(value: Int) {\n    root = removeHelper(root, value)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfun removeHelper(n: TreeNode?, value: Int): TreeNode? {\n    var node = n ?: return null\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (value &lt; node.value) node.left = removeHelper(node.left, value)\n    else if (value &gt; node.value) node.right = removeHelper(node.right, value)\n    else {\n        if (node.left == null || node.right == null) {\n            val child = if (node.left != null) node.left else node.right\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null) return null\n            else node = child\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            var temp = node.right\n            while (temp!!.left != null) {\n                temp = temp.left\n            }\n            node.right = removeHelper(node.right, temp.value)\n            node.value = temp.value\n        }\n    }\n    updateHeight(node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{remove}\n\n[class]{AVLTree}-[func]{remove_helper}\n</code></pre> avl_tree.zig<pre><code>// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, val: T) void {\n   self.root = self.removeHelper(self.root, val).?;\n}\n\n// \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {\n    var node = node_;\n    if (node == null) return null;\n    // 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664\n    if (val &lt; node.?.val) {\n        node.?.left = self.removeHelper(node.?.left, val);\n    } else if (val &gt; node.?.val) {\n        node.?.right = self.removeHelper(node.?.right, val);\n    } else {\n        if (node.?.left == null or node.?.right == null) {\n            var child = if (node.?.left != null) node.?.left else node.?.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null) {\n                return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            } else {\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            var temp = node.?.right;\n            while (temp.?.left != null) {\n                temp = temp.?.left;\n            }\n            node.?.right = self.removeHelper(node.?.right, temp.?.val);\n            node.?.val = temp.?.val;\n        }\n    }\n    self.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    // 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    node = self.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3_1","title":"3. \u00a0 \u67e5\u627e\u8282\u70b9","text":"<p>AVL \u6811\u7684\u8282\u70b9\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u53c9\u641c\u7d22\u6811\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002</p>"},{"location":"chapter_tree/avl_tree/#754-avl","title":"7.5.4 \u00a0 AVL \u6811\u5178\u578b\u5e94\u7528","text":"<ul> <li>\u7ec4\u7ec7\u548c\u5b58\u50a8\u5927\u578b\u6570\u636e\uff0c\u9002\u7528\u4e8e\u9ad8\u9891\u67e5\u627e\u3001\u4f4e\u9891\u589e\u5220\u7684\u573a\u666f\u3002</li> <li>\u7528\u4e8e\u6784\u5efa\u6570\u636e\u5e93\u4e2d\u7684\u7d22\u5f15\u7cfb\u7edf\u3002</li> <li>\u7ea2\u9ed1\u6811\u4e5f\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\u3002\u76f8\u8f83\u4e8e AVL \u6811\uff0c\u7ea2\u9ed1\u6811\u7684\u5e73\u8861\u6761\u4ef6\u66f4\u5bbd\u677e\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u6240\u9700\u7684\u65cb\u8f6c\u64cd\u4f5c\u66f4\u5c11\uff0c\u8282\u70b9\u589e\u5220\u64cd\u4f5c\u7684\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002</li> </ul>"},{"location":"chapter_tree/binary_search_tree/","title":"7.4 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811","text":"<p>\u5982\u56fe 7-16 \u6240\u793a\uff0c\u300c\u4e8c\u53c9\u641c\u7d22\u6811 binary search tree\u300d\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\u3002</p> <ol> <li>\u5bf9\u4e8e\u6839\u8282\u70b9\uff0c\u5de6\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c \\(&lt;\\) \u6839\u8282\u70b9\u7684\u503c \\(&lt;\\) \u53f3\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c\u3002</li> <li>\u4efb\u610f\u8282\u70b9\u7684\u5de6\u3001\u53f3\u5b50\u6811\u4e5f\u662f\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5373\u540c\u6837\u6ee1\u8db3\u6761\u4ef6 <code>1.</code> \u3002</li> </ol> <p></p> <p> \u56fe 7-16 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811 </p>"},{"location":"chapter_tree/binary_search_tree/#741","title":"7.4.1 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u64cd\u4f5c","text":"<p>\u6211\u4eec\u5c06\u4e8c\u53c9\u641c\u7d22\u6811\u5c01\u88c5\u4e3a\u4e00\u4e2a\u7c7b <code>BinarySearchTree</code> \uff0c\u5e76\u58f0\u660e\u4e00\u4e2a\u6210\u5458\u53d8\u91cf <code>root</code> \uff0c\u6307\u5411\u6811\u7684\u6839\u8282\u70b9\u3002</p>"},{"location":"chapter_tree/binary_search_tree/#1","title":"1. \u00a0 \u67e5\u627e\u8282\u70b9","text":"<p>\u7ed9\u5b9a\u76ee\u6807\u8282\u70b9\u503c <code>num</code> \uff0c\u53ef\u4ee5\u6839\u636e\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u6765\u67e5\u627e\u3002\u5982\u56fe 7-17 \u6240\u793a\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u8282\u70b9 <code>cur</code> \uff0c\u4ece\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9 <code>root</code> \u51fa\u53d1\uff0c\u5faa\u73af\u6bd4\u8f83\u8282\u70b9\u503c <code>cur.val</code> \u548c <code>num</code> \u4e4b\u95f4\u7684\u5927\u5c0f\u5173\u7cfb\u3002</p> <ul> <li>\u82e5 <code>cur.val &lt; num</code> \uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728 <code>cur</code> \u7684\u53f3\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c <code>cur = cur.right</code> \u3002</li> <li>\u82e5 <code>cur.val &gt; num</code> \uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728 <code>cur</code> \u7684\u5de6\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c <code>cur = cur.left</code> \u3002</li> <li>\u82e5 <code>cur.val = num</code> \uff0c\u8bf4\u660e\u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\u5e76\u8fd4\u56de\u8be5\u8282\u70b9\u3002</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-17 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u67e5\u627e\u8282\u70b9\u793a\u4f8b </p> <p>\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u4e00\u81f4\uff0c\u90fd\u662f\u6bcf\u8f6e\u6392\u9664\u4e00\u534a\u60c5\u51b5\u3002\u5faa\u73af\u6b21\u6570\u6700\u591a\u4e3a\u4e8c\u53c9\u6811\u7684\u9ad8\u5ea6\uff0c\u5f53\u4e8c\u53c9\u6811\u5e73\u8861\u65f6\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def search(self, num: int) -&gt; TreeNode | None:\n    \"\"\"\u67e5\u627e\u8282\u70b9\"\"\"\n    cur = self._root\n    # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur is not None:\n        # \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        elif cur.val &gt; num:\n            cur = cur.left\n        # \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else:\n            break\n    return cur\n</code></pre> binary_search_tree.cpp<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(int num) {\n    TreeNode *cur = root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur-&gt;val &gt; num)\n            cur = cur-&gt;left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode search(int num) {\n    TreeNode cur = root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? Search(int num) {\n    TreeNode? cur = root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur =\n            cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\n    node := bst.root\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    for node != nil {\n        if node.Val.(int) &lt; num {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            node = node.Right\n        } else if node.Val.(int) &gt; num {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            node = node.Left\n        } else {\n            // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            break\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return node\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfunc search(num: Int) -&gt; TreeNode? {\n    var cur = root\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur != nil {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if cur!.val &gt; num {\n            cur = cur?.left\n        }\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else {\n            break\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nsearch(num) {\n    let cur = this.root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num) cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nsearch(num: number): TreeNode | null {\n    let cur = this.root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num) cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? search(int _num) {\n  TreeNode? cur = _root;\n  // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n  while (cur != null) {\n    // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n    else if (cur.val &gt; _num)\n      cur = cur.left;\n    // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n    else\n      break;\n  }\n  // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n  return cur;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u67e5\u627e\u8282\u70b9 */\npub fn search(&amp;self, num: i32) -&gt; OptionTreeNodeRc {\n    let mut cur = self.root.clone();\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            Ordering::Greater =&gt; cur = node.borrow().right.clone(),\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            Ordering::Less =&gt; cur = node.borrow().left.clone(),\n            // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            Ordering::Equal =&gt; break,\n        }\n    }\n\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    cur\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(BinarySearchTree *bst, int num) {\n    TreeNode *cur = bst-&gt;root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != NULL) {\n        if (cur-&gt;val &lt; num) {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            cur = cur-&gt;right;\n        } else if (cur-&gt;val &gt; num) {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            cur = cur-&gt;left;\n        } else {\n            // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfun search(num: Int): TreeNode? {\n    var cur = root\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        cur = if (cur.value &lt; num) cur.right\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.value &gt; num) cur.left\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else break\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur\n}\n</code></pre> binary_search_tree.rb<pre><code>[class]{BinarySearchTree}-[func]{search}\n</code></pre> binary_search_tree.zig<pre><code>// \u67e5\u627e\u8282\u70b9\nfn search(self: *Self, num: T) ?*inc.TreeNode(T) {\n    var cur = self.root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        } else if (cur.?.val &gt; num) {\n            cur = cur.?.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        } else {\n            break;\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_tree/binary_search_tree/#2","title":"2. \u00a0 \u63d2\u5165\u8282\u70b9","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u5f85\u63d2\u5165\u5143\u7d20 <code>num</code> \uff0c\u4e3a\u4e86\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 &lt; \u6839\u8282\u70b9 &lt; \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u63d2\u5165\u64cd\u4f5c\u6d41\u7a0b\u5982\u56fe 7-18 \u6240\u793a\u3002</p> <ol> <li>\u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff1a\u4e0e\u67e5\u627e\u64cd\u4f5c\u76f8\u4f3c\uff0c\u4ece\u6839\u8282\u70b9\u51fa\u53d1\uff0c\u6839\u636e\u5f53\u524d\u8282\u70b9\u503c\u548c <code>num</code> \u7684\u5927\u5c0f\u5173\u7cfb\u5faa\u73af\u5411\u4e0b\u641c\u7d22\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\uff08\u904d\u5386\u81f3 <code>None</code> \uff09\u65f6\u8df3\u51fa\u5faa\u73af\u3002</li> <li>\u5728\u8be5\u4f4d\u7f6e\u63d2\u5165\u8282\u70b9\uff1a\u521d\u59cb\u5316\u8282\u70b9 <code>num</code> \uff0c\u5c06\u8be5\u8282\u70b9\u7f6e\u4e8e <code>None</code> \u7684\u4f4d\u7f6e\u3002</li> </ol> <p></p> <p> \u56fe 7-18 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u63d2\u5165\u8282\u70b9 </p> <p>\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u4e24\u70b9\u3002</p> <ul> <li>\u4e8c\u53c9\u641c\u7d22\u6811\u4e0d\u5141\u8bb8\u5b58\u5728\u91cd\u590d\u8282\u70b9\uff0c\u5426\u5219\u5c06\u8fdd\u53cd\u5176\u5b9a\u4e49\u3002\u56e0\u6b64\uff0c\u82e5\u5f85\u63d2\u5165\u8282\u70b9\u5728\u6811\u4e2d\u5df2\u5b58\u5728\uff0c\u5219\u4e0d\u6267\u884c\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\u3002</li> <li>\u4e3a\u4e86\u5b9e\u73b0\u63d2\u5165\u8282\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u8282\u70b9 <code>pre</code> \u4fdd\u5b58\u4e0a\u4e00\u8f6e\u5faa\u73af\u7684\u8282\u70b9\u3002\u8fd9\u6837\u5728\u904d\u5386\u81f3 <code>None</code> \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u83b7\u53d6\u5230\u5176\u7236\u8282\u70b9\uff0c\u4ece\u800c\u5b8c\u6210\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def insert(self, num: int):\n    \"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n    # \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if self._root is None:\n        self._root = TreeNode(num)\n        return\n    # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    cur, pre = self._root, None\n    while cur is not None:\n        # \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if cur.val == num:\n            return\n        pre = cur\n        # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else:\n            cur = cur.left\n    # \u63d2\u5165\u8282\u70b9\n    node = TreeNode(num)\n    if pre.val &lt; num:\n        pre.right = node\n    else:\n        pre.left = node\n</code></pre> binary_search_tree.cpp<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == nullptr) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode *cur = root, *pre = nullptr;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur-&gt;val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur-&gt;left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode *node = new TreeNode(num);\n    if (pre-&gt;val &lt; num)\n        pre-&gt;right = node;\n    else\n        pre-&gt;left = node;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode node = new TreeNode(num);\n    if (pre.val &lt; num)\n        pre.right = node;\n    else\n        pre.left = node;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode? cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode node = new(num);\n    if (pre != null) {\n        if (pre.val &lt; num)\n            pre.right = node;\n        else\n            pre.left = node;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc (bst *binarySearchTree) insert(num int) {\n    cur := bst.root\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if cur == nil {\n        bst.root = NewTreeNode(num)\n        return\n    }\n    // \u5f85\u63d2\u5165\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\n    var pre *TreeNode = nil\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    for cur != nil {\n        if cur.Val == num {\n            return\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            cur = cur.Right\n        } else {\n            cur = cur.Left\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    node := NewTreeNode(num)\n    if pre.Val.(int) &lt; num {\n        pre.Right = node\n    } else {\n        pre.Left = node\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if root == nil {\n        root = TreeNode(x: num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur != nil {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if cur!.val == num {\n            return\n        }\n        pre = cur\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else {\n            cur = cur?.left\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    let node = TreeNode(x: num)\n    if pre!.val &lt; num {\n        pre?.right = node\n    } else {\n        pre?.left = node\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur = this.root,\n        pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val === num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    const node = new TreeNode(num);\n    if (pre.val &lt; num) pre.right = node;\n    else pre.left = node;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(num: number): void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val === num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    const node = new TreeNode(num);\n    if (pre!.val &lt; num) pre!.right = node;\n    else pre!.left = node;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int _num) {\n  // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n  if (_root == null) {\n    _root = TreeNode(_num);\n    return;\n  }\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n  while (cur != null) {\n    // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n    if (cur.val == _num) return;\n    pre = cur;\n    // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n    else\n      cur = cur.left;\n  }\n  // \u63d2\u5165\u8282\u70b9\n  TreeNode? node = TreeNode(_num);\n  if (pre!.val &lt; _num)\n    pre.right = node;\n  else\n    pre.left = node;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\npub fn insert(&amp;mut self, num: i32) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if self.root.is_none() {\n        self.root = Some(TreeNode::new(num));\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n            Ordering::Equal =&gt; return,\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    let pre = pre.unwrap();\n    let node = Some(TreeNode::new(num));\n    if num &gt; pre.borrow().val {\n        pre.borrow_mut().right = node;\n    } else {\n        pre.borrow_mut().left = node;\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(BinarySearchTree *bst, int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (bst-&gt;root == NULL) {\n        bst-&gt;root = newTreeNode(num);\n        return;\n    }\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != NULL) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur-&gt;val == num) {\n            return;\n        }\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            cur = cur-&gt;right;\n        } else {\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            cur = cur-&gt;left;\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode *node = newTreeNode(num);\n    if (pre-&gt;val &lt; num) {\n        pre-&gt;right = node;\n    } else {\n        pre-&gt;left = node;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfun insert(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == null) {\n        root = TreeNode(num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode? = null\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.value == num) return\n        pre = cur\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        cur = if (cur.value &lt; num) cur.right\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur.left\n    }\n    // \u63d2\u5165\u8282\u70b9\n    val node = TreeNode(num)\n    if (pre?.value!! &lt; num) pre.right = node\n    else pre.left = node\n}\n</code></pre> binary_search_tree.rb<pre><code>[class]{BinarySearchTree}-[func]{insert}\n</code></pre> binary_search_tree.zig<pre><code>// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, num: T) !void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (self.root == null) {\n        self.root = try self.mem_allocator.create(inc.TreeNode(T));\n        return;\n    }\n    var cur = self.root;\n    var pre: ?*inc.TreeNode(T) = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.?.val == num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        } else {\n            cur = cur.?.left;\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    var node = try self.mem_allocator.create(inc.TreeNode(T));\n    node.init(num);\n    if (pre.?.val &lt; num) {\n        pre.?.right = node;\n    } else {\n        pre.?.left = node;\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u4e0e\u67e5\u627e\u8282\u70b9\u76f8\u540c\uff0c\u63d2\u5165\u8282\u70b9\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002</p>"},{"location":"chapter_tree/binary_search_tree/#3","title":"3. \u00a0 \u5220\u9664\u8282\u70b9","text":"<p>\u5148\u5728\u4e8c\u53c9\u6811\u4e2d\u67e5\u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u518d\u5c06\u5176\u5220\u9664\u3002\u4e0e\u63d2\u5165\u8282\u70b9\u7c7b\u4f3c\uff0c\u6211\u4eec\u9700\u8981\u4fdd\u8bc1\u5728\u5220\u9664\u64cd\u4f5c\u5b8c\u6210\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u201c\u5de6\u5b50\u6811 &lt; \u6839\u8282\u70b9 &lt; \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\u4ecd\u7136\u6ee1\u8db3\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u6839\u636e\u76ee\u6807\u8282\u70b9\u7684\u5b50\u8282\u70b9\u6570\u91cf\uff0c\u5206 0\u30011 \u548c 2 \u4e09\u79cd\u60c5\u51b5\uff0c\u6267\u884c\u5bf9\u5e94\u7684\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u3002</p> <p>\u5982\u56fe 7-19 \u6240\u793a\uff0c\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \u65f6\uff0c\u8868\u793a\u8be5\u8282\u70b9\u662f\u53f6\u8282\u70b9\uff0c\u53ef\u4ee5\u76f4\u63a5\u5220\u9664\u3002</p> <p></p> <p> \u56fe 7-19 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 0 \uff09 </p> <p>\u5982\u56fe 7-20 \u6240\u793a\uff0c\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(1\\) \u65f6\uff0c\u5c06\u5f85\u5220\u9664\u8282\u70b9\u66ff\u6362\u4e3a\u5176\u5b50\u8282\u70b9\u5373\u53ef\u3002</p> <p></p> <p> \u56fe 7-20 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 1 \uff09 </p> <p>\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(2\\) \u65f6\uff0c\u6211\u4eec\u65e0\u6cd5\u76f4\u63a5\u5220\u9664\u5b83\uff0c\u800c\u9700\u8981\u4f7f\u7528\u4e00\u4e2a\u8282\u70b9\u66ff\u6362\u8be5\u8282\u70b9\u3002\u7531\u4e8e\u8981\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 \\(&lt;\\) \u6839\u8282\u70b9 \\(&lt;\\) \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u56e0\u6b64\u8fd9\u4e2a\u8282\u70b9\u53ef\u4ee5\u662f\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\u6216\u5de6\u5b50\u6811\u7684\u6700\u5927\u8282\u70b9\u3002</p> <p>\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\uff08\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\uff09\uff0c\u5219\u5220\u9664\u64cd\u4f5c\u6d41\u7a0b\u5982\u56fe 7-21 \u6240\u793a\u3002</p> <ol> <li>\u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\u5728\u201c\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u201d\u4e2d\u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\uff0c\u8bb0\u4e3a <code>tmp</code> \u3002</li> <li>\u7528 <code>tmp</code> \u7684\u503c\u8986\u76d6\u5f85\u5220\u9664\u8282\u70b9\u7684\u503c\uff0c\u5e76\u5728\u6811\u4e2d\u9012\u5f52\u5220\u9664\u8282\u70b9 <code>tmp</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-21 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 2 \uff09 </p> <p>\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u540c\u6837\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d\u67e5\u627e\u5f85\u5220\u9664\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u540e\u7ee7\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def remove(self, num: int):\n    \"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n    # \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if self._root is None:\n        return\n    # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    cur, pre = self._root, None\n    while cur is not None:\n        # \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if cur.val == num:\n            break\n        pre = cur\n        # \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else:\n            cur = cur.left\n    # \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur is None:\n        return\n\n    # \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if cur.left is None or cur.right is None:\n        # \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        child = cur.left or cur.right\n        # \u5220\u9664\u8282\u70b9 cur\n        if cur != self._root:\n            if pre.left == cur:\n                pre.left = child\n            else:\n                pre.right = child\n        else:\n            # \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            self._root = child\n    # \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else:\n        # \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        tmp: TreeNode = cur.right\n        while tmp.left is not None:\n            tmp = tmp.left\n        # \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        self.remove(tmp.val)\n        # \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val\n</code></pre> binary_search_tree.cpp<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == nullptr)\n        return;\n    TreeNode *cur = root, *pre = nullptr;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur-&gt;left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == nullptr)\n        return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur-&gt;left == nullptr || cur-&gt;right == nullptr) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n        TreeNode *child = cur-&gt;left != nullptr ? cur-&gt;left : cur-&gt;right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre-&gt;left == cur)\n                pre-&gt;left = child;\n            else\n                pre-&gt;right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child;\n        }\n        // \u91ca\u653e\u5185\u5b58\n        delete cur;\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != nullptr) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(tmp-&gt;val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return;\n    TreeNode cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        TreeNode child = cur.left != null ? cur.left : cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(tmp.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return;\n    TreeNode? cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        TreeNode? child = cur.left ?? cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre!.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode? tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        Remove(tmp.val!.Value);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc (bst *binarySearchTree) remove(num int) {\n    cur := bst.root\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5f85\u5220\u9664\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\n    var pre *TreeNode = nil\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    for cur != nil {\n        if cur.Val == num {\n            break\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728\u53f3\u5b50\u6811\u4e2d\n            cur = cur.Right\n        } else {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728\u5de6\u5b50\u6811\u4e2d\n            cur = cur.Left\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5b50\u8282\u70b9\u6570\u4e3a 0 \u6216 1\n    if cur.Left == nil || cur.Right == nil {\n        var child *TreeNode = nil\n        // \u53d6\u51fa\u5f85\u5220\u9664\u8282\u70b9\u7684\u5b50\u8282\u70b9\n        if cur.Left != nil {\n            child = cur.Left\n        } else {\n            child = cur.Right\n        }\n        // \u5220\u9664\u8282\u70b9 cur\n        if cur != bst.root {\n            if pre.Left == cur {\n                pre.Left = child\n            } else {\n                pre.Right = child\n            }\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            bst.root = child\n        }\n        // \u5b50\u8282\u70b9\u6570\u4e3a 2\n    } else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u5f85\u5220\u9664\u8282\u70b9 cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        tmp := cur.Right\n        for tmp.Left != nil {\n            tmp = tmp.Left\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        bst.remove(tmp.Val.(int))\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.Val = tmp.Val\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc remove(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if root == nil {\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur != nil {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if cur!.val == num {\n            break\n        }\n        pre = cur\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else {\n            cur = cur?.left\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if cur?.left == nil || cur?.right == nil {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        let child = cur?.left ?? cur?.right\n        // \u5220\u9664\u8282\u70b9 cur\n        if cur !== root {\n            if pre?.left === cur {\n                pre?.left = child\n            } else {\n                pre?.right = child\n            }\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        var tmp = cur?.right\n        while tmp?.left != nil {\n            tmp = tmp?.left\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(num: tmp!.val)\n        // \u7528 tmp \u8986\u76d6 cur\n        cur?.val = tmp!.val\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (this.root === null) return;\n    let cur = this.root,\n        pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val === num) break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur === null) return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        const child = cur.left !== null ? cur.left : cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur !== this.root) {\n            if (pre.left === cur) pre.left = child;\n            else pre.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            this.root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        let tmp = cur.right;\n        while (tmp.left !== null) {\n            tmp = tmp.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        this.remove(tmp.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(num: number): void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (this.root === null) return;\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val === num) break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur === null) return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        const child: TreeNode | null =\n            cur.left !== null ? cur.left : cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur !== this.root) {\n            if (pre!.left === cur) pre!.left = child;\n            else pre!.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            this.root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        let tmp: TreeNode | null = cur.right;\n        while (tmp!.left !== null) {\n            tmp = tmp!.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        this.remove(tmp!.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp!.val;\n    }\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int _num) {\n  // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n  if (_root == null) return;\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n  while (cur != null) {\n    // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n    if (cur.val == _num) break;\n    pre = cur;\n    // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n    else\n      cur = cur.left;\n  }\n  // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n  if (cur == null) return;\n  // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n  if (cur.left == null || cur.right == null) {\n    // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n    TreeNode? child = cur.left ?? cur.right;\n    // \u5220\u9664\u8282\u70b9 cur\n    if (cur != _root) {\n      if (pre!.left == cur)\n        pre.left = child;\n      else\n        pre.right = child;\n    } else {\n      // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n      _root = child;\n    }\n  } else {\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n    TreeNode? tmp = cur.right;\n    while (tmp!.left != null) {\n      tmp = tmp.left;\n    }\n    // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n    remove(tmp.val);\n    // \u7528 tmp \u8986\u76d6 cur\n    cur.val = tmp.val;\n  }\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u5220\u9664\u8282\u70b9 */\npub fn remove(&amp;mut self, num: i32) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if self.root.is_none() {\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            Ordering::Equal =&gt; break,\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur.is_none() {\n        return;\n    }\n    let cur = cur.unwrap();\n    let (left_child, right_child) = (cur.borrow().left.clone(), cur.borrow().right.clone());\n    match (left_child.clone(), right_child.clone()) {\n        // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n        (None, None) | (Some(_), None) | (None, Some(_)) =&gt; {\n            // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n            let child = left_child.or(right_child);\n            let pre = pre.unwrap();\n            // \u5220\u9664\u8282\u70b9 cur\n            if !Rc::ptr_eq(&amp;cur, self.root.as_ref().unwrap()) {\n                let left = pre.borrow().left.clone();\n                if left.is_some() &amp;&amp; Rc::ptr_eq(&amp;left.as_ref().unwrap(), &amp;cur) {\n                    pre.borrow_mut().left = child;\n                } else {\n                    pre.borrow_mut().right = child;\n                }\n            } else {\n                // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n                self.root = child;\n            }\n        }\n        // \u5b50\u8282\u70b9\u6570\u91cf = 2\n        (Some(_), Some(_)) =&gt; {\n            // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n            let mut tmp = cur.borrow().right.clone();\n            while let Some(node) = tmp.clone() {\n                if node.borrow().left.is_some() {\n                    tmp = node.borrow().left.clone();\n                } else {\n                    break;\n                }\n            }\n            let tmpval = tmp.unwrap().borrow().val;\n            // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n            self.remove(tmpval);\n            // \u7528 tmp \u8986\u76d6 cur\n            cur.borrow_mut().val = tmpval;\n        }\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(BinarySearchTree *bst, int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (bst-&gt;root == NULL)\n        return;\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != NULL) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u53f3\u5b50\u6811\u4e2d\n            cur = cur-&gt;right;\n        } else {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u5de6\u5b50\u6811\u4e2d\n            cur = cur-&gt;left;\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == NULL)\n        return;\n    // \u5224\u65ad\u5f85\u5220\u9664\u8282\u70b9\u662f\u5426\u5b58\u5728\u5b50\u8282\u70b9\n    if (cur-&gt;left == NULL || cur-&gt;right == NULL) {\n        /* \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1 */\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n        TreeNode *child = cur-&gt;left != NULL ? cur-&gt;left : cur-&gt;right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (pre-&gt;left == cur) {\n            pre-&gt;left = child;\n        } else {\n            pre-&gt;right = child;\n        }\n        // \u91ca\u653e\u5185\u5b58\n        free(cur);\n    } else {\n        /* \u5b50\u8282\u70b9\u6570\u91cf = 2 */\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != NULL) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        removeItem(bst, tmp-&gt;val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfun remove(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null) return\n    var cur = root\n    var pre: TreeNode? = null\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.value == num) break\n        pre = cur\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        cur = if (cur.value &lt; num) cur.right\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur.left\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null) return\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        val child = if (cur.left != null) cur.left else cur.right\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre!!.left == cur) pre.left = child\n            else pre.right = child\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child\n        }\n        // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    } else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        var tmp = cur.right\n        while (tmp!!.left != null) {\n            tmp = tmp.left\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(tmp.value)\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.value = tmp.value\n    }\n}\n</code></pre> binary_search_tree.rb<pre><code>[class]{BinarySearchTree}-[func]{remove}\n</code></pre> binary_search_tree.zig<pre><code>// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, num: T) void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (self.root == null) return;\n    var cur = self.root;\n    var pre: ?*inc.TreeNode(T) = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.?.val == num) break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        } else {\n            cur = cur.?.left;\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null) return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.?.left == null or cur.?.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        var child = if (cur.?.left != null) cur.?.left else cur.?.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (pre.?.left == cur) {\n            pre.?.left = child;\n        } else {\n            pre.?.right = child;\n        }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    } else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        var tmp = cur.?.right;\n        while (tmp.?.left != null) {\n            tmp = tmp.?.left;\n        }\n        var tmp_val = tmp.?.val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        self.remove(tmp.?.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.?.val = tmp_val;\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_tree/binary_search_tree/#4","title":"4. \u00a0 \u4e2d\u5e8f\u904d\u5386\u6709\u5e8f","text":"<p>\u5982\u56fe 7-22 \u6240\u793a\uff0c\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u9075\u5faa\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u904d\u5386\u987a\u5e8f\uff0c\u800c\u4e8c\u53c9\u641c\u7d22\u6811\u6ee1\u8db3\u201c\u5de6\u5b50\u8282\u70b9 \\(&lt;\\) \u6839\u8282\u70b9 \\(&lt;\\) \u53f3\u5b50\u8282\u70b9\u201d\u7684\u5927\u5c0f\u5173\u7cfb\u3002</p> <p>\u8fd9\u610f\u5473\u7740\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u8fdb\u884c\u4e2d\u5e8f\u904d\u5386\u65f6\uff0c\u603b\u662f\u4f1a\u4f18\u5148\u904d\u5386\u4e0b\u4e00\u4e2a\u6700\u5c0f\u8282\u70b9\uff0c\u4ece\u800c\u5f97\u51fa\u4e00\u4e2a\u91cd\u8981\u6027\u8d28\uff1a\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u662f\u5347\u5e8f\u7684\u3002</p> <p>\u5229\u7528\u4e2d\u5e8f\u904d\u5386\u5347\u5e8f\u7684\u6027\u8d28\uff0c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u83b7\u53d6\u6709\u5e8f\u6570\u636e\u4ec5\u9700 \\(O(n)\\) \u65f6\u95f4\uff0c\u65e0\u987b\u8fdb\u884c\u989d\u5916\u7684\u6392\u5e8f\u64cd\u4f5c\uff0c\u975e\u5e38\u9ad8\u6548\u3002</p> <p></p> <p> \u56fe 7-22 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217 </p>"},{"location":"chapter_tree/binary_search_tree/#742","title":"7.4.2 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387","text":"<p>\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u8003\u8651\u4f7f\u7528\u6570\u7ec4\u6216\u4e8c\u53c9\u641c\u7d22\u6811\u5b58\u50a8\u3002\u89c2\u5bdf\u8868 7-2 \uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5404\u9879\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f\u5bf9\u6570\u9636\uff0c\u5177\u6709\u7a33\u5b9a\u4e14\u9ad8\u6548\u7684\u6027\u80fd\u3002\u53ea\u6709\u5728\u9ad8\u9891\u6dfb\u52a0\u3001\u4f4e\u9891\u67e5\u627e\u5220\u9664\u6570\u636e\u7684\u573a\u666f\u4e0b\uff0c\u6570\u7ec4\u6bd4\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387\u66f4\u9ad8\u3002</p> <p> \u8868 7-2 \u00a0 \u6570\u7ec4\u4e0e\u641c\u7d22\u6811\u7684\u6548\u7387\u5bf9\u6bd4 </p> \u65e0\u5e8f\u6570\u7ec4 \u4e8c\u53c9\u641c\u7d22\u6811 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(\\log n)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) <p>\u5728\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u662f\u201c\u5e73\u8861\u201d\u7684\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5728 \\(\\log n\\) \u8f6e\u5faa\u73af\u5185\u67e5\u627e\u4efb\u610f\u8282\u70b9\u3002</p> <p>\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u4e0d\u65ad\u5730\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\uff0c\u53ef\u80fd\u5bfc\u81f4\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u56fe 7-23 \u6240\u793a\u7684\u94fe\u8868\uff0c\u8fd9\u65f6\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4f1a\u9000\u5316\u4e3a \\(O(n)\\) \u3002</p> <p></p> <p> \u56fe 7-23 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316 </p>"},{"location":"chapter_tree/binary_search_tree/#743","title":"7.4.3 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u5e38\u89c1\u5e94\u7528","text":"<ul> <li>\u7528\u4f5c\u7cfb\u7edf\u4e2d\u7684\u591a\u7ea7\u7d22\u5f15\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u3002</li> <li>\u4f5c\u4e3a\u67d0\u4e9b\u641c\u7d22\u7b97\u6cd5\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u7528\u4e8e\u5b58\u50a8\u6570\u636e\u6d41\uff0c\u4ee5\u4fdd\u6301\u5176\u6709\u5e8f\u72b6\u6001\u3002</li> </ul>"},{"location":"chapter_tree/binary_tree/","title":"7.1 \u00a0 \u4e8c\u53c9\u6811","text":"<p>\u300c\u4e8c\u53c9\u6811 binary tree\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4ee3\u8868\u201c\u7956\u5148\u201d\u4e0e\u201c\u540e\u4ee3\u201d\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff0c\u4f53\u73b0\u4e86\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u4e8c\u53c9\u6811\u7684\u57fa\u672c\u5355\u5143\u662f\u8282\u70b9\uff0c\u6bcf\u4e2a\u8282\u70b9\u5305\u542b\u503c\u3001\u5de6\u5b50\u8282\u70b9\u5f15\u7528\u548c\u53f3\u5b50\u8282\u70b9\u5f15\u7528\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class TreeNode:\n    \"\"\"\u4e8c\u53c9\u6811\u8282\u70b9\u7c7b\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # \u8282\u70b9\u503c\n        self.left: TreeNode | None = None  # \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n        self.right: TreeNode | None = None # \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\n    int val;          // \u8282\u70b9\u503c\n    TreeNode *left;   // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n    TreeNode *right;  // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n    TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    int val;         // \u8282\u70b9\u503c\n    TreeNode left;   // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    TreeNode right;  // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n    TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode(int? x) {\n    public int? val = x;    // \u8282\u70b9\u503c\n    public TreeNode? left;  // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    public TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\n    Val   int\n    Left  *TreeNode\n    Right *TreeNode\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc NewTreeNode(v int) *TreeNode {\n    return &amp;TreeNode{\n        Left:  nil, // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n        Right: nil, // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n        Val:   v,   // \u8282\u70b9\u503c\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    var val: Int // \u8282\u70b9\u503c\n    var left: TreeNode? // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    var right: TreeNode? // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n\n    init(x: Int) {\n        val = x\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    val; // \u8282\u70b9\u503c\n    left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n    right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n    constructor(val, left, right) {\n        this.val = val === undefined ? 0 : val;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    val: number;\n    left: TreeNode | null;\n    right: TreeNode | null;\n\n    constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n        this.left = left === undefined ? null : left; // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n        this.right = right === undefined ? null : right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n  int val;         // \u8282\u70b9\u503c\n  TreeNode? left;  // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n  TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n  TreeNode(this.val, [this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\n    val: i32,                               // \u8282\u70b9\u503c\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n\nimpl TreeNode {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct TreeNode {\n    int val;                // \u8282\u70b9\u503c\n    int height;             // \u8282\u70b9\u9ad8\u5ea6\n    struct TreeNode *left;  // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n    struct TreeNode *right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n} TreeNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode(val _val: Int) {  // \u8282\u70b9\u503c\n    val left: TreeNode? = null   // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    val right: TreeNode? = null  // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>\u6bcf\u4e2a\u8282\u70b9\u90fd\u6709\u4e24\u4e2a\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u5206\u522b\u6307\u5411\u300c\u5de6\u5b50\u8282\u70b9 left-child node\u300d\u548c\u300c\u53f3\u5b50\u8282\u70b9 right-child node\u300d\uff0c\u8be5\u8282\u70b9\u88ab\u79f0\u4e3a\u8fd9\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u300c\u7236\u8282\u70b9 parent node\u300d\u3002\u5f53\u7ed9\u5b9a\u4e00\u4e2a\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u65f6\uff0c\u6211\u4eec\u5c06\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u79f0\u4e3a\u8be5\u8282\u70b9\u7684\u300c\u5de6\u5b50\u6811 left subtree\u300d\uff0c\u540c\u7406\u53ef\u5f97\u300c\u53f3\u5b50\u6811 right subtree\u300d\u3002</p> <p>\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u9664\u53f6\u8282\u70b9\u5916\uff0c\u5176\u4ed6\u6240\u6709\u8282\u70b9\u90fd\u5305\u542b\u5b50\u8282\u70b9\u548c\u975e\u7a7a\u5b50\u6811\u3002\u5982\u56fe 7-1 \u6240\u793a\uff0c\u5982\u679c\u5c06\u201c\u8282\u70b9 2\u201d\u89c6\u4e3a\u7236\u8282\u70b9\uff0c\u5219\u5176\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\u5206\u522b\u662f\u201c\u8282\u70b9 4\u201d\u548c\u201c\u8282\u70b9 5\u201d\uff0c\u5de6\u5b50\u6811\u662f\u201c\u8282\u70b9 4 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\uff0c\u53f3\u5b50\u6811\u662f\u201c\u8282\u70b9 5 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\u3002</p> <p></p> <p> \u56fe 7-1 \u00a0 \u7236\u8282\u70b9\u3001\u5b50\u8282\u70b9\u3001\u5b50\u6811 </p>"},{"location":"chapter_tree/binary_tree/#711","title":"7.1.1 \u00a0 \u4e8c\u53c9\u6811\u5e38\u89c1\u672f\u8bed","text":"<p>\u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed\u5982\u56fe 7-2 \u6240\u793a\u3002</p> <ul> <li>\u300c\u6839\u8282\u70b9 root node\u300d\uff1a\u4f4d\u4e8e\u4e8c\u53c9\u6811\u9876\u5c42\u7684\u8282\u70b9\uff0c\u6ca1\u6709\u7236\u8282\u70b9\u3002</li> <li>\u300c\u53f6\u8282\u70b9 leaf node\u300d\uff1a\u6ca1\u6709\u5b50\u8282\u70b9\u7684\u8282\u70b9\uff0c\u5176\u4e24\u4e2a\u6307\u9488\u5747\u6307\u5411 <code>None</code> \u3002</li> <li>\u300c\u8fb9 edge\u300d\uff1a\u8fde\u63a5\u4e24\u4e2a\u8282\u70b9\u7684\u7ebf\u6bb5\uff0c\u5373\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\u3002</li> <li>\u8282\u70b9\u6240\u5728\u7684\u300c\u5c42 level\u300d\uff1a\u4ece\u9876\u81f3\u5e95\u9012\u589e\uff0c\u6839\u8282\u70b9\u6240\u5728\u5c42\u4e3a 1 \u3002</li> <li>\u8282\u70b9\u7684\u300c\u5ea6 degree\u300d\uff1a\u8282\u70b9\u7684\u5b50\u8282\u70b9\u7684\u6570\u91cf\u3002\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u5ea6\u7684\u53d6\u503c\u8303\u56f4\u662f 0\u30011\u30012 \u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u300c\u9ad8\u5ea6 height\u300d\uff1a\u4ece\u6839\u8282\u70b9\u5230\u6700\u8fdc\u53f6\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002</li> <li>\u8282\u70b9\u7684\u300c\u6df1\u5ea6 depth\u300d\uff1a\u4ece\u6839\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002</li> <li>\u8282\u70b9\u7684\u300c\u9ad8\u5ea6 height\u300d\uff1a\u4ece\u8ddd\u79bb\u8be5\u8282\u70b9\u6700\u8fdc\u7684\u53f6\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002</li> </ul> <p></p> <p> \u56fe 7-2 \u00a0 \u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed </p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u6211\u4eec\u901a\u5e38\u5c06\u201c\u9ad8\u5ea6\u201d\u548c\u201c\u6df1\u5ea6\u201d\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u201d\uff0c\u4f46\u6709\u4e9b\u9898\u76ee\u6216\u6559\u6750\u53ef\u80fd\u4f1a\u5c06\u5176\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8282\u70b9\u7684\u6570\u91cf\u201d\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u9ad8\u5ea6\u548c\u6df1\u5ea6\u90fd\u9700\u8981\u52a0 1 \u3002</p>"},{"location":"chapter_tree/binary_tree/#712","title":"7.1.2 \u00a0 \u4e8c\u53c9\u6811\u57fa\u672c\u64cd\u4f5c","text":""},{"location":"chapter_tree/binary_tree/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u4e8c\u53c9\u6811","text":"<p>\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u9996\u5148\u521d\u59cb\u5316\u8282\u70b9\uff0c\u7136\u540e\u6784\u5efa\u5f15\u7528\uff08\u6307\u9488\uff09\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree.py<pre><code># \u521d\u59cb\u5316\u4e8c\u53c9\u6811\n# \u521d\u59cb\u5316\u8282\u70b9\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.cpp<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.java<pre><code>// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.cs<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new(1);\nTreeNode n2 = new(2);\nTreeNode n3 = new(3);\nTreeNode n4 = new(4);\nTreeNode n5 = new(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.go<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n</code></pre> binary_tree.swift<pre><code>// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.js<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.ts<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.dart<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.rs<pre><code>// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = TreeNode::new(1);\nlet n2 = TreeNode::new(2);\nlet n3 = TreeNode::new(3);\nlet n4 = TreeNode::new(4);\nlet n5 = TreeNode::new(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.borrow_mut().left = Some(n2.clone());\nn1.borrow_mut().right = Some(n3);\nn2.borrow_mut().left = Some(n4);\nn2.borrow_mut().right = Some(n5);\n</code></pre> binary_tree.c<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.kt<pre><code>// \u521d\u59cb\u5316\u8282\u70b9\nval n1 = TreeNode(1)\nval n2 = TreeNode(2)\nval n3 = TreeNode(3)\nval n4 = TreeNode(4)\nval n5 = TreeNode(5)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.rb<pre><code>\n</code></pre> binary_tree.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_tree/binary_tree/#2","title":"2. \u00a0 \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9","text":"<p>\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u53ef\u4ee5\u901a\u8fc7\u4fee\u6539\u6307\u9488\u6765\u5b9e\u73b0\u3002\u56fe 7-3 \u7ed9\u51fa\u4e86\u4e00\u4e2a\u793a\u4f8b\u3002</p> <p></p> <p> \u56fe 7-3 \u00a0 \u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree.py<pre><code># \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\np = TreeNode(0)\n# \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = p\np.left = n2\n# \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n</code></pre> binary_tree.cpp<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode* P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1-&gt;left = P;\nP-&gt;left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1-&gt;left = n2;\n</code></pre> binary_tree.java<pre><code>TreeNode P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.cs<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.go<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.Left = n2\n</code></pre> binary_tree.swift<pre><code>let P = TreeNode(x: 0)\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P\nP.left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n</code></pre> binary_tree.js<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nlet P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.ts<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nconst P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.dart<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.rs<pre><code>let p = TreeNode::new(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.borrow_mut().left = Some(p.clone());\np.borrow_mut().left = Some(n2.clone());\n// \u5220\u9664\u8282\u70b9 p\nn1.borrow_mut().left = Some(n2);\n</code></pre> binary_tree.c<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode *P = newTreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1-&gt;left = P;\nP-&gt;left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1-&gt;left = n2;\n</code></pre> binary_tree.kt<pre><code>val P = TreeNode(0)\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P\nP.left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n</code></pre> binary_tree.rb<pre><code>\n</code></pre> binary_tree.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>Note</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u63d2\u5165\u8282\u70b9\u53ef\u80fd\u4f1a\u6539\u53d8\u4e8c\u53c9\u6811\u7684\u539f\u6709\u903b\u8f91\u7ed3\u6784\uff0c\u800c\u5220\u9664\u8282\u70b9\u901a\u5e38\u610f\u5473\u7740\u5220\u9664\u8be5\u8282\u70b9\u53ca\u5176\u6240\u6709\u5b50\u6811\u3002\u56e0\u6b64\uff0c\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u901a\u5e38\u662f\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\u7684\uff0c\u4ee5\u5b9e\u73b0\u6709\u5b9e\u9645\u610f\u4e49\u7684\u64cd\u4f5c\u3002</p>"},{"location":"chapter_tree/binary_tree/#713","title":"7.1.3 \u00a0 \u5e38\u89c1\u4e8c\u53c9\u6811\u7c7b\u578b","text":""},{"location":"chapter_tree/binary_tree/#1_1","title":"1. \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-4 \u6240\u793a\uff0c\u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811 perfect binary tree\u300d\u6240\u6709\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u5b8c\u5168\u586b\u6ee1\u3002\u5728\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\uff0c\u53f6\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u7684\u5ea6\u90fd\u4e3a \\(2\\) \uff1b\u82e5\u6811\u7684\u9ad8\u5ea6\u4e3a \\(h\\) \uff0c\u5219\u8282\u70b9\u603b\u6570\u4e3a \\(2^{h+1} - 1\\) \uff0c\u5448\u73b0\u6807\u51c6\u7684\u6307\u6570\u7ea7\u5173\u7cfb\uff0c\u53cd\u6620\u4e86\u81ea\u7136\u754c\u4e2d\u5e38\u89c1\u7684\u7ec6\u80de\u5206\u88c2\u73b0\u8c61\u3002</p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u5728\u4e2d\u6587\u793e\u533a\u4e2d\uff0c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u5e38\u88ab\u79f0\u4e3a\u300c\u6ee1\u4e8c\u53c9\u6811\u300d\u3002</p> <p></p> <p> \u56fe 7-4 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#2_1","title":"2. \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-5 \u6240\u793a\uff0c\u300c\u5b8c\u5168\u4e8c\u53c9\u6811 complete binary tree\u300d\u53ea\u6709\u6700\u5e95\u5c42\u7684\u8282\u70b9\u672a\u88ab\u586b\u6ee1\uff0c\u4e14\u6700\u5e95\u5c42\u8282\u70b9\u5c3d\u91cf\u9760\u5de6\u586b\u5145\u3002</p> <p></p> <p> \u56fe 7-5 \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#3","title":"3. \u00a0 \u5b8c\u6ee1\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-6 \u6240\u793a\uff0c\u300c\u5b8c\u6ee1\u4e8c\u53c9\u6811 full binary tree\u300d\u9664\u4e86\u53f6\u8282\u70b9\u4e4b\u5916\uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u90fd\u6709\u4e24\u4e2a\u5b50\u8282\u70b9\u3002</p> <p></p> <p> \u56fe 7-6 \u00a0 \u5b8c\u6ee1\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#4","title":"4. \u00a0 \u5e73\u8861\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-7 \u6240\u793a\uff0c\u300c\u5e73\u8861\u4e8c\u53c9\u6811 balanced binary tree\u300d\u4e2d\u4efb\u610f\u8282\u70b9\u7684\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\u4e4b\u5dee\u7684\u7edd\u5bf9\u503c\u4e0d\u8d85\u8fc7 1 \u3002</p> <p></p> <p> \u56fe 7-7 \u00a0 \u5e73\u8861\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#714","title":"7.1.4 \u00a0 \u4e8c\u53c9\u6811\u7684\u9000\u5316","text":"<p>\u56fe 7-8 \u5c55\u793a\u4e86\u4e8c\u53c9\u6811\u7684\u7406\u60f3\u7ed3\u6784\u4e0e\u9000\u5316\u7ed3\u6784\u3002\u5f53\u4e8c\u53c9\u6811\u7684\u6bcf\u5c42\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u65f6\uff0c\u8fbe\u5230\u201c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u201d\uff1b\u800c\u5f53\u6240\u6709\u8282\u70b9\u90fd\u504f\u5411\u4e00\u4fa7\u65f6\uff0c\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u201c\u94fe\u8868\u201d\u3002</p> <ul> <li>\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u7406\u60f3\u60c5\u51b5\uff0c\u53ef\u4ee5\u5145\u5206\u53d1\u6325\u4e8c\u53c9\u6811\u201c\u5206\u6cbb\u201d\u7684\u4f18\u52bf\u3002</li> <li>\u94fe\u8868\u5219\u662f\u53e6\u4e00\u4e2a\u6781\u7aef\uff0c\u5404\u9879\u64cd\u4f5c\u90fd\u53d8\u4e3a\u7ebf\u6027\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002</li> </ul> <p></p> <p> \u56fe 7-8 \u00a0 \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u7ed3\u6784\u4e0e\u6700\u5dee\u7ed3\u6784 </p> <p>\u5982\u8868 7-1 \u6240\u793a\uff0c\u5728\u6700\u4f73\u7ed3\u6784\u548c\u6700\u5dee\u7ed3\u6784\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u53f6\u8282\u70b9\u6570\u91cf\u3001\u8282\u70b9\u603b\u6570\u3001\u9ad8\u5ea6\u7b49\u8fbe\u5230\u6781\u5927\u503c\u6216\u6781\u5c0f\u503c\u3002</p> <p> \u8868 7-1 \u00a0 \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u7ed3\u6784\u4e0e\u6700\u5dee\u7ed3\u6784 </p> \u5b8c\u7f8e\u4e8c\u53c9\u6811 \u94fe\u8868 \u7b2c \\(i\\) \u5c42\u7684\u8282\u70b9\u6570\u91cf \\(2^{i-1}\\) \\(1\\) \u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u6811\u7684\u53f6\u8282\u70b9\u6570\u91cf \\(2^h\\) \\(1\\) \u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u6811\u7684\u8282\u70b9\u603b\u6570 \\(2^{h+1} - 1\\) \\(h + 1\\) \u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \u7684\u6811\u7684\u9ad8\u5ea6 \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)"},{"location":"chapter_tree/binary_tree_traversal/","title":"7.2 \u00a0 \u4e8c\u53c9\u6811\u904d\u5386","text":"<p>\u4ece\u7269\u7406\u7ed3\u6784\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u6811\u662f\u4e00\u79cd\u57fa\u4e8e\u94fe\u8868\u7684\u6570\u636e\u7ed3\u6784\uff0c\u56e0\u6b64\u5176\u904d\u5386\u65b9\u5f0f\u662f\u901a\u8fc7\u6307\u9488\u9010\u4e2a\u8bbf\u95ee\u8282\u70b9\u3002\u7136\u800c\uff0c\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u8fd9\u4f7f\u5f97\u904d\u5386\u6811\u6bd4\u904d\u5386\u94fe\u8868\u66f4\u52a0\u590d\u6742\uff0c\u9700\u8981\u501f\u52a9\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u3002</p> <p>\u4e8c\u53c9\u6811\u5e38\u89c1\u7684\u904d\u5386\u65b9\u5f0f\u5305\u62ec\u5c42\u5e8f\u904d\u5386\u3001\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u7b49\u3002</p>"},{"location":"chapter_tree/binary_tree_traversal/#721","title":"7.2.1 \u00a0 \u5c42\u5e8f\u904d\u5386","text":"<p>\u5982\u56fe 7-9 \u6240\u793a\uff0c\u300c\u5c42\u5e8f\u904d\u5386 level-order traversal\u300d\u4ece\u9876\u90e8\u5230\u5e95\u90e8\u9010\u5c42\u904d\u5386\u4e8c\u53c9\u6811\uff0c\u5e76\u5728\u6bcf\u4e00\u5c42\u6309\u7167\u4ece\u5de6\u5230\u53f3\u7684\u987a\u5e8f\u8bbf\u95ee\u8282\u70b9\u3002</p> <p>\u5c42\u5e8f\u904d\u5386\u672c\u8d28\u4e0a\u5c5e\u4e8e\u300c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 breadth-first traversal\u300d\uff0c\u4e5f\u79f0\u300c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 breadth-first search, BFS\u300d\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u6269\u5c55\u201d\u7684\u9010\u5c42\u904d\u5386\u65b9\u5f0f\u3002</p> <p></p> <p> \u56fe 7-9 \u00a0 \u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386 </p>"},{"location":"chapter_tree/binary_tree_traversal/#1","title":"1. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u901a\u5e38\u501f\u52a9\u201c\u961f\u5217\u201d\u6765\u5b9e\u73b0\u3002\u961f\u5217\u9075\u5faa\u201c\u5148\u8fdb\u5148\u51fa\u201d\u7684\u89c4\u5219\uff0c\u800c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u5219\u9075\u5faa\u201c\u9010\u5c42\u63a8\u8fdb\u201d\u7684\u89c4\u5219\uff0c\u4e24\u8005\u80cc\u540e\u7684\u601d\u60f3\u662f\u4e00\u81f4\u7684\u3002\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree_bfs.py<pre><code>def level_order(root: TreeNode | None) -&gt; list[int]:\n    \"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n    # \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    queue: deque[TreeNode] = deque()\n    queue.append(root)\n    # \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    res = []\n    while queue:\n        node: TreeNode = queue.popleft()  # \u961f\u5217\u51fa\u961f\n        res.append(node.val)  # \u4fdd\u5b58\u8282\u70b9\u503c\n        if node.left is not None:\n            queue.append(node.left)  # \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if node.right is not None:\n            queue.append(node.right)  # \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    return res\n</code></pre> binary_tree_bfs.cpp<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nvector&lt;int&gt; levelOrder(TreeNode *root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    queue&lt;TreeNode *&gt; queue;\n    queue.push(root);\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    vector&lt;int&gt; vec;\n    while (!queue.empty()) {\n        TreeNode *node = queue.front();\n        queue.pop();              // \u961f\u5217\u51fa\u961f\n        vec.push_back(node-&gt;val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node-&gt;left != nullptr)\n            queue.push(node-&gt;left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node-&gt;right != nullptr)\n            queue.push(node-&gt;right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return vec;\n}\n</code></pre> binary_tree_bfs.java<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;Integer&gt; levelOrder(TreeNode root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    Queue&lt;TreeNode&gt; queue = new LinkedList&lt;&gt;();\n    queue.add(root);\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    List&lt;Integer&gt; list = new ArrayList&lt;&gt;();\n    while (!queue.isEmpty()) {\n        TreeNode node = queue.poll(); // \u961f\u5217\u51fa\u961f\n        list.add(node.val);           // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null)\n            queue.offer(node.left);   // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node.right != null)\n            queue.offer(node.right);  // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.cs<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;int&gt; LevelOrder(TreeNode root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    Queue&lt;TreeNode&gt; queue = new();\n    queue.Enqueue(root);\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    List&lt;int&gt; list = [];\n    while (queue.Count != 0) {\n        TreeNode node = queue.Dequeue(); // \u961f\u5217\u51fa\u961f\n        list.Add(node.val!.Value);       // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null)\n            queue.Enqueue(node.left);    // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node.right != null)\n            queue.Enqueue(node.right);   // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.go<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root *TreeNode) []any {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    queue := list.New()\n    queue.PushBack(root)\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5207\u7247\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    nums := make([]any, 0)\n    for queue.Len() &gt; 0 {\n        // \u961f\u5217\u51fa\u961f\n        node := queue.Remove(queue.Front()).(*TreeNode)\n        // \u4fdd\u5b58\u8282\u70b9\u503c\n        nums = append(nums, node.Val)\n        if node.Left != nil {\n            // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n            queue.PushBack(node.Left)\n        }\n        if node.Right != nil {\n            // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n            queue.PushBack(node.Right)\n        }\n    }\n    return nums\n}\n</code></pre> binary_tree_bfs.swift<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root: TreeNode) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    var queue: [TreeNode] = [root]\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    var list: [Int] = []\n    while !queue.isEmpty {\n        let node = queue.removeFirst() // \u961f\u5217\u51fa\u961f\n        list.append(node.val) // \u4fdd\u5b58\u8282\u70b9\u503c\n        if let left = node.left {\n            queue.append(left) // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if let right = node.right {\n            queue.append(right) // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        }\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.js<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    const queue = [root];\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    const list = [];\n    while (queue.length) {\n        let node = queue.shift(); // \u961f\u5217\u51fa\u961f\n        list.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left) queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node.right) queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.ts<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root: TreeNode | null): number[] {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    const queue = [root];\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    const list: number[] = [];\n    while (queue.length) {\n        let node = queue.shift() as TreeNode; // \u961f\u5217\u51fa\u961f\n        list.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left) {\n            queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if (node.right) {\n            queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        }\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.dart<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;int&gt; levelOrder(TreeNode? root) {\n  // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n  Queue&lt;TreeNode?&gt; queue = Queue();\n  queue.add(root);\n  // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n  List&lt;int&gt; res = [];\n  while (queue.isNotEmpty) {\n    TreeNode? node = queue.removeFirst(); // \u961f\u5217\u51fa\u961f\n    res.add(node!.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n    if (node.left != null) queue.add(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n    if (node.right != null) queue.add(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n  }\n  return res;\n}\n</code></pre> binary_tree_bfs.rs<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfn level_order(root: &amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    let mut que = VecDeque::new();\n    que.push_back(Rc::clone(&amp;root));\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    let mut vec = Vec::new();\n\n    while let Some(node) = que.pop_front() {\n        // \u961f\u5217\u51fa\u961f\n        vec.push(node.borrow().val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if let Some(left) = node.borrow().left.as_ref() {\n            que.push_back(Rc::clone(left)); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if let Some(right) = node.borrow().right.as_ref() {\n            que.push_back(Rc::clone(right)); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        };\n    }\n    vec\n}\n</code></pre> binary_tree_bfs.c<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(TreeNode *root, int *size) {\n    /* \u8f85\u52a9\u961f\u5217 */\n    int front, rear;\n    int index, *arr;\n    TreeNode *node;\n    TreeNode **queue;\n\n    /* \u8f85\u52a9\u961f\u5217 */\n    queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);\n    // \u961f\u5217\u6307\u9488\n    front = 0, rear = 0;\n    // \u52a0\u5165\u6839\u8282\u70b9\n    queue[rear++] = root;\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    /* \u8f85\u52a9\u6570\u7ec4 */\n    arr = (int *)malloc(sizeof(int) * MAX_SIZE);\n    // \u6570\u7ec4\u6307\u9488\n    index = 0;\n    while (front &lt; rear) {\n        // \u961f\u5217\u51fa\u961f\n        node = queue[front++];\n        // \u4fdd\u5b58\u8282\u70b9\u503c\n        arr[index++] = node-&gt;val;\n        if (node-&gt;left != NULL) {\n            // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n            queue[rear++] = node-&gt;left;\n        }\n        if (node-&gt;right != NULL) {\n            // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n            queue[rear++] = node-&gt;right;\n        }\n    }\n    // \u66f4\u65b0\u6570\u7ec4\u957f\u5ea6\u7684\u503c\n    *size = index;\n    arr = realloc(arr, sizeof(int) * (*size));\n\n    // \u91ca\u653e\u8f85\u52a9\u6570\u7ec4\u7a7a\u95f4\n    free(queue);\n    return arr;\n}\n</code></pre> binary_tree_bfs.kt<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfun levelOrder(root: TreeNode?): MutableList&lt;Int&gt; {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    val queue = LinkedList&lt;TreeNode?&gt;()\n    queue.add(root)\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    val list = ArrayList&lt;Int&gt;()\n    while (!queue.isEmpty()) {\n        val node = queue.poll() // \u961f\u5217\u51fa\u961f\n        list.add(node?.value!!) // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null) queue.offer(node.left) // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n\n        if (node.right != null) queue.offer(node.right) // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.rb<pre><code>[class]{}-[func]{level_order}\n</code></pre> binary_tree_bfs.zig<pre><code>// \u5c42\u5e8f\u904d\u5386\nfn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    const L = std.TailQueue(*inc.TreeNode(T));\n    var queue = L{};\n    var root_node = try mem_allocator.create(L.Node);\n    root_node.data = root;\n    queue.append(root_node); \n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    var list = std.ArrayList(T).init(std.heap.page_allocator);\n    while (queue.len &gt; 0) {\n        var queue_node = queue.popFirst().?;    // \u961f\u5217\u51fa\u961f\n        var node = queue_node.data;\n        try list.append(node.val);              // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null) {\n            var tmp_node = try mem_allocator.create(L.Node);\n            tmp_node.data = node.left.?;\n            queue.append(tmp_node);             // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if (node.right != null) {\n            var tmp_node = try mem_allocator.create(L.Node);\n            tmp_node.data = node.right.?;\n            queue.append(tmp_node);             // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        }        \n    }\n    return list;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_tree/binary_tree_traversal/#2","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1a\u6240\u6709\u8282\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d \\(n\\) \u4e3a\u8282\u70b9\u6570\u91cf\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u6ee1\u4e8c\u53c9\u6811\u65f6\uff0c\u904d\u5386\u5230\u6700\u5e95\u5c42\u4e4b\u524d\uff0c\u961f\u5217\u4e2d\u6700\u591a\u540c\u65f6\u5b58\u5728 \\((n + 1) / 2\\) \u4e2a\u8282\u70b9\uff0c\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_tree/binary_tree_traversal/#722","title":"7.2.2 \u00a0 \u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386","text":"<p>\u76f8\u5e94\u5730\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 depth-first traversal\u300d\uff0c\u4e5f\u79f0\u300c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22 depth-first search, DFS\u300d\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u5148\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u6eaf\u7ee7\u7eed\u201d\u7684\u904d\u5386\u65b9\u5f0f\u3002</p> <p>\u56fe 7-10 \u5c55\u793a\u4e86\u5bf9\u4e8c\u53c9\u6811\u8fdb\u884c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5de5\u4f5c\u539f\u7406\u3002\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5c31\u50cf\u662f\u7ed5\u7740\u6574\u68f5\u4e8c\u53c9\u6811\u7684\u5916\u56f4\u201c\u8d70\u201d\u4e00\u5708\uff0c\u5728\u6bcf\u4e2a\u8282\u70b9\u90fd\u4f1a\u9047\u5230\u4e09\u4e2a\u4f4d\u7f6e\uff0c\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u3002</p> <p></p> <p> \u56fe 7-10 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386 </p>"},{"location":"chapter_tree/binary_tree_traversal/#1_1","title":"1. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree_dfs.py<pre><code>def pre_order(root: TreeNode | None):\n    \"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\n    if root is None:\n        return\n    # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    res.append(root.val)\n    pre_order(root=root.left)\n    pre_order(root=root.right)\n\ndef in_order(root: TreeNode | None):\n    \"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\n    if root is None:\n        return\n    # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    in_order(root=root.left)\n    res.append(root.val)\n    in_order(root=root.right)\n\ndef post_order(root: TreeNode | None):\n    \"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\n    if root is None:\n        return\n    # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    post_order(root=root.left)\n    post_order(root=root.right)\n    res.append(root.val)\n</code></pre> binary_tree_dfs.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    vec.push_back(root-&gt;val);\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root-&gt;left);\n    vec.push_back(root-&gt;val);\n    inOrder(root-&gt;right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root-&gt;left);\n    postOrder(root-&gt;right);\n    vec.push_back(root-&gt;val);\n}\n</code></pre> binary_tree_dfs.java<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.add(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left);\n    list.add(root.val);\n    inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left);\n    postOrder(root.right);\n    list.add(root.val);\n}\n</code></pre> binary_tree_dfs.cs<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.Add(root.val!.Value);\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid InOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    InOrder(root.left);\n    list.Add(root.val!.Value);\n    InOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid PostOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    PostOrder(root.left);\n    PostOrder(root.right);\n    list.Add(root.val!.Value);\n}\n</code></pre> binary_tree_dfs.go<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    nums = append(nums, node.Val)\n    preOrder(node.Left)\n    preOrder(node.Right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(node.Left)\n    nums = append(nums, node.Val)\n    inOrder(node.Right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(node.Left)\n    postOrder(node.Right)\n    nums = append(nums, node.Val)\n}\n</code></pre> binary_tree_dfs.swift<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.append(root.val)\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root: root.left)\n    list.append(root.val)\n    inOrder(root: root.right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root: root.left)\n    postOrder(root: root.right)\n    list.append(root.val)\n}\n</code></pre> binary_tree_dfs.js<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root) {\n    if (root === null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root) {\n    if (root === null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root) {\n    if (root === null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.ts<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.dart<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n  list.add(node.val);\n  preOrder(node.left);\n  preOrder(node.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n  inOrder(node.left);\n  list.add(node.val);\n  inOrder(node.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n  postOrder(node.left);\n  postOrder(node.right);\n  list.add(node.val);\n}\n</code></pre> binary_tree_dfs.rs<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfn pre_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    if let Some(node) = root {\n        // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n        result.push(node.borrow().val);\n        result.append(&amp;mut pre_order(node.borrow().left.as_ref()));\n        result.append(&amp;mut pre_order(node.borrow().right.as_ref()));\n    }\n    result\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfn in_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    if let Some(node) = root {\n        // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n        result.append(&amp;mut in_order(node.borrow().left.as_ref()));\n        result.push(node.borrow().val);\n        result.append(&amp;mut in_order(node.borrow().right.as_ref()));\n    }\n    result\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfn post_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    if let Some(node) = root {\n        // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n        result.append(&amp;mut post_order(node.borrow().left.as_ref()));\n        result.append(&amp;mut post_order(node.borrow().right.as_ref()));\n        result.push(node.borrow().val);\n    }\n    result\n}\n</code></pre> binary_tree_dfs.c<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    arr[(*size)++] = root-&gt;val;\n    preOrder(root-&gt;left, size);\n    preOrder(root-&gt;right, size);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root-&gt;left, size);\n    arr[(*size)++] = root-&gt;val;\n    inOrder(root-&gt;right, size);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root-&gt;left, size);\n    postOrder(root-&gt;right, size);\n    arr[(*size)++] = root-&gt;val;\n}\n</code></pre> binary_tree_dfs.kt<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.add(root.value)\n    preOrder(root.left)\n    preOrder(root.right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfun inOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left)\n    list.add(root.value)\n    inOrder(root.right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfun postOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left)\n    postOrder(root.right)\n    list.add(root.value)\n}\n</code></pre> binary_tree_dfs.rb<pre><code>[class]{}-[func]{pre_order}\n\n[class]{}-[func]{in_order}\n\n[class]{}-[func]{post_order}\n</code></pre> binary_tree_dfs.zig<pre><code>// \u524d\u5e8f\u904d\u5386\nfn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    try list.append(root.?.val);\n    try preOrder(T, root.?.left);\n    try preOrder(T, root.?.right);\n}\n\n// \u4e2d\u5e8f\u904d\u5386\nfn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    try inOrder(T, root.?.left);\n    try list.append(root.?.val);\n    try inOrder(T, root.?.right);\n}\n\n// \u540e\u5e8f\u904d\u5386\nfn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    try postOrder(T, root.?.left);\n    try postOrder(T, root.?.right);\n    try list.append(root.?.val);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>Tip</p> <p>\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u4e5f\u53ef\u4ee5\u57fa\u4e8e\u8fed\u4ee3\u5b9e\u73b0\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u7814\u7a76\u3002</p> <p>\u56fe 7-11 \u5c55\u793a\u4e86\u524d\u5e8f\u904d\u5386\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b\uff0c\u5176\u53ef\u5206\u4e3a\u201c\u9012\u201d\u548c\u201c\u5f52\u201d\u4e24\u4e2a\u9006\u5411\u7684\u90e8\u5206\u3002</p> <ol> <li>\u201c\u9012\u201d\u8868\u793a\u5f00\u542f\u65b0\u65b9\u6cd5\uff0c\u7a0b\u5e8f\u5728\u6b64\u8fc7\u7a0b\u4e2d\u8bbf\u95ee\u4e0b\u4e00\u4e2a\u8282\u70b9\u3002</li> <li>\u201c\u5f52\u201d\u8868\u793a\u51fd\u6570\u8fd4\u56de\uff0c\u4ee3\u8868\u5f53\u524d\u8282\u70b9\u5df2\u7ecf\u8bbf\u95ee\u5b8c\u6bd5\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-11 \u00a0 \u524d\u5e8f\u904d\u5386\u7684\u9012\u5f52\u8fc7\u7a0b </p>"},{"location":"chapter_tree/binary_tree_traversal/#2_1","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1a\u6240\u6709\u8282\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u9012\u5f52\u6df1\u5ea6\u8fbe\u5230 \\(n\\) \uff0c\u7cfb\u7edf\u5360\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_tree/summary/","title":"7.6 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_tree/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u4e8c\u53c9\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4f53\u73b0\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u6bcf\u4e2a\u4e8c\u53c9\u6811\u8282\u70b9\u5305\u542b\u4e00\u4e2a\u503c\u4ee5\u53ca\u4e24\u4e2a\u6307\u9488\uff0c\u5206\u522b\u6307\u5411\u5176\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\u3002</li> <li>\u5bf9\u4e8e\u4e8c\u53c9\u6811\u4e2d\u7684\u67d0\u4e2a\u8282\u70b9\uff0c\u5176\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u53ca\u5176\u4ee5\u4e0b\u5f62\u6210\u7684\u6811\u88ab\u79f0\u4e3a\u8be5\u8282\u70b9\u7684\u5de6\uff08\u53f3\uff09\u5b50\u6811\u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u76f8\u5173\u672f\u8bed\u5305\u62ec\u6839\u8282\u70b9\u3001\u53f6\u8282\u70b9\u3001\u5c42\u3001\u5ea6\u3001\u8fb9\u3001\u9ad8\u5ea6\u548c\u6df1\u5ea6\u7b49\u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u521d\u59cb\u5316\u3001\u8282\u70b9\u63d2\u5165\u548c\u8282\u70b9\u5220\u9664\u64cd\u4f5c\u4e0e\u94fe\u8868\u64cd\u4f5c\u65b9\u6cd5\u7c7b\u4f3c\u3002</li> <li>\u5e38\u89c1\u7684\u4e8c\u53c9\u6811\u7c7b\u578b\u6709\u5b8c\u7f8e\u4e8c\u53c9\u6811\u3001\u5b8c\u5168\u4e8c\u53c9\u6811\u3001\u5b8c\u6ee1\u4e8c\u53c9\u6811\u548c\u5e73\u8861\u4e8c\u53c9\u6811\u3002\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u6700\u7406\u60f3\u7684\u72b6\u6001\uff0c\u800c\u94fe\u8868\u662f\u9000\u5316\u540e\u7684\u6700\u5dee\u72b6\u6001\u3002</li> <li>\u4e8c\u53c9\u6811\u53ef\u4ee5\u7528\u6570\u7ec4\u8868\u793a\uff0c\u65b9\u6cd5\u662f\u5c06\u8282\u70b9\u503c\u548c\u7a7a\u4f4d\u6309\u5c42\u5e8f\u904d\u5386\u987a\u5e8f\u6392\u5217\uff0c\u5e76\u6839\u636e\u7236\u8282\u70b9\u4e0e\u5b50\u8282\u70b9\u4e4b\u95f4\u7684\u7d22\u5f15\u6620\u5c04\u5173\u7cfb\u6765\u5b9e\u73b0\u6307\u9488\u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386\u662f\u4e00\u79cd\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u65b9\u6cd5\uff0c\u5b83\u4f53\u73b0\u4e86\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u6269\u5c55\u201d\u7684\u9010\u5c42\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u901a\u8fc7\u961f\u5217\u6765\u5b9e\u73b0\u3002</li> <li>\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\u7686\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u5b83\u4eec\u4f53\u73b0\u4e86\u201c\u5148\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u6eaf\u7ee7\u7eed\u201d\u7684\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u6765\u5b9e\u73b0\u3002</li> <li>\u4e8c\u53c9\u641c\u7d22\u6811\u662f\u4e00\u79cd\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u627e\u6570\u636e\u7ed3\u6784\uff0c\u5176\u67e5\u627e\u3001\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \u3002\u5f53\u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u5404\u9879\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u52a3\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>AVL \u6811\uff0c\u4e5f\u79f0\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5b83\u901a\u8fc7\u65cb\u8f6c\u64cd\u4f5c\u786e\u4fdd\u5728\u4e0d\u65ad\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\u540e\u6811\u4ecd\u7136\u4fdd\u6301\u5e73\u8861\u3002</li> <li>AVL \u6811\u7684\u65cb\u8f6c\u64cd\u4f5c\u5305\u62ec\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u518d\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u518d\u53f3\u65cb\u3002\u5728\u63d2\u5165\u6216\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4f1a\u4ece\u5e95\u5411\u9876\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002</li> </ul>"},{"location":"chapter_tree/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u5bf9\u4e8e\u53ea\u6709\u4e00\u4e2a\u8282\u70b9\u7684\u4e8c\u53c9\u6811\uff0c\u6811\u7684\u9ad8\u5ea6\u548c\u6839\u8282\u70b9\u7684\u6df1\u5ea6\u90fd\u662f \\(0\\) \u5417\uff1f</p> <p>\u662f\u7684\uff0c\u56e0\u4e3a\u9ad8\u5ea6\u548c\u6df1\u5ea6\u901a\u5e38\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u201d\u3002</p> <p>Q\uff1a\u4e8c\u53c9\u6811\u4e2d\u7684\u63d2\u5165\u4e0e\u5220\u9664\u4e00\u822c\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\uff0c\u8fd9\u91cc\u7684\u201c\u4e00\u5957\u64cd\u4f5c\u201d\u6307\u4ec0\u4e48\u5462\uff1f\u53ef\u4ee5\u7406\u89e3\u4e3a\u8d44\u6e90\u7684\u5b50\u8282\u70b9\u7684\u8d44\u6e90\u91ca\u653e\u5417\uff1f</p> <p>\u62ff\u4e8c\u53c9\u641c\u7d22\u6811\u6765\u4e3e\u4f8b\uff0c\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u8981\u5206\u4e09\u79cd\u60c5\u51b5\u5904\u7406\uff0c\u5176\u4e2d\u6bcf\u79cd\u60c5\u51b5\u90fd\u9700\u8981\u8fdb\u884c\u591a\u4e2a\u6b65\u9aa4\u7684\u8282\u70b9\u64cd\u4f5c\u3002</p> <p>Q\uff1a\u4e3a\u4ec0\u4e48 DFS \u904d\u5386\u4e8c\u53c9\u6811\u6709\u524d\u3001\u4e2d\u3001\u540e\u4e09\u79cd\u987a\u5e8f\uff0c\u5206\u522b\u6709\u4ec0\u4e48\u7528\u5462\uff1f</p> <p>\u4e0e\u987a\u5e8f\u548c\u9006\u5e8f\u904d\u5386\u6570\u7ec4\u7c7b\u4f3c\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\u662f\u4e09\u79cd\u4e8c\u53c9\u6811\u904d\u5386\u65b9\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u5b83\u4eec\u5f97\u5230\u4e00\u4e2a\u7279\u5b9a\u987a\u5e8f\u7684\u904d\u5386\u7ed3\u679c\u3002\u4f8b\u5982\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\uff0c\u7531\u4e8e\u8282\u70b9\u5927\u5c0f\u6ee1\u8db3 <code>\u5de6\u5b50\u8282\u70b9\u503c &lt; \u6839\u8282\u70b9\u503c &lt; \u53f3\u5b50\u8282\u70b9\u503c</code> \uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u8981\u6309\u7167\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u4f18\u5148\u7ea7\u904d\u5386\u6811\uff0c\u5c31\u53ef\u4ee5\u83b7\u5f97\u6709\u5e8f\u7684\u8282\u70b9\u5e8f\u5217\u3002</p> <p>Q\uff1a\u53f3\u65cb\u64cd\u4f5c\u662f\u5904\u7406\u5931\u8861\u8282\u70b9 <code>node</code>\u3001<code>child</code>\u3001<code>grand_child</code> \u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u90a3 <code>node</code> \u7684\u7236\u8282\u70b9\u548c <code>node</code> \u539f\u6765\u7684\u8fde\u63a5\u4e0d\u9700\u8981\u7ef4\u62a4\u5417\uff1f\u53f3\u65cb\u64cd\u4f5c\u540e\u5c82\u4e0d\u662f\u65ad\u6389\u4e86\uff1f</p> <p>\u6211\u4eec\u9700\u8981\u4ece\u9012\u5f52\u7684\u89c6\u89d2\u6765\u770b\u8fd9\u4e2a\u95ee\u9898\u3002\u53f3\u65cb\u64cd\u4f5c <code>right_rotate(root)</code> \u4f20\u5165\u7684\u662f\u5b50\u6811\u7684\u6839\u8282\u70b9\uff0c\u6700\u7ec8 <code>return child</code> \u8fd4\u56de\u65cb\u8f6c\u4e4b\u540e\u7684\u5b50\u6811\u7684\u6839\u8282\u70b9\u3002\u5b50\u6811\u7684\u6839\u8282\u70b9\u548c\u5176\u7236\u8282\u70b9\u7684\u8fde\u63a5\u662f\u5728\u8be5\u51fd\u6570\u8fd4\u56de\u540e\u5b8c\u6210\u7684\uff0c\u4e0d\u5c5e\u4e8e\u53f3\u65cb\u64cd\u4f5c\u7684\u7ef4\u62a4\u8303\u56f4\u3002</p> <p>Q\uff1a\u5728 C++ \u4e2d\uff0c\u51fd\u6570\u88ab\u5212\u5206\u5230 <code>private</code> \u548c <code>public</code> \u4e2d\uff0c\u8fd9\u65b9\u9762\u6709\u4ec0\u4e48\u8003\u91cf\u5417\uff1f\u4e3a\u4ec0\u4e48\u8981\u5c06 <code>height()</code> \u51fd\u6570\u548c <code>updateHeight()</code> \u51fd\u6570\u5206\u522b\u653e\u5728 <code>public</code> \u548c <code>private</code> \u4e2d\u5462\uff1f</p> <p>\u4e3b\u8981\u770b\u65b9\u6cd5\u7684\u4f7f\u7528\u8303\u56f4\uff0c\u5982\u679c\u65b9\u6cd5\u53ea\u5728\u7c7b\u5185\u90e8\u4f7f\u7528\uff0c\u90a3\u4e48\u5c31\u8bbe\u8ba1\u4e3a <code>private</code> \u3002\u4f8b\u5982\uff0c\u7528\u6237\u5355\u72ec\u8c03\u7528 <code>updateHeight()</code> \u662f\u6ca1\u6709\u610f\u4e49\u7684\uff0c\u5b83\u53ea\u662f\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u4e2d\u7684\u4e00\u6b65\u3002\u800c <code>height()</code> \u662f\u8bbf\u95ee\u8282\u70b9\u9ad8\u5ea6\uff0c\u7c7b\u4f3c\u4e8e <code>vector.size()</code> \uff0c\u56e0\u6b64\u8bbe\u7f6e\u6210 <code>public</code> \u4ee5\u4fbf\u4f7f\u7528\u3002</p> <p>Q\uff1a\u5982\u4f55\u4ece\u4e00\u7ec4\u8f93\u5165\u6570\u636e\u6784\u5efa\u4e00\u68f5\u4e8c\u53c9\u641c\u7d22\u6811\uff1f\u6839\u8282\u70b9\u7684\u9009\u62e9\u662f\u4e0d\u662f\u5f88\u91cd\u8981\uff1f</p> <p>\u662f\u7684\uff0c\u6784\u5efa\u6811\u7684\u65b9\u6cd5\u5df2\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4ee3\u7801\u4e2d\u7684 <code>build_tree()</code> \u65b9\u6cd5\u4e2d\u7ed9\u51fa\u3002\u81f3\u4e8e\u6839\u8282\u70b9\u7684\u9009\u62e9\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u8f93\u5165\u6570\u636e\u6392\u5e8f\uff0c\u7136\u540e\u5c06\u4e2d\u70b9\u5143\u7d20\u4f5c\u4e3a\u6839\u8282\u70b9\uff0c\u518d\u9012\u5f52\u5730\u6784\u5efa\u5de6\u53f3\u5b50\u6811\u3002\u8fd9\u6837\u505a\u53ef\u4ee5\u6700\u5927\u7a0b\u5ea6\u4fdd\u8bc1\u6811\u7684\u5e73\u8861\u6027\u3002</p> <p>Q\uff1a\u5728 Java \u4e2d\uff0c\u5b57\u7b26\u4e32\u5bf9\u6bd4\u662f\u5426\u4e00\u5b9a\u8981\u7528 <code>equals()</code> \u65b9\u6cd5\uff1f</p> <p>\u5728 Java \u4e2d\uff0c\u5bf9\u4e8e\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c<code>==</code> \u7528\u4e8e\u5bf9\u6bd4\u4e24\u4e2a\u53d8\u91cf\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002\u5bf9\u4e8e\u5f15\u7528\u7c7b\u578b\uff0c\u4e24\u79cd\u7b26\u53f7\u7684\u5de5\u4f5c\u539f\u7406\u662f\u4e0d\u540c\u7684\u3002</p> <ul> <li><code>==</code> \uff1a\u7528\u6765\u6bd4\u8f83\u4e24\u4e2a\u53d8\u91cf\u662f\u5426\u6307\u5411\u540c\u4e00\u4e2a\u5bf9\u8c61\uff0c\u5373\u5b83\u4eec\u5728\u5185\u5b58\u4e2d\u7684\u4f4d\u7f6e\u662f\u5426\u76f8\u540c\u3002</li> <li><code>equals()</code>\uff1a\u7528\u6765\u5bf9\u6bd4\u4e24\u4e2a\u5bf9\u8c61\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002</li> </ul> <p>\u56e0\u6b64\uff0c\u5982\u679c\u8981\u5bf9\u6bd4\u503c\uff0c\u6211\u4eec\u5e94\u8be5\u4f7f\u7528 <code>equals()</code> \u3002\u7136\u800c\uff0c\u901a\u8fc7 <code>String a = \"hi\"; String b = \"hi\";</code> \u521d\u59cb\u5316\u7684\u5b57\u7b26\u4e32\u90fd\u5b58\u50a8\u5728\u5b57\u7b26\u4e32\u5e38\u91cf\u6c60\u4e2d\uff0c\u5b83\u4eec\u6307\u5411\u540c\u4e00\u4e2a\u5bf9\u8c61\uff0c\u56e0\u6b64\u4e5f\u53ef\u4ee5\u7528 <code>a == b</code> \u6765\u6bd4\u8f83\u4e24\u4e2a\u5b57\u7b26\u4e32\u7684\u5185\u5bb9\u3002</p> <p>Q\uff1a\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u5230\u6700\u5e95\u5c42\u4e4b\u524d\uff0c\u961f\u5217\u4e2d\u7684\u8282\u70b9\u6570\u91cf\u662f \\(2^h\\) \u5417\uff1f</p> <p>\u662f\u7684\uff0c\u4f8b\u5982\u9ad8\u5ea6 \\(h = 2\\) \u7684\u6ee1\u4e8c\u53c9\u6811\uff0c\u5176\u8282\u70b9\u603b\u6570 \\(n = 7\\) \uff0c\u5219\u5e95\u5c42\u8282\u70b9\u6570\u91cf \\(4 = 2^h = (n + 1) / 2\\) \u3002</p>"}]}
\ No newline at end of file
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\u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c","text":"<p>\u7531\u4e8e\u7b14\u8005\u80fd\u529b\u6709\u9650\uff0c\u4e66\u4e2d\u96be\u514d\u5b58\u5728\u4e00\u4e9b\u9057\u6f0f\u548c\u9519\u8bef\uff0c\u8bf7\u60a8\u8c05\u89e3\u3002\u5982\u679c\u60a8\u53d1\u73b0\u4e86\u7b14\u8bef\u3001\u94fe\u63a5\u5931\u6548\u3001\u5185\u5bb9\u7f3a\u5931\u3001\u6587\u5b57\u6b67\u4e49\u3001\u89e3\u91ca\u4e0d\u6e05\u6670\u6216\u884c\u6587\u7ed3\u6784\u4e0d\u5408\u7406\u7b49\u95ee\u9898\uff0c\u8bf7\u534f\u52a9\u6211\u4eec\u8fdb\u884c\u4fee\u6b63\uff0c\u4ee5\u7ed9\u8bfb\u8005\u63d0\u4f9b\u66f4\u4f18\u8d28\u7684\u5b66\u4e60\u8d44\u6e90\u3002</p> <p>\u6240\u6709\u64b0\u7a3f\u4eba\u7684 GitHub ID \u5c06\u5728\u672c\u4e66\u4ed3\u5e93\u3001\u7f51\u9875\u7248\u548c PDF \u7248\u7684\u4e3b\u9875\u4e0a\u8fdb\u884c\u5c55\u793a\uff0c\u4ee5\u611f\u8c22\u4ed6\u4eec\u5bf9\u5f00\u6e90\u793e\u533a\u7684\u65e0\u79c1\u5949\u732e\u3002</p> <p>\u5f00\u6e90\u7684\u9b45\u529b</p> <p>\u7eb8\u8d28\u56fe\u4e66\u7684\u4e24\u6b21\u5370\u5237\u7684\u95f4\u9694\u65f6\u95f4\u5f80\u5f80\u8f83\u4e45\uff0c\u5185\u5bb9\u66f4\u65b0\u975e\u5e38\u4e0d\u65b9\u4fbf\u3002</p> <p>\u800c\u5728\u672c\u5f00\u6e90\u4e66\u4e2d\uff0c\u5185\u5bb9\u66f4\u8fed\u7684\u65f6\u95f4\u88ab\u7f29\u77ed\u81f3\u6570\u65e5\u751a\u81f3\u51e0\u4e2a\u5c0f\u65f6\u3002</p>"},{"location":"chapter_appendix/contribution/#1","title":"1. \u00a0 \u5185\u5bb9\u5fae\u8c03","text":"<p>\u5982\u56fe 16-3 \u6240\u793a\uff0c\u6bcf\u4e2a\u9875\u9762\u7684\u53f3\u4e0a\u89d2\u90fd\u6709\u201c\u7f16\u8f91\u56fe\u6807\u201d\u3002\u60a8\u53ef\u4ee5\u6309\u7167\u4ee5\u4e0b\u6b65\u9aa4\u4fee\u6539\u6587\u672c\u6216\u4ee3\u7801\u3002</p> <ol> <li>\u70b9\u51fb\u201c\u7f16\u8f91\u56fe\u6807\u201d\uff0c\u5982\u679c\u9047\u5230\u201c\u9700\u8981 Fork \u6b64\u4ed3\u5e93\u201d\u7684\u63d0\u793a\uff0c\u8bf7\u540c\u610f\u8be5\u64cd\u4f5c\u3002</li> <li>\u4fee\u6539 Markdown \u6e90\u6587\u4ef6\u5185\u5bb9\uff0c\u68c0\u67e5\u5185\u5bb9\u7684\u6b63\u786e\u6027\uff0c\u5e76\u5c3d\u91cf\u4fdd\u6301\u6392\u7248\u683c\u5f0f\u7684\u7edf\u4e00\u3002</li> <li>\u5728\u9875\u9762\u5e95\u90e8\u586b\u5199\u4fee\u6539\u8bf4\u660e\uff0c\u7136\u540e\u70b9\u51fb\u201cPropose file change\u201d\u6309\u94ae\u3002\u9875\u9762\u8df3\u8f6c\u540e\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u3002</li> </ol> <p></p> <p> \u56fe 16-3 \u00a0 \u9875\u9762\u7f16\u8f91\u6309\u952e </p> <p>\u56fe\u7247\u65e0\u6cd5\u76f4\u63a5\u4fee\u6539\uff0c\u9700\u8981\u901a\u8fc7\u65b0\u5efa Issue \u6216\u8bc4\u8bba\u7559\u8a00\u6765\u63cf\u8ff0\u95ee\u9898\uff0c\u6211\u4eec\u4f1a\u5c3d\u5feb\u91cd\u65b0\u7ed8\u5236\u5e76\u66ff\u6362\u56fe\u7247\u3002</p>"},{"location":"chapter_appendix/contribution/#2","title":"2. \u00a0 \u5185\u5bb9\u521b\u4f5c","text":"<p>\u5982\u679c\u60a8\u6709\u5174\u8da3\u53c2\u4e0e\u6b64\u5f00\u6e90\u9879\u76ee\uff0c\u5305\u62ec\u5c06\u4ee3\u7801\u7ffb\u8bd1\u6210\u5176\u4ed6\u7f16\u7a0b\u8bed\u8a00\u3001\u6269\u5c55\u6587\u7ae0\u5185\u5bb9\u7b49\uff0c\u90a3\u4e48\u9700\u8981\u5b9e\u65bd\u4ee5\u4e0b Pull Request \u5de5\u4f5c\u6d41\u7a0b\u3002</p> <ol> <li>\u767b\u5f55 GitHub \uff0c\u5c06\u672c\u4e66\u7684\u4ee3\u7801\u4ed3\u5e93 Fork \u5230\u4e2a\u4eba\u8d26\u53f7\u4e0b\u3002</li> <li>\u8fdb\u5165\u60a8\u7684 Fork \u4ed3\u5e93\u7f51\u9875\uff0c\u4f7f\u7528 <code>git clone</code> \u547d\u4ee4\u5c06\u4ed3\u5e93\u514b\u9686\u81f3\u672c\u5730\u3002</li> <li>\u5728\u672c\u5730\u8fdb\u884c\u5185\u5bb9\u521b\u4f5c\uff0c\u5e76\u8fdb\u884c\u5b8c\u6574\u6d4b\u8bd5\uff0c\u9a8c\u8bc1\u4ee3\u7801\u7684\u6b63\u786e\u6027\u3002</li> <li>\u5c06\u672c\u5730\u6240\u505a\u66f4\u6539 Commit \uff0c\u7136\u540e Push \u81f3\u8fdc\u7a0b\u4ed3\u5e93\u3002</li> <li>\u5237\u65b0\u4ed3\u5e93\u7f51\u9875\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u3002</li> </ol>"},{"location":"chapter_appendix/contribution/#3-docker","title":"3. \u00a0 Docker \u90e8\u7f72","text":"<p>\u5728 <code>hello-algo</code> \u6839\u76ee\u5f55\u4e0b\uff0c\u6267\u884c\u4ee5\u4e0b Docker \u811a\u672c\uff0c\u5373\u53ef\u5728 <code>http://localhost:8000</code> \u8bbf\u95ee\u672c\u9879\u76ee\uff1a</p> <pre><code>docker-compose up -d\n</code></pre> <p>\u4f7f\u7528\u4ee5\u4e0b\u547d\u4ee4\u5373\u53ef\u5220\u9664\u90e8\u7f72\uff1a</p> <pre><code>docker-compose down\n</code></pre>"},{"location":"chapter_appendix/installation/","title":"16.1 \u00a0 \u7f16\u7a0b\u73af\u5883\u5b89\u88c5","text":""},{"location":"chapter_appendix/installation/#1611-ide","title":"16.1.1 \u00a0 \u5b89\u88c5 IDE","text":"<p>\u63a8\u8350\u4f7f\u7528\u5f00\u6e90\u3001\u8f7b\u91cf\u7684 VS Code \u4f5c\u4e3a\u672c\u5730\u96c6\u6210\u5f00\u53d1\u73af\u5883\uff08IDE\uff09\u3002\u8bbf\u95ee VS Code \u5b98\u7f51\uff0c\u6839\u636e\u64cd\u4f5c\u7cfb\u7edf\u9009\u62e9\u76f8\u5e94\u7248\u672c\u7684 VS Code \u8fdb\u884c\u4e0b\u8f7d\u548c\u5b89\u88c5\u3002</p> <p></p> <p> \u56fe 16-1 \u00a0 \u4ece\u5b98\u7f51\u4e0b\u8f7d VS Code </p> <p>VS Code \u62e5\u6709\u5f3a\u5927\u7684\u6269\u5c55\u5305\u751f\u6001\u7cfb\u7edf\uff0c\u652f\u6301\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u7684\u8fd0\u884c\u548c\u8c03\u8bd5\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u5b89\u88c5\u201cPython Extension Pack\u201d\u6269\u5c55\u5305\u4e4b\u540e\uff0c\u5373\u53ef\u8fdb\u884c Python \u4ee3\u7801\u8c03\u8bd5\u3002\u5b89\u88c5\u6b65\u9aa4\u5982\u56fe 16-2 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 16-2 \u00a0 \u5b89\u88c5 VS Code \u6269\u5c55\u5305 </p>"},{"location":"chapter_appendix/installation/#1612","title":"16.1.2 \u00a0 \u5b89\u88c5\u8bed\u8a00\u73af\u5883","text":""},{"location":"chapter_appendix/installation/#1-python","title":"1. \u00a0 Python \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Miniconda3 \uff0c\u9700\u8981 Python 3.10 \u6216\u66f4\u65b0\u7248\u672c\u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>python</code> \uff0c\u5b89\u88c5 Python Extension Pack \u3002</li> <li>\uff08\u53ef\u9009\uff09\u5728\u547d\u4ee4\u884c\u8f93\u5165 <code>pip install black</code> \uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#2-cc","title":"2. \u00a0 C/C++ \u73af\u5883","text":"<ol> <li>Windows \u7cfb\u7edf\u9700\u8981\u5b89\u88c5 MinGW\uff08\u914d\u7f6e\u6559\u7a0b\uff09\uff1bMacOS \u81ea\u5e26 Clang \uff0c\u65e0\u987b\u5b89\u88c5\u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>c++</code> \uff0c\u5b89\u88c5 C/C++ Extension Pack \u3002</li> <li>\uff08\u53ef\u9009\uff09\u6253\u5f00 Settings \u9875\u9762\uff0c\u641c\u7d22 <code>Clang_format_fallback Style</code> \u4ee3\u7801\u683c\u5f0f\u5316\u9009\u9879\uff0c\u8bbe\u7f6e\u4e3a <code>{ BasedOnStyle: Microsoft, BreakBeforeBraces: Attach }</code> \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#3-java","title":"3. \u00a0 Java \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 OpenJDK\uff08\u7248\u672c\u9700\u6ee1\u8db3 &gt; JDK 9\uff09\u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>java</code> \uff0c\u5b89\u88c5 Extension Pack for Java \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#4-c","title":"4. \u00a0 C# \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 .Net 8.0 \u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>C# Dev Kit</code> \uff0c\u5b89\u88c5 C# Dev Kit \uff08\u914d\u7f6e\u6559\u7a0b\uff09\u3002</li> <li>\u4e5f\u53ef\u4f7f\u7528 Visual Studio\uff08\u5b89\u88c5\u6559\u7a0b\uff09\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#5-go","title":"5. \u00a0 Go \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 go \u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>go</code> \uff0c\u5b89\u88c5 Go \u3002</li> <li>\u6309\u5feb\u6377\u952e <code>Ctrl + Shift + P</code> \u547c\u51fa\u547d\u4ee4\u680f\uff0c\u8f93\u5165 go \uff0c\u9009\u62e9 <code>Go: Install/Update Tools</code> \uff0c\u5168\u90e8\u52fe\u9009\u5e76\u5b89\u88c5\u5373\u53ef\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#6-swift","title":"6. \u00a0 Swift \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Swift \u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>swift</code> \uff0c\u5b89\u88c5 Swift for Visual Studio Code \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#7-javascript","title":"7. \u00a0 JavaScript \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 node.js \u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>javascript</code> \uff0c\u5b89\u88c5 JavaScript (ES6) code snippets \u3002</li> <li>\uff08\u53ef\u9009\uff09\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>Prettier</code> \uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#8-dart","title":"8. \u00a0 Dart \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Dart \u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>dart</code> \uff0c\u5b89\u88c5 Dart \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#9-rust","title":"9. \u00a0 Rust \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Rust \u3002</li> <li>\u5728 VS Code \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>rust</code> \uff0c\u5b89\u88c5 rust-analyzer \u3002</li> </ol>"},{"location":"chapter_appendix/terminology/","title":"16.3 \u00a0 \u672f\u8bed\u8868","text":"<p>\u8868 16-1 \u5217\u51fa\u4e86\u4e66\u4e2d\u51fa\u73b0\u7684\u91cd\u8981\u672f\u8bed\uff0c\u503c\u5f97\u6ce8\u610f\u4ee5\u4e0b\u51e0\u70b9\u3002</p> <ul> <li>\u5efa\u8bae\u8bb0\u4f4f\u540d\u8bcd\u7684\u82f1\u6587\u53eb\u6cd5\uff0c\u4ee5\u4fbf\u9605\u8bfb\u82f1\u6587\u6587\u732e\u3002</li> <li>\u90e8\u5206\u540d\u8bcd\u5728\u7b80\u4f53\u4e2d\u6587\u548c\u7e41\u4f53\u4e2d\u6587\u4e0b\u7684\u53eb\u6cd5\u4e0d\u540c\u3002</li> </ul> <p> \u8868 16-1 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u91cd\u8981\u540d\u8bcd </p> English \u7b80\u4f53\u4e2d\u6587 \u7e41\u4f53\u4e2d\u6587 algorithm \u7b97\u6cd5 \u6f14\u7b97\u6cd5 data structure \u6570\u636e\u7ed3\u6784 \u8cc7\u6599\u7d50\u69cb code \u4ee3\u7801 \u7a0b\u5f0f\u78bc file \u6587\u4ef6 \u6a94\u6848 function \u51fd\u6570 \u51fd\u5f0f method \u65b9\u6cd5 \u65b9\u6cd5 variable \u53d8\u91cf \u8b8a\u6578 asymptotic complexity analysis \u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 \u6f38\u8fd1\u8907\u96dc\u5ea6\u5206\u6790 time complexity \u65f6\u95f4\u590d\u6742\u5ea6 \u6642\u9593\u8907\u96dc\u5ea6 space complexity \u7a7a\u95f4\u590d\u6742\u5ea6 \u7a7a\u9593\u8907\u96dc\u5ea6 loop \u5faa\u73af \u8ff4\u5708 iteration \u8fed\u4ee3 \u8fed\u4ee3 recursion \u9012\u5f52 \u905e\u8ff4 tail recursion \u5c3e\u9012\u5f52 \u5c3e\u905e\u8ff4 recursion tree \u9012\u5f52\u6811 \u905e\u8ff4\u6a39 big-\\(O\\) notation \u5927 \\(O\\) \u8bb0\u53f7 \u5927 \\(O\\) \u8a18\u865f asymptotic upper bound \u6e10\u8fd1\u4e0a\u754c \u6f38\u8fd1\u4e0a\u754c sign-magnitude \u539f\u7801 \u539f\u78bc 1\u2019s complement \u53cd\u7801 \u53cd\u78bc 2\u2019s complement \u8865\u7801 \u88dc\u78bc array \u6570\u7ec4 \u9663\u5217 index \u7d22\u5f15 \u7d22\u5f15 linked list \u94fe\u8868 \u93c8\u7d50\u4e32\u5217 linked list node, list node \u94fe\u8868\u8282\u70b9 \u93c8\u7d50\u4e32\u5217\u7bc0\u9ede head node \u5934\u8282\u70b9 \u982d\u7bc0\u9ede tail node \u5c3e\u8282\u70b9 \u5c3e\u7bc0\u9ede list \u5217\u8868 \u5217\u8868 dynamic array \u52a8\u6001\u6570\u7ec4 \u52d5\u614b\u9663\u5217 hard disk \u786c\u76d8 \u786c\u789f random-access memory (RAM) \u5185\u5b58 \u5167\u5b58 cache memory \u7f13\u5b58 \u5feb\u53d6 cache miss \u7f13\u5b58\u672a\u547d\u4e2d \u5feb\u53d6\u672a\u547d\u4e2d cache hit rate \u7f13\u5b58\u547d\u4e2d\u7387 \u5feb\u53d6\u547d\u4e2d\u7387 stack \u6808 \u5806\u758a top of the stack \u6808\u9876 \u5806\u758a\u9802 bottom of the stack \u6808\u5e95 \u5806\u758a\u5e95 queue \u961f\u5217 \u4f47\u5217 double-ended queue \u53cc\u5411\u961f\u5217 \u96d9\u5411\u4f47\u5217 front of the queue \u961f\u9996 \u4f47\u5217\u9996 rear of the queue \u961f\u5c3e \u4f47\u5217\u5c3e hash table \u54c8\u5e0c\u8868 \u96dc\u6e4a\u8868 bucket \u6876 \u6876 hash function \u54c8\u5e0c\u51fd\u6570 \u96dc\u6e4a\u51fd\u5f0f hash collision \u54c8\u5e0c\u51b2\u7a81 \u96dc\u6e4a\u885d\u7a81 load factor \u8d1f\u8f7d\u56e0\u5b50 \u8ca0\u8f09\u56e0\u5b50 separate chaining \u94fe\u5f0f\u5730\u5740 \u93c8\u7d50\u4f4d\u5740 open addressing \u5f00\u653e\u5bfb\u5740 \u958b\u653e\u5b9a\u5740 linear probing \u7ebf\u6027\u63a2\u6d4b \u7dda\u6027\u63a2\u67e5 lazy deletion \u61d2\u5220\u9664 \u61f6\u522a\u9664 binary tree \u4e8c\u53c9\u6811 \u4e8c\u5143\u6a39 tree node \u6811\u8282\u70b9 \u6a39\u7bc0\u9ede left-child node \u5de6\u5b50\u8282\u70b9 \u5de6\u5b50\u7bc0\u9ede right-child node \u53f3\u5b50\u8282\u70b9 \u53f3\u5b50\u7bc0\u9ede parent node \u7236\u8282\u70b9 \u7236\u7bc0\u9ede left subtree \u5de6\u5b50\u6811 \u5de6\u5b50\u6a39 right subtree \u53f3\u5b50\u6811 \u53f3\u5b50\u6a39 root node \u6839\u8282\u70b9 \u6839\u7bc0\u9ede leaf node \u53f6\u8282\u70b9 \u8449\u7bc0\u9ede edge \u8fb9 \u908a level \u5c42 \u5c64 degree \u5ea6 \u5ea6 height \u9ad8\u5ea6 \u9ad8\u5ea6 depth \u6df1\u5ea6 \u6df1\u5ea6 perfect binary tree \u5b8c\u7f8e\u4e8c\u53c9\u6811 \u5b8c\u7f8e\u4e8c\u5143\u6a39 complete binary tree \u5b8c\u5168\u4e8c\u53c9\u6811 \u5b8c\u5168\u4e8c\u5143\u6a39 full binary tree \u5b8c\u6ee1\u4e8c\u53c9\u6811 \u5b8c\u6eff\u4e8c\u5143\u6a39 balanced binary tree \u5e73\u8861\u4e8c\u53c9\u6811 \u5e73\u8861\u4e8c\u5143\u6a39 binary search tree \u4e8c\u53c9\u641c\u7d22\u6811 \u4e8c\u5143\u641c\u5c0b\u6a39 AVL tree AVL \u6811 AVL \u6a39 red-black tree \u7ea2\u9ed1\u6811 \u7d05\u9ed1\u6a39 level-order traversal \u5c42\u5e8f\u904d\u5386 \u5c64\u5e8f\u8d70\u8a2a breadth-first traversal \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 \u5ee3\u5ea6\u512a\u5148\u8d70\u8a2a depth-first traversal \u6df1\u5ea6\u4f18\u5148\u904d\u5386 \u6df1\u5ea6\u512a\u5148\u8d70\u8a2a binary search tree \u4e8c\u53c9\u641c\u7d22\u6811 \u4e8c\u5143\u641c\u5c0b\u6a39 balanced binary search tree \u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811 \u5e73\u8861\u4e8c\u5143\u641c\u5c0b\u6a39 balance factor \u5e73\u8861\u56e0\u5b50 \u5e73\u8861\u56e0\u5b50 heap \u5806 \u5806\u7a4d max heap \u5927\u9876\u5806 \u5927\u9802\u5806\u7a4d min heap \u5c0f\u9876\u5806 \u5c0f\u9802\u5806\u7a4d priority queue \u4f18\u5148\u961f\u5217 \u512a\u5148\u4f47\u5217 heapify \u5806\u5316 \u5806\u7a4d\u5316 top-\\(k\\) problem Top-\\(k\\) \u95ee\u9898 Top-\\(k\\) \u554f\u984c graph \u56fe \u5716 vertex \u9876\u70b9 \u9802\u9ede undirected graph \u65e0\u5411\u56fe \u7121\u5411\u5716 directed graph \u6709\u5411\u56fe \u6709\u5411\u5716 connected graph \u8fde\u901a\u56fe \u9023\u901a\u5716 disconnected graph \u975e\u8fde\u901a\u56fe \u975e\u9023\u901a\u5716 weighted graph \u6709\u6743\u56fe \u6709\u6b0a\u5716 adjacency \u90bb\u63a5 \u9130\u63a5 path \u8def\u5f84 \u8def\u5f91 in-degree \u5165\u5ea6 \u5165\u5ea6 out-degree \u51fa\u5ea6 \u51fa\u5ea6 adjacency matrix \u90bb\u63a5\u77e9\u9635 \u9130\u63a5\u77e9\u9663 adjacency list \u90bb\u63a5\u8868 \u9130\u63a5\u8868 breadth-first search \u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 \u5ee3\u5ea6\u512a\u5148\u641c\u5c0b depth-first search \u6df1\u5ea6\u4f18\u5148\u641c\u7d22 \u6df1\u5ea6\u512a\u5148\u641c\u5c0b binary search \u4e8c\u5206\u67e5\u627e \u4e8c\u5206\u641c\u5c0b searching algorithm \u641c\u7d22\u7b97\u6cd5 \u641c\u5c0b\u6f14\u7b97\u6cd5 sorting algorithm \u6392\u5e8f\u7b97\u6cd5 \u6392\u5e8f\u6f14\u7b97\u6cd5 selection sort \u9009\u62e9\u6392\u5e8f \u9078\u64c7\u6392\u5e8f bubble sort \u5192\u6ce1\u6392\u5e8f \u6ce1\u6cab\u6392\u5e8f insertion sort \u63d2\u5165\u6392\u5e8f \u63d2\u5165\u6392\u5e8f quick sort \u5feb\u901f\u6392\u5e8f \u5feb\u901f\u6392\u5e8f merge sort \u5f52\u5e76\u6392\u5e8f \u5408\u4f75\u6392\u5e8f heap sort \u5806\u6392\u5e8f \u5806\u7a4d\u6392\u5e8f bucket sort \u6876\u6392\u5e8f \u6876\u6392\u5e8f counting sort \u8ba1\u6570\u6392\u5e8f \u8a08\u6578\u6392\u5e8f radix sort \u57fa\u6570\u6392\u5e8f \u57fa\u6578\u6392\u5e8f divide and conquer \u5206\u6cbb \u5206\u6cbb hanota problem \u6c49\u8bfa\u5854\u95ee\u9898 \u6cb3\u5167\u5854\u554f\u984c backtracking algorithm \u56de\u6eaf\u7b97\u6cd5 \u56de\u6eaf\u6f14\u7b97\u6cd5 constraint \u7ea6\u675f \u7d04\u675f solution \u89e3 \u89e3 state \u72b6\u6001 \u72c0\u614b pruning \u526a\u679d \u526a\u679d permutations problem \u5168\u6392\u5217\u95ee\u9898 \u5168\u6392\u5217\u554f\u984c subset-sum problem \u5b50\u96c6\u548c\u95ee\u9898 \u5b50\u96c6\u5408\u554f\u984c \\(n\\)-queens problem \\(n\\) \u7687\u540e\u95ee\u9898 \\(n\\) \u7687\u540e\u554f\u984c dynamic programming \u52a8\u6001\u89c4\u5212 \u52d5\u614b\u898f\u5283 initial state \u521d\u59cb\u72b6\u6001 \u521d\u59cb\u72c0\u614b state-transition equation \u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b \u72c0\u614b\u8f49\u79fb\u65b9\u7a0b knapsack problem \u80cc\u5305\u95ee\u9898 \u80cc\u5305\u554f\u984c edit distance problem \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898 \u7de8\u8f2f\u8ddd\u96e2\u554f\u984c greedy algorithm \u8d2a\u5fc3\u7b97\u6cd5 \u8caa\u5a6a\u6f14\u7b97\u6cd5"},{"location":"chapter_array_and_linkedlist/","title":"\u7b2c 4 \u7ae0 \u00a0 \u6570\u7ec4\u4e0e\u94fe\u8868","text":"<p>Abstract</p> <p>\u6570\u636e\u7ed3\u6784\u7684\u4e16\u754c\u5982\u540c\u4e00\u5835\u539a\u5b9e\u7684\u7816\u5899\u3002</p> <p>\u6570\u7ec4\u7684\u7816\u5757\u6574\u9f50\u6392\u5217\uff0c\u9010\u4e2a\u7d27\u8d34\u3002\u94fe\u8868\u7684\u7816\u5757\u5206\u6563\u5404\u5904\uff0c\u8fde\u63a5\u7684\u85e4\u8513\u81ea\u7531\u5730\u7a7f\u68ad\u4e8e\u7816\u7f1d\u4e4b\u95f4\u3002</p>"},{"location":"chapter_array_and_linkedlist/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>4.1 \u00a0 \u6570\u7ec4</li> <li>4.2 \u00a0 \u94fe\u8868</li> <li>4.3 \u00a0 \u5217\u8868</li> <li>4.4 \u00a0 \u5185\u5b58\u4e0e\u7f13\u5b58 *</li> <li>4.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/","title":"4.1 \u00a0 \u6570\u7ec4","text":"<p>\u6570\u7ec4\uff08array\uff09\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5c06\u76f8\u540c\u7c7b\u578b\u7684\u5143\u7d20\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\u3002\u6211\u4eec\u5c06\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u4f4d\u7f6e\u79f0\u4e3a\u8be5\u5143\u7d20\u7684\u7d22\u5f15\uff08index\uff09\u3002\u56fe 4-1 \u5c55\u793a\u4e86\u6570\u7ec4\u7684\u4e3b\u8981\u6982\u5ff5\u548c\u5b58\u50a8\u65b9\u5f0f\u3002</p> <p></p> <p> \u56fe 4-1 \u00a0 \u6570\u7ec4\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f </p>"},{"location":"chapter_array_and_linkedlist/array/#411","title":"4.1.1 \u00a0 \u6570\u7ec4\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/array/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u6570\u7ec4","text":"<p>\u6211\u4eec\u53ef\u4ee5\u6839\u636e\u9700\u6c42\u9009\u7528\u6570\u7ec4\u7684\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u5f0f\uff1a\u65e0\u521d\u59cb\u503c\u3001\u7ed9\u5b9a\u521d\u59cb\u503c\u3002\u5728\u672a\u6307\u5b9a\u521d\u59cb\u503c\u7684\u60c5\u51b5\u4e0b\uff0c\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4f1a\u5c06\u6570\u7ec4\u5143\u7d20\u521d\u59cb\u5316\u4e3a \\(0\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code># \u521d\u59cb\u5316\u6570\u7ec4\narr: list[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: list[int] = [1, 3, 2, 5, 4]  \n</code></pre> array.cpp<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\n// \u5b58\u50a8\u5728\u6808\u4e0a\nint arr[5];\nint nums[5] = { 1, 3, 2, 5, 4 };\n// \u5b58\u50a8\u5728\u5806\u4e0a\uff08\u9700\u8981\u624b\u52a8\u91ca\u653e\u7a7a\u95f4\uff09\nint* arr1 = new int[5];\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n</code></pre> array.java<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n</code></pre> array.cs<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // [ 0, 0, 0, 0, 0 ]\nint[] nums = [1, 3, 2, 5, 4];\n</code></pre> array.go<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr [5]int\n// \u5728 Go \u4e2d\uff0c\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[5]int\uff09\u4e3a\u6570\u7ec4\uff0c\u4e0d\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[]int\uff09\u4e3a\u5207\u7247\n// \u7531\u4e8e Go \u7684\u6570\u7ec4\u88ab\u8bbe\u8ba1\u4e3a\u5728\u7f16\u8bd1\u671f\u786e\u5b9a\u957f\u5ea6\uff0c\u56e0\u6b64\u53ea\u80fd\u4f7f\u7528\u5e38\u91cf\u6765\u6307\u5b9a\u957f\u5ea6\n// \u4e3a\u4e86\u65b9\u4fbf\u5b9e\u73b0\u6269\u5bb9 extend() \u65b9\u6cd5\uff0c\u4ee5\u4e0b\u5c06\u5207\u7247\uff08Slice\uff09\u770b\u4f5c\u6570\u7ec4\uff08Array\uff09\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> array.swift<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n</code></pre> array.js<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n</code></pre> array.ts<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> array.dart<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nList&lt;int&gt; arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> array.rs<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: Vec&lt;i32&gt; = vec![0; 5]; // [0, 0, 0, 0, 0]\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> array.c<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n</code></pre> array.kt<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr = IntArray(5) // { 0, 0, 0, 0, 0 }\nvar nums = intArrayOf(1, 3, 2, 5, 4)\n</code></pre> array.rb<pre><code># \u521d\u59cb\u5316\u6570\u7ec4\narr = Array.new(5, 0)\nnums = [1, 3, 2, 5, 4]\n</code></pre> array.zig<pre><code>// \u521d\u59cb\u5316\u6570\u7ec4\nvar arr = [_]i32{0} ** 5; // { 0, 0, 0, 0, 0 }\nvar nums = [_]i32{ 1, 3, 2, 5, 4 };\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#2","title":"2. \u00a0 \u8bbf\u95ee\u5143\u7d20","text":"<p>\u6570\u7ec4\u5143\u7d20\u88ab\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u8fd9\u610f\u5473\u7740\u8ba1\u7b97\u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u975e\u5e38\u5bb9\u6613\u3002\u7ed9\u5b9a\u6570\u7ec4\u5185\u5b58\u5730\u5740\uff08\u9996\u5143\u7d20\u5185\u5b58\u5730\u5740\uff09\u548c\u67d0\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u56fe 4-2 \u6240\u793a\u7684\u516c\u5f0f\u8ba1\u7b97\u5f97\u5230\u8be5\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u76f4\u63a5\u8bbf\u95ee\u8be5\u5143\u7d20\u3002</p> <p></p> <p> \u56fe 4-2 \u00a0 \u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u8ba1\u7b97 </p> <p>\u89c2\u5bdf\u56fe 4-2 \uff0c\u6211\u4eec\u53d1\u73b0\u6570\u7ec4\u9996\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u4e3a \\(0\\) \uff0c\u8fd9\u4f3c\u4e4e\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u4ece \\(1\\) \u5f00\u59cb\u8ba1\u6570\u4f1a\u66f4\u81ea\u7136\u3002\u4f46\u4ece\u5730\u5740\u8ba1\u7b97\u516c\u5f0f\u7684\u89d2\u5ea6\u770b\uff0c\u7d22\u5f15\u672c\u8d28\u4e0a\u662f\u5185\u5b58\u5730\u5740\u7684\u504f\u79fb\u91cf\u3002\u9996\u4e2a\u5143\u7d20\u7684\u5730\u5740\u504f\u79fb\u91cf\u662f \\(0\\) \uff0c\u56e0\u6b64\u5b83\u7684\u7d22\u5f15\u4e3a \\(0\\) \u662f\u5408\u7406\u7684\u3002</p> <p>\u5728\u6570\u7ec4\u4e2d\u8bbf\u95ee\u5143\u7d20\u975e\u5e38\u9ad8\u6548\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u968f\u673a\u8bbf\u95ee\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\u5143\u7d20\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def random_access(nums: list[int]) -&gt; int:\n    \"\"\"\u968f\u673a\u8bbf\u95ee\u5143\u7d20\"\"\"\n    # \u5728\u533a\u95f4 [0, len(nums)-1] \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    random_index = random.randint(0, len(nums) - 1)\n    # \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    random_num = nums[random_index]\n    return random_num\n</code></pre> array.cpp<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n    // \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = rand() % size;\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.java<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int[] nums) {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.cs<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint RandomAccess(int[] nums) {\n    Random random = new();\n    // \u5728\u533a\u95f4 [0, nums.Length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = random.Next(nums.Length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.go<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums []int) (randomNum int) {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    randomIndex := rand.Intn(len(nums))\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    randomNum = nums[randomIndex]\n    return\n}\n</code></pre> array.swift<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums: [Int]) -&gt; Int {\n    // \u5728\u533a\u95f4 [0, nums.count) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    let randomIndex = nums.indices.randomElement()!\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    let randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.js<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums) {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    const random_index = Math.floor(Math.random() * nums.length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.ts<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums: number[]): number {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    const random_index = Math.floor(Math.random() * nums.length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.dart<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(List&lt;int&gt; nums) {\n  // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n  int randomIndex = Random().nextInt(nums.length);\n  // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n  int randomNum = nums[randomIndex];\n  return randomNum;\n}\n</code></pre> array.rs<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfn random_access(nums: &amp;[i32]) -&gt; i32 {\n    // \u5728\u533a\u95f4 [0, nums.len()) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    let random_index = rand::thread_rng().gen_range(0..nums.len());\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    let random_num = nums[random_index];\n    random_num\n}\n</code></pre> array.c<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n    // \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = rand() % size;\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.kt<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfun randomAccess(nums: IntArray): Int {\n    // \u5728\u533a\u95f4 [0, nums.size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    val randomIndex = ThreadLocalRandom.current().nextInt(0, nums.size)\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    val randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.rb<pre><code>### \u968f\u673a\u8bbf\u95ee\u5143\u7d20 ###\ndef random_access(nums)\n  # \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n  random_index = Random.rand 0...(nums.length - 1)\n\n  # \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n  nums[random_index]\nend\n</code></pre> array.zig<pre><code>// \u968f\u673a\u8bbf\u95ee\u5143\u7d20\nfn randomAccess(nums: []i32) i32 {\n    // \u5728\u533a\u95f4 [0, nums.len) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6574\u6570\n    var randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    var randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#3","title":"3. \u00a0 \u63d2\u5165\u5143\u7d20","text":"<p>\u6570\u7ec4\u5143\u7d20\u5728\u5185\u5b58\u4e2d\u662f\u201c\u7d27\u6328\u7740\u7684\u201d\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u7a7a\u95f4\u518d\u5b58\u653e\u4efb\u4f55\u6570\u636e\u3002\u5982\u56fe 4-3 \u6240\u793a\uff0c\u5982\u679c\u60f3\u5728\u6570\u7ec4\u4e2d\u95f4\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\uff0c\u5219\u9700\u8981\u5c06\u8be5\u5143\u7d20\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u4e4b\u540e\u518d\u628a\u5143\u7d20\u8d4b\u503c\u7ed9\u8be5\u7d22\u5f15\u3002</p> <p></p> <p> \u56fe 4-3 \u00a0 \u6570\u7ec4\u63d2\u5165\u5143\u7d20\u793a\u4f8b </p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u662f\u56fa\u5b9a\u7684\uff0c\u56e0\u6b64\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u4f1a\u5bfc\u81f4\u6570\u7ec4\u5c3e\u90e8\u5143\u7d20\u201c\u4e22\u5931\u201d\u3002\u6211\u4eec\u5c06\u8fd9\u4e2a\u95ee\u9898\u7684\u89e3\u51b3\u65b9\u6848\u7559\u5728\u201c\u5217\u8868\u201d\u7ae0\u8282\u4e2d\u8ba8\u8bba\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def insert(nums: list[int], num: int, index: int):\n    \"\"\"\u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\"\"\"\n    # \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i in range(len(nums) - 1, index, -1):\n        nums[i] = nums[i - 1]\n    # \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n</code></pre> array.cpp<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.java<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid Insert(int[] nums, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = nums.Length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.go<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums []int, num int, index int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i := len(nums) - 1; i &gt; index; i-- {\n        nums[i] = nums[i-1]\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.swift<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i in nums.indices.dropFirst(index).reversed() {\n        nums[i] = nums[i - 1]\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.js<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums, num, index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums: number[], num: number, index: number): void {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 _num */\nvoid insert(List&lt;int&gt; nums, int _num, int index) {\n  // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n  for (var i = nums.length - 1; i &gt; index; i--) {\n    nums[i] = nums[i - 1];\n  }\n  // \u5c06 _num \u8d4b\u7ed9 index \u5904\u5143\u7d20\n  nums[index] = _num;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfn insert(nums: &amp;mut Vec&lt;i32&gt;, num: i32, index: usize) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i in (index + 1..nums.len()).rev() {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.c<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.kt<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfun insert(nums: IntArray, num: Int, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (i in nums.size - 1 downTo index + 1) {\n        nums[i] = nums[i - 1]\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.rb<pre><code>### \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num ###\ndef insert(nums, num, index)\n  # \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n  for i in (nums.length - 1).downto(index + 1)\n    nums[i] = nums[i - 1]\n  end\n\n  # \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n  nums[index] = num\nend\n</code></pre> array.zig<pre><code>// \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\nfn insert(nums: []i32, num: i32, index: usize) void {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    var i = nums.len - 1;\n    while (i &gt; index) : (i -= 1) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#4","title":"4. \u00a0 \u5220\u9664\u5143\u7d20","text":"<p>\u540c\u7406\uff0c\u5982\u56fe 4-4 \u6240\u793a\uff0c\u82e5\u60f3\u5220\u9664\u7d22\u5f15 \\(i\\) \u5904\u7684\u5143\u7d20\uff0c\u5219\u9700\u8981\u628a\u7d22\u5f15 \\(i\\) \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\u3002</p> <p></p> <p> \u56fe 4-4 \u00a0 \u6570\u7ec4\u5220\u9664\u5143\u7d20\u793a\u4f8b </p> <p>\u8bf7\u6ce8\u610f\uff0c\u5220\u9664\u5143\u7d20\u5b8c\u6210\u540e\uff0c\u539f\u5148\u672b\u5c3e\u7684\u5143\u7d20\u53d8\u5f97\u201c\u65e0\u610f\u4e49\u201d\u4e86\uff0c\u6240\u4ee5\u6211\u4eec\u65e0\u987b\u7279\u610f\u53bb\u4fee\u6539\u5b83\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def remove(nums: list[int], index: int):\n    \"\"\"\u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\"\"\"\n    # \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i in range(index, len(nums) - 1):\n        nums[i] = nums[i + 1]\n</code></pre> array.cpp<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int *nums, int size, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.java<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.cs<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid Remove(int[] nums, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; nums.Length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.go<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums []int, index int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i := index; i &lt; len(nums)-1; i++ {\n        nums[i] = nums[i+1]\n    }\n}\n</code></pre> array.swift<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums: inout [Int], index: Int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i in nums.indices.dropFirst(index).dropLast() {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.js<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums, index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.ts<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums: number[], index: number): void {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.dart<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(List&lt;int&gt; nums, int index) {\n  // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n  for (var i = index; i &lt; nums.length - 1; i++) {\n    nums[i] = nums[i + 1];\n  }\n}\n</code></pre> array.rs<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfn remove(nums: &amp;mut Vec&lt;i32&gt;, index: usize) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i in index..nums.len() - 1 {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.c<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(int *nums, int size, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.kt<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfun remove(nums: IntArray, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (i in index..&lt;nums.size - 1) {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.rb<pre><code>### \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 ###\ndef remove(nums, index)\n  # \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n  for i in index...nums.length\n    nums[i] = nums[i + 1] || 0\n  end\nend\n</code></pre> array.zig<pre><code>// \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\nfn remove(nums: []i32, index: usize) void {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    var i = index;\n    while (i &lt; nums.len - 1) : (i += 1) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u603b\u7684\u6765\u770b\uff0c\u6570\u7ec4\u7684\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u6709\u4ee5\u4e0b\u7f3a\u70b9\u3002</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u9ad8\uff1a\u6570\u7ec4\u7684\u63d2\u5165\u548c\u5220\u9664\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u957f\u5ea6\u3002</li> <li>\u4e22\u5931\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u5728\u63d2\u5165\u5143\u7d20\u540e\uff0c\u8d85\u51fa\u6570\u7ec4\u957f\u5ea6\u8303\u56f4\u7684\u5143\u7d20\u4f1a\u4e22\u5931\u3002</li> <li>\u5185\u5b58\u6d6a\u8d39\uff1a\u6211\u4eec\u53ef\u4ee5\u521d\u59cb\u5316\u4e00\u4e2a\u6bd4\u8f83\u957f\u7684\u6570\u7ec4\uff0c\u53ea\u7528\u524d\u9762\u4e00\u90e8\u5206\uff0c\u8fd9\u6837\u5728\u63d2\u5165\u6570\u636e\u65f6\uff0c\u4e22\u5931\u7684\u672b\u5c3e\u5143\u7d20\u90fd\u662f\u201c\u65e0\u610f\u4e49\u201d\u7684\uff0c\u4f46\u8fd9\u6837\u505a\u4f1a\u9020\u6210\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#5","title":"5. \u00a0 \u904d\u5386\u6570\u7ec4","text":"<p>\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u65e2\u53ef\u4ee5\u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u904d\u5386\u83b7\u53d6\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def traverse(nums: list[int]):\n    \"\"\"\u904d\u5386\u6570\u7ec4\"\"\"\n    count = 0\n    # \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i in range(len(nums)):\n        count += nums[i]\n    # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for num in nums:\n        count += num\n    # \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n    for i, num in enumerate(nums):\n        count += nums[i]\n        count += num\n</code></pre> array.cpp<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.java<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (int num : nums) {\n        count += num;\n    }\n}\n</code></pre> array.cs<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid Traverse(int[] nums) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; nums.Length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    foreach (int num in nums) {\n        count += num;\n    }\n}\n</code></pre> array.go<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums []int) {\n    count := 0\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i := 0; i &lt; len(nums); i++ {\n        count += nums[i]\n    }\n    count = 0\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for _, num := range nums {\n        count += num\n    }\n    // \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n    for i, num := range nums {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.swift<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums: [Int]) {\n    var count = 0\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i in nums.indices {\n        count += nums[i]\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for num in nums {\n        count += num\n    }\n    // \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n    for (i, num) in nums.enumerated() {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.js<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums) {\n    let count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.ts<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums: number[]): void {\n    let count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.dart<pre><code>/* \u904d\u5386\u6570\u7ec4\u5143\u7d20 */\nvoid traverse(List&lt;int&gt; nums) {\n  int count = 0;\n  // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n  for (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n  }\n  // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n  for (int _num in nums) {\n    count += _num;\n  }\n  // \u901a\u8fc7 forEach \u65b9\u6cd5\u904d\u5386\u6570\u7ec4\n  nums.forEach((_num) {\n    count += _num;\n  });\n}\n</code></pre> array.rs<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfn traverse(nums: &amp;[i32]) {\n    let mut _count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i in 0..nums.len() {\n        _count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for num in nums {\n        _count += num;\n    }\n}\n</code></pre> array.c<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.kt<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfun traverse(nums: IntArray) {\n    var count = 0\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (i in nums.indices) {\n        count += nums[i]\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (j: Int in nums) {\n        count += j\n    }\n}\n</code></pre> array.rb<pre><code>### \u904d\u5386\u6570\u7ec4 ###\ndef traverse(nums)\n  count = 0\n\n  # \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n  for i in 0...nums.length\n    count += nums[i]\n  end\n\n  # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n  for num in nums\n    count += num\n  end\nend\n</code></pre> array.zig<pre><code>// \u904d\u5386\u6570\u7ec4\nfn traverse(nums: []i32) void {\n    var count: i32 = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    var i: i32 = 0;\n    while (i &lt; nums.len) : (i += 1) {\n        count += nums[i];\n    }\n    count = 0;\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (nums) |num| {\n        count += num;\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#6","title":"6. \u00a0 \u67e5\u627e\u5143\u7d20","text":"<p>\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\u9700\u8981\u904d\u5386\u6570\u7ec4\uff0c\u6bcf\u8f6e\u5224\u65ad\u5143\u7d20\u503c\u662f\u5426\u5339\u914d\uff0c\u82e5\u5339\u914d\u5219\u8f93\u51fa\u5bf9\u5e94\u7d22\u5f15\u3002</p> <p>\u56e0\u4e3a\u6570\u7ec4\u662f\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u6240\u4ee5\u4e0a\u8ff0\u67e5\u627e\u64cd\u4f5c\u88ab\u79f0\u4e3a\u201c\u7ebf\u6027\u67e5\u627e\u201d\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def find(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\"\"\"\n    for i in range(len(nums)):\n        if nums[i] == target:\n            return i\n    return -1\n</code></pre> array.cpp<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.java<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint Find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.go<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums []int, target int) (index int) {\n    index = -1\n    for i := 0; i &lt; len(nums); i++ {\n        if nums[i] == target {\n            index = i\n            break\n        }\n    }\n    return\n}\n</code></pre> array.swift<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums: [Int], target: Int) -&gt; Int {\n    for i in nums.indices {\n        if nums[i] == target {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> array.js<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums, target) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) return i;\n    }\n    return -1;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums: number[], target: number): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(List&lt;int&gt; nums, int target) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    if (nums[i] == target) return i;\n  }\n  return -1;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfn find(nums: &amp;[i32], target: i32) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        if nums[i] == target {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> array.c<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.kt<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfun find(nums: IntArray, target: Int): Int {\n    for (i in nums.indices) {\n        if (nums[i] == target) return i\n    }\n    return -1\n}\n</code></pre> array.rb<pre><code>### \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 ###\ndef find(nums, target)\n  for i in 0...nums.length\n    return i if nums[i] == target\n  end\n\n  -1\nend\n</code></pre> array.zig<pre><code>// \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\nfn find(nums: []i32, target: i32) i32 {\n    for (nums, 0..) |num, i| {\n        if (num == target) return @intCast(i);\n    }\n    return -1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#7","title":"7. \u00a0 \u6269\u5bb9\u6570\u7ec4","text":"<p>\u5728\u590d\u6742\u7684\u7cfb\u7edf\u73af\u5883\u4e2d\uff0c\u7a0b\u5e8f\u96be\u4ee5\u4fdd\u8bc1\u6570\u7ec4\u4e4b\u540e\u7684\u5185\u5b58\u7a7a\u95f4\u662f\u53ef\u7528\u7684\uff0c\u4ece\u800c\u65e0\u6cd5\u5b89\u5168\u5730\u6269\u5c55\u6570\u7ec4\u5bb9\u91cf\u3002\u56e0\u6b64\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6570\u7ec4\u7684\u957f\u5ea6\u662f\u4e0d\u53ef\u53d8\u7684\u3002</p> <p>\u5982\u679c\u6211\u4eec\u5e0c\u671b\u6269\u5bb9\u6570\u7ec4\uff0c\u5219\u9700\u91cd\u65b0\u5efa\u7acb\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u7136\u540e\u628a\u539f\u6570\u7ec4\u5143\u7d20\u4f9d\u6b21\u590d\u5236\u5230\u65b0\u6570\u7ec4\u3002\u8fd9\u662f\u4e00\u4e2a \\(O(n)\\) \u7684\u64cd\u4f5c\uff0c\u5728\u6570\u7ec4\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\u975e\u5e38\u8017\u65f6\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array.py<pre><code>def extend(nums: list[int], enlarge: int) -&gt; list[int]:\n    \"\"\"\u6269\u5c55\u6570\u7ec4\u957f\u5ea6\"\"\"\n    # \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    res = [0] * (len(nums) + enlarge)\n    # \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for i in range(len(nums)):\n        res[i] = nums[i]\n    # \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n</code></pre> array.cpp<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int *res = new int[size + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // \u91ca\u653e\u5185\u5b58\n    delete[] nums;\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.java<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int[] res = new int[nums.length + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.cs<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] Extend(int[] nums, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int[] res = new int[nums.Length + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; nums.Length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.go<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums []int, enlarge int) []int {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    res := make([]int, len(nums)+enlarge)\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for i, num := range nums {\n        res[i] = num\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n}\n</code></pre> array.swift<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums: [Int], enlarge: Int) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    var res = Array(repeating: 0, count: nums.count + enlarge)\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for i in nums.indices {\n        res[i] = nums[i]\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n}\n</code></pre> array.js<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cJavaScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums, enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    const res = new Array(nums.length + enlarge).fill(0);\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.ts<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cTypeScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums: number[], enlarge: number): number[] {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    const res = new Array(nums.length + enlarge).fill(0);\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.dart<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nList&lt;int&gt; extend(List&lt;int&gt; nums, int enlarge) {\n  // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n  List&lt;int&gt; res = List.filled(nums.length + enlarge, 0);\n  // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n  for (var i = 0; i &lt; nums.length; i++) {\n    res[i] = nums[i];\n  }\n  // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n  return res;\n}\n</code></pre> array.rs<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfn extend(nums: Vec&lt;i32&gt;, enlarge: usize) -&gt; Vec&lt;i32&gt; {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    let mut res: Vec&lt;i32&gt; = vec![0; nums.len() + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\n    for i in 0..nums.len() {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    res\n}\n</code></pre> array.c<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int *res = (int *)malloc(sizeof(int) * (size + enlarge));\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // \u521d\u59cb\u5316\u6269\u5c55\u540e\u7684\u7a7a\u95f4\n    for (int i = size; i &lt; size + enlarge; i++) {\n        res[i] = 0;\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.kt<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfun extend(nums: IntArray, enlarge: Int): IntArray {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    val res = IntArray(nums.size + enlarge)\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (i in nums.indices) {\n        res[i] = nums[i]\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n}\n</code></pre> array.rb<pre><code>### \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 ###\n# \u8bf7\u6ce8\u610f\uff0cRuby \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n# \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\ndef extend(nums, enlarge)\n  # \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n  res = Array.new(nums.length + enlarge, 0)\n\n  # \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n  for i in 0...nums.length\n    res[i] = nums[i]\n  end\n\n  # \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n  res\nend\n</code></pre> array.zig<pre><code>// \u6269\u5c55\u6570\u7ec4\u957f\u5ea6\nfn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    var res = try mem_allocator.alloc(i32, nums.len + enlarge);\n    @memset(res, 0);\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    std.mem.copy(i32, res, nums);\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/array/#412","title":"4.1.2 \u00a0 \u6570\u7ec4\u7684\u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u6570\u7ec4\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u5185\uff0c\u4e14\u5143\u7d20\u7c7b\u578b\u76f8\u540c\u3002\u8fd9\u79cd\u505a\u6cd5\u5305\u542b\u4e30\u5bcc\u7684\u5148\u9a8c\u4fe1\u606f\uff0c\u7cfb\u7edf\u53ef\u4ee5\u5229\u7528\u8fd9\u4e9b\u4fe1\u606f\u6765\u4f18\u5316\u6570\u636e\u7ed3\u6784\u7684\u64cd\u4f5c\u6548\u7387\u3002</p> <ul> <li>\u7a7a\u95f4\u6548\u7387\u9ad8\uff1a\u6570\u7ec4\u4e3a\u6570\u636e\u5206\u914d\u4e86\u8fde\u7eed\u7684\u5185\u5b58\u5757\uff0c\u65e0\u987b\u989d\u5916\u7684\u7ed3\u6784\u5f00\u9500\u3002</li> <li>\u652f\u6301\u968f\u673a\u8bbf\u95ee\uff1a\u6570\u7ec4\u5141\u8bb8\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u4efb\u4f55\u5143\u7d20\u3002</li> <li>\u7f13\u5b58\u5c40\u90e8\u6027\uff1a\u5f53\u8bbf\u95ee\u6570\u7ec4\u5143\u7d20\u65f6\uff0c\u8ba1\u7b97\u673a\u4e0d\u4ec5\u4f1a\u52a0\u8f7d\u5b83\uff0c\u8fd8\u4f1a\u7f13\u5b58\u5176\u5468\u56f4\u7684\u5176\u4ed6\u6570\u636e\uff0c\u4ece\u800c\u501f\u52a9\u9ad8\u901f\u7f13\u5b58\u6765\u63d0\u5347\u540e\u7eed\u64cd\u4f5c\u7684\u6267\u884c\u901f\u5ea6\u3002</li> </ul> <p>\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u662f\u4e00\u628a\u53cc\u5203\u5251\uff0c\u5176\u5b58\u5728\u4ee5\u4e0b\u5c40\u9650\u6027\u3002</p> <ul> <li>\u63d2\u5165\u4e0e\u5220\u9664\u6548\u7387\u4f4e\uff1a\u5f53\u6570\u7ec4\u4e2d\u5143\u7d20\u8f83\u591a\u65f6\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u9700\u8981\u79fb\u52a8\u5927\u91cf\u7684\u5143\u7d20\u3002</li> <li>\u957f\u5ea6\u4e0d\u53ef\u53d8\uff1a\u6570\u7ec4\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u5c31\u56fa\u5b9a\u4e86\uff0c\u6269\u5bb9\u6570\u7ec4\u9700\u8981\u5c06\u6240\u6709\u6570\u636e\u590d\u5236\u5230\u65b0\u6570\u7ec4\uff0c\u5f00\u9500\u5f88\u5927\u3002</li> <li>\u7a7a\u95f4\u6d6a\u8d39\uff1a\u5982\u679c\u6570\u7ec4\u5206\u914d\u7684\u5927\u5c0f\u8d85\u8fc7\u5b9e\u9645\u6240\u9700\uff0c\u90a3\u4e48\u591a\u4f59\u7684\u7a7a\u95f4\u5c31\u88ab\u6d6a\u8d39\u4e86\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#413","title":"4.1.3 \u00a0 \u6570\u7ec4\u5178\u578b\u5e94\u7528","text":"<p>\u6570\u7ec4\u662f\u4e00\u79cd\u57fa\u7840\u4e14\u5e38\u89c1\u7684\u6570\u636e\u7ed3\u6784\uff0c\u65e2\u9891\u7e41\u5e94\u7528\u5728\u5404\u7c7b\u7b97\u6cd5\u4e4b\u4e2d\uff0c\u4e5f\u53ef\u7528\u4e8e\u5b9e\u73b0\u5404\u79cd\u590d\u6742\u6570\u636e\u7ed3\u6784\u3002</p> <ul> <li>\u968f\u673a\u8bbf\u95ee\uff1a\u5982\u679c\u6211\u4eec\u60f3\u968f\u673a\u62bd\u53d6\u4e00\u4e9b\u6837\u672c\uff0c\u90a3\u4e48\u53ef\u4ee5\u7528\u6570\u7ec4\u5b58\u50a8\uff0c\u5e76\u751f\u6210\u4e00\u4e2a\u968f\u673a\u5e8f\u5217\uff0c\u6839\u636e\u7d22\u5f15\u5b9e\u73b0\u968f\u673a\u62bd\u6837\u3002</li> <li>\u6392\u5e8f\u548c\u641c\u7d22\uff1a\u6570\u7ec4\u662f\u6392\u5e8f\u548c\u641c\u7d22\u7b97\u6cd5\u6700\u5e38\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u4e8c\u5206\u67e5\u627e\u7b49\u90fd\u4e3b\u8981\u5728\u6570\u7ec4\u4e0a\u8fdb\u884c\u3002</li> <li>\u67e5\u627e\u8868\uff1a\u5f53\u9700\u8981\u5feb\u901f\u67e5\u627e\u4e00\u4e2a\u5143\u7d20\u6216\u5176\u5bf9\u5e94\u5173\u7cfb\u65f6\uff0c\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u4f5c\u4e3a\u67e5\u627e\u8868\u3002\u5047\u5982\u6211\u4eec\u60f3\u5b9e\u73b0\u5b57\u7b26\u5230 ASCII \u7801\u7684\u6620\u5c04\uff0c\u5219\u53ef\u4ee5\u5c06\u5b57\u7b26\u7684 ASCII \u7801\u503c\u4f5c\u4e3a\u7d22\u5f15\uff0c\u5bf9\u5e94\u7684\u5143\u7d20\u5b58\u653e\u5728\u6570\u7ec4\u4e2d\u7684\u5bf9\u5e94\u4f4d\u7f6e\u3002</li> <li>\u673a\u5668\u5b66\u4e60\uff1a\u795e\u7ecf\u7f51\u7edc\u4e2d\u5927\u91cf\u4f7f\u7528\u4e86\u5411\u91cf\u3001\u77e9\u9635\u3001\u5f20\u91cf\u4e4b\u95f4\u7684\u7ebf\u6027\u4ee3\u6570\u8fd0\u7b97\uff0c\u8fd9\u4e9b\u6570\u636e\u90fd\u662f\u4ee5\u6570\u7ec4\u7684\u5f62\u5f0f\u6784\u5efa\u7684\u3002\u6570\u7ec4\u662f\u795e\u7ecf\u7f51\u7edc\u7f16\u7a0b\u4e2d\u6700\u5e38\u4f7f\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff1a\u6570\u7ec4\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u3002\u4f8b\u5982\uff0c\u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a\u5b9e\u9645\u4e0a\u662f\u4e00\u4e2a\u4e8c\u7ef4\u6570\u7ec4\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/linked_list/","title":"4.2 \u00a0 \u94fe\u8868","text":"<p>\u5185\u5b58\u7a7a\u95f4\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u516c\u5171\u8d44\u6e90\uff0c\u5728\u4e00\u4e2a\u590d\u6742\u7684\u7cfb\u7edf\u8fd0\u884c\u73af\u5883\u4e0b\uff0c\u7a7a\u95f2\u7684\u5185\u5b58\u7a7a\u95f4\u53ef\u80fd\u6563\u843d\u5728\u5185\u5b58\u5404\u5904\u3002\u6211\u4eec\u77e5\u9053\uff0c\u5b58\u50a8\u6570\u7ec4\u7684\u5185\u5b58\u7a7a\u95f4\u5fc5\u987b\u662f\u8fde\u7eed\u7684\uff0c\u800c\u5f53\u6570\u7ec4\u975e\u5e38\u5927\u65f6\uff0c\u5185\u5b58\u53ef\u80fd\u65e0\u6cd5\u63d0\u4f9b\u5982\u6b64\u5927\u7684\u8fde\u7eed\u7a7a\u95f4\u3002\u6b64\u65f6\u94fe\u8868\u7684\u7075\u6d3b\u6027\u4f18\u52bf\u5c31\u4f53\u73b0\u51fa\u6765\u4e86\u3002</p> <p>\u94fe\u8868\uff08linked list\uff09\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u662f\u4e00\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u5404\u4e2a\u8282\u70b9\u901a\u8fc7\u201c\u5f15\u7528\u201d\u76f8\u8fde\u63a5\u3002\u5f15\u7528\u8bb0\u5f55\u4e86\u4e0b\u4e00\u4e2a\u8282\u70b9\u7684\u5185\u5b58\u5730\u5740\uff0c\u901a\u8fc7\u5b83\u53ef\u4ee5\u4ece\u5f53\u524d\u8282\u70b9\u8bbf\u95ee\u5230\u4e0b\u4e00\u4e2a\u8282\u70b9\u3002</p> <p>\u94fe\u8868\u7684\u8bbe\u8ba1\u4f7f\u5f97\u5404\u4e2a\u8282\u70b9\u53ef\u4ee5\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\uff0c\u5b83\u4eec\u7684\u5185\u5b58\u5730\u5740\u65e0\u987b\u8fde\u7eed\u3002</p> <p></p> <p> \u56fe 4-5 \u00a0 \u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f </p> <p>\u89c2\u5bdf\u56fe 4-5 \uff0c\u94fe\u8868\u7684\u7ec4\u6210\u5355\u4f4d\u662f\u8282\u70b9\uff08node\uff09\u5bf9\u8c61\u3002\u6bcf\u4e2a\u8282\u70b9\u90fd\u5305\u542b\u4e24\u9879\u6570\u636e\uff1a\u8282\u70b9\u7684\u201c\u503c\u201d\u548c\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u201c\u5f15\u7528\u201d\u3002</p> <ul> <li>\u94fe\u8868\u7684\u9996\u4e2a\u8282\u70b9\u88ab\u79f0\u4e3a\u201c\u5934\u8282\u70b9\u201d\uff0c\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u88ab\u79f0\u4e3a\u201c\u5c3e\u8282\u70b9\u201d\u3002</li> <li>\u5c3e\u8282\u70b9\u6307\u5411\u7684\u662f\u201c\u7a7a\u201d\uff0c\u5b83\u5728 Java\u3001C++ \u548c Python \u4e2d\u5206\u522b\u88ab\u8bb0\u4e3a <code>null</code>\u3001<code>nullptr</code> \u548c <code>None</code> \u3002</li> <li>\u5728 C\u3001C++\u3001Go \u548c Rust \u7b49\u652f\u6301\u6307\u9488\u7684\u8bed\u8a00\u4e2d\uff0c\u4e0a\u8ff0\u201c\u5f15\u7528\u201d\u5e94\u88ab\u66ff\u6362\u4e3a\u201c\u6307\u9488\u201d\u3002</li> </ul> <p>\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u94fe\u8868\u8282\u70b9 <code>ListNode</code> \u9664\u4e86\u5305\u542b\u503c\uff0c\u8fd8\u9700\u989d\u5916\u4fdd\u5b58\u4e00\u4e2a\u5f15\u7528\uff08\u6307\u9488\uff09\u3002\u56e0\u6b64\u5728\u76f8\u540c\u6570\u636e\u91cf\u4e0b\uff0c\u94fe\u8868\u6bd4\u6570\u7ec4\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class ListNode:\n    \"\"\"\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val               # \u8282\u70b9\u503c\n        self.next: ListNode | None = None # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\n    int val;         // \u8282\u70b9\u503c\n    ListNode *next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n    ListNode(int x) : val(x), next(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    int val;        // \u8282\u70b9\u503c\n    ListNode next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n    ListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode(int x) {  //\u6784\u9020\u51fd\u6570\n    int val = x;         // \u8282\u70b9\u503c\n    ListNode? next;      // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype ListNode struct {\n    Val  int       // \u8282\u70b9\u503c\n    Next *ListNode // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n}\n\n// NewListNode \u6784\u9020\u51fd\u6570\uff0c\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u94fe\u8868\nfunc NewListNode(val int) *ListNode {\n    return &amp;ListNode{\n        Val:  val,\n        Next: nil,\n    }\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    var val: Int // \u8282\u70b9\u503c\n    var next: ListNode? // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n\n    init(x: Int) { // \u6784\u9020\u51fd\u6570\n        val = x\n    }\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    constructor(val, next) {\n        this.val = (val === undefined ? 0 : val);       // \u8282\u70b9\u503c\n        this.next = (next === undefined ? null : next); // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    constructor(val?: number, next?: ListNode | null) {\n        this.val = val === undefined ? 0 : val;        // \u8282\u70b9\u503c\n        this.next = next === undefined ? null : next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n  int val; // \u8282\u70b9\u503c\n  ListNode? next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n  ListNode(this.val, [this.next]); // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // \u8282\u70b9\u503c\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct ListNode {\n    int val;               // \u8282\u70b9\u503c\n    struct ListNode *next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n} ListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\n// \u6784\u9020\u65b9\u6cd5\nclass ListNode(x: Int) {\n    val _val: Int = x          // \u8282\u70b9\u503c\n    val next: ListNode? = null // \u6307\u5411\u4e0b\u4e00\u4e2a\u8282\u70b9\u7684\u5f15\u7528\n}\n</code></pre> <pre><code># \u94fe\u8868\u8282\u70b9\u7c7b\nclass ListNode\n  attr_accessor :val  # \u8282\u70b9\u503c\n  attr_accessor :next # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n\n  def initialize(val=0, next_node=nil)\n    @val = val\n    @next = next_node\n  end\nend\n</code></pre> <pre><code>// \u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = 0, // \u8282\u70b9\u503c\n        next: ?*Self = null, // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n\n        // \u6784\u9020\u51fd\u6570\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#421","title":"4.2.1 \u00a0 \u94fe\u8868\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/linked_list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u94fe\u8868","text":"<p>\u5efa\u7acb\u94fe\u8868\u5206\u4e3a\u4e24\u6b65\uff0c\u7b2c\u4e00\u6b65\u662f\u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u7b2c\u4e8c\u6b65\u662f\u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\u5173\u7cfb\u3002\u521d\u59cb\u5316\u5b8c\u6210\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u4ece\u94fe\u8868\u7684\u5934\u8282\u70b9\u51fa\u53d1\uff0c\u901a\u8fc7\u5f15\u7528\u6307\u5411 <code>next</code> \u4f9d\u6b21\u8bbf\u95ee\u6240\u6709\u8282\u70b9\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code># \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.cpp<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.java<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.cs<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = new(1);\nListNode n1 = new(3);\nListNode n2 = new(2);\nListNode n3 = new(5);\nListNode n4 = new(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.go<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n</code></pre> linked_list.swift<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.js<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.ts<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.dart<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\\\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rs<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nlet n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None }));\nlet n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None }));\nlet n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None }));\nlet n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None }));\nlet n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None }));\n\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.borrow_mut().next = Some(n1.clone());\nn1.borrow_mut().next = Some(n2.clone());\nn2.borrow_mut().next = Some(n3.clone());\nn3.borrow_mut().next = Some(n4.clone());\n</code></pre> linked_list.c<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.kt<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nval n0 = ListNode(1)\nval n1 = ListNode(3)\nval n2 = ListNode(2)\nval n3 = ListNode(5)\nval n4 = ListNode(4)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rb<pre><code># \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 = ListNode.new 1\nn1 = ListNode.new 3\nn2 = ListNode.new 2\nn3 = ListNode.new 5\nn4 = ListNode.new 4\n# \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.zig<pre><code>// \u521d\u59cb\u5316\u94fe\u8868\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nvar n0 = inc.ListNode(i32){.val = 1};\nvar n1 = inc.ListNode(i32){.val = 3};\nvar n2 = inc.ListNode(i32){.val = 2};\nvar n3 = inc.ListNode(i32){.val = 5};\nvar n4 = inc.ListNode(i32){.val = 4};\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = &amp;n1;\nn1.next = &amp;n2;\nn2.next = &amp;n3;\nn3.next = &amp;n4;\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u6570\u7ec4\u6574\u4f53\u662f\u4e00\u4e2a\u53d8\u91cf\uff0c\u6bd4\u5982\u6570\u7ec4 <code>nums</code> \u5305\u542b\u5143\u7d20 <code>nums[0]</code> \u548c <code>nums[1]</code> \u7b49\uff0c\u800c\u94fe\u8868\u662f\u7531\u591a\u4e2a\u72ec\u7acb\u7684\u8282\u70b9\u5bf9\u8c61\u7ec4\u6210\u7684\u3002\u6211\u4eec\u901a\u5e38\u5c06\u5934\u8282\u70b9\u5f53\u4f5c\u94fe\u8868\u7684\u4ee3\u79f0\uff0c\u6bd4\u5982\u4ee5\u4e0a\u4ee3\u7801\u4e2d\u7684\u94fe\u8868\u53ef\u8bb0\u4f5c\u94fe\u8868 <code>n0</code> \u3002</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#2","title":"2. \u00a0 \u63d2\u5165\u8282\u70b9","text":"<p>\u5728\u94fe\u8868\u4e2d\u63d2\u5165\u8282\u70b9\u975e\u5e38\u5bb9\u6613\u3002\u5982\u56fe 4-6 \u6240\u793a\uff0c\u5047\u8bbe\u6211\u4eec\u60f3\u5728\u76f8\u90bb\u7684\u4e24\u4e2a\u8282\u70b9 <code>n0</code> \u548c <code>n1</code> \u4e4b\u95f4\u63d2\u5165\u4e00\u4e2a\u65b0\u8282\u70b9 <code>P</code> \uff0c\u5219\u53ea\u9700\u6539\u53d8\u4e24\u4e2a\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</p> <p>\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5728\u6570\u7ec4\u4e2d\u63d2\u5165\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u7684\u6548\u7387\u8f83\u4f4e\u3002</p> <p></p> <p> \u56fe 4-6 \u00a0 \u94fe\u8868\u63d2\u5165\u8282\u70b9\u793a\u4f8b </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def insert(n0: ListNode, P: ListNode):\n    \"\"\"\u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\"\"\"\n    n1 = n0.next\n    P.next = n1\n    n0.next = P\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\n    ListNode n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid Insert(ListNode n0, ListNode P) {\n    ListNode? n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insertNode(n0 *ListNode, P *ListNode) {\n    n1 := n0.Next\n    P.Next = n1\n    n0.Next = P\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insert(n0: ListNode, P: ListNode) {\n    let n1 = n0.next\n    P.next = n1\n    n0.next = P\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0, P) {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0: ListNode, P: ListNode): void {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\n  ListNode? n1 = n0.next;\n  P.next = n1;\n  n0.next = P;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\n#[allow(non_snake_case)]\npub fn insert&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, P: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    let n1 = n0.borrow_mut().next.take();\n    P.borrow_mut().next = n1;\n    n0.borrow_mut().next = Some(P);\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.kt<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9p */\nfun insert(n0: ListNode?, p: ListNode?) {\n    val n1 = n0?.next\n    p?.next = n1\n    n0?.next = p\n}\n</code></pre> linked_list.rb<pre><code>### \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 _p ###\n# Ruby \u7684 `p` \u662f\u4e00\u4e2a\u5185\u7f6e\u51fd\u6570\uff0c `P` \u662f\u4e00\u4e2a\u5e38\u91cf\uff0c\u6240\u4ee5\u53ef\u4ee5\u4f7f\u7528 `_p` \u4ee3\u66ff\ndef insert(n0, _p)\n  n1 = n0.next\n  _p.next = n1\n  n0.next = _p\nend\n</code></pre> linked_list.zig<pre><code>// \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\nfn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {\n    var n1 = n0.?.next;\n    P.?.next = n1;\n    n0.?.next = P;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#3","title":"3. \u00a0 \u5220\u9664\u8282\u70b9","text":"<p>\u5982\u56fe 4-7 \u6240\u793a\uff0c\u5728\u94fe\u8868\u4e2d\u5220\u9664\u8282\u70b9\u4e5f\u975e\u5e38\u65b9\u4fbf\uff0c\u53ea\u9700\u6539\u53d8\u4e00\u4e2a\u8282\u70b9\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\u3002</p> <p>\u8bf7\u6ce8\u610f\uff0c\u5c3d\u7ba1\u5728\u5220\u9664\u64cd\u4f5c\u5b8c\u6210\u540e\u8282\u70b9 <code>P</code> \u4ecd\u7136\u6307\u5411 <code>n1</code> \uff0c\u4f46\u5b9e\u9645\u4e0a\u904d\u5386\u6b64\u94fe\u8868\u5df2\u7ecf\u65e0\u6cd5\u8bbf\u95ee\u5230 <code>P</code> \uff0c\u8fd9\u610f\u5473\u7740 <code>P</code> \u5df2\u7ecf\u4e0d\u518d\u5c5e\u4e8e\u8be5\u94fe\u8868\u4e86\u3002</p> <p></p> <p> \u56fe 4-7 \u00a0 \u94fe\u8868\u5220\u9664\u8282\u70b9 </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def remove(n0: ListNode):\n    \"\"\"\u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\"\"\"\n    if not n0.next:\n        return\n    # n0 -&gt; P -&gt; n1\n    P = n0.next\n    n1 = P.next\n    n0.next = n1\n</code></pre> linked_list.cpp<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode *n0) {\n    if (n0-&gt;next == nullptr)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // \u91ca\u653e\u5185\u5b58\n    delete P;\n}\n</code></pre> linked_list.java<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid Remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode? n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.go<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc removeItem(n0 *ListNode) {\n    if n0.Next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    P := n0.Next\n    n1 := P.Next\n    n0.Next = n1\n}\n</code></pre> linked_list.swift<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc remove(n0: ListNode) {\n    if n0.next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    let P = n0.next\n    let n1 = P?.next\n    n0.next = n1\n}\n</code></pre> linked_list.js<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0) {\n    if (!n0.next) return;\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0: ListNode): void {\n    if (!n0.next) {\n        return;\n    }\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\n  if (n0.next == null) return;\n  // n0 -&gt; P -&gt; n1\n  ListNode P = n0.next!;\n  ListNode? n1 = P.next;\n  n0.next = n1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n#[allow(non_snake_case)]\npub fn remove&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    if n0.borrow().next.is_none() {\n        return;\n    };\n    // n0 -&gt; P -&gt; n1\n    let P = n0.borrow_mut().next.take();\n    if let Some(node) = P {\n        let n1 = node.borrow_mut().next.take();\n        n0.borrow_mut().next = n1;\n    }\n}\n</code></pre> linked_list.c<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(ListNode *n0) {\n    if (!n0-&gt;next)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // \u91ca\u653e\u5185\u5b58\n    free(P);\n}\n</code></pre> linked_list.kt<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfun remove(n0: ListNode?) {\n    val p = n0?.next\n    val n1 = p?.next\n    n0?.next = n1\n}\n</code></pre> linked_list.rb<pre><code>### \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 ###\ndef remove(n0)\n  return if n0.next.nil?\n\n  # n0 -&gt; remove_node -&gt; n1\n  remove_node = n0.next\n  n1 = remove_node.next\n  n0.next = n1\nend\n</code></pre> linked_list.zig<pre><code>// \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\nfn remove(n0: ?*inc.ListNode(i32)) void {\n    if (n0.?.next == null) return;\n    // n0 -&gt; P -&gt; n1\n    var P = n0.?.next;\n    var n1 = P.?.next;\n    n0.?.next = n1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#4","title":"4. \u00a0 \u8bbf\u95ee\u8282\u70b9","text":"<p>\u5728\u94fe\u8868\u4e2d\u8bbf\u95ee\u8282\u70b9\u7684\u6548\u7387\u8f83\u4f4e\u3002\u5982\u4e0a\u4e00\u8282\u6240\u8ff0\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u4e0b\u8bbf\u95ee\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u5143\u7d20\u3002\u94fe\u8868\u5219\u4e0d\u7136\uff0c\u7a0b\u5e8f\u9700\u8981\u4ece\u5934\u8282\u70b9\u51fa\u53d1\uff0c\u9010\u4e2a\u5411\u540e\u904d\u5386\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6807\u8282\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u8bbf\u95ee\u94fe\u8868\u7684\u7b2c \\(i\\) \u4e2a\u8282\u70b9\u9700\u8981\u5faa\u73af \\(i - 1\\) \u8f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def access(head: ListNode, index: int) -&gt; ListNode | None:\n    \"\"\"\u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\"\"\"\n    for _ in range(index):\n        if not head:\n            return None\n        head = head.next\n    return head\n</code></pre> linked_list.cpp<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == nullptr)\n            return nullptr;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.java<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode access(ListNode head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.cs<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? Access(ListNode? head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.go<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head *ListNode, index int) *ListNode {\n    for i := 0; i &lt; index; i++ {\n        if head == nil {\n            return nil\n        }\n        head = head.Next\n    }\n    return head\n}\n</code></pre> linked_list.swift<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head: ListNode, index: Int) -&gt; ListNode? {\n    var head: ListNode? = head\n    for _ in 0 ..&lt; index {\n        if head == nil {\n            return nil\n        }\n        head = head?.next\n    }\n    return head\n}\n</code></pre> linked_list.js<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head, index) {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.ts<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head: ListNode | null, index: number): ListNode | null {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.dart<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? access(ListNode? head, int index) {\n  for (var i = 0; i &lt; index; i++) {\n    if (head == null) return null;\n    head = head.next;\n  }\n  return head;\n}\n</code></pre> linked_list.rs<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\npub fn access&lt;T&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, index: i32) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n    if index &lt;= 0 {\n        return head;\n    };\n    if let Some(node) = &amp;head.borrow().next {\n        return access(node.clone(), index - 1);\n    }\n\n    return head;\n}\n</code></pre> linked_list.c<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == NULL)\n            return NULL;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.kt<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfun access(head: ListNode?, index: Int): ListNode? {\n    var h = head\n    for (i in 0..&lt;index) {\n        h = h?.next\n    }\n    return h\n}\n</code></pre> linked_list.rb<pre><code>### \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 ###\ndef access(head, index)\n  for i in 0...index\n    return nil if head.nil?\n    head = head.next\n  end\n\n  head\nend\n</code></pre> linked_list.zig<pre><code>// \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\nfn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {\n    var head = node;\n    var i: i32 = 0;\n    while (i &lt; index) : (i += 1) {\n        head = head.?.next;\n        if (head == null) return null;\n    }\n    return head;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#5","title":"5. \u00a0 \u67e5\u627e\u8282\u70b9","text":"<p>\u904d\u5386\u94fe\u8868\uff0c\u67e5\u627e\u5176\u4e2d\u503c\u4e3a <code>target</code> \u7684\u8282\u70b9\uff0c\u8f93\u51fa\u8be5\u8282\u70b9\u5728\u94fe\u8868\u4e2d\u7684\u7d22\u5f15\u3002\u6b64\u8fc7\u7a0b\u4e5f\u5c5e\u4e8e\u7ebf\u6027\u67e5\u627e\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linked_list.py<pre><code>def find(head: ListNode, target: int) -&gt; int:\n    \"\"\"\u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\"\"\"\n    index = 0\n    while head:\n        if head.val == target:\n            return index\n        head = head.next\n        index += 1\n    return -1\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head != nullptr) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint Find(ListNode? head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc findNode(head *ListNode, target int) int {\n    index := 0\n    for head != nil {\n        if head.Val == target {\n            return index\n        }\n        head = head.Next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc find(head: ListNode, target: Int) -&gt; Int {\n    var head: ListNode? = head\n    var index = 0\n    while head != nil {\n        if head?.val == target {\n            return index\n        }\n        head = head?.next\n        index += 1\n    }\n    return -1\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head, target) {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head: ListNode | null, target: number): number {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode? head, int target) {\n  int index = 0;\n  while (head != null) {\n    if (head.val == target) {\n      return index;\n    }\n    head = head.next;\n    index++;\n  }\n  return -1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\npub fn find&lt;T: PartialEq&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, target: T, index: i32) -&gt; i32 {\n    if head.borrow().val == target {\n        return index;\n    };\n    if let Some(node) = &amp;head.borrow_mut().next {\n        return find(node.clone(), target, index + 1);\n    }\n    return -1;\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.kt<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfun find(head: ListNode?, target: Int): Int {\n    var index = 0\n    var h = head\n    while (h != null) {\n        if (h.value == target) return index\n        h = h.next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.rb<pre><code>### \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 ###\ndef find(head, target)\n  index = 0\n  while head\n    return index if head.val == target\n    head = head.next\n    index += 1\n  end\n\n  -1\nend\n</code></pre> linked_list.zig<pre><code>// \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\nfn find(node: ?*inc.ListNode(i32), target: i32) i32 {\n    var head = node;\n    var index: i32 = 0;\n    while (head != null) {\n        if (head.?.val == target) return index;\n        head = head.?.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#422-vs","title":"4.2.2 \u00a0 \u6570\u7ec4 vs. \u94fe\u8868","text":"<p>\u8868 4-1 \u603b\u7ed3\u4e86\u6570\u7ec4\u548c\u94fe\u8868\u7684\u5404\u9879\u7279\u70b9\u5e76\u5bf9\u6bd4\u4e86\u64cd\u4f5c\u6548\u7387\u3002\u7531\u4e8e\u5b83\u4eec\u91c7\u7528\u4e24\u79cd\u76f8\u53cd\u7684\u5b58\u50a8\u7b56\u7565\uff0c\u56e0\u6b64\u5404\u79cd\u6027\u8d28\u548c\u64cd\u4f5c\u6548\u7387\u4e5f\u5448\u73b0\u5bf9\u7acb\u7684\u7279\u70b9\u3002</p> <p> \u8868 4-1 \u00a0 \u6570\u7ec4\u4e0e\u94fe\u8868\u7684\u6548\u7387\u5bf9\u6bd4 </p> \u6570\u7ec4 \u94fe\u8868 \u5b58\u50a8\u65b9\u5f0f \u8fde\u7eed\u5185\u5b58\u7a7a\u95f4 \u5206\u6563\u5185\u5b58\u7a7a\u95f4 \u5bb9\u91cf\u6269\u5c55 \u957f\u5ea6\u4e0d\u53ef\u53d8 \u53ef\u7075\u6d3b\u6269\u5c55 \u5185\u5b58\u6548\u7387 \u5143\u7d20\u5360\u7528\u5185\u5b58\u5c11\u3001\u4f46\u53ef\u80fd\u6d6a\u8d39\u7a7a\u95f4 \u5143\u7d20\u5360\u7528\u5185\u5b58\u591a \u8bbf\u95ee\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(1)\\)"},{"location":"chapter_array_and_linkedlist/linked_list/#423","title":"4.2.3 \u00a0 \u5e38\u89c1\u94fe\u8868\u7c7b\u578b","text":"<p>\u5982\u56fe 4-8 \u6240\u793a\uff0c\u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u5305\u62ec\u4e09\u79cd\u3002</p> <ul> <li>\u5355\u5411\u94fe\u8868\uff1a\u5373\u524d\u9762\u4ecb\u7ecd\u7684\u666e\u901a\u94fe\u8868\u3002\u5355\u5411\u94fe\u8868\u7684\u8282\u70b9\u5305\u542b\u503c\u548c\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\u4e24\u9879\u6570\u636e\u3002\u6211\u4eec\u5c06\u9996\u4e2a\u8282\u70b9\u79f0\u4e3a\u5934\u8282\u70b9\uff0c\u5c06\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u79f0\u4e3a\u5c3e\u8282\u70b9\uff0c\u5c3e\u8282\u70b9\u6307\u5411\u7a7a <code>None</code> \u3002</li> <li>\u73af\u5f62\u94fe\u8868\uff1a\u5982\u679c\u6211\u4eec\u4ee4\u5355\u5411\u94fe\u8868\u7684\u5c3e\u8282\u70b9\u6307\u5411\u5934\u8282\u70b9\uff08\u9996\u5c3e\u76f8\u63a5\uff09\uff0c\u5219\u5f97\u5230\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\u3002\u5728\u73af\u5f62\u94fe\u8868\u4e2d\uff0c\u4efb\u610f\u8282\u70b9\u90fd\u53ef\u4ee5\u89c6\u4f5c\u5934\u8282\u70b9\u3002</li> <li>\u53cc\u5411\u94fe\u8868\uff1a\u4e0e\u5355\u5411\u94fe\u8868\u76f8\u6bd4\uff0c\u53cc\u5411\u94fe\u8868\u8bb0\u5f55\u4e86\u4e24\u4e2a\u65b9\u5411\u7684\u5f15\u7528\u3002\u53cc\u5411\u94fe\u8868\u7684\u8282\u70b9\u5b9a\u4e49\u540c\u65f6\u5305\u542b\u6307\u5411\u540e\u7ee7\u8282\u70b9\uff08\u4e0b\u4e00\u4e2a\u8282\u70b9\uff09\u548c\u524d\u9a71\u8282\u70b9\uff08\u4e0a\u4e00\u4e2a\u8282\u70b9\uff09\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\u3002\u76f8\u8f83\u4e8e\u5355\u5411\u94fe\u8868\uff0c\u53cc\u5411\u94fe\u8868\u66f4\u5177\u7075\u6d3b\u6027\uff0c\u53ef\u4ee5\u671d\u4e24\u4e2a\u65b9\u5411\u904d\u5386\u94fe\u8868\uff0c\u4f46\u76f8\u5e94\u5730\u4e5f\u9700\u8981\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class ListNode:\n    \"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # \u8282\u70b9\u503c\n        self.next: ListNode | None = None  # \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n        self.prev: ListNode | None = None  # \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\n    int val;         // \u8282\u70b9\u503c\n    ListNode *next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n    ListNode *prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n    ListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    int val;        // \u8282\u70b9\u503c\n    ListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n    ListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n    ListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode(int x) {  // \u6784\u9020\u51fd\u6570\n    int val = x;    // \u8282\u70b9\u503c\n    ListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n    ListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype DoublyListNode struct {\n    Val  int             // \u8282\u70b9\u503c\n    Next *DoublyListNode // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n    Prev *DoublyListNode // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n}\n\n// NewDoublyListNode \u521d\u59cb\u5316\nfunc NewDoublyListNode(val int) *DoublyListNode {\n    return &amp;DoublyListNode{\n        Val:  val,\n        Next: nil,\n        Prev: nil,\n    }\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    var val: Int // \u8282\u70b9\u503c\n    var next: ListNode? // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n    var prev: ListNode? // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n\n    init(x: Int) { // \u6784\u9020\u51fd\u6570\n        val = x\n    }\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    constructor(val, next, prev) {\n        this.val = val  ===  undefined ? 0 : val;        // \u8282\u70b9\u503c\n        this.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n        this.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    prev: ListNode | null;\n    constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\n        this.val = val  ===  undefined ? 0 : val;        // \u8282\u70b9\u503c\n        this.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n        this.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    int val;        // \u8282\u70b9\u503c\n    ListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n    ListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n    ListNode(this.val, [this.next, this.prev]);  // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\u578b */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // \u8282\u70b9\u503c\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n    prev: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nimpl ListNode {\n    fn new(val: i32) -&gt; Self {\n        ListNode {\n            val,\n            next: None,\n            prev: None,\n        }\n    }\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct ListNode {\n    int val;               // \u8282\u70b9\u503c\n    struct ListNode *next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n    struct ListNode *prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n} ListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    node-&gt;prev = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\n// \u6784\u9020\u65b9\u6cd5\nclass ListNode(x: Int) {\n    val _val: Int = x           // \u8282\u70b9\u503c\n    val next: ListNode? = null  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n    val prev: ListNode? = null  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n}\n</code></pre> <pre><code># \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\nclass ListNode\n  attr_accessor :val    # \u8282\u70b9\u503c\n  attr_accessor :next   # \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n  attr_accessor :prev   # \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n\n  def initialize(val=0, next_node=nil, prev_node=nil)\n    @val = val\n    @next = next_node\n    @prev = prev_node\n  end\nend\n</code></pre> <pre><code>// \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = 0, // \u8282\u70b9\u503c\n        next: ?*Self = null, // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n        prev: ?*Self = null, // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n\n        // \u6784\u9020\u51fd\u6570\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n            self.prev = null;\n        }\n    };\n}\n</code></pre> <p></p> <p> \u56fe 4-8 \u00a0 \u5e38\u89c1\u94fe\u8868\u79cd\u7c7b </p>"},{"location":"chapter_array_and_linkedlist/linked_list/#424","title":"4.2.4 \u00a0 \u94fe\u8868\u5178\u578b\u5e94\u7528","text":"<p>\u5355\u5411\u94fe\u8868\u901a\u5e38\u7528\u4e8e\u5b9e\u73b0\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u548c\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u3002</p> <ul> <li>\u6808\u4e0e\u961f\u5217\uff1a\u5f53\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u90fd\u5728\u94fe\u8868\u7684\u4e00\u7aef\u8fdb\u884c\u65f6\uff0c\u5b83\u8868\u73b0\u51fa\u5148\u8fdb\u540e\u51fa\u7684\u7279\u6027\uff0c\u5bf9\u5e94\u6808\uff1b\u5f53\u63d2\u5165\u64cd\u4f5c\u5728\u94fe\u8868\u7684\u4e00\u7aef\u8fdb\u884c\uff0c\u5220\u9664\u64cd\u4f5c\u5728\u94fe\u8868\u7684\u53e6\u4e00\u7aef\u8fdb\u884c\uff0c\u5b83\u8868\u73b0\u51fa\u5148\u8fdb\u5148\u51fa\u7684\u7279\u6027\uff0c\u5bf9\u5e94\u961f\u5217\u3002</li> <li>\u54c8\u5e0c\u8868\uff1a\u94fe\u5f0f\u5730\u5740\u662f\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\u7684\u4e3b\u6d41\u65b9\u6848\u4e4b\u4e00\uff0c\u5728\u8be5\u65b9\u6848\u4e2d\uff0c\u6240\u6709\u51b2\u7a81\u7684\u5143\u7d20\u90fd\u4f1a\u88ab\u653e\u5230\u4e00\u4e2a\u94fe\u8868\u4e2d\u3002</li> <li>\u56fe\uff1a\u90bb\u63a5\u8868\u662f\u8868\u793a\u56fe\u7684\u4e00\u79cd\u5e38\u7528\u65b9\u5f0f\uff0c\u5176\u4e2d\u56fe\u7684\u6bcf\u4e2a\u9876\u70b9\u90fd\u4e0e\u4e00\u4e2a\u94fe\u8868\u76f8\u5173\u8054\uff0c\u94fe\u8868\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u4ee3\u8868\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u5176\u4ed6\u9876\u70b9\u3002</li> </ul> <p>\u53cc\u5411\u94fe\u8868\u5e38\u7528\u4e8e\u9700\u8981\u5feb\u901f\u67e5\u627e\u524d\u4e00\u4e2a\u548c\u540e\u4e00\u4e2a\u5143\u7d20\u7684\u573a\u666f\u3002</p> <ul> <li>\u9ad8\u7ea7\u6570\u636e\u7ed3\u6784\uff1a\u6bd4\u5982\u5728\u7ea2\u9ed1\u6811\u3001B \u6811\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u8bbf\u95ee\u8282\u70b9\u7684\u7236\u8282\u70b9\uff0c\u8fd9\u53ef\u4ee5\u901a\u8fc7\u5728\u8282\u70b9\u4e2d\u4fdd\u5b58\u4e00\u4e2a\u6307\u5411\u7236\u8282\u70b9\u7684\u5f15\u7528\u6765\u5b9e\u73b0\uff0c\u7c7b\u4f3c\u4e8e\u53cc\u5411\u94fe\u8868\u3002</li> <li>\u6d4f\u89c8\u5668\u5386\u53f2\uff1a\u5728\u7f51\u9875\u6d4f\u89c8\u5668\u4e2d\uff0c\u5f53\u7528\u6237\u70b9\u51fb\u524d\u8fdb\u6216\u540e\u9000\u6309\u94ae\u65f6\uff0c\u6d4f\u89c8\u5668\u9700\u8981\u77e5\u9053\u7528\u6237\u8bbf\u95ee\u8fc7\u7684\u524d\u4e00\u4e2a\u548c\u540e\u4e00\u4e2a\u7f51\u9875\u3002\u53cc\u5411\u94fe\u8868\u7684\u7279\u6027\u4f7f\u5f97\u8fd9\u79cd\u64cd\u4f5c\u53d8\u5f97\u7b80\u5355\u3002</li> <li>LRU \u7b97\u6cd5\uff1a\u5728\u7f13\u5b58\u6dd8\u6c70\uff08LRU\uff09\u7b97\u6cd5\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u5feb\u901f\u627e\u5230\u6700\u8fd1\u6700\u5c11\u4f7f\u7528\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u652f\u6301\u5feb\u901f\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u3002\u8fd9\u65f6\u5019\u4f7f\u7528\u53cc\u5411\u94fe\u8868\u5c31\u975e\u5e38\u5408\u9002\u3002</li> </ul> <p>\u73af\u5f62\u94fe\u8868\u5e38\u7528\u4e8e\u9700\u8981\u5468\u671f\u6027\u64cd\u4f5c\u7684\u573a\u666f\uff0c\u6bd4\u5982\u64cd\u4f5c\u7cfb\u7edf\u7684\u8d44\u6e90\u8c03\u5ea6\u3002</p> <ul> <li>\u65f6\u95f4\u7247\u8f6e\u8f6c\u8c03\u5ea6\u7b97\u6cd5\uff1a\u5728\u64cd\u4f5c\u7cfb\u7edf\u4e2d\uff0c\u65f6\u95f4\u7247\u8f6e\u8f6c\u8c03\u5ea6\u7b97\u6cd5\u662f\u4e00\u79cd\u5e38\u89c1\u7684 CPU \u8c03\u5ea6\u7b97\u6cd5\uff0c\u5b83\u9700\u8981\u5bf9\u4e00\u7ec4\u8fdb\u7a0b\u8fdb\u884c\u5faa\u73af\u3002\u6bcf\u4e2a\u8fdb\u7a0b\u88ab\u8d4b\u4e88\u4e00\u4e2a\u65f6\u95f4\u7247\uff0c\u5f53\u65f6\u95f4\u7247\u7528\u5b8c\u65f6\uff0cCPU \u5c06\u5207\u6362\u5230\u4e0b\u4e00\u4e2a\u8fdb\u7a0b\u3002\u8fd9\u79cd\u5faa\u73af\u64cd\u4f5c\u53ef\u4ee5\u901a\u8fc7\u73af\u5f62\u94fe\u8868\u6765\u5b9e\u73b0\u3002</li> <li>\u6570\u636e\u7f13\u51b2\u533a\uff1a\u5728\u67d0\u4e9b\u6570\u636e\u7f13\u51b2\u533a\u7684\u5b9e\u73b0\u4e2d\uff0c\u4e5f\u53ef\u80fd\u4f1a\u4f7f\u7528\u73af\u5f62\u94fe\u8868\u3002\u6bd4\u5982\u5728\u97f3\u9891\u3001\u89c6\u9891\u64ad\u653e\u5668\u4e2d\uff0c\u6570\u636e\u6d41\u53ef\u80fd\u4f1a\u88ab\u5206\u6210\u591a\u4e2a\u7f13\u51b2\u5757\u5e76\u653e\u5165\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\uff0c\u4ee5\u4fbf\u5b9e\u73b0\u65e0\u7f1d\u64ad\u653e\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/list/","title":"4.3 \u00a0 \u5217\u8868","text":"<p>\u5217\u8868\uff08list\uff09\u662f\u4e00\u4e2a\u62bd\u8c61\u7684\u6570\u636e\u7ed3\u6784\u6982\u5ff5\uff0c\u5b83\u8868\u793a\u5143\u7d20\u7684\u6709\u5e8f\u96c6\u5408\uff0c\u652f\u6301\u5143\u7d20\u8bbf\u95ee\u3001\u4fee\u6539\u3001\u6dfb\u52a0\u3001\u5220\u9664\u548c\u904d\u5386\u7b49\u64cd\u4f5c\uff0c\u65e0\u987b\u4f7f\u7528\u8005\u8003\u8651\u5bb9\u91cf\u9650\u5236\u7684\u95ee\u9898\u3002\u5217\u8868\u53ef\u4ee5\u57fa\u4e8e\u94fe\u8868\u6216\u6570\u7ec4\u5b9e\u73b0\u3002</p> <ul> <li>\u94fe\u8868\u5929\u7136\u53ef\u4ee5\u770b\u4f5c\u4e00\u4e2a\u5217\u8868\uff0c\u5176\u652f\u6301\u5143\u7d20\u589e\u5220\u67e5\u6539\u64cd\u4f5c\uff0c\u5e76\u4e14\u53ef\u4ee5\u7075\u6d3b\u52a8\u6001\u6269\u5bb9\u3002</li> <li>\u6570\u7ec4\u4e5f\u652f\u6301\u5143\u7d20\u589e\u5220\u67e5\u6539\uff0c\u4f46\u7531\u4e8e\u5176\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u53ea\u80fd\u770b\u4f5c\u4e00\u4e2a\u5177\u6709\u957f\u5ea6\u9650\u5236\u7684\u5217\u8868\u3002</li> </ul> <p>\u5f53\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u5217\u8868\u65f6\uff0c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6027\u8d28\u4f1a\u5bfc\u81f4\u5217\u8868\u7684\u5b9e\u7528\u6027\u964d\u4f4e\u3002\u8fd9\u662f\u56e0\u4e3a\u6211\u4eec\u901a\u5e38\u65e0\u6cd5\u4e8b\u5148\u786e\u5b9a\u9700\u8981\u5b58\u50a8\u591a\u5c11\u6570\u636e\uff0c\u4ece\u800c\u96be\u4ee5\u9009\u62e9\u5408\u9002\u7684\u5217\u8868\u957f\u5ea6\u3002\u82e5\u957f\u5ea6\u8fc7\u5c0f\uff0c\u5219\u5f88\u53ef\u80fd\u65e0\u6cd5\u6ee1\u8db3\u4f7f\u7528\u9700\u6c42\uff1b\u82e5\u957f\u5ea6\u8fc7\u5927\uff0c\u5219\u4f1a\u9020\u6210\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002</p> <p>\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u52a8\u6001\u6570\u7ec4\uff08dynamic array\uff09\u6765\u5b9e\u73b0\u5217\u8868\u3002\u5b83\u7ee7\u627f\u4e86\u6570\u7ec4\u7684\u5404\u9879\u4f18\u70b9\uff0c\u5e76\u4e14\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u8fdb\u884c\u52a8\u6001\u6269\u5bb9\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u6807\u51c6\u5e93\u63d0\u4f9b\u7684\u5217\u8868\u662f\u57fa\u4e8e\u52a8\u6001\u6570\u7ec4\u5b9e\u73b0\u7684\uff0c\u4f8b\u5982 Python \u4e2d\u7684 <code>list</code> \u3001Java \u4e2d\u7684 <code>ArrayList</code> \u3001C++ \u4e2d\u7684 <code>vector</code> \u548c C# \u4e2d\u7684 <code>List</code> \u7b49\u3002\u5728\u63a5\u4e0b\u6765\u7684\u8ba8\u8bba\u4e2d\uff0c\u6211\u4eec\u5c06\u628a\u201c\u5217\u8868\u201d\u548c\u201c\u52a8\u6001\u6570\u7ec4\u201d\u89c6\u4e3a\u7b49\u540c\u7684\u6982\u5ff5\u3002</p>"},{"location":"chapter_array_and_linkedlist/list/#431","title":"4.3.1 \u00a0 \u5217\u8868\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u5217\u8868","text":"<p>\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u201c\u65e0\u521d\u59cb\u503c\u201d\u548c\u201c\u6709\u521d\u59cb\u503c\u201d\u8fd9\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u521d\u59cb\u5316\u5217\u8868\n# \u65e0\u521d\u59cb\u503c\nnums1: list[int] = []\n# \u6709\u521d\u59cb\u503c\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> list.cpp<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u9700\u6ce8\u610f\uff0cC++ \u4e2d vector \u5373\u662f\u672c\u6587\u63cf\u8ff0\u7684 nums\n// \u65e0\u521d\u59cb\u503c\nvector&lt;int&gt; nums1;\n// \u6709\u521d\u59cb\u503c\nvector&lt;int&gt; nums = { 1, 3, 2, 5, 4 };\n</code></pre> list.java<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;();\n// \u6709\u521d\u59cb\u503c\uff08\u6ce8\u610f\u6570\u7ec4\u7684\u5143\u7d20\u7c7b\u578b\u9700\u4e3a int[] \u7684\u5305\u88c5\u7c7b Integer[]\uff09\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList&lt;Integer&gt; nums = new ArrayList&lt;&gt;(Arrays.asList(numbers));\n</code></pre> list.cs<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList&lt;int&gt; nums1 = [];\n// \u6709\u521d\u59cb\u503c\nint[] numbers = [1, 3, 2, 5, 4];\nList&lt;int&gt; nums = [.. numbers];\n</code></pre> list_test.go<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nnums1 := []int{}\n// \u6709\u521d\u59cb\u503c\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> list.swift<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet nums1: [Int] = []\n// \u6709\u521d\u59cb\u503c\nvar nums = [1, 3, 2, 5, 4]\n</code></pre> list.js<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst nums1 = [];\n// \u6709\u521d\u59cb\u503c\nconst nums = [1, 3, 2, 5, 4];\n</code></pre> list.ts<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst nums1: number[] = [];\n// \u6709\u521d\u59cb\u503c\nconst nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> list.dart<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList&lt;int&gt; nums1 = [];\n// \u6709\u521d\u59cb\u503c\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> list.rs<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet nums1: Vec&lt;i32&gt; = Vec::new();\n// \u6709\u521d\u59cb\u503c\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.kt<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nvar nums1 = listOf&lt;Int&gt;()\n// \u6709\u521d\u59cb\u503c\nvar numbers = arrayOf(1, 3, 2, 5, 4)\nvar nums = numbers.toMutableList()\n</code></pre> list.rb<pre><code># \u521d\u59cb\u5316\u5217\u8868\n# \u65e0\u521d\u59cb\u503c\nnums1 = []\n# \u6709\u521d\u59cb\u503c\nnums = [1, 3, 2, 5, 4]\n</code></pre> list.zig<pre><code>// \u521d\u59cb\u5316\u5217\u8868\nvar nums = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums.deinit();\ntry nums.appendSlice(&amp;[_]i32{ 1, 3, 2, 5, 4 });\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#2","title":"2. \u00a0 \u8bbf\u95ee\u5143\u7d20","text":"<p>\u5217\u8868\u672c\u8d28\u4e0a\u662f\u6570\u7ec4\uff0c\u56e0\u6b64\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u548c\u66f4\u65b0\u5143\u7d20\uff0c\u6548\u7387\u5f88\u9ad8\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u8bbf\u95ee\u5143\u7d20\nnum: int = nums[1]  # \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n# \u66f4\u65b0\u5143\u7d20\nnums[1] = 0    # \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.cpp<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.java<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums.get(1);  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums.set(1, 0);  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.cs<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list_test.go<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nnum := nums[1]  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0     // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.swift<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nlet num = nums[1] // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0 // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.js<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nconst num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.ts<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nconst num: number = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.dart<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.rs<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nlet num: i32 = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;             // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.kt<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nval num = nums[1]       // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0             // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.rb<pre><code># \u8bbf\u95ee\u5143\u7d20\nnum = nums[1] # \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n# \u66f4\u65b0\u5143\u7d20\nnums[1] = 0 # \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.zig<pre><code>// \u8bbf\u95ee\u5143\u7d20\nvar num = nums.items[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n// \u66f4\u65b0\u5143\u7d20\nnums.items[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#3","title":"3. \u00a0 \u63d2\u5165\u4e0e\u5220\u9664\u5143\u7d20","text":"<p>\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5730\u6dfb\u52a0\u4e0e\u5220\u9664\u5143\u7d20\u3002\u5728\u5217\u8868\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u7684\u6548\u7387\u4ecd\u4e0e\u6570\u7ec4\u76f8\u540c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u6e05\u7a7a\u5217\u8868\nnums.clear()\n\n# \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n# \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\nnums.insert(3, 6)  # \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n# \u5220\u9664\u5143\u7d20\nnums.pop(3)        # \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.cpp<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push_back(1);\nnums.push_back(3);\nnums.push_back(2);\nnums.push_back(5);\nnums.push_back(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(nums.begin() + 3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.erase(nums.begin() + 3);      // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.java<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.add(3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(3);  // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.cs<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.Clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.Add(1);\nnums.Add(3);\nnums.Add(2);\nnums.Add(5);\nnums.Add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.Insert(3, 6);\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.RemoveAt(3);\n</code></pre> list_test.go<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums = nil\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums = append(nums, 1)\nnums = append(nums, 3)\nnums = append(nums, 2)\nnums = append(nums, 5)\nnums = append(nums, 4)\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums = append(nums[:3], append([]int{6}, nums[3:]...)...) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums = append(nums[:3], nums[4:]...) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.swift<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.removeAll()\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(6, at: 3) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(at: 3) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.js<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.length = 0;\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6);\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.splice(3, 1);\n</code></pre> list.ts<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.length = 0;\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6);\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.splice(3, 1);\n</code></pre> list.dart<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.removeAt(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.rs<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(3);    // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.kt<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.add(3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(3);  // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.rb<pre><code># \u6e05\u7a7a\u5217\u8868\nnums.clear\n\n# \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\nnums &lt;&lt; 1\nnums &lt;&lt; 3\nnums &lt;&lt; 2\nnums &lt;&lt; 5\nnums &lt;&lt; 4\n\n# \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\nnums.insert 3, 6 # \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n# \u5220\u9664\u5143\u7d20\nnums.delete_at 3 # \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.zig<pre><code>// \u6e05\u7a7a\u5217\u8868\nnums.clearRetainingCapacity();\n\n// \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\ntry nums.append(1);\ntry nums.append(3);\ntry nums.append(2);\ntry nums.append(5);\ntry nums.append(4);\n\n// \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\ntry nums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n// \u5220\u9664\u5143\u7d20\n_ = nums.orderedRemove(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#4","title":"4. \u00a0 \u904d\u5386\u5217\u8868","text":"<p>\u4e0e\u6570\u7ec4\u4e00\u6837\uff0c\u5217\u8868\u53ef\u4ee5\u6839\u636e\u7d22\u5f15\u904d\u5386\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u904d\u5386\u5404\u5143\u7d20\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\ncount = 0\nfor i in range(len(nums)):\n    count += nums[i]\n\n# \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\nfor num in nums:\n    count += num\n</code></pre> list.cpp<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.java<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums.get(i);\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.cs<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i &lt; nums.Count; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nforeach (int num in nums) {\n    count += num;\n}\n</code></pre> list_test.go<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\ncount := 0\nfor i := 0; i &lt; len(nums); i++ {\n    count += nums[i]\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor _, num := range nums {\n    count += num\n}\n</code></pre> list.swift<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nvar count = 0\nfor i in nums.indices {\n    count += nums[i]\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor num in nums {\n    count += num\n}\n</code></pre> list.js<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.ts<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.dart<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (var num in nums) {\n    count += num;\n}\n</code></pre> list.rs<pre><code>// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nlet mut _count = 0;\nfor i in 0..nums.len() {\n    _count += nums[i];\n}\n\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\n_count = 0;\nfor num in &amp;nums {\n    _count += num;\n}\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.kt<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nvar count = 0\nfor (i in nums.indices) {\n    count += nums[i]\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\nfor (num in nums) {\n    count += num\n}\n</code></pre> list.rb<pre><code># \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\ncount = 0\nfor i in 0...nums.length\n    count += nums[i]\nend\n\n# \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\ncount = 0\nfor num in nums\n    count += num\nend\n</code></pre> list.zig<pre><code>// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i &lt; nums.items.len) : (i += 1) {\n    count += nums[i];\n}\n\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\ncount = 0;\nfor (nums.items) |num| {\n    count += num;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#5","title":"5. \u00a0 \u62fc\u63a5\u5217\u8868","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u65b0\u5217\u8868 <code>nums1</code> \uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u62fc\u63a5\u5230\u539f\u5217\u8868\u7684\u5c3e\u90e8\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nnums1: list[int] = [6, 8, 7, 10, 9]\nnums += nums1  # \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.cpp<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nvector&lt;int&gt; nums1 = { 6, 8, 7, 10, 9 };\n// \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\nnums.insert(nums.end(), nums1.begin(), nums1.end());\n</code></pre> list.java<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nnums.addAll(nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.cs<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.AddRange(nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list_test.go<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nnums1 := []int{6, 8, 7, 10, 9}\nnums = append(nums, nums1...)  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.swift<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet nums1 = [6, 8, 7, 10, 9]\nnums.append(contentsOf: nums1) // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.js<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst nums1 = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.ts<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst nums1: number[] = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.dart<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.addAll(nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.rs<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet nums1: Vec&lt;i32&gt; = vec![6, 8, 7, 10, 9];\nnums.extend(nums1);\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.kt<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nval nums1 = intArrayOf(6, 8, 7, 10, 9).toMutableList()\nnums.addAll(nums1)  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.rb<pre><code># \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nnums1 = [6, 8, 7, 10, 9]\nnums += nums1\n</code></pre> list.zig<pre><code>// \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nvar nums1 = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums1.deinit();\ntry nums1.appendSlice(&amp;[_]i32{ 6, 8, 7, 10, 9 });\ntry nums.insertSlice(nums.items.len, nums1.items); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#6","title":"6. \u00a0 \u6392\u5e8f\u5217\u8868","text":"<p>\u5b8c\u6210\u5217\u8868\u6392\u5e8f\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u4ee5\u4f7f\u7528\u5728\u6570\u7ec4\u7c7b\u7b97\u6cd5\u9898\u4e2d\u7ecf\u5e38\u8003\u67e5\u7684\u201c\u4e8c\u5206\u67e5\u627e\u201d\u548c\u201c\u53cc\u6307\u9488\u201d\u7b97\u6cd5\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig list.py<pre><code># \u6392\u5e8f\u5217\u8868\nnums.sort()  # \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.cpp<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nsort(nums.begin(), nums.end());  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.java<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nCollections.sort(nums);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.cs<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.Sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list_test.go<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nsort.Ints(nums)  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.swift<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort() // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.js<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort((a, b) =&gt; a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.ts<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort((a, b) =&gt; a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.dart<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.rs<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.kt<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort() // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.rb<pre><code># \u6392\u5e8f\u5217\u8868\nnums = nums.sort { |a, b| a &lt;=&gt; b } # \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.zig<pre><code>// \u6392\u5e8f\u5217\u8868\nstd.sort.sort(i32, nums.items, {}, comptime std.sort.asc(i32));\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/list/#432","title":"4.3.2 \u00a0 \u5217\u8868\u5b9e\u73b0","text":"<p>\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u4e86\u5217\u8868\uff0c\u4f8b\u5982 Java\u3001C++\u3001Python \u7b49\u3002\u5b83\u4eec\u7684\u5b9e\u73b0\u6bd4\u8f83\u590d\u6742\uff0c\u5404\u4e2a\u53c2\u6570\u7684\u8bbe\u5b9a\u4e5f\u975e\u5e38\u8003\u7a76\uff0c\u4f8b\u5982\u521d\u59cb\u5bb9\u91cf\u3001\u6269\u5bb9\u500d\u6570\u7b49\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u6e90\u7801\u8fdb\u884c\u5b66\u4e60\u3002</p> <p>\u4e3a\u4e86\u52a0\u6df1\u5bf9\u5217\u8868\u5de5\u4f5c\u539f\u7406\u7684\u7406\u89e3\uff0c\u6211\u4eec\u5c1d\u8bd5\u5b9e\u73b0\u4e00\u4e2a\u7b80\u6613\u7248\u5217\u8868\uff0c\u5305\u62ec\u4ee5\u4e0b\u4e09\u4e2a\u91cd\u70b9\u8bbe\u8ba1\u3002</p> <ul> <li>\u521d\u59cb\u5bb9\u91cf\uff1a\u9009\u53d6\u4e00\u4e2a\u5408\u7406\u7684\u6570\u7ec4\u521d\u59cb\u5bb9\u91cf\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u9009\u62e9 10 \u4f5c\u4e3a\u521d\u59cb\u5bb9\u91cf\u3002</li> <li>\u6570\u91cf\u8bb0\u5f55\uff1a\u58f0\u660e\u4e00\u4e2a\u53d8\u91cf <code>size</code> \uff0c\u7528\u4e8e\u8bb0\u5f55\u5217\u8868\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff0c\u5e76\u968f\u7740\u5143\u7d20\u63d2\u5165\u548c\u5220\u9664\u5b9e\u65f6\u66f4\u65b0\u3002\u6839\u636e\u6b64\u53d8\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4f4d\u5217\u8868\u5c3e\u90e8\uff0c\u4ee5\u53ca\u5224\u65ad\u662f\u5426\u9700\u8981\u6269\u5bb9\u3002</li> <li>\u6269\u5bb9\u673a\u5236\uff1a\u82e5\u63d2\u5165\u5143\u7d20\u65f6\u5217\u8868\u5bb9\u91cf\u5df2\u6ee1\uff0c\u5219\u9700\u8981\u8fdb\u884c\u6269\u5bb9\u3002\u5148\u6839\u636e\u6269\u5bb9\u500d\u6570\u521b\u5efa\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u518d\u5c06\u5f53\u524d\u6570\u7ec4\u7684\u6240\u6709\u5143\u7d20\u4f9d\u6b21\u79fb\u52a8\u81f3\u65b0\u6570\u7ec4\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u89c4\u5b9a\u6bcf\u6b21\u5c06\u6570\u7ec4\u6269\u5bb9\u81f3\u4e4b\u524d\u7684 2 \u500d\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_list.py<pre><code>class MyList:\n    \"\"\"\u5217\u8868\u7c7b\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._capacity: int = 10  # \u5217\u8868\u5bb9\u91cf\n        self._arr: list[int] = [0] * self._capacity  # \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n        self._size: int = 0  # \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        self._extend_ratio: int = 2  # \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\"\"\"\n        return self._size\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u5217\u8868\u5bb9\u91cf\"\"\"\n        return self._capacity\n\n    def get(self, index: int) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u5143\u7d20\"\"\"\n        # \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        return self._arr[index]\n\n    def set(self, num: int, index: int):\n        \"\"\"\u66f4\u65b0\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        self._arr[index] = num\n\n    def add(self, num: int):\n        \"\"\"\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\"\"\"\n        # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self.size() == self.capacity():\n            self.extend_capacity()\n        self._arr[self._size] = num\n        self._size += 1\n\n    def insert(self, num: int, index: int):\n        \"\"\"\u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self._size == self.capacity():\n            self.extend_capacity()\n        # \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for j in range(self._size - 1, index - 1, -1):\n            self._arr[j + 1] = self._arr[j]\n        self._arr[index] = num\n        # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self._size += 1\n\n    def remove(self, index: int) -&gt; int:\n        \"\"\"\u5220\u9664\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        num = self._arr[index]\n        # \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for j in range(index, self._size - 1):\n            self._arr[j] = self._arr[j + 1]\n        # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self._size -= 1\n        # \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num\n\n    def extend_capacity(self):\n        \"\"\"\u5217\u8868\u6269\u5bb9\"\"\"\n        # \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 _extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        self._arr = self._arr + [0] * self.capacity() * (self._extend_ratio - 1)\n        # \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        self._capacity = len(self._arr)\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868\"\"\"\n        return self._arr[: self._size]\n</code></pre> my_list.cpp<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n  private:\n    int *arr;             // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    int arrCapacity = 10; // \u5217\u8868\u5bb9\u91cf\n    int arrSize = 0;      // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    int extendRatio = 2;   // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~MyList() {\n        delete[] arr;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    int size() {\n        return arrSize;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    int capacity() {\n        return arrCapacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    int get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    void add(int num) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size() == capacity())\n            extendCapacity();\n        arr[size()] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size() == capacity())\n            extendCapacity();\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (int j = size() - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    int remove(int index) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (int j = index; j &lt; size() - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    void extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\n        int newCapacity = capacity() * extendRatio;\n        int *tmp = arr;\n        arr = new int[newCapacity];\n        // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        for (int i = 0; i &lt; size(); i++) {\n            arr[i] = tmp[i];\n        }\n        // \u91ca\u653e\u5185\u5b58\n        delete[] tmp;\n        arrCapacity = newCapacity;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Vector \u7528\u4e8e\u6253\u5370 */\n    vector&lt;int&gt; toVector() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        vector&lt;int&gt; vec(size());\n        for (int i = 0; i &lt; size(); i++) {\n            vec[i] = arr[i];\n        }\n        return vec;\n    }\n};\n</code></pre> my_list.java<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private int[] arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private int capacity = 10; // \u5217\u8868\u5bb9\u91cf\n    private int size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private int extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public MyList() {\n        arr = new int[capacity];\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\n    public int size() {\n        return size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    public int capacity() {\n        return capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    public int get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    public void add(int num) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity();\n        arr[size] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    public void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity();\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (int j = size - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    public int remove(int index) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (int j = index; j &lt; size - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    public void extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        arr = Arrays.copyOf(arr, capacity() * extendRatio);\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        capacity = arr.length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    public int[] toArray() {\n        int size = size();\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] arr = new int[size];\n        for (int i = 0; i &lt; size; i++) {\n            arr[i] = get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.cs<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private int[] arr;           // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private int arrCapacity = 10;    // \u5217\u8868\u5bb9\u91cf\n    private int arrSize = 0;         // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private readonly int extendRatio = 2;  // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    public int Size() {\n        return arrSize;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    public int Capacity() {\n        return arrCapacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    public int Get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public void Set(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    public void Add(int num) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        arr[arrSize] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    public void Insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (int j = arrSize - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    public int Remove(int index) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (int j = index; j &lt; arrSize - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    public void ExtendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a arrCapacity * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        Array.Resize(ref arr, arrCapacity * extendRatio);\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        arrCapacity = arr.Length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    public int[] ToArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] arr = new int[arrSize];\n        for (int i = 0; i &lt; arrSize; i++) {\n            arr[i] = Get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.go<pre><code>/* \u5217\u8868\u7c7b */\ntype myList struct {\n    arrCapacity int\n    arr         []int\n    arrSize     int\n    extendRatio int\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newMyList() *myList {\n    return &amp;myList{\n        arrCapacity: 10,              // \u5217\u8868\u5bb9\u91cf\n        arr:         make([]int, 10), // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n        arrSize:     0,               // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        extendRatio: 2,               // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n    }\n}\n\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\nfunc (l *myList) size() int {\n    return l.arrSize\n}\n\n/*  \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc (l *myList) capacity() int {\n    return l.arrCapacity\n}\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc (l *myList) get(index int) int {\n    // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    return l.arr[index]\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc (l *myList) set(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    l.arr[index] = num\n}\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc (l *myList) add(num int) {\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    l.arr[l.arrSize] = num\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    l.arrSize++\n}\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc (l *myList) insert(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for j := l.arrSize - 1; j &gt;= index; j-- {\n        l.arr[j+1] = l.arr[j]\n    }\n    l.arr[index] = num\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    l.arrSize++\n}\n\n/* \u5220\u9664\u5143\u7d20 */\nfunc (l *myList) remove(index int) int {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    num := l.arr[index]\n    // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for j := index; j &lt; l.arrSize-1; j++ {\n        l.arr[j] = l.arr[j+1]\n    }\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    l.arrSize--\n    // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n    return num\n}\n\n/* \u5217\u8868\u6269\u5bb9 */\nfunc (l *myList) extendCapacity() {\n    // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    l.arr = append(l.arr, make([]int, l.arrCapacity*(l.extendRatio-1))...)\n    // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n    l.arrCapacity = len(l.arr)\n}\n\n/* \u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868 */\nfunc (l *myList) toArray() []int {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    return l.arr[:l.arrSize]\n}\n</code></pre> my_list.swift<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private var arr: [Int] // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private var _capacity: Int // \u5217\u8868\u5bb9\u91cf\n    private var _size: Int // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private let extendRatio: Int // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init() {\n        _capacity = 10\n        _size = 0\n        extendRatio = 2\n        arr = Array(repeating: 0, count: _capacity)\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        _capacity\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    func get(index: Int) -&gt; Int {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u9519\u8bef\uff0c\u4e0b\u540c\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        return arr[index]\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    func set(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        arr[index] = num\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    func add(num: Int) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if size() == capacity() {\n            extendCapacity()\n        }\n        arr[size()] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        _size += 1\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    func insert(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if size() == capacity() {\n            extendCapacity()\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for j in (index ..&lt; size()).reversed() {\n            arr[j + 1] = arr[j]\n        }\n        arr[index] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        _size += 1\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    @discardableResult\n    func remove(index: Int) -&gt; Int {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        let num = arr[index]\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for j in index ..&lt; (size() - 1) {\n            arr[j] = arr[j + 1]\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        _size -= 1\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    func extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        arr = arr + Array(repeating: 0, count: capacity() * (extendRatio - 1))\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        _capacity = arr.count\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    func toArray() -&gt; [Int] {\n        Array(arr.prefix(size()))\n    }\n}\n</code></pre> my_list.js<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    #arr = new Array(); // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    #capacity = 10; // \u5217\u8868\u5bb9\u91cf\n    #size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    #extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#arr = new Array(this.#capacity);\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    size() {\n        return this.#size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    capacity() {\n        return this.#capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    get(index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        return this.#arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    set(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        this.#arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    add(num) {\n        // \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\n        this.#arr[this.#size] = num;\n        this.#size++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    insert(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (let j = this.#size - 1; j &gt;= index; j--) {\n            this.#arr[j + 1] = this.#arr[j];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this.#arr[index] = num;\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    remove(index) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        let num = this.#arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (let j = index; j &lt; this.#size - 1; j++) {\n            this.#arr[j] = this.#arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this.#size--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        this.#arr = this.#arr.concat(\n            new Array(this.capacity() * (this.#extendRatio - 1))\n        );\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        this.#capacity = this.#arr.length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    toArray() {\n        let size = this.size();\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.ts<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private arr: Array&lt;number&gt;; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private _capacity: number = 10; // \u5217\u8868\u5bb9\u91cf\n    private _size: number = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private extendRatio: number = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.arr = new Array(this._capacity);\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    public size(): number {\n        return this._size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    public capacity(): number {\n        return this._capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    public get(index: number): number {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        return this.arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public set(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        this.arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    public add(num: number): void {\n        // \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\n        if (this._size === this._capacity) this.extendCapacity();\n        // \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\n        this.arr[this._size] = num;\n        this._size++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    public insert(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (this._size === this._capacity) {\n            this.extendCapacity();\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (let j = this._size - 1; j &gt;= index; j--) {\n            this.arr[j + 1] = this.arr[j];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this.arr[index] = num;\n        this._size++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    public remove(index: number): number {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        let num = this.arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (let j = index; j &lt; this._size - 1; j++) {\n            this.arr[j] = this.arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this._size--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    public extendCapacity(): void {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        this.arr = this.arr.concat(\n            new Array(this.capacity() * (this.extendRatio - 1))\n        );\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        this._capacity = this.arr.length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    public toArray(): number[] {\n        let size = this.size();\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.dart<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n  late List&lt;int&gt; _arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n  int _capacity = 10; // \u5217\u8868\u5bb9\u91cf\n  int _size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n  int _extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  MyList() {\n    _arr = List.filled(_capacity, 0);\n  }\n\n  /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n  int size() =&gt; _size;\n\n  /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n  int capacity() =&gt; _capacity;\n\n  /* \u8bbf\u95ee\u5143\u7d20 */\n  int get(int index) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    return _arr[index];\n  }\n\n  /* \u66f4\u65b0\u5143\u7d20 */\n  void set(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    _arr[index] = _num;\n  }\n\n  /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n  void add(int _num) {\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if (_size == _capacity) extendCapacity();\n    _arr[_size] = _num;\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    _size++;\n  }\n\n  /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n  void insert(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if (_size == _capacity) extendCapacity();\n    // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (var j = _size - 1; j &gt;= index; j--) {\n      _arr[j + 1] = _arr[j];\n    }\n    _arr[index] = _num;\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    _size++;\n  }\n\n  /* \u5220\u9664\u5143\u7d20 */\n  int remove(int index) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    int _num = _arr[index];\n    // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (var j = index; j &lt; _size - 1; j++) {\n      _arr[j] = _arr[j + 1];\n    }\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    _size--;\n    // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n    return _num;\n  }\n\n  /* \u5217\u8868\u6269\u5bb9 */\n  void extendCapacity() {\n    // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 _extendRatio \u500d\u7684\u65b0\u6570\u7ec4\n    final _newNums = List.filled(_capacity * _extendRatio, 0);\n    // \u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    List.copyRange(_newNums, 0, _arr);\n    // \u66f4\u65b0 _arr \u7684\u5f15\u7528\n    _arr = _newNums;\n    // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n    _capacity = _arr.length;\n  }\n\n  /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n  List&lt;int&gt; toArray() {\n    List&lt;int&gt; arr = [];\n    for (var i = 0; i &lt; _size; i++) {\n      arr.add(get(i));\n    }\n    return arr;\n  }\n}\n</code></pre> my_list.rs<pre><code>/* \u5217\u8868\u7c7b */\n#[allow(dead_code)]\nstruct MyList {\n    arr: Vec&lt;i32&gt;,       // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    capacity: usize,     // \u5217\u8868\u5bb9\u91cf\n    size: usize,         // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    extend_ratio: usize, // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n}\n\n#[allow(unused, unused_comparisons)]\nimpl MyList {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(capacity: usize) -&gt; Self {\n        let mut vec = Vec::new();\n        vec.resize(capacity, 0);\n        Self {\n            arr: vec,\n            capacity,\n            size: 0,\n            extend_ratio: 2,\n        }\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    pub fn size(&amp;self) -&gt; usize {\n        return self.size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    pub fn capacity(&amp;self) -&gt; usize {\n        return self.capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    pub fn get(&amp;self, index: usize) -&gt; i32 {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if index &gt;= self.size {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        return self.arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    pub fn set(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        self.arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    pub fn add(&amp;mut self, num: i32) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        self.arr[self.size] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self.size += 1;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    pub fn insert(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size() {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for j in (index..self.size).rev() {\n            self.arr[j + 1] = self.arr[j];\n        }\n        self.arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self.size += 1;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    pub fn remove(&amp;mut self, index: usize) -&gt; i32 {\n        if index &gt;= self.size() {\n            panic!(\"\u7d22\u5f15\u8d8a\u754c\")\n        };\n        let num = self.arr[index];\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for j in (index..self.size - 1) {\n            self.arr[j] = self.arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self.size -= 1;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    pub fn extend_capacity(&amp;mut self) {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        let new_capacity = self.capacity * self.extend_ratio;\n        self.arr.resize(new_capacity, 0);\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        self.capacity = new_capacity;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    pub fn to_array(&amp;mut self) -&gt; Vec&lt;i32&gt; {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        let mut arr = Vec::new();\n        for i in 0..self.size {\n            arr.push(self.get(i));\n        }\n        arr\n    }\n}\n</code></pre> my_list.c<pre><code>/* \u5217\u8868\u7c7b */\ntypedef struct {\n    int *arr;        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    int capacity;    // \u5217\u8868\u5bb9\u91cf\n    int size;        // \u5217\u8868\u5927\u5c0f\n    int extendRatio; // \u5217\u8868\u6bcf\u6b21\u6269\u5bb9\u7684\u500d\u6570\n} MyList;\n\n/* \u6784\u9020\u51fd\u6570 */\nMyList *newMyList() {\n    MyList *nums = malloc(sizeof(MyList));\n    nums-&gt;capacity = 10;\n    nums-&gt;arr = malloc(sizeof(int) * nums-&gt;capacity);\n    nums-&gt;size = 0;\n    nums-&gt;extendRatio = 2;\n    return nums;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delMyList(MyList *nums) {\n    free(nums-&gt;arr);\n    free(nums);\n}\n\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6 */\nint size(MyList *nums) {\n    return nums-&gt;size;\n}\n\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity(MyList *nums) {\n    return nums-&gt;capacity;\n}\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    return nums-&gt;arr[index];\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    nums-&gt;arr[index] = num;\n}\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(MyList *nums, int num) {\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // \u6269\u5bb9\n    }\n    nums-&gt;arr[size(nums)] = num;\n    nums-&gt;size++;\n}\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // \u6269\u5bb9\n    }\n    for (int i = size(nums); i &gt; index; --i) {\n        nums-&gt;arr[i] = nums-&gt;arr[i - 1];\n    }\n    nums-&gt;arr[index] = num;\n    nums-&gt;size++;\n}\n\n/* \u5220\u9664\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nint removeItem(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    int num = nums-&gt;arr[index];\n    for (int i = index; i &lt; size(nums) - 1; i++) {\n        nums-&gt;arr[i] = nums-&gt;arr[i + 1];\n    }\n    nums-&gt;size--;\n    return num;\n}\n\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity(MyList *nums) {\n    // \u5148\u5206\u914d\u7a7a\u95f4\n    int newCapacity = capacity(nums) * nums-&gt;extendRatio;\n    int *extend = (int *)malloc(sizeof(int) * newCapacity);\n    int *temp = nums-&gt;arr;\n\n    // \u62f7\u8d1d\u65e7\u6570\u636e\u5230\u65b0\u6570\u636e\n    for (int i = 0; i &lt; size(nums); i++)\n        extend[i] = nums-&gt;arr[i];\n\n    // \u91ca\u653e\u65e7\u6570\u636e\n    free(temp);\n\n    // \u66f4\u65b0\u65b0\u6570\u636e\n    nums-&gt;arr = extend;\n    nums-&gt;capacity = newCapacity;\n}\n\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Array \u7528\u4e8e\u6253\u5370 */\nint *toArray(MyList *nums) {\n    return nums-&gt;arr;\n}\n</code></pre> my_list.kt<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private var arr: IntArray = intArrayOf() // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private var capacity = 10 // \u5217\u8868\u5bb9\u91cf\n    private var size = 0 // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private var extendRatio = 2 // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u51fd\u6570 */\n    init {\n        arr = IntArray(capacity)\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\n    fun size(): Int {\n        return size\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    fun capacity(): Int {\n        return capacity\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    fun get(index: Int): Int {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException()\n        return arr[index]\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    fun set(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        arr[index] = num\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    fun add(num: Int) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity()\n        arr[size] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    fun insert(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity()\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (j in size - 1 downTo index)\n            arr[j + 1] = arr[j]\n        arr[index] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    fun remove(index: Int): Int {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\")\n        val num: Int = arr[index]\n        // \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (j in index..&lt;size - 1)\n            arr[j] = arr[j + 1]\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size--\n        // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n        return num\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    fun extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        arr = arr.copyOf(capacity() * extendRatio)\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        capacity = arr.size\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    fun toArray(): IntArray {\n        val size = size()\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        val arr = IntArray(size)\n        for (i in 0..&lt;size) {\n            arr[i] = get(i)\n        }\n        return arr\n    }\n}\n</code></pre> my_list.rb<pre><code>### \u5217\u8868\u7c7b ###\nclass MyList\n  attr_reader :size       # \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n  attr_reader :capacity   # \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\n\n  ### \u6784\u9020\u65b9\u6cd5 ###\n  def initialize\n    @capacity = 10\n    @size = 0\n    @extend_ratio = 2\n    @arr = Array.new capacity\n  end\n\n  ### \u8bbf\u95ee\u5143\u7d20 ###\n  def get(index)\n    # \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    @arr[index]\n  end\n\n  ### \u8bbf\u95ee\u5143\u7d20 ###\n  def set(index, num)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    @arr[index] = num\n  end\n\n  ### \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 ###\n  def add(num)\n    # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    extend_capacity if size == capacity\n    @arr[size] = num\n\n    # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    @size += 1\n  end\n\n  ### \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 ###\n  def insert(index, num)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n\n    # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    extend_capacity if size == capacity\n\n    # \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for j in (size - 1).downto(index)\n      @arr[j + 1] = @arr[j] \n    end\n    @arr[index] = num\n\n    # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    @size += 1\n  end\n\n  ### \u5220\u9664\u5143\u7d20 ###\n  def remove(index)\n    raise IndexError, \"\u7d22\u5f15\u8d8a\u754c\" if index &lt; 0 || index &gt;= size\n    num = @arr[index]\n\n    # \u5c06\u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for j in index...size\n      @arr[j] = @arr[j + 1]\n    end\n\n    # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    @size -= 1\n\n    # \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n    num\n  end\n\n  ### \u5217\u8868\u6269\u5bb9 ###\n  def extend_capacity\n    # \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    arr = @arr.dup + Array.new(capacity * (@extend_ratio - 1))\n    # \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n    @capacity = arr.length\n  end\n\n  ### \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 ###\n  def to_array\n    sz = size\n    # \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    arr = Array.new sz\n    for i in 0...sz\n      arr[i] = get i\n    end\n    arr\n  end\nend\n</code></pre> my_list.zig<pre><code>// \u5217\u8868\u7c7b\nfn MyList(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        arr: []T = undefined,                        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n        arrCapacity: usize = 10,                     // \u5217\u8868\u5bb9\u91cf\n        numSize: usize = 0,                           // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        extendRatio: usize = 2,                       // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u5217\u8868\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.arr = try self.mem_allocator.alloc(T, self.arrCapacity);\n            @memset(self.arr, @as(T, 0));\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        pub fn size(self: *Self) usize {\n            return self.numSize;\n        }\n\n        // \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\n        pub fn capacity(self: *Self) usize {\n            return self.arrCapacity;\n        }\n\n        // \u8bbf\u95ee\u5143\u7d20\n        pub fn get(self: *Self, index: usize) T {\n            // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            return self.arr[index];\n        }  \n\n        // \u66f4\u65b0\u5143\u7d20\n        pub fn set(self: *Self, index: usize, num: T) void {\n            // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\uff0c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            self.arr[index] = num;\n        }  \n\n        // \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\n        pub fn add(self: *Self, num: T) !void {\n            // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n            if (self.size() == self.capacity()) try self.extendCapacity();\n            self.arr[self.size()] = num;\n            // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n            self.numSize += 1;\n        }  \n\n        // \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\n        pub fn insert(self: *Self, index: usize, num: T) !void {\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n            if (self.size() == self.capacity()) try self.extendCapacity();\n            // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n            var j = self.size() - 1;\n            while (j &gt;= index) : (j -= 1) {\n                self.arr[j + 1] = self.arr[j];\n            }\n            self.arr[index] = num;\n            // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n            self.numSize += 1;\n        }\n\n        // \u5220\u9664\u5143\u7d20\n        pub fn remove(self: *Self, index: usize) T {\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            var num = self.arr[index];\n            // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n            var j = index;\n            while (j &lt; self.size() - 1) : (j += 1) {\n                self.arr[j] = self.arr[j + 1];\n            }\n            // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n            self.numSize -= 1;\n            // \u8fd4\u56de\u88ab\u5220\u9664\u7684\u5143\u7d20\n            return num;\n        }\n\n        // \u5217\u8868\u6269\u5bb9\n        pub fn extendCapacity(self: *Self) !void {\n            // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u590d\u5236\u5230\u65b0\u6570\u7ec4\n            var newCapacity = self.capacity() * self.extendRatio;\n            var extend = try self.mem_allocator.alloc(T, newCapacity);\n            @memset(extend, @as(T, 0));\n            // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n            std.mem.copy(T, extend, self.arr);\n            self.arr = extend;\n            // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n            self.arrCapacity = newCapacity;\n        }\n\n        // \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n            var arr = try self.mem_allocator.alloc(T, self.size());\n           @memset(arr, @as(T, 0));\n            for (arr, 0..) |*num, i| {\n                num.* = self.get(i);\n            }\n            return arr;\n        }\n    };\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/","title":"4.4 \u00a0 \u5185\u5b58\u4e0e\u7f13\u5b58 *","text":"<p>\u5728\u672c\u7ae0\u7684\u524d\u4e24\u8282\u4e2d\uff0c\u6211\u4eec\u63a2\u8ba8\u4e86\u6570\u7ec4\u548c\u94fe\u8868\u8fd9\u4e24\u79cd\u57fa\u7840\u4e14\u91cd\u8981\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5b83\u4eec\u5206\u522b\u4ee3\u8868\u4e86\u201c\u8fde\u7eed\u5b58\u50a8\u201d\u548c\u201c\u5206\u6563\u5b58\u50a8\u201d\u4e24\u79cd\u7269\u7406\u7ed3\u6784\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u7269\u7406\u7ed3\u6784\u5728\u5f88\u5927\u7a0b\u5ea6\u4e0a\u51b3\u5b9a\u4e86\u7a0b\u5e8f\u5bf9\u5185\u5b58\u548c\u7f13\u5b58\u7684\u4f7f\u7528\u6548\u7387\uff0c\u8fdb\u800c\u5f71\u54cd\u7b97\u6cd5\u7a0b\u5e8f\u7684\u6574\u4f53\u6027\u80fd\u3002</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#441","title":"4.4.1 \u00a0 \u8ba1\u7b97\u673a\u5b58\u50a8\u8bbe\u5907","text":"<p>\u8ba1\u7b97\u673a\u4e2d\u5305\u62ec\u4e09\u79cd\u7c7b\u578b\u7684\u5b58\u50a8\u8bbe\u5907\uff1a\u786c\u76d8\uff08hard disk\uff09\u3001\u5185\u5b58\uff08random-access memory, RAM\uff09\u3001\u7f13\u5b58\uff08cache memory\uff09\u3002\u8868 4-2 \u5c55\u793a\u4e86\u5b83\u4eec\u5728\u8ba1\u7b97\u673a\u7cfb\u7edf\u4e2d\u7684\u4e0d\u540c\u89d2\u8272\u548c\u6027\u80fd\u7279\u70b9\u3002</p> <p> \u8868 4-2 \u00a0 \u8ba1\u7b97\u673a\u7684\u5b58\u50a8\u8bbe\u5907 </p> \u786c\u76d8 \u5185\u5b58 \u7f13\u5b58 \u7528\u9014 \u957f\u671f\u5b58\u50a8\u6570\u636e\uff0c\u5305\u62ec\u64cd\u4f5c\u7cfb\u7edf\u3001\u7a0b\u5e8f\u3001\u6587\u4ef6\u7b49 \u4e34\u65f6\u5b58\u50a8\u5f53\u524d\u8fd0\u884c\u7684\u7a0b\u5e8f\u548c\u6b63\u5728\u5904\u7406\u7684\u6570\u636e \u5b58\u50a8\u7ecf\u5e38\u8bbf\u95ee\u7684\u6570\u636e\u548c\u6307\u4ee4\uff0c\u51cf\u5c11 CPU \u8bbf\u95ee\u5185\u5b58\u7684\u6b21\u6570 \u6613\u5931\u6027 \u65ad\u7535\u540e\u6570\u636e\u4e0d\u4f1a\u4e22\u5931 \u65ad\u7535\u540e\u6570\u636e\u4f1a\u4e22\u5931 \u65ad\u7535\u540e\u6570\u636e\u4f1a\u4e22\u5931 \u5bb9\u91cf \u8f83\u5927\uff0cTB \u7ea7\u522b \u8f83\u5c0f\uff0cGB \u7ea7\u522b \u975e\u5e38\u5c0f\uff0cMB \u7ea7\u522b \u901f\u5ea6 \u8f83\u6162\uff0c\u51e0\u767e\u5230\u51e0\u5343 MB/s \u8f83\u5feb\uff0c\u51e0\u5341 GB/s \u975e\u5e38\u5feb\uff0c\u51e0\u5341\u5230\u51e0\u767e GB/s \u4ef7\u683c \u8f83\u4fbf\u5b9c\uff0c\u51e0\u6bdb\u5230\u51e0\u5143 / GB \u8f83\u8d35\uff0c\u51e0\u5341\u5230\u51e0\u767e\u5143 / GB \u975e\u5e38\u8d35\uff0c\u968f CPU \u6253\u5305\u8ba1\u4ef7 <p>\u6211\u4eec\u53ef\u4ee5\u5c06\u8ba1\u7b97\u673a\u5b58\u50a8\u7cfb\u7edf\u60f3\u8c61\u4e3a\u56fe 4-9 \u6240\u793a\u7684\u91d1\u5b57\u5854\u7ed3\u6784\u3002\u8d8a\u9760\u8fd1\u91d1\u5b57\u5854\u9876\u7aef\u7684\u5b58\u50a8\u8bbe\u5907\u7684\u901f\u5ea6\u8d8a\u5feb\u3001\u5bb9\u91cf\u8d8a\u5c0f\u3001\u6210\u672c\u8d8a\u9ad8\u3002\u8fd9\u79cd\u591a\u5c42\u7ea7\u7684\u8bbe\u8ba1\u5e76\u975e\u5076\u7136\uff0c\u800c\u662f\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6\u548c\u5de5\u7a0b\u5e08\u4eec\u7ecf\u8fc7\u6df1\u601d\u719f\u8651\u7684\u7ed3\u679c\u3002</p> <ul> <li>\u786c\u76d8\u96be\u4ee5\u88ab\u5185\u5b58\u53d6\u4ee3\u3002\u9996\u5148\uff0c\u5185\u5b58\u4e2d\u7684\u6570\u636e\u5728\u65ad\u7535\u540e\u4f1a\u4e22\u5931\uff0c\u56e0\u6b64\u5b83\u4e0d\u9002\u5408\u957f\u671f\u5b58\u50a8\u6570\u636e\uff1b\u5176\u6b21\uff0c\u5185\u5b58\u7684\u6210\u672c\u662f\u786c\u76d8\u7684\u51e0\u5341\u500d\uff0c\u8fd9\u4f7f\u5f97\u5b83\u96be\u4ee5\u5728\u6d88\u8d39\u8005\u5e02\u573a\u666e\u53ca\u3002</li> <li>\u7f13\u5b58\u7684\u5927\u5bb9\u91cf\u548c\u9ad8\u901f\u5ea6\u96be\u4ee5\u517c\u5f97\u3002\u968f\u7740 L1\u3001L2\u3001L3 \u7f13\u5b58\u7684\u5bb9\u91cf\u9010\u6b65\u589e\u5927\uff0c\u5176\u7269\u7406\u5c3a\u5bf8\u4f1a\u53d8\u5927\uff0c\u4e0e CPU \u6838\u5fc3\u4e4b\u95f4\u7684\u7269\u7406\u8ddd\u79bb\u4f1a\u53d8\u8fdc\uff0c\u4ece\u800c\u5bfc\u81f4\u6570\u636e\u4f20\u8f93\u65f6\u95f4\u589e\u52a0\uff0c\u5143\u7d20\u8bbf\u95ee\u5ef6\u8fdf\u53d8\u9ad8\u3002\u5728\u5f53\u524d\u6280\u672f\u4e0b\uff0c\u591a\u5c42\u7ea7\u7684\u7f13\u5b58\u7ed3\u6784\u662f\u5bb9\u91cf\u3001\u901f\u5ea6\u548c\u6210\u672c\u4e4b\u95f4\u7684\u6700\u4f73\u5e73\u8861\u70b9\u3002</li> </ul> <p></p> <p> \u56fe 4-9 \u00a0 \u8ba1\u7b97\u673a\u5b58\u50a8\u7cfb\u7edf </p> <p>Note</p> <p>\u8ba1\u7b97\u673a\u7684\u5b58\u50a8\u5c42\u6b21\u7ed3\u6784\u4f53\u73b0\u4e86\u901f\u5ea6\u3001\u5bb9\u91cf\u548c\u6210\u672c\u4e09\u8005\u4e4b\u95f4\u7684\u7cbe\u5999\u5e73\u8861\u3002\u5b9e\u9645\u4e0a\uff0c\u8fd9\u79cd\u6743\u8861\u666e\u904d\u5b58\u5728\u4e8e\u6240\u6709\u5de5\u4e1a\u9886\u57df\uff0c\u5b83\u8981\u6c42\u6211\u4eec\u5728\u4e0d\u540c\u7684\u4f18\u52bf\u548c\u9650\u5236\u4e4b\u95f4\u627e\u5230\u6700\u4f73\u5e73\u8861\u70b9\u3002</p> <p>\u603b\u7684\u6765\u8bf4\uff0c\u786c\u76d8\u7528\u4e8e\u957f\u671f\u5b58\u50a8\u5927\u91cf\u6570\u636e\uff0c\u5185\u5b58\u7528\u4e8e\u4e34\u65f6\u5b58\u50a8\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u6b63\u5728\u5904\u7406\u7684\u6570\u636e\uff0c\u800c\u7f13\u5b58\u5219\u7528\u4e8e\u5b58\u50a8\u7ecf\u5e38\u8bbf\u95ee\u7684\u6570\u636e\u548c\u6307\u4ee4\uff0c\u4ee5\u63d0\u9ad8\u7a0b\u5e8f\u8fd0\u884c\u6548\u7387\u3002\u4e09\u8005\u5171\u540c\u534f\u4f5c\uff0c\u786e\u4fdd\u8ba1\u7b97\u673a\u7cfb\u7edf\u9ad8\u6548\u8fd0\u884c\u3002</p> <p>\u5982\u56fe 4-10 \u6240\u793a\uff0c\u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u4f1a\u4ece\u786c\u76d8\u4e2d\u88ab\u8bfb\u53d6\u5230\u5185\u5b58\u4e2d\uff0c\u4f9b CPU \u8ba1\u7b97\u4f7f\u7528\u3002\u7f13\u5b58\u53ef\u4ee5\u770b\u4f5c CPU \u7684\u4e00\u90e8\u5206\uff0c\u5b83\u901a\u8fc7\u667a\u80fd\u5730\u4ece\u5185\u5b58\u52a0\u8f7d\u6570\u636e\uff0c\u7ed9 CPU \u63d0\u4f9b\u9ad8\u901f\u7684\u6570\u636e\u8bfb\u53d6\uff0c\u4ece\u800c\u663e\u8457\u63d0\u5347\u7a0b\u5e8f\u7684\u6267\u884c\u6548\u7387\uff0c\u51cf\u5c11\u5bf9\u8f83\u6162\u7684\u5185\u5b58\u7684\u4f9d\u8d56\u3002</p> <p></p> <p> \u56fe 4-10 \u00a0 \u786c\u76d8\u3001\u5185\u5b58\u548c\u7f13\u5b58\u4e4b\u95f4\u7684\u6570\u636e\u6d41\u901a </p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#442","title":"4.4.2 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u5185\u5b58\u6548\u7387","text":"<p>\u5728\u5185\u5b58\u7a7a\u95f4\u5229\u7528\u65b9\u9762\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u5404\u81ea\u5177\u6709\u4f18\u52bf\u548c\u5c40\u9650\u6027\u3002</p> <p>\u4e00\u65b9\u9762\uff0c\u5185\u5b58\u662f\u6709\u9650\u7684\uff0c\u4e14\u540c\u4e00\u5757\u5185\u5b58\u4e0d\u80fd\u88ab\u591a\u4e2a\u7a0b\u5e8f\u5171\u4eab\uff0c\u56e0\u6b64\u6211\u4eec\u5e0c\u671b\u6570\u636e\u7ed3\u6784\u80fd\u591f\u5c3d\u53ef\u80fd\u9ad8\u6548\u5730\u5229\u7528\u7a7a\u95f4\u3002\u6570\u7ec4\u7684\u5143\u7d20\u7d27\u5bc6\u6392\u5217\uff0c\u4e0d\u9700\u8981\u989d\u5916\u7684\u7a7a\u95f4\u6765\u5b58\u50a8\u94fe\u8868\u8282\u70b9\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u56e0\u6b64\u7a7a\u95f4\u6548\u7387\u66f4\u9ad8\u3002\u7136\u800c\uff0c\u6570\u7ec4\u9700\u8981\u4e00\u6b21\u6027\u5206\u914d\u8db3\u591f\u7684\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\uff0c\u8fd9\u53ef\u80fd\u5bfc\u81f4\u5185\u5b58\u6d6a\u8d39\uff0c\u6570\u7ec4\u6269\u5bb9\u4e5f\u9700\u8981\u989d\u5916\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u6210\u672c\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u94fe\u8868\u4ee5\u201c\u8282\u70b9\u201d\u4e3a\u5355\u4f4d\u8fdb\u884c\u52a8\u6001\u5185\u5b58\u5206\u914d\u548c\u56de\u6536\uff0c\u63d0\u4f9b\u4e86\u66f4\u5927\u7684\u7075\u6d3b\u6027\u3002</p> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u968f\u7740\u53cd\u590d\u7533\u8bf7\u4e0e\u91ca\u653e\u5185\u5b58\uff0c\u7a7a\u95f2\u5185\u5b58\u7684\u788e\u7247\u5316\u7a0b\u5ea6\u4f1a\u8d8a\u6765\u8d8a\u9ad8\uff0c\u4ece\u800c\u5bfc\u81f4\u5185\u5b58\u7684\u5229\u7528\u6548\u7387\u964d\u4f4e\u3002\u6570\u7ec4\u7531\u4e8e\u5176\u8fde\u7eed\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u76f8\u5bf9\u4e0d\u5bb9\u6613\u5bfc\u81f4\u5185\u5b58\u788e\u7247\u5316\u3002\u76f8\u53cd\uff0c\u94fe\u8868\u7684\u5143\u7d20\u662f\u5206\u6563\u5b58\u50a8\u7684\uff0c\u5728\u9891\u7e41\u7684\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u4e2d\uff0c\u66f4\u5bb9\u6613\u5bfc\u81f4\u5185\u5b58\u788e\u7247\u5316\u3002</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#443","title":"4.4.3 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u7f13\u5b58\u6548\u7387","text":"<p>\u7f13\u5b58\u867d\u7136\u5728\u7a7a\u95f4\u5bb9\u91cf\u4e0a\u8fdc\u5c0f\u4e8e\u5185\u5b58\uff0c\u4f46\u5b83\u6bd4\u5185\u5b58\u5feb\u5f97\u591a\uff0c\u5728\u7a0b\u5e8f\u6267\u884c\u901f\u5ea6\u4e0a\u8d77\u7740\u81f3\u5173\u91cd\u8981\u7684\u4f5c\u7528\u3002\u7531\u4e8e\u7f13\u5b58\u7684\u5bb9\u91cf\u6709\u9650\uff0c\u53ea\u80fd\u5b58\u50a8\u4e00\u5c0f\u90e8\u5206\u9891\u7e41\u8bbf\u95ee\u7684\u6570\u636e\uff0c\u56e0\u6b64\u5f53 CPU \u5c1d\u8bd5\u8bbf\u95ee\u7684\u6570\u636e\u4e0d\u5728\u7f13\u5b58\u4e2d\u65f6\uff0c\u5c31\u4f1a\u53d1\u751f\u7f13\u5b58\u672a\u547d\u4e2d\uff08cache miss\uff09\uff0c\u6b64\u65f6 CPU \u4e0d\u5f97\u4e0d\u4ece\u901f\u5ea6\u8f83\u6162\u7684\u5185\u5b58\u4e2d\u52a0\u8f7d\u6240\u9700\u6570\u636e\u3002</p> <p>\u663e\u7136\uff0c\u201c\u7f13\u5b58\u672a\u547d\u4e2d\u201d\u8d8a\u5c11\uff0cCPU \u8bfb\u5199\u6570\u636e\u7684\u6548\u7387\u5c31\u8d8a\u9ad8\uff0c\u7a0b\u5e8f\u6027\u80fd\u4e5f\u5c31\u8d8a\u597d\u3002\u6211\u4eec\u5c06 CPU \u4ece\u7f13\u5b58\u4e2d\u6210\u529f\u83b7\u53d6\u6570\u636e\u7684\u6bd4\u4f8b\u79f0\u4e3a\u7f13\u5b58\u547d\u4e2d\u7387\uff08cache hit rate\uff09\uff0c\u8fd9\u4e2a\u6307\u6807\u901a\u5e38\u7528\u6765\u8861\u91cf\u7f13\u5b58\u6548\u7387\u3002</p> <p>\u4e3a\u4e86\u5c3d\u53ef\u80fd\u8fbe\u5230\u66f4\u9ad8\u7684\u6548\u7387\uff0c\u7f13\u5b58\u4f1a\u91c7\u53d6\u4ee5\u4e0b\u6570\u636e\u52a0\u8f7d\u673a\u5236\u3002</p> <ul> <li>\u7f13\u5b58\u884c\uff1a\u7f13\u5b58\u4e0d\u662f\u5355\u4e2a\u5b57\u8282\u5730\u5b58\u50a8\u4e0e\u52a0\u8f7d\u6570\u636e\uff0c\u800c\u662f\u4ee5\u7f13\u5b58\u884c\u4e3a\u5355\u4f4d\u3002\u76f8\u6bd4\u4e8e\u5355\u4e2a\u5b57\u8282\u7684\u4f20\u8f93\uff0c\u7f13\u5b58\u884c\u7684\u4f20\u8f93\u5f62\u5f0f\u66f4\u52a0\u9ad8\u6548\u3002</li> <li>\u9884\u53d6\u673a\u5236\uff1a\u5904\u7406\u5668\u4f1a\u5c1d\u8bd5\u9884\u6d4b\u6570\u636e\u8bbf\u95ee\u6a21\u5f0f\uff08\u4f8b\u5982\u987a\u5e8f\u8bbf\u95ee\u3001\u56fa\u5b9a\u6b65\u957f\u8df3\u8dc3\u8bbf\u95ee\u7b49\uff09\uff0c\u5e76\u6839\u636e\u7279\u5b9a\u6a21\u5f0f\u5c06\u6570\u636e\u52a0\u8f7d\u81f3\u7f13\u5b58\u4e4b\u4e2d\uff0c\u4ece\u800c\u63d0\u5347\u547d\u4e2d\u7387\u3002</li> <li>\u7a7a\u95f4\u5c40\u90e8\u6027\uff1a\u5982\u679c\u4e00\u4e2a\u6570\u636e\u88ab\u8bbf\u95ee\uff0c\u90a3\u4e48\u5b83\u9644\u8fd1\u7684\u6570\u636e\u53ef\u80fd\u8fd1\u671f\u4e5f\u4f1a\u88ab\u8bbf\u95ee\u3002\u56e0\u6b64\uff0c\u7f13\u5b58\u5728\u52a0\u8f7d\u67d0\u4e00\u6570\u636e\u65f6\uff0c\u4e5f\u4f1a\u52a0\u8f7d\u5176\u9644\u8fd1\u7684\u6570\u636e\uff0c\u4ee5\u63d0\u9ad8\u547d\u4e2d\u7387\u3002</li> <li>\u65f6\u95f4\u5c40\u90e8\u6027\uff1a\u5982\u679c\u4e00\u4e2a\u6570\u636e\u88ab\u8bbf\u95ee\uff0c\u90a3\u4e48\u5b83\u5728\u4e0d\u4e45\u7684\u5c06\u6765\u5f88\u53ef\u80fd\u518d\u6b21\u88ab\u8bbf\u95ee\u3002\u7f13\u5b58\u5229\u7528\u8fd9\u4e00\u539f\u7406\uff0c\u901a\u8fc7\u4fdd\u7559\u6700\u8fd1\u8bbf\u95ee\u8fc7\u7684\u6570\u636e\u6765\u63d0\u9ad8\u547d\u4e2d\u7387\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u5bf9\u7f13\u5b58\u7684\u5229\u7528\u6548\u7387\u662f\u4e0d\u540c\u7684\uff0c\u4e3b\u8981\u4f53\u73b0\u5728\u4ee5\u4e0b\u51e0\u4e2a\u65b9\u9762\u3002</p> <ul> <li>\u5360\u7528\u7a7a\u95f4\uff1a\u94fe\u8868\u5143\u7d20\u6bd4\u6570\u7ec4\u5143\u7d20\u5360\u7528\u7a7a\u95f4\u66f4\u591a\uff0c\u5bfc\u81f4\u7f13\u5b58\u4e2d\u5bb9\u7eb3\u7684\u6709\u6548\u6570\u636e\u91cf\u66f4\u5c11\u3002</li> <li>\u7f13\u5b58\u884c\uff1a\u94fe\u8868\u6570\u636e\u5206\u6563\u5728\u5185\u5b58\u5404\u5904\uff0c\u800c\u7f13\u5b58\u662f\u201c\u6309\u884c\u52a0\u8f7d\u201d\u7684\uff0c\u56e0\u6b64\u52a0\u8f7d\u5230\u65e0\u6548\u6570\u636e\u7684\u6bd4\u4f8b\u66f4\u9ad8\u3002</li> <li>\u9884\u53d6\u673a\u5236\uff1a\u6570\u7ec4\u6bd4\u94fe\u8868\u7684\u6570\u636e\u8bbf\u95ee\u6a21\u5f0f\u66f4\u5177\u201c\u53ef\u9884\u6d4b\u6027\u201d\uff0c\u5373\u7cfb\u7edf\u66f4\u5bb9\u6613\u731c\u51fa\u5373\u5c06\u88ab\u52a0\u8f7d\u7684\u6570\u636e\u3002</li> <li>\u7a7a\u95f4\u5c40\u90e8\u6027\uff1a\u6570\u7ec4\u88ab\u5b58\u50a8\u5728\u96c6\u4e2d\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u56e0\u6b64\u88ab\u52a0\u8f7d\u6570\u636e\u9644\u8fd1\u7684\u6570\u636e\u66f4\u6709\u53ef\u80fd\u5373\u5c06\u88ab\u8bbf\u95ee\u3002</li> </ul> <p>\u603b\u4f53\u800c\u8a00\uff0c\u6570\u7ec4\u5177\u6709\u66f4\u9ad8\u7684\u7f13\u5b58\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u5728\u64cd\u4f5c\u6548\u7387\u4e0a\u901a\u5e38\u4f18\u4e8e\u94fe\u8868\u3002\u8fd9\u4f7f\u5f97\u5728\u89e3\u51b3\u7b97\u6cd5\u95ee\u9898\u65f6\uff0c\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u5f80\u5f80\u66f4\u53d7\u6b22\u8fce\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u9ad8\u7f13\u5b58\u6548\u7387\u5e76\u4e0d\u610f\u5473\u7740\u6570\u7ec4\u5728\u6240\u6709\u60c5\u51b5\u4e0b\u90fd\u4f18\u4e8e\u94fe\u8868\u3002\u5b9e\u9645\u5e94\u7528\u4e2d\u9009\u62e9\u54ea\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u5e94\u6839\u636e\u5177\u4f53\u9700\u6c42\u6765\u51b3\u5b9a\u3002\u4f8b\u5982\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u90fd\u53ef\u4ee5\u5b9e\u73b0\u201c\u6808\u201d\u6570\u636e\u7ed3\u6784\uff08\u4e0b\u4e00\u7ae0\u4f1a\u8be6\u7ec6\u4ecb\u7ecd\uff09\uff0c\u4f46\u5b83\u4eec\u9002\u7528\u4e8e\u4e0d\u540c\u573a\u666f\u3002</p> <ul> <li>\u5728\u505a\u7b97\u6cd5\u9898\u65f6\uff0c\u6211\u4eec\u4f1a\u503e\u5411\u4e8e\u9009\u62e9\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\uff0c\u56e0\u4e3a\u5b83\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u64cd\u4f5c\u6548\u7387\u548c\u968f\u673a\u8bbf\u95ee\u7684\u80fd\u529b\uff0c\u4ee3\u4ef7\u4ec5\u662f\u9700\u8981\u9884\u5148\u4e3a\u6570\u7ec4\u5206\u914d\u4e00\u5b9a\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u5982\u679c\u6570\u636e\u91cf\u975e\u5e38\u5927\u3001\u52a8\u6001\u6027\u5f88\u9ad8\u3001\u6808\u7684\u9884\u671f\u5927\u5c0f\u96be\u4ee5\u4f30\u8ba1\uff0c\u90a3\u4e48\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u66f4\u52a0\u5408\u9002\u3002\u94fe\u8868\u80fd\u591f\u5c06\u5927\u91cf\u6570\u636e\u5206\u6563\u5b58\u50a8\u4e8e\u5185\u5b58\u7684\u4e0d\u540c\u90e8\u5206\uff0c\u5e76\u4e14\u907f\u514d\u4e86\u6570\u7ec4\u6269\u5bb9\u4ea7\u751f\u7684\u989d\u5916\u5f00\u9500\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/","title":"4.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_array_and_linkedlist/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u6570\u7ec4\u548c\u94fe\u8868\u662f\u4e24\u79cd\u57fa\u672c\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5206\u522b\u4ee3\u8868\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u4e24\u79cd\u5b58\u50a8\u65b9\u5f0f\uff1a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\u3002\u4e24\u8005\u7684\u7279\u70b9\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u6027\u3002</li> <li>\u6570\u7ec4\u652f\u6301\u968f\u673a\u8bbf\u95ee\u3001\u5360\u7528\u5185\u5b58\u8f83\u5c11\uff1b\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u6548\u7387\u4f4e\uff0c\u4e14\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002</li> <li>\u94fe\u8868\u901a\u8fc7\u66f4\u6539\u5f15\u7528\uff08\u6307\u9488\uff09\u5b9e\u73b0\u9ad8\u6548\u7684\u8282\u70b9\u63d2\u5165\u4e0e\u5220\u9664\uff0c\u4e14\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\uff1b\u4f46\u8282\u70b9\u8bbf\u95ee\u6548\u7387\u4f4e\u3001\u5360\u7528\u5185\u5b58\u8f83\u591a\u3002\u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u5305\u62ec\u5355\u5411\u94fe\u8868\u3001\u73af\u5f62\u94fe\u8868\u3001\u53cc\u5411\u94fe\u8868\u3002</li> <li>\u5217\u8868\u662f\u4e00\u79cd\u652f\u6301\u589e\u5220\u67e5\u6539\u7684\u5143\u7d20\u6709\u5e8f\u96c6\u5408\uff0c\u901a\u5e38\u57fa\u4e8e\u52a8\u6001\u6570\u7ec4\u5b9e\u73b0\u3002\u5b83\u4fdd\u7559\u4e86\u6570\u7ec4\u7684\u4f18\u52bf\uff0c\u540c\u65f6\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\u3002</li> <li>\u5217\u8868\u7684\u51fa\u73b0\u5927\u5e45\u63d0\u9ad8\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5bfc\u81f4\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002</li> <li>\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u4e3b\u8981\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u6570\u7ec4\u53ef\u63d0\u4f9b\u66f4\u9ad8\u7684\u5185\u5b58\u7a7a\u95f4\u6548\u7387\uff0c\u800c\u94fe\u8868\u5219\u5728\u5185\u5b58\u4f7f\u7528\u4e0a\u66f4\u52a0\u7075\u6d3b\u3002</li> <li>\u7f13\u5b58\u901a\u8fc7\u7f13\u5b58\u884c\u3001\u9884\u53d6\u673a\u5236\u4ee5\u53ca\u7a7a\u95f4\u5c40\u90e8\u6027\u548c\u65f6\u95f4\u5c40\u90e8\u6027\u7b49\u6570\u636e\u52a0\u8f7d\u673a\u5236\uff0c\u4e3a CPU \u63d0\u4f9b\u5feb\u901f\u6570\u636e\u8bbf\u95ee\uff0c\u663e\u8457\u63d0\u5347\u7a0b\u5e8f\u7684\u6267\u884c\u6548\u7387\u3002</li> <li>\u7531\u4e8e\u6570\u7ec4\u5177\u6709\u66f4\u9ad8\u7684\u7f13\u5b58\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u901a\u5e38\u6bd4\u94fe\u8868\u66f4\u9ad8\u6548\u3002\u5728\u9009\u62e9\u6570\u636e\u7ed3\u6784\u65f6\uff0c\u5e94\u6839\u636e\u5177\u4f53\u9700\u6c42\u548c\u573a\u666f\u505a\u51fa\u6070\u5f53\u9009\u62e9\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u6570\u7ec4\u5b58\u50a8\u5728\u6808\u4e0a\u548c\u5b58\u50a8\u5728\u5806\u4e0a\uff0c\u5bf9\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u5426\u6709\u5f71\u54cd\uff1f</p> <p>\u5b58\u50a8\u5728\u6808\u4e0a\u548c\u5806\u4e0a\u7684\u6570\u7ec4\u90fd\u88ab\u5b58\u50a8\u5728\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\u5185\uff0c\u6570\u636e\u64cd\u4f5c\u6548\u7387\u57fa\u672c\u4e00\u81f4\u3002\u7136\u800c\uff0c\u6808\u548c\u5806\u5177\u6709\u5404\u81ea\u7684\u7279\u70b9\uff0c\u4ece\u800c\u5bfc\u81f4\u4ee5\u4e0b\u4e0d\u540c\u70b9\u3002</p> <ol> <li>\u5206\u914d\u548c\u91ca\u653e\u6548\u7387\uff1a\u6808\u662f\u4e00\u5757\u8f83\u5c0f\u7684\u5185\u5b58\uff0c\u5206\u914d\u7531\u7f16\u8bd1\u5668\u81ea\u52a8\u5b8c\u6210\uff1b\u800c\u5806\u5185\u5b58\u76f8\u5bf9\u66f4\u5927\uff0c\u53ef\u4ee5\u5728\u4ee3\u7801\u4e2d\u52a8\u6001\u5206\u914d\uff0c\u66f4\u5bb9\u6613\u788e\u7247\u5316\u3002\u56e0\u6b64\uff0c\u5806\u4e0a\u7684\u5206\u914d\u548c\u91ca\u653e\u64cd\u4f5c\u901a\u5e38\u6bd4\u6808\u4e0a\u7684\u6162\u3002</li> <li>\u5927\u5c0f\u9650\u5236\uff1a\u6808\u5185\u5b58\u76f8\u5bf9\u8f83\u5c0f\uff0c\u5806\u7684\u5927\u5c0f\u4e00\u822c\u53d7\u9650\u4e8e\u53ef\u7528\u5185\u5b58\u3002\u56e0\u6b64\u5806\u66f4\u52a0\u9002\u5408\u5b58\u50a8\u5927\u578b\u6570\u7ec4\u3002</li> <li>\u7075\u6d3b\u6027\uff1a\u6808\u4e0a\u7684\u6570\u7ec4\u7684\u5927\u5c0f\u9700\u8981\u5728\u7f16\u8bd1\u65f6\u786e\u5b9a\uff0c\u800c\u5806\u4e0a\u7684\u6570\u7ec4\u7684\u5927\u5c0f\u53ef\u4ee5\u5728\u8fd0\u884c\u65f6\u52a8\u6001\u786e\u5b9a\u3002</li> </ol> <p>Q\uff1a\u4e3a\u4ec0\u4e48\u6570\u7ec4\u8981\u6c42\u76f8\u540c\u7c7b\u578b\u7684\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u5374\u6ca1\u6709\u5f3a\u8c03\u76f8\u540c\u7c7b\u578b\u5462\uff1f</p> <p>\u94fe\u8868\u7531\u8282\u70b9\u7ec4\u6210\uff0c\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u5f15\u7528\uff08\u6307\u9488\uff09\u8fde\u63a5\uff0c\u5404\u4e2a\u8282\u70b9\u53ef\u4ee5\u5b58\u50a8\u4e0d\u540c\u7c7b\u578b\u7684\u6570\u636e\uff0c\u4f8b\u5982 <code>int</code>\u3001<code>double</code>\u3001<code>string</code>\u3001<code>object</code> \u7b49\u3002</p> <p>\u76f8\u5bf9\u5730\uff0c\u6570\u7ec4\u5143\u7d20\u5219\u5fc5\u987b\u662f\u76f8\u540c\u7c7b\u578b\u7684\uff0c\u8fd9\u6837\u624d\u80fd\u901a\u8fc7\u8ba1\u7b97\u504f\u79fb\u91cf\u6765\u83b7\u53d6\u5bf9\u5e94\u5143\u7d20\u4f4d\u7f6e\u3002\u4f8b\u5982\uff0c\u6570\u7ec4\u540c\u65f6\u5305\u542b <code>int</code> \u548c <code>long</code> \u4e24\u79cd\u7c7b\u578b\uff0c\u5355\u4e2a\u5143\u7d20\u5206\u522b\u5360\u7528 4 \u5b57\u8282 \u548c 8 \u5b57\u8282 \uff0c\u6b64\u65f6\u5c31\u4e0d\u80fd\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u504f\u79fb\u91cf\u4e86\uff0c\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5305\u542b\u4e86\u4e24\u79cd\u201c\u5143\u7d20\u957f\u5ea6\u201d\u3002</p> <pre><code># \u5143\u7d20\u5185\u5b58\u5730\u5740 = \u6570\u7ec4\u5185\u5b58\u5730\u5740\uff08\u9996\u5143\u7d20\u5185\u5b58\u5730\u5740\uff09 + \u5143\u7d20\u957f\u5ea6 * \u5143\u7d20\u7d22\u5f15\n</code></pre> <p>Q\uff1a\u5220\u9664\u8282\u70b9 <code>P</code> \u540e\uff0c\u662f\u5426\u9700\u8981\u628a <code>P.next</code> \u8bbe\u4e3a <code>None</code> \u5462\uff1f</p> <p>\u4e0d\u4fee\u6539 <code>P.next</code> \u4e5f\u53ef\u4ee5\u3002\u4ece\u8be5\u94fe\u8868\u7684\u89d2\u5ea6\u770b\uff0c\u4ece\u5934\u8282\u70b9\u904d\u5386\u5230\u5c3e\u8282\u70b9\u5df2\u7ecf\u4e0d\u4f1a\u9047\u5230 <code>P</code> \u4e86\u3002\u8fd9\u610f\u5473\u7740\u8282\u70b9 <code>P</code> \u5df2\u7ecf\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u4e86\uff0c\u6b64\u65f6\u8282\u70b9 <code>P</code> \u6307\u5411\u54ea\u91cc\u90fd\u4e0d\u4f1a\u5bf9\u8be5\u94fe\u8868\u4ea7\u751f\u5f71\u54cd\u3002</p> <p>\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\uff08\u505a\u9898\uff09\u7684\u89d2\u5ea6\u770b\uff0c\u4e0d\u65ad\u5f00\u6ca1\u6709\u5173\u7cfb\uff0c\u53ea\u8981\u4fdd\u8bc1\u7a0b\u5e8f\u7684\u903b\u8f91\u662f\u6b63\u786e\u7684\u5c31\u884c\u3002\u4ece\u6807\u51c6\u5e93\u7684\u89d2\u5ea6\u770b\uff0c\u65ad\u5f00\u66f4\u52a0\u5b89\u5168\u3001\u903b\u8f91\u66f4\u52a0\u6e05\u6670\u3002\u5982\u679c\u4e0d\u65ad\u5f00\uff0c\u5047\u8bbe\u88ab\u5220\u9664\u8282\u70b9\u672a\u88ab\u6b63\u5e38\u56de\u6536\uff0c\u90a3\u4e48\u5b83\u4f1a\u5f71\u54cd\u540e\u7ee7\u8282\u70b9\u7684\u5185\u5b58\u56de\u6536\u3002</p> <p>Q\uff1a\u5728\u94fe\u8868\u4e2d\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u4f46\u662f\u589e\u5220\u4e4b\u524d\u90fd\u9700\u8981 \\(O(n)\\) \u7684\u65f6\u95f4\u67e5\u627e\u5143\u7d20\uff0c\u90a3\u4e3a\u4ec0\u4e48\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u662f \\(O(n)\\) \u5462\uff1f</p> <p>\u5982\u679c\u662f\u5148\u67e5\u627e\u5143\u7d20\u3001\u518d\u5220\u9664\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u786e\u5b9e\u662f \\(O(n)\\) \u3002\u7136\u800c\uff0c\u94fe\u8868\u7684 \\(O(1)\\) \u589e\u5220\u7684\u4f18\u52bf\u53ef\u4ee5\u5728\u5176\u4ed6\u5e94\u7528\u4e0a\u5f97\u5230\u4f53\u73b0\u3002\u4f8b\u5982\uff0c\u53cc\u5411\u961f\u5217\u9002\u5408\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff0c\u6211\u4eec\u7ef4\u62a4\u4e00\u4e2a\u6307\u9488\u53d8\u91cf\u59cb\u7ec8\u6307\u5411\u5934\u8282\u70b9\u3001\u5c3e\u8282\u70b9\uff0c\u6bcf\u6b21\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u90fd\u662f \\(O(1)\\) \u3002</p> <p>Q\uff1a\u56fe\u201c\u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f\u201d\u4e2d\uff0c\u6d45\u84dd\u8272\u7684\u5b58\u50a8\u8282\u70b9\u6307\u9488\u662f\u5360\u7528\u4e00\u5757\u5185\u5b58\u5730\u5740\u5417\uff1f\u8fd8\u662f\u548c\u8282\u70b9\u503c\u5404\u5360\u4e00\u534a\u5462\uff1f</p> <p>\u8be5\u793a\u610f\u56fe\u53ea\u662f\u5b9a\u6027\u8868\u793a\uff0c\u5b9a\u91cf\u8868\u793a\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u8fdb\u884c\u5206\u6790\u3002</p> <ul> <li>\u4e0d\u540c\u7c7b\u578b\u7684\u8282\u70b9\u503c\u5360\u7528\u7684\u7a7a\u95f4\u662f\u4e0d\u540c\u7684\uff0c\u6bd4\u5982 <code>int</code>\u3001<code>long</code>\u3001<code>double</code> \u548c\u5b9e\u4f8b\u5bf9\u8c61\u7b49\u3002</li> <li>\u6307\u9488\u53d8\u91cf\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u5927\u5c0f\u6839\u636e\u6240\u4f7f\u7528\u7684\u64cd\u4f5c\u7cfb\u7edf\u53ca\u7f16\u8bd1\u73af\u5883\u800c\u5b9a\uff0c\u5927\u591a\u4e3a 8 \u5b57\u8282\u6216 4 \u5b57\u8282\u3002</li> </ul> <p>Q\uff1a\u5728\u5217\u8868\u672b\u5c3e\u6dfb\u52a0\u5143\u7d20\u662f\u5426\u65f6\u65f6\u523b\u523b\u90fd\u4e3a \\(O(1)\\) \uff1f</p> <p>\u5982\u679c\u6dfb\u52a0\u5143\u7d20\u65f6\u8d85\u51fa\u5217\u8868\u957f\u5ea6\uff0c\u5219\u9700\u8981\u5148\u6269\u5bb9\u5217\u8868\u518d\u6dfb\u52a0\u3002\u7cfb\u7edf\u4f1a\u7533\u8bf7\u4e00\u5757\u65b0\u7684\u5185\u5b58\uff0c\u5e76\u5c06\u539f\u5217\u8868\u7684\u6240\u6709\u5143\u7d20\u642c\u8fd0\u8fc7\u53bb\uff0c\u8fd9\u65f6\u5019\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u4f1a\u662f \\(O(n)\\) \u3002</p> <p>Q\uff1a\u201c\u5217\u8868\u7684\u51fa\u73b0\u6781\u5927\u5730\u63d0\u9ad8\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5bfc\u81f4\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u201d\uff0c\u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u662f\u6307\u989d\u5916\u589e\u52a0\u7684\u53d8\u91cf\u5982\u5bb9\u91cf\u3001\u957f\u5ea6\u3001\u6269\u5bb9\u500d\u6570\u6240\u5360\u7684\u5185\u5b58\u5417\uff1f</p> <p>\u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u4e3b\u8981\u6709\u4e24\u65b9\u9762\u542b\u4e49\uff1a\u4e00\u65b9\u9762\uff0c\u5217\u8868\u90fd\u4f1a\u8bbe\u5b9a\u4e00\u4e2a\u521d\u59cb\u957f\u5ea6\uff0c\u6211\u4eec\u4e0d\u4e00\u5b9a\u9700\u8981\u7528\u8fd9\u4e48\u591a\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u4e3a\u4e86\u9632\u6b62\u9891\u7e41\u6269\u5bb9\uff0c\u6269\u5bb9\u4e00\u822c\u4f1a\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u6bd4\u5982 \\(\\times 1.5\\) \u3002\u8fd9\u6837\u4e00\u6765\uff0c\u4e5f\u4f1a\u51fa\u73b0\u5f88\u591a\u7a7a\u4f4d\uff0c\u6211\u4eec\u901a\u5e38\u4e0d\u80fd\u5b8c\u5168\u586b\u6ee1\u5b83\u4eec\u3002</p> <p>Q\uff1a\u5728 Python \u4e2d\u521d\u59cb\u5316 <code>n = [1, 2, 3]</code> \u540e\uff0c\u8fd9 3 \u4e2a\u5143\u7d20\u7684\u5730\u5740\u662f\u76f8\u8fde\u7684\uff0c\u4f46\u662f\u521d\u59cb\u5316 <code>m = [2, 1, 3]</code> \u4f1a\u53d1\u73b0\u5b83\u4eec\u6bcf\u4e2a\u5143\u7d20\u7684 id \u5e76\u4e0d\u662f\u8fde\u7eed\u7684\uff0c\u800c\u662f\u5206\u522b\u8ddf <code>n</code> \u4e2d\u7684\u76f8\u540c\u3002\u8fd9\u4e9b\u5143\u7d20\u7684\u5730\u5740\u4e0d\u8fde\u7eed\uff0c\u90a3\u4e48 <code>m</code> \u8fd8\u662f\u6570\u7ec4\u5417\uff1f</p> <p>\u5047\u5982\u628a\u5217\u8868\u5143\u7d20\u6362\u6210\u94fe\u8868\u8282\u70b9 <code>n = [n1, n2, n3, n4, n5]</code> \uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u8fd9 5 \u4e2a\u8282\u70b9\u5bf9\u8c61\u4e5f\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\u3002\u7136\u800c\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u5217\u8868\u7d22\u5f15\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u8282\u70b9\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u8bbf\u95ee\u5230\u5bf9\u5e94\u7684\u8282\u70b9\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5b58\u50a8\u7684\u662f\u8282\u70b9\u7684\u5f15\u7528\uff0c\u800c\u975e\u8282\u70b9\u672c\u8eab\u3002</p> <p>\u4e0e\u8bb8\u591a\u8bed\u8a00\u4e0d\u540c\uff0cPython \u4e2d\u7684\u6570\u5b57\u4e5f\u88ab\u5305\u88c5\u4e3a\u5bf9\u8c61\uff0c\u5217\u8868\u4e2d\u5b58\u50a8\u7684\u4e0d\u662f\u6570\u5b57\u672c\u8eab\uff0c\u800c\u662f\u5bf9\u6570\u5b57\u7684\u5f15\u7528\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u4f1a\u53d1\u73b0\u4e24\u4e2a\u6570\u7ec4\u4e2d\u7684\u76f8\u540c\u6570\u5b57\u62e5\u6709\u540c\u4e00\u4e2a id \uff0c\u5e76\u4e14\u8fd9\u4e9b\u6570\u5b57\u7684\u5185\u5b58\u5730\u5740\u65e0\u987b\u8fde\u7eed\u3002</p> <p>Q\uff1aC++ STL \u91cc\u9762\u7684 <code>std::list</code> \u5df2\u7ecf\u5b9e\u73b0\u4e86\u53cc\u5411\u94fe\u8868\uff0c\u4f46\u597d\u50cf\u4e00\u4e9b\u7b97\u6cd5\u4e66\u4e0a\u4e0d\u600e\u4e48\u76f4\u63a5\u4f7f\u7528\u5b83\uff0c\u662f\u4e0d\u662f\u56e0\u4e3a\u6709\u4ec0\u4e48\u5c40\u9650\u6027\u5462\uff1f</p> <p>\u4e00\u65b9\u9762\uff0c\u6211\u4eec\u5f80\u5f80\u66f4\u9752\u7750\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u7b97\u6cd5\uff0c\u800c\u53ea\u5728\u5fc5\u8981\u65f6\u624d\u4f7f\u7528\u94fe\u8868\uff0c\u4e3b\u8981\u6709\u4e24\u4e2a\u539f\u56e0\u3002</p> <ul> <li>\u7a7a\u95f4\u5f00\u9500\uff1a\u7531\u4e8e\u6bcf\u4e2a\u5143\u7d20\u9700\u8981\u4e24\u4e2a\u989d\u5916\u7684\u6307\u9488\uff08\u4e00\u4e2a\u7528\u4e8e\u524d\u4e00\u4e2a\u5143\u7d20\uff0c\u4e00\u4e2a\u7528\u4e8e\u540e\u4e00\u4e2a\u5143\u7d20\uff09\uff0c\u6240\u4ee5 <code>std::list</code> \u901a\u5e38\u6bd4 <code>std::vector</code> \u66f4\u5360\u7528\u7a7a\u95f4\u3002</li> <li>\u7f13\u5b58\u4e0d\u53cb\u597d\uff1a\u7531\u4e8e\u6570\u636e\u4e0d\u662f\u8fde\u7eed\u5b58\u653e\u7684\uff0c\u56e0\u6b64 <code>std::list</code> \u5bf9\u7f13\u5b58\u7684\u5229\u7528\u7387\u8f83\u4f4e\u3002\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c<code>std::vector</code> \u7684\u6027\u80fd\u4f1a\u66f4\u597d\u3002</li> </ul> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5fc5\u8981\u4f7f\u7528\u94fe\u8868\u7684\u60c5\u51b5\u4e3b\u8981\u662f\u4e8c\u53c9\u6811\u548c\u56fe\u3002\u6808\u548c\u961f\u5217\u5f80\u5f80\u4f1a\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684 <code>stack</code> \u548c <code>queue</code> \uff0c\u800c\u975e\u94fe\u8868\u3002</p> <p>Q\uff1a\u521d\u59cb\u5316\u5217\u8868 <code>res = [0] * self.size()</code> \u64cd\u4f5c\uff0c\u4f1a\u5bfc\u81f4 <code>res</code> \u7684\u6bcf\u4e2a\u5143\u7d20\u5f15\u7528\u76f8\u540c\u7684\u5730\u5740\u5417\uff1f</p> <p>\u4e0d\u4f1a\u3002\u4f46\u4e8c\u7ef4\u6570\u7ec4\u4f1a\u6709\u8fd9\u4e2a\u95ee\u9898\uff0c\u4f8b\u5982\u521d\u59cb\u5316\u4e8c\u7ef4\u5217\u8868 <code>res = [[0] * self.size()]</code> \uff0c\u5219\u591a\u6b21\u5f15\u7528\u4e86\u540c\u4e00\u4e2a\u5217\u8868 <code>[0]</code> \u3002</p>"},{"location":"chapter_backtracking/","title":"\u7b2c 13 \u7ae0 \u00a0 \u56de\u6eaf","text":"<p>Abstract</p> <p>\u6211\u4eec\u5982\u540c\u8ff7\u5bab\u4e2d\u7684\u63a2\u7d22\u8005\uff0c\u5728\u524d\u8fdb\u7684\u9053\u8def\u4e0a\u53ef\u80fd\u4f1a\u9047\u5230\u56f0\u96be\u3002</p> <p>\u56de\u6eaf\u7684\u529b\u91cf\u8ba9\u6211\u4eec\u80fd\u591f\u91cd\u65b0\u5f00\u59cb\uff0c\u4e0d\u65ad\u5c1d\u8bd5\uff0c\u6700\u7ec8\u627e\u5230\u901a\u5f80\u5149\u660e\u7684\u51fa\u53e3\u3002</p>"},{"location":"chapter_backtracking/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>13.1 \u00a0 \u56de\u6eaf\u7b97\u6cd5</li> <li>13.2 \u00a0 \u5168\u6392\u5217\u95ee\u9898</li> <li>13.3 \u00a0 \u5b50\u96c6\u548c\u95ee\u9898</li> <li>13.4 \u00a0 N \u7687\u540e\u95ee\u9898</li> <li>13.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_backtracking/backtracking_algorithm/","title":"13.1 \u00a0 \u56de\u6eaf\u7b97\u6cd5","text":"<p>\u56de\u6eaf\u7b97\u6cd5\uff08backtracking algorithm\uff09\u662f\u4e00\u79cd\u901a\u8fc7\u7a77\u4e3e\u6765\u89e3\u51b3\u95ee\u9898\u7684\u65b9\u6cd5\uff0c\u5b83\u7684\u6838\u5fc3\u601d\u60f3\u662f\u4ece\u4e00\u4e2a\u521d\u59cb\u72b6\u6001\u51fa\u53d1\uff0c\u66b4\u529b\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u5f53\u9047\u5230\u6b63\u786e\u7684\u89e3\u5219\u5c06\u5176\u8bb0\u5f55\uff0c\u76f4\u5230\u627e\u5230\u89e3\u6216\u8005\u5c1d\u8bd5\u4e86\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u90fd\u65e0\u6cd5\u627e\u5230\u89e3\u4e3a\u6b62\u3002</p> <p>\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u201c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u6765\u904d\u5386\u89e3\u7a7a\u95f4\u3002\u5728\u201c\u4e8c\u53c9\u6811\u201d\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5229\u7528\u524d\u5e8f\u904d\u5386\u6784\u9020\u4e00\u4e2a\u56de\u6eaf\u95ee\u9898\uff0c\u9010\u6b65\u4e86\u89e3\u56de\u6eaf\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002</p> <p>\u4f8b\u9898\u4e00</p> <p>\u7ed9\u5b9a\u4e00\u68f5\u4e8c\u53c9\u6811\uff0c\u641c\u7d22\u5e76\u8bb0\u5f55\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u8282\u70b9\u5217\u8868\u3002</p> <p>\u5bf9\u4e8e\u6b64\u9898\uff0c\u6211\u4eec\u524d\u5e8f\u904d\u5386\u8fd9\u68f5\u6811\uff0c\u5e76\u5224\u65ad\u5f53\u524d\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \uff0c\u82e5\u662f\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u7684\u503c\u52a0\u5165\u7ed3\u679c\u5217\u8868 <code>res</code> \u4e4b\u4e2d\u3002\u76f8\u5173\u8fc7\u7a0b\u5b9e\u73b0\u5982\u56fe 13-1 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_i_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00\"\"\"\n    if root is None:\n        return\n    if root.val == 7:\n        # \u8bb0\u5f55\u89e3\n        res.append(root)\n    pre_order(root.left)\n    pre_order(root.right)\n</code></pre> preorder_traversal_i_compact.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push_back(root);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.java<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.add(root);\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.cs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.Add(root);\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.go<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrderI(root *TreeNode, res *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    if (root.Val).(int) == 7 {\n        // \u8bb0\u5f55\u89e3\n        *res = append(*res, root)\n    }\n    preOrderI(root.Left, res)\n    preOrderI(root.Right, res)\n}\n</code></pre> preorder_traversal_i_compact.swift<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    if root.val == 7 {\n        // \u8bb0\u5f55\u89e3\n        res.append(root)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n</code></pre> preorder_traversal_i_compact.js<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root, res) {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.ts<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root: TreeNode | null, res: TreeNode[]): void {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.dart<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode? root, List&lt;TreeNode&gt; res) {\n  if (root == null) {\n    return;\n  }\n  if (root.val == 7) {\n    // \u8bb0\u5f55\u89e3\n    res.add(root);\n  }\n  preOrder(root.left, res);\n  preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.rs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfn pre_order(res: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, root: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        if node.borrow().val == 7 {\n            // \u8bb0\u5f55\u89e3\n            res.push(node.clone());\n        }\n        pre_order(res, node.borrow().left.clone());\n        pre_order(res, node.borrow().right.clone());\n    }\n}\n</code></pre> preorder_traversal_i_compact.c<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res[resSize++] = root;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.kt<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    if (root.value == 7) {\n        // \u8bb0\u5f55\u89e3\n        res!!.add(root)\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n}\n</code></pre> preorder_traversal_i_compact.rb<pre><code>[class]{}-[func]{pre_order}\n</code></pre> preorder_traversal_i_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 13-1 \u00a0 \u5728\u524d\u5e8f\u904d\u5386\u4e2d\u641c\u7d22\u8282\u70b9 </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1311","title":"13.1.1 \u00a0 \u5c1d\u8bd5\u4e0e\u56de\u9000","text":"<p>\u4e4b\u6240\u4ee5\u79f0\u4e4b\u4e3a\u56de\u6eaf\u7b97\u6cd5\uff0c\u662f\u56e0\u4e3a\u8be5\u7b97\u6cd5\u5728\u641c\u7d22\u89e3\u7a7a\u95f4\u65f6\u4f1a\u91c7\u7528\u201c\u5c1d\u8bd5\u201d\u4e0e\u201c\u56de\u9000\u201d\u7684\u7b56\u7565\u3002\u5f53\u7b97\u6cd5\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u9047\u5230\u67d0\u4e2a\u72b6\u6001\u65e0\u6cd5\u7ee7\u7eed\u524d\u8fdb\u6216\u65e0\u6cd5\u5f97\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u65f6\uff0c\u5b83\u4f1a\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u5c1d\u8bd5\u5176\u4ed6\u53ef\u80fd\u7684\u9009\u62e9\u3002</p> <p>\u5bf9\u4e8e\u4f8b\u9898\u4e00\uff0c\u8bbf\u95ee\u6bcf\u4e2a\u8282\u70b9\u90fd\u4ee3\u8868\u4e00\u6b21\u201c\u5c1d\u8bd5\u201d\uff0c\u800c\u8d8a\u8fc7\u53f6\u8282\u70b9\u6216\u8fd4\u56de\u7236\u8282\u70b9\u7684 <code>return</code> \u5219\u8868\u793a\u201c\u56de\u9000\u201d\u3002</p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u56de\u9000\u5e76\u4e0d\u4ec5\u4ec5\u5305\u62ec\u51fd\u6570\u8fd4\u56de\u3002\u4e3a\u89e3\u91ca\u8fd9\u4e00\u70b9\uff0c\u6211\u4eec\u5bf9\u4f8b\u9898\u4e00\u7a0d\u4f5c\u62d3\u5c55\u3002</p> <p>\u4f8b\u9898\u4e8c</p> <p>\u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\u3002</p> <p>\u5728\u4f8b\u9898\u4e00\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u5217\u8868 <code>path</code> \u8bb0\u5f55\u8bbf\u95ee\u8fc7\u7684\u8282\u70b9\u8def\u5f84\u3002\u5f53\u8bbf\u95ee\u5230\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u590d\u5236 <code>path</code> \u5e76\u6dfb\u52a0\u8fdb\u7ed3\u679c\u5217\u8868 <code>res</code> \u3002\u904d\u5386\u5b8c\u6210\u540e\uff0c<code>res</code> \u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6240\u6709\u7684\u89e3\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_ii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c\"\"\"\n    if root is None:\n        return\n    # \u5c1d\u8bd5\n    path.append(root)\n    if root.val == 7:\n        # \u8bb0\u5f55\u89e3\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # \u56de\u9000\n    path.pop()\n</code></pre> preorder_traversal_ii_compact.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    path.pop_back();\n}\n</code></pre> preorder_traversal_ii_compact.java<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.add(root);\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // \u56de\u9000\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_ii_compact.cs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.Add(root);\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // \u56de\u9000\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_ii_compact.go<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrderII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    // \u5c1d\u8bd5\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // \u8bb0\u5f55\u89e3\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderII(root.Left, res, path)\n    preOrderII(root.Right, res, path)\n    // \u56de\u9000\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_ii_compact.swift<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u5c1d\u8bd5\n    path.append(root)\n    if root.val == 7 {\n        // \u8bb0\u5f55\u89e3\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // \u56de\u9000\n    path.removeLast()\n}\n</code></pre> preorder_traversal_ii_compact.js<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(root, path, res) {\n    if (root === null) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push(root);\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.ts<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    if (root === null) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push(root);\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.dart<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null) {\n    return;\n  }\n\n  // \u5c1d\u8bd5\n  path.add(root);\n  if (root.val == 7) {\n    // \u8bb0\u5f55\u89e3\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // \u56de\u9000\n  path.removeLast();\n}\n</code></pre> preorder_traversal_ii_compact.rs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        // \u5c1d\u8bd5\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // \u8bb0\u5f55\u89e3\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.clone());\n        pre_order(res, path, node.borrow().right.clone());\n        // \u56de\u9000\n        path.remove(path.len() - 1);\n    }\n}\n</code></pre> preorder_traversal_ii_compact.c<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        for (int i = 0; i &lt; pathSize; ++i) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    pathSize--;\n}\n</code></pre> preorder_traversal_ii_compact.kt<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    // \u5c1d\u8bd5\n    path!!.add(root)\n    if (root.value == 7) {\n        // \u8bb0\u5f55\u89e3\n        res!!.add(ArrayList(path!!))\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // \u56de\u9000\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_ii_compact.rb<pre><code>[class]{}-[func]{pre_order}\n</code></pre> preorder_traversal_ii_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5728\u6bcf\u6b21\u201c\u5c1d\u8bd5\u201d\u4e2d\uff0c\u6211\u4eec\u901a\u8fc7\u5c06\u5f53\u524d\u8282\u70b9\u6dfb\u52a0\u8fdb <code>path</code> \u6765\u8bb0\u5f55\u8def\u5f84\uff1b\u800c\u5728\u201c\u56de\u9000\u201d\u524d\uff0c\u6211\u4eec\u9700\u8981\u5c06\u8be5\u8282\u70b9\u4ece <code>path</code> \u4e2d\u5f39\u51fa\uff0c\u4ee5\u6062\u590d\u672c\u6b21\u5c1d\u8bd5\u4e4b\u524d\u7684\u72b6\u6001\u3002</p> <p>\u89c2\u5bdf\u56fe 13-2 \u6240\u793a\u7684\u8fc7\u7a0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5c1d\u8bd5\u548c\u56de\u9000\u7406\u89e3\u4e3a\u201c\u524d\u8fdb\u201d\u4e0e\u201c\u64a4\u9500\u201d\uff0c\u4e24\u4e2a\u64cd\u4f5c\u4e92\u4e3a\u9006\u5411\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 13-2 \u00a0 \u5c1d\u8bd5\u4e0e\u56de\u9000 </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1312","title":"13.1.2 \u00a0 \u526a\u679d","text":"<p>\u590d\u6742\u7684\u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u4e00\u4e2a\u6216\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u7ea6\u675f\u6761\u4ef6\u901a\u5e38\u53ef\u7528\u4e8e\u201c\u526a\u679d\u201d\u3002</p> <p>\u4f8b\u9898\u4e09</p> <p>\u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\uff0c\u5e76\u8981\u6c42\u8def\u5f84\u4e2d\u4e0d\u5305\u542b\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u3002</p> <p>\u4e3a\u4e86\u6ee1\u8db3\u4ee5\u4e0a\u7ea6\u675f\u6761\u4ef6\uff0c\u6211\u4eec\u9700\u8981\u6dfb\u52a0\u526a\u679d\u64cd\u4f5c\uff1a\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u82e5\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\uff0c\u5219\u63d0\u524d\u8fd4\u56de\uff0c\u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_iii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09\"\"\"\n    # \u526a\u679d\n    if root is None or root.val == 3:\n        return\n    # \u5c1d\u8bd5\n    path.append(root)\n    if root.val == 7:\n        # \u8bb0\u5f55\u89e3\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # \u56de\u9000\n    path.pop()\n</code></pre> preorder_traversal_iii_compact.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode *root) {\n    // \u526a\u679d\n    if (root == nullptr || root-&gt;val == 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    path.pop_back();\n}\n</code></pre> preorder_traversal_iii_compact.java<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode root) {\n    // \u526a\u679d\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.add(root);\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // \u56de\u9000\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_iii_compact.cs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid PreOrder(TreeNode? root) {\n    // \u526a\u679d\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.Add(root);\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // \u56de\u9000\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_iii_compact.go<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrderIII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    // \u526a\u679d\n    if root == nil || root.Val == 3 {\n        return\n    }\n    // \u5c1d\u8bd5\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // \u8bb0\u5f55\u89e3\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderIII(root.Left, res, path)\n    preOrderIII(root.Right, res, path)\n    // \u56de\u9000\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_iii_compact.swift<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrder(root: TreeNode?) {\n    // \u526a\u679d\n    guard let root = root, root.val != 3 else {\n        return\n    }\n    // \u5c1d\u8bd5\n    path.append(root)\n    if root.val == 7 {\n        // \u8bb0\u5f55\u89e3\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // \u56de\u9000\n    path.removeLast()\n}\n</code></pre> preorder_traversal_iii_compact.js<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(root, path, res) {\n    // \u526a\u679d\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push(root);\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.ts<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // \u526a\u679d\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push(root);\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.dart<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null || root.val == 3) {\n    return;\n  }\n\n  // \u5c1d\u8bd5\n  path.add(root);\n  if (root.val == 7) {\n    // \u8bb0\u5f55\u89e3\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // \u56de\u9000\n  path.removeLast();\n}\n</code></pre> preorder_traversal_iii_compact.rs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    // \u526a\u679d\n    if root.is_none() || root.as_ref().unwrap().borrow().val == 3 {\n        return;\n    }\n    if let Some(node) = root {\n        // \u5c1d\u8bd5\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // \u8bb0\u5f55\u89e3\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.clone());\n        pre_order(res, path, node.borrow().right.clone());\n        // \u56de\u9000\n        path.remove(path.len() - 1);\n    }\n}\n</code></pre> preorder_traversal_iii_compact.c<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode *root) {\n    // \u526a\u679d\n    if (root == NULL || root-&gt;val == 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        for (int i = 0; i &lt; pathSize; i++) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    pathSize--;\n}\n</code></pre> preorder_traversal_iii_compact.kt<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfun preOrder(root: TreeNode?) {\n    // \u526a\u679d\n    if (root == null || root.value == 3) {\n        return\n    }\n    // \u5c1d\u8bd5\n    path!!.add(root)\n    if (root.value == 7) {\n        // \u8bb0\u5f55\u89e3\n        res!!.add(ArrayList(path!!))\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // \u56de\u9000\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_iii_compact.rb<pre><code>[class]{}-[func]{pre_order}\n</code></pre> preorder_traversal_iii_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u201c\u526a\u679d\u201d\u662f\u4e00\u4e2a\u975e\u5e38\u5f62\u8c61\u7684\u540d\u8bcd\u3002\u5982\u56fe 13-3 \u6240\u793a\uff0c\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u201c\u526a\u6389\u201d\u4e86\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u641c\u7d22\u5206\u652f\uff0c\u907f\u514d\u8bb8\u591a\u65e0\u610f\u4e49\u7684\u5c1d\u8bd5\uff0c\u4ece\u800c\u63d0\u9ad8\u4e86\u641c\u7d22\u6548\u7387\u3002</p> <p></p> <p> \u56fe 13-3 \u00a0 \u6839\u636e\u7ea6\u675f\u6761\u4ef6\u526a\u679d </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1313","title":"13.1.3 \u00a0 \u6846\u67b6\u4ee3\u7801","text":"<p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c1d\u8bd5\u5c06\u56de\u6eaf\u7684\u201c\u5c1d\u8bd5\u3001\u56de\u9000\u3001\u526a\u679d\u201d\u7684\u4e3b\u4f53\u6846\u67b6\u63d0\u70bc\u51fa\u6765\uff0c\u63d0\u5347\u4ee3\u7801\u7684\u901a\u7528\u6027\u3002</p> <p>\u5728\u4ee5\u4e0b\u6846\u67b6\u4ee3\u7801\u4e2d\uff0c<code>state</code> \u8868\u793a\u95ee\u9898\u7684\u5f53\u524d\u72b6\u6001\uff0c<code>choices</code> \u8868\u793a\u5f53\u524d\u72b6\u6001\u4e0b\u53ef\u4ee5\u505a\u51fa\u7684\u9009\u62e9\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def backtrack(state: State, choices: list[choice], res: list[state]):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\"\"\"\n    # \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if is_solution(state):\n        # \u8bb0\u5f55\u89e3\n        record_solution(state, res)\n        # \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices:\n        # \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice):\n            # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            make_choice(state, choice)\n            backtrack(state, choices, res)\n            # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undo_choice(state, choice)\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, vector&lt;Choice *&gt; &amp;choices, vector&lt;State *&gt; &amp;res) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (Choice choice : choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (Choice choice : choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid Backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (IsSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        RecordSolution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    foreach (Choice choice in choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (IsValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            MakeChoice(state, choice);\n            Backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state *State, choices []Choice, res *[]State) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if isSolution(state) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res)\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for _, choice := range choices {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if isValid(state, choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state: inout State, choices: [Choice], res: inout [State]) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if isSolution(state: state) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state: state, res: &amp;res)\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if isValid(state: state, choice: choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state: &amp;state, choice: choice)\n            backtrack(state: &amp;state, choices: choices, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state, choices, res) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (let choice of choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state: State, choices: Choice[], res: State[]): void {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (let choice of choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List&lt;Choice&gt;, List&lt;State&gt; res) {\n  // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n  if (isSolution(state)) {\n    // \u8bb0\u5f55\u89e3\n    recordSolution(state, res);\n    // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  for (Choice choice in choices) {\n    // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n    if (isValid(state, choice)) {\n      // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n      makeChoice(state, choice);\n      backtrack(state, choices, res);\n      // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfn backtrack(state: &amp;mut State, choices: &amp;Vec&lt;Choice&gt;, res: &amp;mut Vec&lt;State&gt;) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if is_solution(state) {\n        // \u8bb0\u5f55\u89e3\n        record_solution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            make_choice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undo_choice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res, numRes);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; numChoices; i++) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, &amp;choices[i])) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, &amp;choices[i]);\n            backtrack(state, choices, numChoices, res, numRes);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, &amp;choices[i]);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfun backtrack(state: State?, choices: List&lt;Choice?&gt;, res: List&lt;State?&gt;?) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res)\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (choice in choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u57fa\u4e8e\u6846\u67b6\u4ee3\u7801\u6765\u89e3\u51b3\u4f8b\u9898\u4e09\u3002\u72b6\u6001 <code>state</code> \u4e3a\u8282\u70b9\u904d\u5386\u8def\u5f84\uff0c\u9009\u62e9 <code>choices</code> \u4e3a\u5f53\u524d\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\uff0c\u7ed3\u679c <code>res</code> \u662f\u8def\u5f84\u5217\u8868\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig preorder_traversal_iii_template.py<pre><code>def is_solution(state: list[TreeNode]) -&gt; bool:\n    \"\"\"\u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3\"\"\"\n    return state and state[-1].val == 7\n\ndef record_solution(state: list[TreeNode], res: list[list[TreeNode]]):\n    \"\"\"\u8bb0\u5f55\u89e3\"\"\"\n    res.append(list(state))\n\ndef is_valid(state: list[TreeNode], choice: TreeNode) -&gt; bool:\n    \"\"\"\u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5\"\"\"\n    return choice is not None and choice.val != 3\n\ndef make_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"\u66f4\u65b0\u72b6\u6001\"\"\"\n    state.append(choice)\n\ndef undo_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"\u6062\u590d\u72b6\u6001\"\"\"\n    state.pop()\n\ndef backtrack(\n    state: list[TreeNode], choices: list[TreeNode], res: list[list[TreeNode]]\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09\"\"\"\n    # \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if is_solution(state):\n        # \u8bb0\u5f55\u89e3\n        record_solution(state, res)\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices:\n        # \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice):\n            # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            make_choice(state, choice)\n            # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, [choice.left, choice.right], res)\n            # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undo_choice(state, choice)\n</code></pre> preorder_traversal_iii_template.cpp<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(vector&lt;TreeNode *&gt; &amp;state) {\n    return !state.empty() &amp;&amp; state.back()-&gt;val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(vector&lt;TreeNode *&gt; &amp;state, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    res.push_back(state);\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    return choice != nullptr &amp;&amp; choice-&gt;val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.push_back(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.pop_back();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(vector&lt;TreeNode *&gt; &amp;state, vector&lt;TreeNode *&gt; &amp;choices, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (TreeNode *choice : choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            vector&lt;TreeNode *&gt; nextChoices{choice-&gt;left, choice-&gt;right};\n            backtrack(state, nextChoices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.java<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nboolean isSolution(List&lt;TreeNode&gt; state) {\n    return !state.isEmpty() &amp;&amp; state.get(state.size() - 1).val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.add(new ArrayList&lt;&gt;(state));\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nboolean isValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.add(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.remove(state.size() - 1);\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (TreeNode choice : choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, Arrays.asList(choice.left, choice.right), res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.cs<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool IsSolution(List&lt;TreeNode&gt; state) {\n    return state.Count != 0 &amp;&amp; state[^1].val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid RecordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.Add(new List&lt;TreeNode&gt;(state));\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool IsValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid MakeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.Add(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid UndoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.RemoveAt(state.Count - 1);\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid Backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (IsSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        RecordSolution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    foreach (TreeNode choice in choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (IsValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            MakeChoice(state, choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            Backtrack(state, [choice.left!, choice.right!], res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.go<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state *[]*TreeNode) bool {\n    return len(*state) != 0 &amp;&amp; (*state)[len(*state)-1].Val == 7\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state *[]*TreeNode, res *[][]*TreeNode) {\n    *res = append(*res, append([]*TreeNode{}, *state...))\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state *[]*TreeNode, choice *TreeNode) bool {\n    return choice != nil &amp;&amp; choice.Val != 3\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = append(*state, choice)\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = (*state)[:len(*state)-1]\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrackIII(state *[]*TreeNode, choices *[]*TreeNode, res *[][]*TreeNode) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if isSolution(state) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res)\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for _, choice := range *choices {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if isValid(state, choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            temp := make([]*TreeNode, 0)\n            temp = append(temp, choice.Left, choice.Right)\n            backtrackIII(state, &amp;temp, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.swift<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state: [TreeNode]) -&gt; Bool {\n    !state.isEmpty &amp;&amp; state.last!.val == 7\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state: [TreeNode], res: inout [[TreeNode]]) {\n    res.append(state)\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state: [TreeNode], choice: TreeNode?) -&gt; Bool {\n    choice != nil &amp;&amp; choice!.val != 3\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.append(choice)\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.removeLast()\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrack(state: inout [TreeNode], choices: [TreeNode], res: inout [[TreeNode]]) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if isSolution(state: state) {\n        recordSolution(state: state, res: &amp;res)\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if isValid(state: state, choice: choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state: &amp;state, choice: choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state: &amp;state, choices: [choice.left, choice.right].compactMap { $0 }, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.js<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state) {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state, res) {\n    res.push([...state]);\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state, choice) {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state, choice) {\n    state.push(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state) {\n    state.pop();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(state, choices, res) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, [choice.left, choice.right], res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.ts<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state: TreeNode[]): boolean {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state: TreeNode[], res: TreeNode[][]): void {\n    res.push([...state]);\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state: TreeNode[], choice: TreeNode): boolean {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state: TreeNode[], choice: TreeNode): void {\n    state.push(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state: TreeNode[]): void {\n    state.pop();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(\n    state: TreeNode[],\n    choices: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, [choice.left, choice.right], res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.dart<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(List&lt;TreeNode&gt; state) {\n  return state.isNotEmpty &amp;&amp; state.last.val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n  res.add(List.from(state));\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.add(choice!);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.removeLast();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(\n  List&lt;TreeNode&gt; state,\n  List&lt;TreeNode?&gt; choices,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n  if (isSolution(state)) {\n    // \u8bb0\u5f55\u89e3\n    recordSolution(state, res);\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  for (TreeNode? choice in choices) {\n    // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n    if (isValid(state, choice)) {\n      // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n      makeChoice(state, choice);\n      // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n      backtrack(state, [choice!.left, choice.right], res);\n      // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> preorder_traversal_iii_template.rs<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfn is_solution(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\n    return !state.is_empty() &amp;&amp; state.get(state.len() - 1).unwrap().borrow().val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfn record_solution(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    res.push(state.clone());\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfn is_valid(_: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; bool {\n    return choice.borrow().val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfn make_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.push(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfn undo_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, _: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.remove(state.len() - 1);\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfn backtrack(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    choices: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if is_solution(state) {\n        // \u8bb0\u5f55\u89e3\n        record_solution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice.clone()) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            make_choice(state, choice.clone());\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(\n                state,\n                &amp;mut vec![\n                    choice.borrow().left.clone().unwrap(),\n                    choice.borrow().right.clone().unwrap(),\n                ],\n                res,\n            );\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undo_choice(state, choice.clone());\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.c<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(void) {\n    return pathSize &gt; 0 &amp;&amp; path[pathSize - 1]-&gt;val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(void) {\n    for (int i = 0; i &lt; pathSize; i++) {\n        res[resSize][i] = path[i];\n    }\n    resSize++;\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(TreeNode *choice) {\n    return choice != NULL &amp;&amp; choice-&gt;val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(TreeNode *choice) {\n    path[pathSize++] = choice;\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(void) {\n    pathSize--;\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(TreeNode *choices[2]) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution()) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution();\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; 2; i++) {\n        TreeNode *choice = choices[i];\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            TreeNode *nextChoices[2] = {choice-&gt;left, choice-&gt;right};\n            backtrack(nextChoices);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice();\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.kt<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfun isSolution(state: List&lt;TreeNode?&gt;): Boolean {\n    return state.isNotEmpty() &amp;&amp; state[state.size - 1]?.value == 7\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfun recordSolution(state: MutableList&lt;TreeNode?&gt;?, res: MutableList&lt;List&lt;TreeNode?&gt;?&gt;) {\n    res.add(state?.let { ArrayList(it) })\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfun isValid(state: List&lt;TreeNode?&gt;?, choice: TreeNode?): Boolean {\n    return choice != null &amp;&amp; choice.value != 3\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfun makeChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.add(choice)\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfun undoChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.removeLast()\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfun backtrack(\n    state: MutableList&lt;TreeNode?&gt;,\n    choices: List&lt;TreeNode?&gt;,\n    res: MutableList&lt;List&lt;TreeNode?&gt;?&gt;\n) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res)\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (choice in choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, listOf(choice!!.left, choice.right), res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.rb<pre><code>[class]{}-[func]{is_solution}\n\n[class]{}-[func]{record_solution}\n\n[class]{}-[func]{is_valid}\n\n[class]{}-[func]{make_choice}\n\n[class]{}-[func]{undo_choice}\n\n[class]{}-[func]{backtrack}\n</code></pre> preorder_traversal_iii_template.zig<pre><code>[class]{}-[func]{isSolution}\n\n[class]{}-[func]{recordSolution}\n\n[class]{}-[func]{isValid}\n\n[class]{}-[func]{makeChoice}\n\n[class]{}-[func]{undoChoice}\n\n[class]{}-[func]{backtrack}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u6839\u636e\u9898\u610f\uff0c\u6211\u4eec\u5728\u627e\u5230\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\u540e\u5e94\u8be5\u7ee7\u7eed\u641c\u7d22\uff0c\u56e0\u6b64\u9700\u8981\u5c06\u8bb0\u5f55\u89e3\u4e4b\u540e\u7684 <code>return</code> \u8bed\u53e5\u5220\u9664\u3002\u56fe 13-4 \u5bf9\u6bd4\u4e86\u4fdd\u7559\u6216\u5220\u9664 <code>return</code> \u8bed\u53e5\u7684\u641c\u7d22\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 13-4 \u00a0 \u4fdd\u7559\u4e0e\u5220\u9664 return \u7684\u641c\u7d22\u8fc7\u7a0b\u5bf9\u6bd4 </p> <p>\u76f8\u6bd4\u57fa\u4e8e\u524d\u5e8f\u904d\u5386\u7684\u4ee3\u7801\u5b9e\u73b0\uff0c\u57fa\u4e8e\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\u7684\u4ee3\u7801\u5b9e\u73b0\u867d\u7136\u663e\u5f97\u5570\u5506\uff0c\u4f46\u901a\u7528\u6027\u66f4\u597d\u3002\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u56de\u6eaf\u95ee\u9898\u53ef\u4ee5\u5728\u8be5\u6846\u67b6\u4e0b\u89e3\u51b3\u3002\u6211\u4eec\u53ea\u9700\u6839\u636e\u5177\u4f53\u95ee\u9898\u6765\u5b9a\u4e49 <code>state</code> \u548c <code>choices</code> \uff0c\u5e76\u5b9e\u73b0\u6846\u67b6\u4e2d\u7684\u5404\u4e2a\u65b9\u6cd5\u5373\u53ef\u3002</p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1314","title":"13.1.4 \u00a0 \u5e38\u7528\u672f\u8bed","text":"<p>\u4e3a\u4e86\u66f4\u6e05\u6670\u5730\u5206\u6790\u7b97\u6cd5\u95ee\u9898\uff0c\u6211\u4eec\u603b\u7ed3\u4e00\u4e0b\u56de\u6eaf\u7b97\u6cd5\u4e2d\u5e38\u7528\u672f\u8bed\u7684\u542b\u4e49\uff0c\u5e76\u5bf9\u7167\u4f8b\u9898\u4e09\u7ed9\u51fa\u5bf9\u5e94\u793a\u4f8b\uff0c\u5982\u8868 13-1 \u6240\u793a\u3002</p> <p> \u8868 13-1 \u00a0 \u5e38\u89c1\u7684\u56de\u6eaf\u7b97\u6cd5\u672f\u8bed </p> \u540d\u8bcd \u5b9a\u4e49 \u4f8b\u9898\u4e09 \u89e3\uff08solution\uff09 \u89e3\u662f\u6ee1\u8db3\u95ee\u9898\u7279\u5b9a\u6761\u4ef6\u7684\u7b54\u6848\uff0c\u53ef\u80fd\u6709\u4e00\u4e2a\u6216\u591a\u4e2a \u6839\u8282\u70b9\u5230\u8282\u70b9 \\(7\\) \u7684\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u6240\u6709\u8def\u5f84 \u7ea6\u675f\u6761\u4ef6\uff08constraint\uff09 \u7ea6\u675f\u6761\u4ef6\u662f\u95ee\u9898\u4e2d\u9650\u5236\u89e3\u7684\u53ef\u884c\u6027\u7684\u6761\u4ef6\uff0c\u901a\u5e38\u7528\u4e8e\u526a\u679d \u8def\u5f84\u4e2d\u4e0d\u5305\u542b\u8282\u70b9 \\(3\\) \u72b6\u6001\uff08state\uff09 \u72b6\u6001\u8868\u793a\u95ee\u9898\u5728\u67d0\u4e00\u65f6\u523b\u7684\u60c5\u51b5\uff0c\u5305\u62ec\u5df2\u7ecf\u505a\u51fa\u7684\u9009\u62e9 \u5f53\u524d\u5df2\u8bbf\u95ee\u7684\u8282\u70b9\u8def\u5f84\uff0c\u5373 <code>path</code> \u8282\u70b9\u5217\u8868 \u5c1d\u8bd5\uff08attempt\uff09 \u5c1d\u8bd5\u662f\u6839\u636e\u53ef\u7528\u9009\u62e9\u6765\u63a2\u7d22\u89e3\u7a7a\u95f4\u7684\u8fc7\u7a0b\uff0c\u5305\u62ec\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\uff0c\u68c0\u67e5\u662f\u5426\u4e3a\u89e3 \u9012\u5f52\u8bbf\u95ee\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\uff0c\u5c06\u8282\u70b9\u6dfb\u52a0\u8fdb <code>path</code> \uff0c\u5224\u65ad\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \u56de\u9000\uff08backtracking\uff09 \u56de\u9000\u6307\u9047\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u72b6\u6001\u65f6\uff0c\u64a4\u9500\u524d\u9762\u505a\u51fa\u7684\u9009\u62e9\uff0c\u56de\u5230\u4e0a\u4e00\u4e2a\u72b6\u6001 \u5f53\u8d8a\u8fc7\u53f6\u8282\u70b9\u3001\u7ed3\u675f\u8282\u70b9\u8bbf\u95ee\u3001\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\u7ec8\u6b62\u641c\u7d22\uff0c\u51fd\u6570\u8fd4\u56de \u526a\u679d\uff08pruning\uff09 \u526a\u679d\u662f\u6839\u636e\u95ee\u9898\u7279\u6027\u548c\u7ea6\u675f\u6761\u4ef6\u907f\u514d\u65e0\u610f\u4e49\u7684\u641c\u7d22\u8def\u5f84\u7684\u65b9\u6cd5\uff0c\u53ef\u63d0\u9ad8\u641c\u7d22\u6548\u7387 \u5f53\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u4e0d\u518d\u7ee7\u7eed\u641c\u7d22 <p>Tip</p> <p>\u95ee\u9898\u3001\u89e3\u3001\u72b6\u6001\u7b49\u6982\u5ff5\u662f\u901a\u7528\u7684\uff0c\u5728\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u8d2a\u5fc3\u7b49\u7b97\u6cd5\u4e2d\u90fd\u6709\u6d89\u53ca\u3002</p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1315","title":"13.1.5 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u4e0a\u662f\u4e00\u79cd\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u7b97\u6cd5\uff0c\u5b83\u5c1d\u8bd5\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\u76f4\u5230\u627e\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u3002\u8fd9\u79cd\u65b9\u6cd5\u7684\u4f18\u70b9\u5728\u4e8e\u80fd\u591f\u627e\u5230\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u800c\u4e14\u5728\u5408\u7406\u7684\u526a\u679d\u64cd\u4f5c\u4e0b\uff0c\u5177\u6709\u5f88\u9ad8\u7684\u6548\u7387\u3002</p> <p>\u7136\u800c\uff0c\u5728\u5904\u7406\u5927\u89c4\u6a21\u6216\u8005\u590d\u6742\u95ee\u9898\u65f6\uff0c\u56de\u6eaf\u7b97\u6cd5\u7684\u8fd0\u884c\u6548\u7387\u53ef\u80fd\u96be\u4ee5\u63a5\u53d7\u3002</p> <ul> <li>\u65f6\u95f4\uff1a\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u9700\u8981\u904d\u5386\u72b6\u6001\u7a7a\u95f4\u7684\u6240\u6709\u53ef\u80fd\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230\u6307\u6570\u9636\u6216\u9636\u4e58\u9636\u3002</li> <li>\u7a7a\u95f4\uff1a\u5728\u9012\u5f52\u8c03\u7528\u4e2d\u9700\u8981\u4fdd\u5b58\u5f53\u524d\u7684\u72b6\u6001\uff08\u4f8b\u5982\u8def\u5f84\u3001\u7528\u4e8e\u526a\u679d\u7684\u8f85\u52a9\u53d8\u91cf\u7b49\uff09\uff0c\u5f53\u6df1\u5ea6\u5f88\u5927\u65f6\uff0c\u7a7a\u95f4\u9700\u6c42\u53ef\u80fd\u4f1a\u53d8\u5f97\u5f88\u5927\u3002</li> </ul> <p>\u5373\u4fbf\u5982\u6b64\uff0c\u56de\u6eaf\u7b97\u6cd5\u4ecd\u7136\u662f\u67d0\u4e9b\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u7684\u6700\u4f73\u89e3\u51b3\u65b9\u6848\u3002\u5bf9\u4e8e\u8fd9\u4e9b\u95ee\u9898\uff0c\u7531\u4e8e\u65e0\u6cd5\u9884\u6d4b\u54ea\u4e9b\u9009\u62e9\u53ef\u751f\u6210\u6709\u6548\u7684\u89e3\uff0c\u56e0\u6b64\u6211\u4eec\u5fc5\u987b\u5bf9\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u8fdb\u884c\u904d\u5386\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5173\u952e\u662f\u5982\u4f55\u4f18\u5316\u6548\u7387\uff0c\u5e38\u89c1\u7684\u6548\u7387\u4f18\u5316\u65b9\u6cd5\u6709\u4e24\u79cd\u3002</p> <ul> <li>\u526a\u679d\uff1a\u907f\u514d\u641c\u7d22\u90a3\u4e9b\u80af\u5b9a\u4e0d\u4f1a\u4ea7\u751f\u89e3\u7684\u8def\u5f84\uff0c\u4ece\u800c\u8282\u7701\u65f6\u95f4\u548c\u7a7a\u95f4\u3002</li> <li>\u542f\u53d1\u5f0f\u641c\u7d22\uff1a\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u5f15\u5165\u4e00\u4e9b\u7b56\u7565\u6216\u8005\u4f30\u8ba1\u503c\uff0c\u4ece\u800c\u4f18\u5148\u641c\u7d22\u6700\u6709\u53ef\u80fd\u4ea7\u751f\u6709\u6548\u89e3\u7684\u8def\u5f84\u3002</li> </ul>"},{"location":"chapter_backtracking/backtracking_algorithm/#1316","title":"13.1.6 \u00a0 \u56de\u6eaf\u5178\u578b\u4f8b\u9898","text":"<p>\u56de\u6eaf\u7b97\u6cd5\u53ef\u7528\u4e8e\u89e3\u51b3\u8bb8\u591a\u641c\u7d22\u95ee\u9898\u3001\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u548c\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u3002</p> <p>\u641c\u7d22\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u89e3\u51b3\u65b9\u6848\u3002</p> <ul> <li>\u5168\u6392\u5217\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff0c\u6c42\u51fa\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u7ec4\u5408\u3002</li> <li>\u5b50\u96c6\u548c\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\u548c\u4e00\u4e2a\u76ee\u6807\u548c\uff0c\u627e\u5230\u96c6\u5408\u4e2d\u6240\u6709\u548c\u4e3a\u76ee\u6807\u548c\u7684\u5b50\u96c6\u3002</li> <li>\u6c49\u8bfa\u5854\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e09\u6839\u67f1\u5b50\u548c\u4e00\u7cfb\u5217\u5927\u5c0f\u4e0d\u540c\u7684\u5706\u76d8\uff0c\u8981\u6c42\u5c06\u6240\u6709\u5706\u76d8\u4ece\u4e00\u6839\u67f1\u5b50\u79fb\u52a8\u5230\u53e6\u4e00\u6839\u67f1\u5b50\uff0c\u6bcf\u6b21\u53ea\u80fd\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\uff0c\u4e14\u4e0d\u80fd\u5c06\u5927\u5706\u76d8\u653e\u5728\u5c0f\u5706\u76d8\u4e0a\u3002</li> </ul> <p>\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u6240\u6709\u7ea6\u675f\u6761\u4ef6\u7684\u89e3\u3002</p> <ul> <li>\\(n\\) \u7687\u540e\uff1a\u5728 \\(n \\times n\\) \u7684\u68cb\u76d8\u4e0a\u653e\u7f6e \\(n\\) \u4e2a\u7687\u540e\uff0c\u4f7f\u5f97\u5b83\u4eec\u4e92\u4e0d\u653b\u51fb\u3002</li> <li>\u6570\u72ec\uff1a\u5728 \\(9 \\times 9\\) \u7684\u7f51\u683c\u4e2d\u586b\u5165\u6570\u5b57 \\(1\\) ~ \\(9\\) \uff0c\u4f7f\u5f97\u6bcf\u884c\u3001\u6bcf\u5217\u548c\u6bcf\u4e2a \\(3 \\times 3\\) \u5b50\u7f51\u683c\u4e2d\u7684\u6570\u5b57\u4e0d\u91cd\u590d\u3002</li> <li>\u56fe\u7740\u8272\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u7528\u6700\u5c11\u7684\u989c\u8272\u7ed9\u56fe\u7684\u6bcf\u4e2a\u9876\u70b9\u7740\u8272\uff0c\u4f7f\u5f97\u76f8\u90bb\u9876\u70b9\u989c\u8272\u4e0d\u540c\u3002</li> </ul> <p>\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u4e00\u4e2a\u7ec4\u5408\u7a7a\u95f4\u4e2d\u627e\u5230\u6ee1\u8db3\u67d0\u4e9b\u6761\u4ef6\u7684\u6700\u4f18\u89e3\u3002</p> <ul> <li>0-1 \u80cc\u5305\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u7269\u54c1\u548c\u4e00\u4e2a\u80cc\u5305\uff0c\u6bcf\u4e2a\u7269\u54c1\u6709\u4e00\u5b9a\u7684\u4ef7\u503c\u548c\u91cd\u91cf\uff0c\u8981\u6c42\u5728\u80cc\u5305\u5bb9\u91cf\u9650\u5236\u5185\uff0c\u9009\u62e9\u7269\u54c1\u4f7f\u5f97\u603b\u4ef7\u503c\u6700\u5927\u3002</li> <li>\u65c5\u884c\u5546\u95ee\u9898\uff1a\u5728\u4e00\u4e2a\u56fe\u4e2d\uff0c\u4ece\u4e00\u4e2a\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u6240\u6709\u5176\u4ed6\u70b9\u6070\u597d\u4e00\u6b21\u540e\u8fd4\u56de\u8d77\u70b9\uff0c\u6c42\u6700\u77ed\u8def\u5f84\u3002</li> <li>\u6700\u5927\u56e2\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u627e\u5230\u6700\u5927\u7684\u5b8c\u5168\u5b50\u56fe\uff0c\u5373\u5b50\u56fe\u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u90fd\u6709\u8fb9\u76f8\u8fde\u3002</li> </ul> <p>\u8bf7\u6ce8\u610f\uff0c\u5bf9\u4e8e\u8bb8\u591a\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff0c\u56de\u6eaf\u4e0d\u662f\u6700\u4f18\u89e3\u51b3\u65b9\u6848\u3002</p> <ul> <li>0-1 \u80cc\u5305\u95ee\u9898\u901a\u5e38\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u89e3\u51b3\uff0c\u4ee5\u8fbe\u5230\u66f4\u9ad8\u7684\u65f6\u95f4\u6548\u7387\u3002</li> <li>\u65c5\u884c\u5546\u662f\u4e00\u4e2a\u8457\u540d\u7684 NP-Hard \u95ee\u9898\uff0c\u5e38\u7528\u89e3\u6cd5\u6709\u9057\u4f20\u7b97\u6cd5\u548c\u8681\u7fa4\u7b97\u6cd5\u7b49\u3002</li> <li>\u6700\u5927\u56e2\u95ee\u9898\u662f\u56fe\u8bba\u4e2d\u7684\u4e00\u4e2a\u7ecf\u5178\u95ee\u9898\uff0c\u53ef\u7528\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u542f\u53d1\u5f0f\u7b97\u6cd5\u6765\u89e3\u51b3\u3002</li> </ul>"},{"location":"chapter_backtracking/n_queens_problem/","title":"13.4 \u00a0 n \u7687\u540e\u95ee\u9898","text":"<p>Question</p> <p>\u6839\u636e\u56fd\u9645\u8c61\u68cb\u7684\u89c4\u5219\uff0c\u7687\u540e\u53ef\u4ee5\u653b\u51fb\u4e0e\u540c\u5904\u4e00\u884c\u3001\u4e00\u5217\u6216\u4e00\u6761\u659c\u7ebf\u4e0a\u7684\u68cb\u5b50\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u7687\u540e\u548c\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5bfb\u627e\u4f7f\u5f97\u6240\u6709\u7687\u540e\u4e4b\u95f4\u65e0\u6cd5\u76f8\u4e92\u653b\u51fb\u7684\u6446\u653e\u65b9\u6848\u3002</p> <p>\u5982\u56fe 13-15 \u6240\u793a\uff0c\u5f53 \\(n = 4\\) \u65f6\uff0c\u5171\u53ef\u4ee5\u627e\u5230\u4e24\u4e2a\u89e3\u3002\u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\u5171\u6709 \\(n^2\\) \u4e2a\u683c\u5b50\uff0c\u7ed9\u51fa\u4e86\u6240\u6709\u7684\u9009\u62e9 <code>choices</code> \u3002\u5728\u9010\u4e2a\u653e\u7f6e\u7687\u540e\u7684\u8fc7\u7a0b\u4e2d\uff0c\u68cb\u76d8\u72b6\u6001\u5728\u4e0d\u65ad\u5730\u53d8\u5316\uff0c\u6bcf\u4e2a\u65f6\u523b\u7684\u68cb\u76d8\u5c31\u662f\u72b6\u6001 <code>state</code> \u3002</p> <p></p> <p> \u56fe 13-15 \u00a0 4 \u7687\u540e\u95ee\u9898\u7684\u89e3 </p> <p>\u56fe 13-16 \u5c55\u793a\u4e86\u672c\u9898\u7684\u4e09\u4e2a\u7ea6\u675f\u6761\u4ef6\uff1a\u591a\u4e2a\u7687\u540e\u4e0d\u80fd\u5728\u540c\u4e00\u884c\u3001\u540c\u4e00\u5217\u3001\u540c\u4e00\u6761\u5bf9\u89d2\u7ebf\u4e0a\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5bf9\u89d2\u7ebf\u5206\u4e3a\u4e3b\u5bf9\u89d2\u7ebf <code>\\</code> \u548c\u6b21\u5bf9\u89d2\u7ebf <code>/</code> \u4e24\u79cd\u3002</p> <p></p> <p> \u56fe 13-16 \u00a0 n \u7687\u540e\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6 </p>"},{"location":"chapter_backtracking/n_queens_problem/#1","title":"1. \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565","text":"<p>\u7687\u540e\u7684\u6570\u91cf\u548c\u68cb\u76d8\u7684\u884c\u6570\u90fd\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u6211\u4eec\u5bb9\u6613\u5f97\u5230\u4e00\u4e2a\u63a8\u8bba\uff1a\u68cb\u76d8\u6bcf\u884c\u90fd\u5141\u8bb8\u4e14\u53ea\u5141\u8bb8\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002</p> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u53d6\u9010\u884c\u653e\u7f6e\u7b56\u7565\uff1a\u4ece\u7b2c\u4e00\u884c\u5f00\u59cb\uff0c\u5728\u6bcf\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\uff0c\u76f4\u81f3\u6700\u540e\u4e00\u884c\u7ed3\u675f\u3002</p> <p>\u56fe 13-17 \u6240\u793a\u4e3a \\(4\\) \u7687\u540e\u95ee\u9898\u7684\u9010\u884c\u653e\u7f6e\u8fc7\u7a0b\u3002\u53d7\u753b\u5e45\u9650\u5236\uff0c\u56fe 13-17 \u4ec5\u5c55\u5f00\u4e86\u7b2c\u4e00\u884c\u7684\u5176\u4e2d\u4e00\u4e2a\u641c\u7d22\u5206\u652f\uff0c\u5e76\u4e14\u5c06\u4e0d\u6ee1\u8db3\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u65b9\u6848\u90fd\u8fdb\u884c\u4e86\u526a\u679d\u3002</p> <p></p> <p> \u56fe 13-17 \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565 </p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9010\u884c\u653e\u7f6e\u7b56\u7565\u8d77\u5230\u4e86\u526a\u679d\u7684\u4f5c\u7528\uff0c\u5b83\u907f\u514d\u4e86\u540c\u4e00\u884c\u51fa\u73b0\u591a\u4e2a\u7687\u540e\u7684\u6240\u6709\u641c\u7d22\u5206\u652f\u3002</p>"},{"location":"chapter_backtracking/n_queens_problem/#2","title":"2. \u00a0 \u5217\u4e0e\u5bf9\u89d2\u7ebf\u526a\u679d","text":"<p>\u4e3a\u4e86\u6ee1\u8db3\u5217\u7ea6\u675f\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5e03\u5c14\u578b\u6570\u7ec4 <code>cols</code> \u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\u3002\u5728\u6bcf\u6b21\u51b3\u5b9a\u653e\u7f6e\u524d\uff0c\u6211\u4eec\u901a\u8fc7 <code>cols</code> \u5c06\u5df2\u6709\u7687\u540e\u7684\u5217\u8fdb\u884c\u526a\u679d\uff0c\u5e76\u5728\u56de\u6eaf\u4e2d\u52a8\u6001\u66f4\u65b0 <code>cols</code> \u7684\u72b6\u6001\u3002</p> <p>\u90a3\u4e48\uff0c\u5982\u4f55\u5904\u7406\u5bf9\u89d2\u7ebf\u7ea6\u675f\u5462\uff1f\u8bbe\u68cb\u76d8\u4e2d\u67d0\u4e2a\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u4e3a \\((row, col)\\) \uff0c\u9009\u5b9a\u77e9\u9635\u4e2d\u7684\u67d0\u6761\u4e3b\u5bf9\u89d2\u7ebf\uff0c\u6211\u4eec\u53d1\u73b0\u8be5\u5bf9\u89d2\u7ebf\u4e0a\u6240\u6709\u683c\u5b50\u7684\u884c\u7d22\u5f15\u51cf\u5217\u7d22\u5f15\u90fd\u76f8\u7b49\uff0c\u5373\u5bf9\u89d2\u7ebf\u4e0a\u6240\u6709\u683c\u5b50\u7684 \\(row - col\\) \u4e3a\u6052\u5b9a\u503c\u3002</p> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u4e24\u4e2a\u683c\u5b50\u6ee1\u8db3 \\(row_1 - col_1 = row_2 - col_2\\) \uff0c\u5219\u5b83\u4eec\u4e00\u5b9a\u5904\u5728\u540c\u4e00\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u3002\u5229\u7528\u8be5\u89c4\u5f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u501f\u52a9\u56fe 13-18 \u6240\u793a\u7684\u6570\u7ec4 <code>diags1</code> \u8bb0\u5f55\u6bcf\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\u3002</p> <p>\u540c\u7406\uff0c\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u7684\u6240\u6709\u683c\u5b50\u7684 \\(row + col\\) \u662f\u6052\u5b9a\u503c\u3002\u6211\u4eec\u540c\u6837\u4e5f\u53ef\u4ee5\u501f\u52a9\u6570\u7ec4 <code>diags2</code> \u6765\u5904\u7406\u6b21\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002</p> <p></p> <p> \u56fe 13-18 \u00a0 \u5904\u7406\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f </p>"},{"location":"chapter_backtracking/n_queens_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u8bf7\u6ce8\u610f\uff0c\\(n\\) \u7ef4\u65b9\u9635\u4e2d \\(row - col\\) \u7684\u8303\u56f4\u662f \\([-n + 1, n - 1]\\) \uff0c\\(row + col\\) \u7684\u8303\u56f4\u662f \\([0, 2n - 2]\\) \uff0c\u6240\u4ee5\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\u7684\u6570\u91cf\u90fd\u4e3a \\(2n - 1\\) \uff0c\u5373\u6570\u7ec4 <code>diags1</code> \u548c <code>diags2</code> \u7684\u957f\u5ea6\u90fd\u4e3a \\(2n - 1\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig n_queens.py<pre><code>def backtrack(\n    row: int,\n    n: int,\n    state: list[list[str]],\n    res: list[list[list[str]]],\n    cols: list[bool],\n    diags1: list[bool],\n    diags2: list[bool],\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e\"\"\"\n    # \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if row == n:\n        res.append([list(row) for row in state])\n        return\n    # \u904d\u5386\u6240\u6709\u5217\n    for col in range(n):\n        # \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        diag1 = row - col + n - 1\n        diag2 = row + col\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if not cols[col] and not diags1[diag1] and not diags2[diag2]:\n            # \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = \"Q\"\n            cols[col] = diags1[diag1] = diags2[diag2] = True\n            # \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            # \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = \"#\"\n            cols[col] = diags1[diag1] = diags2[diag2] = False\n\ndef n_queens(n: int) -&gt; list[list[list[str]]]:\n    \"\"\"\u6c42\u89e3 n \u7687\u540e\"\"\"\n    # \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    state = [[\"#\" for _ in range(n)] for _ in range(n)]\n    cols = [False] * n  # \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    diags1 = [False] * (2 * n - 1)  # \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    diags2 = [False] * (2 * n - 1)  # \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    res = []\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n</code></pre> n_queens.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(int row, int n, vector&lt;vector&lt;string&gt;&gt; &amp;state, vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; &amp;res, vector&lt;bool&gt; &amp;cols,\n               vector&lt;bool&gt; &amp;diags1, vector&lt;bool&gt; &amp;diags2) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row == n) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nvector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; nQueens(int n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    vector&lt;vector&lt;string&gt;&gt; state(n, vector&lt;string&gt;(n, \"#\"));\n    vector&lt;bool&gt; cols(n, false);           // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    vector&lt;bool&gt; diags1(2 * n - 1, false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    vector&lt;bool&gt; diags2(2 * n - 1, false); // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; res;\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(int row, int n, List&lt;List&lt;String&gt;&gt; state, List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n        boolean[] cols, boolean[] diags1, boolean[] diags2) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row == n) {\n        List&lt;List&lt;String&gt;&gt; copyState = new ArrayList&lt;&gt;();\n        for (List&lt;String&gt; sRow : state) {\n            copyState.add(new ArrayList&lt;&gt;(sRow));\n        }\n        res.add(copyState);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state.get(row).set(col, \"Q\");\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state.get(row).set(col, \"#\");\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    List&lt;List&lt;String&gt;&gt; state = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;String&gt; row = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            row.add(\"#\");\n        }\n        state.add(row);\n    }\n    boolean[] cols = new boolean[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    boolean[] diags1 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    boolean[] diags2 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = new ArrayList&lt;&gt;();\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid Backtrack(int row, int n, List&lt;List&lt;string&gt;&gt; state, List&lt;List&lt;List&lt;string&gt;&gt;&gt; res,\n        bool[] cols, bool[] diags1, bool[] diags2) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row == n) {\n        List&lt;List&lt;string&gt;&gt; copyState = [];\n        foreach (List&lt;string&gt; sRow in state) {\n            copyState.Add(new List&lt;string&gt;(sRow));\n        }\n        res.Add(copyState);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            Backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nList&lt;List&lt;List&lt;string&gt;&gt;&gt; NQueens(int n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    List&lt;List&lt;string&gt;&gt; state = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;string&gt; row = [];\n        for (int j = 0; j &lt; n; j++) {\n            row.Add(\"#\");\n        }\n        state.Add(row);\n    }\n    bool[] cols = new bool[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    bool[] diags1 = new bool[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    bool[] diags2 = new bool[2 * n - 1]; // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    List&lt;List&lt;List&lt;string&gt;&gt;&gt; res = [];\n\n    Backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if row == n {\n        newState := make([][]string, len(*state))\n        for i, _ := range newState {\n            newState[i] = make([]string, len((*state)[0]))\n            copy(newState[i], (*state)[i])\n\n        }\n        *res = append(*res, newState)\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for col := 0; col &lt; n; col++ {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        diag1 := row - col + n - 1\n        diag2 := row + col\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if !(*cols)[col] &amp;&amp; !(*diags1)[diag1] &amp;&amp; !(*diags2)[diag2] {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            (*state)[row][col] = \"Q\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row+1, n, state, res, cols, diags1, diags2)\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            (*state)[row][col] = \"#\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunc nQueens(n int) [][][]string {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    state := make([][]string, n)\n    for i := 0; i &lt; n; i++ {\n        row := make([]string, n)\n        for i := 0; i &lt; n; i++ {\n            row[i] = \"#\"\n        }\n        state[i] = row\n    }\n    // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    cols := make([]bool, n)\n    diags1 := make([]bool, 2*n-1)\n    diags2 := make([]bool, 2*n-1)\n    res := make([][][]string, 0)\n    backtrack(0, n, &amp;state, &amp;res, &amp;cols, &amp;diags1, &amp;diags2)\n    return res\n}\n</code></pre> n_queens.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nfunc backtrack(row: Int, n: Int, state: inout [[String]], res: inout [[[String]]], cols: inout [Bool], diags1: inout [Bool], diags2: inout [Bool]) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if row == n {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for col in 0 ..&lt; n {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        let diag1 = row - col + n - 1\n        let diag2 = row + col\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = \"Q\"\n            cols[col] = true\n            diags1[diag1] = true\n            diags2[diag2] = true\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row: row + 1, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = \"#\"\n            cols[col] = false\n            diags1[diag1] = false\n            diags2[diag2] = false\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunc nQueens(n: Int) -&gt; [[[String]]] {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    var state = Array(repeating: Array(repeating: \"#\", count: n), count: n)\n    var cols = Array(repeating: false, count: n) // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    var diags1 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    var diags2 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    var res: [[[String]]] = []\n\n    backtrack(row: 0, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n\n    return res\n}\n</code></pre> n_queens.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nfunction backtrack(row, n, state, res, cols, diags1, diags2) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (let col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunction nQueens(n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    const diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const res = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nfunction backtrack(\n    row: number,\n    n: number,\n    state: string[][],\n    res: string[][][],\n    cols: boolean[],\n    diags1: boolean[],\n    diags2: boolean[]\n): void {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (let col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfunction nQueens(n: number): string[][][] {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    const diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const res: string[][][] = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(\n  int row,\n  int n,\n  List&lt;List&lt;String&gt;&gt; state,\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n  List&lt;bool&gt; cols,\n  List&lt;bool&gt; diags1,\n  List&lt;bool&gt; diags2,\n) {\n  // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (row == n) {\n    List&lt;List&lt;String&gt;&gt; copyState = [];\n    for (List&lt;String&gt; sRow in state) {\n      copyState.add(List.from(sRow));\n    }\n    res.add(copyState);\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u5217\n  for (int col = 0; col &lt; n; col++) {\n    // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n    int diag1 = row - col + n - 1;\n    int diag2 = row + col;\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n    if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n      // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n      state[row][col] = \"Q\";\n      cols[col] = true;\n      diags1[diag1] = true;\n      diags2[diag2] = true;\n      // \u653e\u7f6e\u4e0b\u4e00\u884c\n      backtrack(row + 1, n, state, res, cols, diags1, diags2);\n      // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n      state[row][col] = \"#\";\n      cols[col] = false;\n      diags1[diag1] = false;\n      diags2[diag2] = false;\n    }\n  }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n  // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n  List&lt;List&lt;String&gt;&gt; state = List.generate(n, (index) =&gt; List.filled(n, \"#\"));\n  List&lt;bool&gt; cols = List.filled(n, false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n  List&lt;bool&gt; diags1 = List.filled(2 * n - 1, false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n  List&lt;bool&gt; diags2 = List.filled(2 * n - 1, false); // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = [];\n\n  backtrack(0, n, state, res, cols, diags1, diags2);\n\n  return res;\n}\n</code></pre> n_queens.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nfn backtrack(\n    row: usize,\n    n: usize,\n    state: &amp;mut Vec&lt;Vec&lt;String&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt;,\n    cols: &amp;mut [bool],\n    diags1: &amp;mut [bool],\n    diags2: &amp;mut [bool],\n) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if row == n {\n        let mut copy_state: Vec&lt;Vec&lt;String&gt;&gt; = Vec::new();\n        for s_row in state.clone() {\n            copy_state.push(s_row);\n        }\n        res.push(copy_state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for col in 0..n {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        let diag1 = row + n - 1 - col;\n        let diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state.get_mut(row).unwrap()[col] = \"Q\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (true, true, true);\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state.get_mut(row).unwrap()[col] = \"#\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (false, false, false);\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfn n_queens(n: usize) -&gt; Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    let mut state: Vec&lt;Vec&lt;String&gt;&gt; = Vec::new();\n    for _ in 0..n {\n        let mut row: Vec&lt;String&gt; = Vec::new();\n        for _ in 0..n {\n            row.push(\"#\".into());\n        }\n        state.push(row);\n    }\n    let mut cols = vec![false; n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    let mut diags1 = vec![false; 2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    let mut diags2 = vec![false; 2 * n - 1]; // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    let mut res: Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; = Vec::new();\n\n    backtrack(\n        0,\n        n,\n        &amp;mut state,\n        &amp;mut res,\n        &amp;mut cols,\n        &amp;mut diags1,\n        &amp;mut diags2,\n    );\n\n    res\n}\n</code></pre> n_queens.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nvoid backtrack(int row, int n, char state[MAX_SIZE][MAX_SIZE], char ***res, int *resSize, bool cols[MAX_SIZE],\n               bool diags1[2 * MAX_SIZE - 1], bool diags2[2 * MAX_SIZE - 1]) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row == n) {\n        res[*resSize] = (char **)malloc(sizeof(char *) * n);\n        for (int i = 0; i &lt; n; ++i) {\n            res[*resSize][i] = (char *)malloc(sizeof(char) * (n + 1));\n            strcpy(res[*resSize][i], state[i]);\n        }\n        (*resSize)++;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, resSize, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nchar ***nQueens(int n, int *returnSize) {\n    char state[MAX_SIZE][MAX_SIZE];\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    for (int i = 0; i &lt; n; ++i) {\n        for (int j = 0; j &lt; n; ++j) {\n            state[i][j] = '#';\n        }\n        state[i][n] = '\\0';\n    }\n    bool cols[MAX_SIZE] = {false};           // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    bool diags1[2 * MAX_SIZE - 1] = {false}; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    bool diags2[2 * MAX_SIZE - 1] = {false}; // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n\n    char ***res = (char ***)malloc(sizeof(char **) * MAX_SIZE);\n    *returnSize = 0;\n    backtrack(0, n, state, res, returnSize, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.kt<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1an \u7687\u540e */\nfun backtrack(\n    row: Int,\n    n: Int,\n    state: List&lt;MutableList&lt;String&gt;&gt;,\n    res: MutableList&lt;List&lt;List&lt;String&gt;&gt;?&gt;,\n    cols: BooleanArray,\n    diags1: BooleanArray,\n    diags2: BooleanArray\n) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row == n) {\n        val copyState: MutableList&lt;List&lt;String&gt;&gt; = ArrayList()\n        for (sRow in state) {\n            copyState.add(ArrayList(sRow))\n        }\n        res.add(copyState)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (col in 0..&lt;n) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\n        val diag1 = row - col + n - 1\n        val diag2 = row + col\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = \"Q\"\n            diags2[diag2] = true\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = \"#\"\n            diags2[diag2] = false\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n        }\n    }\n}\n\n/* \u6c42\u89e3 n \u7687\u540e */\nfun nQueens(n: Int): List&lt;List&lt;List&lt;String&gt;&gt;?&gt; {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    val state: MutableList&lt;MutableList&lt;String&gt;&gt; = ArrayList()\n    for (i in 0..&lt;n) {\n        val row: MutableList&lt;String&gt; = ArrayList()\n        for (j in 0..&lt;n) {\n            row.add(\"#\")\n        }\n        state.add(row)\n    }\n    val cols = BooleanArray(n) // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    val diags1 = BooleanArray(2 * n - 1) // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    val diags2 = BooleanArray(2 * n - 1) // \u8bb0\u5f55\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    val res: MutableList&lt;List&lt;List&lt;String&gt;&gt;?&gt; = ArrayList()\n\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n}\n</code></pre> n_queens.rb<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{n_queens}\n</code></pre> n_queens.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{nQueens}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u9010\u884c\u653e\u7f6e \\(n\\) \u6b21\uff0c\u8003\u8651\u5217\u7ea6\u675f\uff0c\u5219\u4ece\u7b2c\u4e00\u884c\u5230\u6700\u540e\u4e00\u884c\u5206\u522b\u6709 \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u4e2a\u9009\u62e9\uff0c\u4f7f\u7528 \\(O(n!)\\) \u65f6\u95f4\u3002\u5f53\u8bb0\u5f55\u89e3\u65f6\uff0c\u9700\u8981\u590d\u5236\u77e9\u9635 <code>state</code> \u5e76\u6dfb\u52a0\u8fdb <code>res</code> \uff0c\u590d\u5236\u64cd\u4f5c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002\u56e0\u6b64\uff0c\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n! \\cdot n^2)\\) \u3002\u5b9e\u9645\u4e0a\uff0c\u6839\u636e\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u526a\u679d\u4e5f\u80fd\u591f\u5927\u5e45\u7f29\u5c0f\u641c\u7d22\u7a7a\u95f4\uff0c\u56e0\u800c\u641c\u7d22\u6548\u7387\u5f80\u5f80\u4f18\u4e8e\u4ee5\u4e0a\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <p>\u6570\u7ec4 <code>state</code> \u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\uff0c\u6570\u7ec4 <code>cols</code>\u3001<code>diags1</code> \u548c <code>diags2</code> \u7686\u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/","title":"13.2 \u00a0 \u5168\u6392\u5217\u95ee\u9898","text":"<p>\u5168\u6392\u5217\u95ee\u9898\u662f\u56de\u6eaf\u7b97\u6cd5\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u7684\u5b9a\u4e49\u662f\u5728\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff08\u5982\u4e00\u4e2a\u6570\u7ec4\u6216\u5b57\u7b26\u4e32\uff09\u7684\u60c5\u51b5\u4e0b\uff0c\u627e\u51fa\u5176\u4e2d\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002</p> <p>\u8868 13-2 \u5217\u4e3e\u4e86\u51e0\u4e2a\u793a\u4f8b\u6570\u636e\uff0c\u5305\u62ec\u8f93\u5165\u6570\u7ec4\u548c\u5bf9\u5e94\u7684\u6240\u6709\u6392\u5217\u3002</p> <p> \u8868 13-2 \u00a0 \u5168\u6392\u5217\u793a\u4f8b </p> \u8f93\u5165\u6570\u7ec4 \u6240\u6709\u6392\u5217 \\([1]\\) \\([1]\\) \\([1, 2]\\) \\([1, 2], [2, 1]\\) \\([1, 2, 3]\\) \\([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\\)"},{"location":"chapter_backtracking/permutations_problem/#1321","title":"13.2.1 \u00a0 \u65e0\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u5176\u4e2d\u4e0d\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002</p> <p>\u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u751f\u6210\u6392\u5217\u7684\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\u3002\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 2, 3]\\) \uff0c\u5982\u679c\u6211\u4eec\u5148\u9009\u62e9 \\(1\\) \uff0c\u518d\u9009\u62e9 \\(3\\) \uff0c\u6700\u540e\u9009\u62e9 \\(2\\) \uff0c\u5219\u83b7\u5f97\u6392\u5217 \\([1, 3, 2]\\) \u3002\u56de\u9000\u8868\u793a\u64a4\u9500\u4e00\u4e2a\u9009\u62e9\uff0c\u4e4b\u540e\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002</p> <p>\u4ece\u56de\u6eaf\u4ee3\u7801\u7684\u89d2\u5ea6\u770b\uff0c\u5019\u9009\u96c6\u5408 <code>choices</code> \u662f\u8f93\u5165\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u72b6\u6001 <code>state</code> \u662f\u76f4\u81f3\u76ee\u524d\u5df2\u88ab\u9009\u62e9\u7684\u5143\u7d20\u3002\u8bf7\u6ce8\u610f\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u5141\u8bb8\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u56e0\u6b64 <code>state</code> \u4e2d\u7684\u6240\u6709\u5143\u7d20\u90fd\u5e94\u8be5\u662f\u552f\u4e00\u7684\u3002</p> <p>\u5982\u56fe 13-5 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u641c\u7d22\u8fc7\u7a0b\u5c55\u5f00\u6210\u4e00\u68f5\u9012\u5f52\u6811\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u5f53\u524d\u72b6\u6001 <code>state</code> \u3002\u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u7ecf\u8fc7\u4e09\u8f6e\u9009\u62e9\u540e\u5230\u8fbe\u53f6\u8282\u70b9\uff0c\u6bcf\u4e2a\u53f6\u8282\u70b9\u90fd\u5bf9\u5e94\u4e00\u4e2a\u6392\u5217\u3002</p> <p></p> <p> \u56fe 13-5 \u00a0 \u5168\u6392\u5217\u7684\u9012\u5f52\u6811 </p>"},{"location":"chapter_backtracking/permutations_problem/#1","title":"1. \u00a0 \u91cd\u590d\u9009\u62e9\u526a\u679d","text":"<p>\u4e3a\u4e86\u5b9e\u73b0\u6bcf\u4e2a\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u6211\u4eec\u8003\u8651\u5f15\u5165\u4e00\u4e2a\u5e03\u5c14\u578b\u6570\u7ec4 <code>selected</code> \uff0c\u5176\u4e2d <code>selected[i]</code> \u8868\u793a <code>choices[i]</code> \u662f\u5426\u5df2\u88ab\u9009\u62e9\uff0c\u5e76\u57fa\u4e8e\u5b83\u5b9e\u73b0\u4ee5\u4e0b\u526a\u679d\u64cd\u4f5c\u3002</p> <ul> <li>\u5728\u505a\u51fa\u9009\u62e9 <code>choice[i]</code> \u540e\uff0c\u6211\u4eec\u5c31\u5c06 <code>selected[i]</code> \u8d4b\u503c\u4e3a \\(\\text{True}\\) \uff0c\u4ee3\u8868\u5b83\u5df2\u88ab\u9009\u62e9\u3002</li> <li>\u904d\u5386\u9009\u62e9\u5217\u8868 <code>choices</code> \u65f6\uff0c\u8df3\u8fc7\u6240\u6709\u5df2\u88ab\u9009\u62e9\u7684\u8282\u70b9\uff0c\u5373\u526a\u679d\u3002</li> </ul> <p>\u5982\u56fe 13-6 \u6240\u793a\uff0c\u5047\u8bbe\u6211\u4eec\u7b2c\u4e00\u8f6e\u9009\u62e9 1 \uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9 3 \uff0c\u7b2c\u4e09\u8f6e\u9009\u62e9 2 \uff0c\u5219\u9700\u8981\u5728\u7b2c\u4e8c\u8f6e\u526a\u6389\u5143\u7d20 1 \u7684\u5206\u652f\uff0c\u5728\u7b2c\u4e09\u8f6e\u526a\u6389\u5143\u7d20 1 \u548c\u5143\u7d20 3 \u7684\u5206\u652f\u3002</p> <p></p> <p> \u56fe 13-6 \u00a0 \u5168\u6392\u5217\u526a\u679d\u793a\u4f8b </p> <p>\u89c2\u5bdf\u56fe 13-6 \u53d1\u73b0\uff0c\u8be5\u526a\u679d\u64cd\u4f5c\u5c06\u641c\u7d22\u7a7a\u95f4\u5927\u5c0f\u4ece \\(O(n^n)\\) \u51cf\u5c0f\u81f3 \\(O(n!)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u60f3\u6e05\u695a\u4ee5\u4e0a\u4fe1\u606f\u4e4b\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5728\u6846\u67b6\u4ee3\u7801\u4e2d\u505a\u201c\u5b8c\u5f62\u586b\u7a7a\u201d\u4e86\u3002\u4e3a\u4e86\u7f29\u77ed\u6574\u4f53\u4ee3\u7801\uff0c\u6211\u4eec\u4e0d\u5355\u72ec\u5b9e\u73b0\u6846\u67b6\u4ee3\u7801\u4e2d\u7684\u5404\u4e2a\u51fd\u6570\uff0c\u800c\u662f\u5c06\u5b83\u4eec\u5c55\u5f00\u5728 <code>backtrack()</code> \u51fd\u6570\u4e2d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig permutations_i.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I\"\"\"\n    # \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i, choice in enumerate(choices):\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if not selected[i]:\n            # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = True\n            state.append(choice)\n            # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res)\n            # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = False\n            state.pop()\n\ndef permutations_i(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"\u5168\u6392\u5217 I\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_i.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.push_back(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nvector&lt;vector&lt;int&gt;&gt; permutationsI(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_i.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.add(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;Integer&gt;&gt; permutationsI(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_i.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.Add(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            Backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;int&gt;&gt; PermutationsI(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_i.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrackI(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if !(*selected)[i] {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrackI(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackI(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_i.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (i, choice) in choices.enumerated() {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if !selected[i] {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true\n            state.append(choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_i.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(state, choices, selected, res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n    if (!selected[i]) {\n      // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n      selected[i] = true;\n      state.add(choice);\n      // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n      backtrack(state, choices, selected, res);\n      // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;int&gt;&gt; permutationsI(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_i.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if !selected[i] {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state.clone(), choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.remove(state.len() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfn permutations_i(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_i.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state[stateSize] = choice;\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nint **permutationsI(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_i.kt<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;List&lt;Int&gt;?&gt;\n) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size == choices.size) {\n        res.add(ArrayList(state))\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true\n            state.add(choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfun permutationsI(nums: IntArray): List&lt;List&lt;Int&gt;?&gt; {\n    val res: MutableList&lt;List&lt;Int&gt;?&gt; = ArrayList()\n    backtrack(ArrayList(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_i.rb<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutations_i}\n</code></pre> permutations_i.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutationsI}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_backtracking/permutations_problem/#1322","title":"13.2.2 \u00a0 \u8003\u8651\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u6570\u7ec4\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u4e0d\u91cd\u590d\u7684\u6392\u5217\u3002</p> <p>\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 1, 2]\\) \u3002\u4e3a\u4e86\u65b9\u4fbf\u533a\u5206\u4e24\u4e2a\u91cd\u590d\u5143\u7d20 \\(1\\) \uff0c\u6211\u4eec\u5c06\u7b2c\u4e8c\u4e2a \\(1\\) \u8bb0\u4e3a \\(\\hat{1}\\) \u3002</p> <p>\u5982\u56fe 13-7 \u6240\u793a\uff0c\u4e0a\u8ff0\u65b9\u6cd5\u751f\u6210\u7684\u6392\u5217\u6709\u4e00\u534a\u662f\u91cd\u590d\u7684\u3002</p> <p></p> <p> \u56fe 13-7 \u00a0 \u91cd\u590d\u6392\u5217 </p> <p>\u90a3\u4e48\u5982\u4f55\u53bb\u9664\u91cd\u590d\u7684\u6392\u5217\u5462\uff1f\u6700\u76f4\u63a5\u5730\uff0c\u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u76f4\u63a5\u5bf9\u6392\u5217\u7ed3\u679c\u8fdb\u884c\u53bb\u91cd\u3002\u7136\u800c\u8fd9\u6837\u505a\u4e0d\u591f\u4f18\u96c5\uff0c\u56e0\u4e3a\u751f\u6210\u91cd\u590d\u6392\u5217\u7684\u641c\u7d22\u5206\u652f\u6ca1\u6709\u5fc5\u8981\uff0c\u5e94\u5f53\u63d0\u524d\u8bc6\u522b\u5e76\u526a\u679d\uff0c\u8fd9\u6837\u53ef\u4ee5\u8fdb\u4e00\u6b65\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"<p>\u89c2\u5bdf\u56fe 13-8 \uff0c\u5728\u7b2c\u4e00\u8f6e\u4e2d\uff0c\u9009\u62e9 \\(1\\) \u6216\u9009\u62e9 \\(\\hat{1}\\) \u662f\u7b49\u4ef7\u7684\uff0c\u5728\u8fd9\u4e24\u4e2a\u9009\u62e9\u4e4b\u4e0b\u751f\u6210\u7684\u6240\u6709\u6392\u5217\u90fd\u662f\u91cd\u590d\u7684\u3002\u56e0\u6b64\u5e94\u8be5\u628a \\(\\hat{1}\\) \u526a\u679d\u3002</p> <p>\u540c\u7406\uff0c\u5728\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(2\\) \u4e4b\u540e\uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9\u4e2d\u7684 \\(1\\) \u548c \\(\\hat{1}\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5206\u652f\uff0c\u56e0\u6b64\u4e5f\u5e94\u5c06\u7b2c\u4e8c\u8f6e\u7684 \\(\\hat{1}\\) \u526a\u679d\u3002</p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u5728\u67d0\u4e00\u8f6e\u9009\u62e9\u4e2d\uff0c\u4fdd\u8bc1\u591a\u4e2a\u76f8\u7b49\u7684\u5143\u7d20\u4ec5\u88ab\u9009\u62e9\u4e00\u6b21\u3002</p> <p></p> <p> \u56fe 13-8 \u00a0 \u91cd\u590d\u6392\u5217\u526a\u679d </p>"},{"location":"chapter_backtracking/permutations_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5728\u4e0a\u4e00\u9898\u7684\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u8003\u8651\u5728\u6bcf\u4e00\u8f6e\u9009\u62e9\u4e2d\u5f00\u542f\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>duplicated</code> \uff0c\u7528\u4e8e\u8bb0\u5f55\u8be5\u8f6e\u4e2d\u5df2\u7ecf\u5c1d\u8bd5\u8fc7\u7684\u5143\u7d20\uff0c\u5e76\u5c06\u91cd\u590d\u5143\u7d20\u526a\u679d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig permutations_ii.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II\"\"\"\n    # \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    duplicated = set[int]()\n    for i, choice in enumerate(choices):\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if not selected[i] and choice not in duplicated:\n            # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.add(choice)  # \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = True\n            state.append(choice)\n            # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res)\n            # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = False\n            state.pop()\n\ndef permutations_ii(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"\u5168\u6392\u5217 II\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_ii.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    unordered_set&lt;int&gt; duplicated;\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; duplicated.find(choice) == duplicated.end()) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.emplace(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push_back(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nvector&lt;vector&lt;int&gt;&gt; permutationsII(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_ii.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    Set&lt;Integer&gt; duplicated = new HashSet&lt;Integer&gt;();\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.add(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;Integer&gt;&gt; permutationsII(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_ii.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    HashSet&lt;int&gt; duplicated = [];\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.Contains(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.Add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.Add(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            Backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;int&gt;&gt; PermutationsII(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_ii.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrackII(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    duplicated := make(map[int]struct{}, 0)\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if _, ok := duplicated[choice]; !ok &amp;&amp; !(*selected)[i] {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            duplicated[choice] = struct{}{}\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrackI(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackII(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_ii.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    var duplicated: Set&lt;Int&gt; = []\n    for (i, choice) in choices.enumerated() {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if !selected[i], !duplicated.contains(choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.insert(choice) // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true\n            state.append(choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_ii.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(state, choices, selected, res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  Set&lt;int&gt; duplicated = {};\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n    if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n      // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n      duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n      selected[i] = true;\n      state.add(choice);\n      // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n      backtrack(state, choices, selected, res);\n      // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;int&gt;&gt; permutationsII(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_ii.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    let mut duplicated = HashSet::&lt;i32&gt;::new();\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if !selected[i] &amp;&amp; !duplicated.contains(&amp;choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.insert(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state.clone(), choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.remove(state.len() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfn permutations_ii(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new();\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_ii.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    bool duplicated[MAX_SIZE] = {false};\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated[choice]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated[choice] = true; // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state[stateSize] = choice;\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nint **permutationsII(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_ii.kt<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size == choices.size) {\n        res.add(ArrayList(state))\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    val duplicated: MutableSet&lt;Int&gt; = HashSet()\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.add(choice) // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true\n            state.add(choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfun permutationsII(nums: IntArray): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val res: MutableList&lt;MutableList&lt;Int&gt;?&gt; = ArrayList()\n    backtrack(ArrayList(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_ii.rb<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutations_ii}\n</code></pre> permutations_ii.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutationsII}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5047\u8bbe\u5143\u7d20\u4e24\u4e24\u4e4b\u95f4\u4e92\u4e0d\u76f8\u540c\uff0c\u5219 \\(n\\) \u4e2a\u5143\u7d20\u5171\u6709 \\(n!\\)  \u79cd\u6392\u5217\uff08\u9636\u4e58\uff09\uff1b\u5728\u8bb0\u5f55\u7ed3\u679c\u65f6\uff0c\u9700\u8981\u590d\u5236\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5217\u8868\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n!n)\\) \u3002</p> <p>\u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002<code>selected</code> \u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u540c\u4e00\u65f6\u523b\u6700\u591a\u5171\u6709 \\(n\\) \u4e2a <code>duplicated</code> \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\u3002\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#3","title":"3. \u00a0 \u4e24\u79cd\u526a\u679d\u5bf9\u6bd4","text":"<p>\u8bf7\u6ce8\u610f\uff0c\u867d\u7136 <code>selected</code> \u548c <code>duplicated</code> \u90fd\u7528\u4e8e\u526a\u679d\uff0c\u4f46\u4e24\u8005\u7684\u76ee\u6807\u4e0d\u540c\u3002</p> <ul> <li>\u91cd\u590d\u9009\u62e9\u526a\u679d\uff1a\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\u4e2d\u53ea\u6709\u4e00\u4e2a <code>selected</code> \u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5f53\u524d\u72b6\u6001\u4e2d\u5305\u542b\u54ea\u4e9b\u5143\u7d20\uff0c\u5176\u4f5c\u7528\u662f\u907f\u514d\u67d0\u4e2a\u5143\u7d20\u5728 <code>state</code> \u4e2d\u91cd\u590d\u51fa\u73b0\u3002</li> <li>\u76f8\u7b49\u5143\u7d20\u526a\u679d\uff1a\u6bcf\u8f6e\u9009\u62e9\uff08\u6bcf\u4e2a\u8c03\u7528\u7684 <code>backtrack</code> \u51fd\u6570\uff09\u90fd\u5305\u542b\u4e00\u4e2a <code>duplicated</code> \u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5728\u672c\u8f6e\u904d\u5386\uff08<code>for</code> \u5faa\u73af\uff09\u4e2d\u54ea\u4e9b\u5143\u7d20\u5df2\u88ab\u9009\u62e9\u8fc7\uff0c\u5176\u4f5c\u7528\u662f\u4fdd\u8bc1\u76f8\u7b49\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002</li> </ul> <p>\u56fe 13-9 \u5c55\u793a\u4e86\u4e24\u4e2a\u526a\u679d\u6761\u4ef6\u7684\u751f\u6548\u8303\u56f4\u3002\u6ce8\u610f\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u9009\u62e9\uff0c\u4ece\u6839\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\u6784\u6210\u4e00\u4e2a\u6392\u5217\u3002</p> <p></p> <p> \u56fe 13-9 \u00a0 \u4e24\u79cd\u526a\u679d\u6761\u4ef6\u7684\u4f5c\u7528\u8303\u56f4 </p>"},{"location":"chapter_backtracking/subset_sum_problem/","title":"13.3 \u00a0 \u5b50\u96c6\u548c\u95ee\u9898","text":""},{"location":"chapter_backtracking/subset_sum_problem/#1331","title":"13.3.1 \u00a0 \u65e0\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570 <code>target</code> \uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8e <code>target</code> \u3002\u7ed9\u5b9a\u6570\u7ec4\u65e0\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ef\u4ee5\u88ab\u9009\u53d6\u591a\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002</p> <p>\u4f8b\u5982\uff0c\u8f93\u5165\u96c6\u5408 \\(\\{3, 4, 5\\}\\) \u548c\u76ee\u6807\u6574\u6570 \\(9\\) \uff0c\u89e3\u4e3a \\(\\{3, 3, 3\\}, \\{4, 5\\}\\) \u3002\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u4e24\u70b9\u3002</p> <ul> <li>\u8f93\u5165\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u65e0\u9650\u6b21\u91cd\u590d\u9009\u53d6\u3002</li> <li>\u5b50\u96c6\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\uff0c\u6bd4\u5982 \\(\\{4, 5\\}\\) \u548c \\(\\{5, 4\\}\\) \u662f\u540c\u4e00\u4e2a\u5b50\u96c6\u3002</li> </ul>"},{"location":"chapter_backtracking/subset_sum_problem/#1","title":"1. \u00a0 \u53c2\u8003\u5168\u6392\u5217\u89e3\u6cd5","text":"<p>\u7c7b\u4f3c\u4e8e\u5168\u6392\u5217\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b50\u96c6\u7684\u751f\u6210\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\uff0c\u5e76\u5728\u9009\u62e9\u8fc7\u7a0b\u4e2d\u5b9e\u65f6\u66f4\u65b0\u201c\u5143\u7d20\u548c\u201d\uff0c\u5f53\u5143\u7d20\u548c\u7b49\u4e8e <code>target</code> \u65f6\uff0c\u5c31\u5c06\u5b50\u96c6\u8bb0\u5f55\u81f3\u7ed3\u679c\u5217\u8868\u3002</p> <p>\u800c\u4e0e\u5168\u6392\u5217\u95ee\u9898\u4e0d\u540c\u7684\u662f\uff0c\u672c\u9898\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u65e0\u9650\u6b21\u9009\u53d6\uff0c\u56e0\u6b64\u65e0\u987b\u501f\u52a9 <code>selected</code> \u5e03\u5c14\u5217\u8868\u6765\u8bb0\u5f55\u5143\u7d20\u662f\u5426\u5df2\u88ab\u9009\u62e9\u3002\u6211\u4eec\u53ef\u4ee5\u5bf9\u5168\u6392\u5217\u4ee3\u7801\u8fdb\u884c\u5c0f\u5e45\u4fee\u6539\uff0c\u521d\u6b65\u5f97\u5230\u89e3\u9898\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig subset_sum_i_naive.py<pre><code>def backtrack(\n    state: list[int],\n    target: int,\n    total: int,\n    choices: list[int],\n    res: list[list[int]],\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n    # \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if total == target:\n        res.append(list(state))\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i in range(len(choices)):\n        # \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if total + choices[i] &gt; target:\n            continue\n        # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.append(choices[i])\n        # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res)\n        # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop()\n\ndef subset_sum_i_naive(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09\"\"\"\n    state = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    total = 0  # \u5b50\u96c6\u548c\n    res = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res)\n    return res\n</code></pre> subset_sum_i_naive.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, int total, vector&lt;int&gt; &amp;choices, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total == target) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (size_t i = 0; i &lt; choices.size(); i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push_back(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop_back();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nvector&lt;vector&lt;int&gt;&gt; subsetSumINaive(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;       // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    int total = 0;           // \u5b50\u96c6\u548c\n    vector&lt;vector&lt;int&gt;&gt; res; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int total, int[] choices, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total == target) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choices.length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.remove(state.size() - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList&lt;List&lt;Integer&gt;&gt; subsetSumINaive(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    int total = 0; // \u5b50\u96c6\u548c\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List&lt;int&gt; state, int target, int total, int[] choices, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total == target) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choices.Length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.Add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        Backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList&lt;List&lt;int&gt;&gt; SubsetSumINaive(int[] nums, int target) {\n    List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    int total = 0; // \u5b50\u96c6\u548c\n    List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    Backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumINaive(total, target int, state, choices *[]int, res *[][]int) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == total {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i := 0; i &lt; len(*choices); i++ {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if total+(*choices)[i] &gt; target {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        *state = append(*state, (*choices)[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrackSubsetSumINaive(total+(*choices)[i], target, state, choices, res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums []int, target int) [][]int {\n    state := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    total := 0              // \u5b50\u96c6\u548c\n    res := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrackSubsetSumINaive(total, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, total: Int, choices: [Int], res: inout [[Int]]) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if total == target {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i in choices.indices {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if total + choices[i] &gt; target {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.append(choices[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state: &amp;state, target: target, total: total + choices[i], choices: choices, res: &amp;res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.removeLast()\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    let total = 0 // \u5b50\u96c6\u548c\n    var res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state: &amp;state, target: target, total: total, choices: nums, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, total, choices, res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (let i = 0; i &lt; choices.length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums, target) {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    const total = 0; // \u5b50\u96c6\u548c\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    total: number,\n    choices: number[],\n    res: number[][]\n): void {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (let i = 0; i &lt; choices.length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums: number[], target: number): number[][] {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    const total = 0; // \u5b50\u96c6\u548c\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  int total,\n  List&lt;int&gt; choices,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (total == target) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  for (int i = 0; i &lt; choices.length; i++) {\n    // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n    if (total + choices[i] &gt; target) {\n      continue;\n    }\n    // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n    state.add(choices[i]);\n    // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n    backtrack(state, target, total + choices[i], choices, res);\n    // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n    state.removeLast();\n  }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList&lt;List&lt;int&gt;&gt; subsetSumINaive(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n  int total = 0; // \u5143\u7d20\u548c\n  List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n  backtrack(state, target, total, nums, res);\n  return res;\n}\n</code></pre> subset_sum_i_naive.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(\n    mut state: Vec&lt;i32&gt;,\n    target: i32,\n    total: i32,\n    choices: &amp;[i32],\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if total == target {\n        res.push(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i in 0..choices.len() {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if total + choices[i] &gt; target {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state.clone(), target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfn subset_sum_i_naive(nums: &amp;[i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    let total = 0; // \u5b50\u96c6\u548c\n    let mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i_naive.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int total, int *choices, int choicesSize) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total == target) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choicesSize; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state[stateSize++] = choices[i];\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(target, total + choices[i], choices, choicesSize);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        stateSize--;\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nvoid subsetSumINaive(int *nums, int numsSize, int target) {\n    resSize = 0; // \u521d\u59cb\u5316\u89e3\u7684\u6570\u91cf\u4e3a0\n    backtrack(target, 0, nums, numsSize);\n}\n</code></pre> subset_sum_i_naive.kt<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    total: Int,\n    choices: IntArray,\n    res: MutableList&lt;List&lt;Int&gt;?&gt;\n) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total == target) {\n        res.add(ArrayList(state))\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (i in choices.indices) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.add(choices[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfun subsetSumINaive(nums: IntArray, target: Int): List&lt;List&lt;Int&gt;?&gt; {\n    val state: MutableList&lt;Int&gt; = ArrayList() // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    val total = 0 // \u5b50\u96c6\u548c\n    val res: MutableList&lt;List&lt;Int&gt;?&gt; = ArrayList() // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res)\n    return res\n}\n</code></pre> subset_sum_i_naive.rb<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subset_sum_i_naive}\n</code></pre> subset_sum_i_naive.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumINaive}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5411\u4ee5\u4e0a\u4ee3\u7801\u8f93\u5165\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u8f93\u51fa\u7ed3\u679c\u4e3a \\([3, 3, 3], [4, 5], [5, 4]\\) \u3002\u867d\u7136\u6210\u529f\u627e\u51fa\u4e86\u6240\u6709\u548c\u4e3a \\(9\\) \u7684\u5b50\u96c6\uff0c\u4f46\u5176\u4e2d\u5b58\u5728\u91cd\u590d\u7684\u5b50\u96c6 \\([4, 5]\\) \u548c \\([5, 4]\\) \u3002</p> <p>\u8fd9\u662f\u56e0\u4e3a\u641c\u7d22\u8fc7\u7a0b\u662f\u533a\u5206\u9009\u62e9\u987a\u5e8f\u7684\uff0c\u7136\u800c\u5b50\u96c6\u4e0d\u533a\u5206\u9009\u62e9\u987a\u5e8f\u3002\u5982\u56fe 13-10 \u6240\u793a\uff0c\u5148\u9009 \\(4\\) \u540e\u9009 \\(5\\) \u4e0e\u5148\u9009 \\(5\\) \u540e\u9009 \\(4\\) \u662f\u4e0d\u540c\u7684\u5206\u652f\uff0c\u4f46\u5bf9\u5e94\u540c\u4e00\u4e2a\u5b50\u96c6\u3002</p> <p></p> <p> \u56fe 13-10 \u00a0 \u5b50\u96c6\u641c\u7d22\u4e0e\u8d8a\u754c\u526a\u679d </p> <p>\u4e3a\u4e86\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u601d\u8def\u662f\u5bf9\u7ed3\u679c\u5217\u8868\u8fdb\u884c\u53bb\u91cd\u3002\u4f46\u8fd9\u4e2a\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\uff0c\u6709\u4e24\u65b9\u9762\u539f\u56e0\u3002</p> <ul> <li>\u5f53\u6570\u7ec4\u5143\u7d20\u8f83\u591a\uff0c\u5c24\u5176\u662f\u5f53 <code>target</code> \u8f83\u5927\u65f6\uff0c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u4ea7\u751f\u5927\u91cf\u7684\u91cd\u590d\u5b50\u96c6\u3002</li> <li>\u6bd4\u8f83\u5b50\u96c6\uff08\u6570\u7ec4\uff09\u7684\u5f02\u540c\u975e\u5e38\u8017\u65f6\uff0c\u9700\u8981\u5148\u6392\u5e8f\u6570\u7ec4\uff0c\u518d\u6bd4\u8f83\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u5143\u7d20\u7684\u5f02\u540c\u3002</li> </ul>"},{"location":"chapter_backtracking/subset_sum_problem/#2","title":"2. \u00a0 \u91cd\u590d\u5b50\u96c6\u526a\u679d","text":"<p>\u6211\u4eec\u8003\u8651\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u901a\u8fc7\u526a\u679d\u8fdb\u884c\u53bb\u91cd\u3002\u89c2\u5bdf\u56fe 13-11 \uff0c\u91cd\u590d\u5b50\u96c6\u662f\u5728\u4ee5\u4e0d\u540c\u987a\u5e8f\u9009\u62e9\u6570\u7ec4\u5143\u7d20\u65f6\u4ea7\u751f\u7684\uff0c\u4f8b\u5982\u4ee5\u4e0b\u60c5\u51b5\u3002</p> <ol> <li>\u5f53\u7b2c\u4e00\u8f6e\u548c\u7b2c\u4e8c\u8f6e\u5206\u522b\u9009\u62e9 \\(3\\) \u548c \\(4\\) \u65f6\uff0c\u4f1a\u751f\u6210\u5305\u542b\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u6240\u6709\u5b50\u96c6\uff0c\u8bb0\u4e3a \\([3, 4, \\dots]\\) \u3002</li> <li>\u4e4b\u540e\uff0c\u5f53\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(4\\) \u65f6\uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \uff0c\u56e0\u4e3a\u8be5\u9009\u62e9\u4ea7\u751f\u7684\u5b50\u96c6 \\([4, 3, \\dots]\\) \u548c\u7b2c <code>1.</code> \u6b65\u4e2d\u751f\u6210\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u590d\u3002</li> </ol> <p>\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e00\u5c42\u7684\u9009\u62e9\u90fd\u662f\u4ece\u5de6\u5230\u53f3\u88ab\u9010\u4e2a\u5c1d\u8bd5\u7684\uff0c\u56e0\u6b64\u8d8a\u9760\u53f3\u7684\u5206\u652f\u88ab\u526a\u6389\u7684\u8d8a\u591a\u3002</p> <ol> <li>\u524d\u4e24\u8f6e\u9009\u62e9 \\(3\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([3, 5, \\dots]\\) \u3002</li> <li>\u524d\u4e24\u8f6e\u9009\u62e9 \\(4\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([4, 5, \\dots]\\) \u3002</li> <li>\u82e5\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(5\\) \uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \u548c \\(4\\) \uff0c\u56e0\u4e3a\u5b50\u96c6 \\([5, 3, \\dots]\\) \u548c \\([5, 4, \\dots]\\) \u4e0e\u7b2c <code>1.</code> \u6b65\u548c\u7b2c <code>2.</code> \u6b65\u4e2d\u63cf\u8ff0\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u590d\u3002</li> </ol> <p></p> <p> \u56fe 13-11 \u00a0 \u4e0d\u540c\u9009\u62e9\u987a\u5e8f\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6 </p> <p>\u603b\u7ed3\u6765\u770b\uff0c\u7ed9\u5b9a\u8f93\u5165\u6570\u7ec4 \\([x_1, x_2, \\dots, x_n]\\) \uff0c\u8bbe\u641c\u7d22\u8fc7\u7a0b\u4e2d\u7684\u9009\u62e9\u5e8f\u5217\u4e3a \\([x_{i_1}, x_{i_2}, \\dots, x_{i_m}]\\) \uff0c\u5219\u8be5\u9009\u62e9\u5e8f\u5217\u9700\u8981\u6ee1\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u4e0d\u6ee1\u8db3\u8be5\u6761\u4ef6\u7684\u9009\u62e9\u5e8f\u5217\u90fd\u4f1a\u9020\u6210\u91cd\u590d\uff0c\u5e94\u5f53\u526a\u679d\u3002</p>"},{"location":"chapter_backtracking/subset_sum_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u4e3a\u5b9e\u73b0\u8be5\u526a\u679d\uff0c\u6211\u4eec\u521d\u59cb\u5316\u53d8\u91cf <code>start</code> \uff0c\u7528\u4e8e\u6307\u793a\u904d\u5386\u8d77\u59cb\u70b9\u3002\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i\\) \u5f00\u59cb\u904d\u5386\u3002\u8fd9\u6837\u505a\u5c31\u53ef\u4ee5\u8ba9\u9009\u62e9\u5e8f\u5217\u6ee1\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u4ece\u800c\u4fdd\u8bc1\u5b50\u96c6\u552f\u4e00\u3002</p> <p>\u9664\u6b64\u4e4b\u5916\uff0c\u6211\u4eec\u8fd8\u5bf9\u4ee3\u7801\u8fdb\u884c\u4e86\u4ee5\u4e0b\u4e24\u9879\u4f18\u5316\u3002</p> <ul> <li>\u5728\u5f00\u542f\u641c\u7d22\u524d\uff0c\u5148\u5c06\u6570\u7ec4 <code>nums</code> \u6392\u5e8f\u3002\u5728\u904d\u5386\u6240\u6709\u9009\u62e9\u65f6\uff0c\u5f53\u5b50\u96c6\u548c\u8d85\u8fc7 <code>target</code> \u65f6\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\uff0c\u56e0\u4e3a\u540e\u8fb9\u7684\u5143\u7d20\u66f4\u5927\uff0c\u5176\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 <code>target</code> \u3002</li> <li>\u7701\u53bb\u5143\u7d20\u548c\u53d8\u91cf <code>total</code> \uff0c\u901a\u8fc7\u5728 <code>target</code> \u4e0a\u6267\u884c\u51cf\u6cd5\u6765\u7edf\u8ba1\u5143\u7d20\u548c\uff0c\u5f53 <code>target</code> \u7b49\u4e8e \\(0\\) \u65f6\u8bb0\u5f55\u89e3\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig subset_sum_i.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n    # \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0:\n        res.append(list(state))\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    # \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for i in range(start, len(choices)):\n        # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        # \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0:\n            break\n        # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res)\n        # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop()\n\ndef subset_sum_i(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\"\"\"\n    state = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort()  # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    start = 0  # \u904d\u5386\u8d77\u59cb\u70b9\n    res = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_i.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (int i = start; i &lt; choices.size(); i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push_back(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop_back();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvector&lt;vector&lt;int&gt;&gt; subsetSumI(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    sort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0;                  // \u904d\u5386\u8d77\u59cb\u70b9\n    vector&lt;vector&lt;int&gt;&gt; res;        // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (int i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.remove(state.size() - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;Integer&gt;&gt; subsetSumI(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Arrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (int i = start; i &lt; choices.Length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.Add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        Backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;int&gt;&gt; SubsetSumI(int[] nums, int target) {\n    List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Array.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumI(start, target int, state, choices *[]int, res *[][]int) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for i := start; i &lt; len(*choices); i++ {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        *state = append(*state, (*choices)[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrackSubsetSumI(i, target-(*choices)[i], state, choices, res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums []int, target int) [][]int {\n    state := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    sort.Ints(nums)         // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    start := 0              // \u904d\u5386\u8d77\u59cb\u70b9\n    res := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrackSubsetSumI(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for i in choices.indices.dropFirst(start) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i, res: &amp;res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.removeLast()\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    let nums = nums.sorted() // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    let start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\n    var res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, choices, start, res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums, target) {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums: number[], target: number): number[][] {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n  for (int i = start; i &lt; choices.length; i++) {\n    // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n    // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n    state.add(choices[i]);\n    // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n    backtrack(state, target - choices[i], choices, i, res);\n    // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n    state.removeLast();\n  }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;int&gt;&gt; subsetSumI(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n  nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n  int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n  List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_i.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(\n    mut state: Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        res.push(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for i in start..choices.len() {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state.clone(), target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfn subset_sum_i(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    let start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    let mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; ++i) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (int i = start; i &lt; choicesSize; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state[stateSize] = choices[i];\n        stateSize++;\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(target - choices[i], choices, choicesSize, i);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        stateSize--;\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvoid subsetSumI(int *nums, int numsSize, int target) {\n    qsort(nums, numsSize, sizeof(int), cmp); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0;                           // \u904d\u5386\u8d77\u59cb\u70b9\n    backtrack(target, nums, numsSize, start);\n}\n</code></pre> subset_sum_i.kt<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;List&lt;Int&gt;?&gt;\n) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.add(ArrayList(state))\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (i in start..&lt;choices.size) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfun subsetSumI(nums: IntArray, target: Int): List&lt;List&lt;Int&gt;?&gt; {\n    val state: MutableList&lt;Int&gt; = ArrayList() // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Arrays.sort(nums) // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    val start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\n    val res: MutableList&lt;List&lt;Int&gt;?&gt; = ArrayList() // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_i.rb<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subset_sum_i}\n</code></pre> subset_sum_i.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumI}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 13-12 \u6240\u793a\u4e3a\u5c06\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u8f93\u5165\u4ee5\u4e0a\u4ee3\u7801\u540e\u7684\u6574\u4f53\u56de\u6eaf\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 13-12 \u00a0 \u5b50\u96c6\u548c I \u56de\u6eaf\u8fc7\u7a0b </p>"},{"location":"chapter_backtracking/subset_sum_problem/#1332","title":"13.3.2 \u00a0 \u8003\u8651\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570 <code>target</code> \uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8e <code>target</code> \u3002\u7ed9\u5b9a\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u53ef\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002</p> <p>\u76f8\u6bd4\u4e8e\u4e0a\u9898\uff0c\u672c\u9898\u7684\u8f93\u5165\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd9\u5f15\u5165\u4e86\u65b0\u7684\u95ee\u9898\u3002\u4f8b\u5982\uff0c\u7ed9\u5b9a\u6570\u7ec4 \\([4, \\hat{4}, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u5219\u73b0\u6709\u4ee3\u7801\u7684\u8f93\u51fa\u7ed3\u679c\u4e3a \\([4, 5], [\\hat{4}, 5]\\) \uff0c\u51fa\u73b0\u4e86\u91cd\u590d\u5b50\u96c6\u3002</p> <p>\u9020\u6210\u8fd9\u79cd\u91cd\u590d\u7684\u539f\u56e0\u662f\u76f8\u7b49\u5143\u7d20\u5728\u67d0\u8f6e\u4e2d\u88ab\u591a\u6b21\u9009\u62e9\u3002\u5728\u56fe 13-13 \u4e2d\uff0c\u7b2c\u4e00\u8f6e\u5171\u6709\u4e09\u4e2a\u9009\u62e9\uff0c\u5176\u4e2d\u4e24\u4e2a\u90fd\u4e3a \\(4\\) \uff0c\u4f1a\u4ea7\u751f\u4e24\u4e2a\u91cd\u590d\u7684\u641c\u7d22\u5206\u652f\uff0c\u4ece\u800c\u8f93\u51fa\u91cd\u590d\u5b50\u96c6\uff1b\u540c\u7406\uff0c\u7b2c\u4e8c\u8f6e\u7684\u4e24\u4e2a \\(4\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002</p> <p></p> <p> \u56fe 13-13 \u00a0 \u76f8\u7b49\u5143\u7d20\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6 </p>"},{"location":"chapter_backtracking/subset_sum_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"<p>\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u9650\u5236\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u4e00\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5b9e\u73b0\u65b9\u5f0f\u6bd4\u8f83\u5de7\u5999\uff1a\u7531\u4e8e\u6570\u7ec4\u662f\u5df2\u6392\u5e8f\u7684\uff0c\u56e0\u6b64\u76f8\u7b49\u5143\u7d20\u90fd\u662f\u76f8\u90bb\u7684\u3002\u8fd9\u610f\u5473\u7740\u5728\u67d0\u8f6e\u9009\u62e9\u4e2d\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u4e0e\u5176\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u5219\u8bf4\u660e\u5b83\u5df2\u7ecf\u88ab\u9009\u62e9\u8fc7\uff0c\u56e0\u6b64\u76f4\u63a5\u8df3\u8fc7\u5f53\u524d\u5143\u7d20\u3002</p> <p>\u4e0e\u6b64\u540c\u65f6\uff0c\u672c\u9898\u89c4\u5b9a\u6bcf\u4e2a\u6570\u7ec4\u5143\u7d20\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5e78\u8fd0\u7684\u662f\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u5229\u7528\u53d8\u91cf <code>start</code> \u6765\u6ee1\u8db3\u8be5\u7ea6\u675f\uff1a\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i + 1\\) \u5f00\u59cb\u5411\u540e\u904d\u5386\u3002\u8fd9\u6837\u65e2\u80fd\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e5f\u80fd\u907f\u514d\u91cd\u590d\u9009\u62e9\u5143\u7d20\u3002</p>"},{"location":"chapter_backtracking/subset_sum_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig subset_sum_ii.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II\"\"\"\n    # \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0:\n        res.append(list(state))\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    # \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    # \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for i in range(start, len(choices)):\n        # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        # \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0:\n            break\n        # \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if i &gt; start and choices[i] == choices[i - 1]:\n            continue\n        # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop()\n\ndef subset_sum_ii(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c II\"\"\"\n    state = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort()  # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    start = 0  # \u904d\u5386\u8d77\u59cb\u70b9\n    res = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_ii.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choices.size(); i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push_back(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop_back();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvector&lt;vector&lt;int&gt;&gt; subsetSumII(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    sort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0;                  // \u904d\u5386\u8d77\u59cb\u70b9\n    vector&lt;vector&lt;int&gt;&gt; res;        // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.remove(state.size() - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;Integer&gt;&gt; subsetSumII(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Arrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choices.Length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.Add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        Backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;int&gt;&gt; SubsetSumII(int[] nums, int target) {\n    List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Array.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrackSubsetSumII(start, target int, state, choices *[]int, res *[][]int) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for i := start; i &lt; len(*choices); i++ {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if i &gt; start &amp;&amp; (*choices)[i] == (*choices)[i-1] {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        *state = append(*state, (*choices)[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrackSubsetSumII(i+1, target-(*choices)[i], state, choices, res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums []int, target int) [][]int {\n    state := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    sort.Ints(nums)         // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    start := 0              // \u904d\u5386\u8d77\u59cb\u70b9\n    res := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrackSubsetSumII(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for i in choices.indices.dropFirst(start) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if i &gt; start, choices[i] == choices[i - 1] {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i + 1, res: &amp;res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.removeLast()\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    let nums = nums.sorted() // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    let start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\n    var res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(state, target, choices, start, res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums, target) {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums: number[], target: number): number[][] {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n  // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n  for (int i = start; i &lt; choices.length; i++) {\n    // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n    // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n    if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n      continue;\n    }\n    // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n    state.add(choices[i]);\n    // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n    backtrack(state, target - choices[i], choices, i + 1, res);\n    // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n    state.removeLast();\n  }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;int&gt;&gt; subsetSumII(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n  nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n  int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n  List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_ii.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfn backtrack(\n    mut state: Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        res.push(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for i in start..choices.len() {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if i &gt; start &amp;&amp; choices[i] == choices[i - 1] {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state.clone(), target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfn subset_sum_ii(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    let start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    let mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_ii.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choicesSize; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\n        if (target - choices[i] &lt; 0) {\n            continue;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state[stateSize] = choices[i];\n        stateSize++;\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(target - choices[i], choices, choicesSize, i + 1);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        stateSize--;\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvoid subsetSumII(int *nums, int numsSize, int target) {\n    // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    qsort(nums, numsSize, sizeof(int), cmp);\n    // \u5f00\u59cb\u56de\u6eaf\n    backtrack(target, nums, numsSize, 0);\n}\n</code></pre> subset_sum_ii.kt<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;List&lt;Int&gt;?&gt;\n) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.add(ArrayList(state))\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (i in start..&lt;choices.size) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfun subsetSumII(nums: IntArray, target: Int): List&lt;List&lt;Int&gt;?&gt; {\n    val state: MutableList&lt;Int&gt; = ArrayList() // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Arrays.sort(nums) // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    val start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\n    val res: MutableList&lt;List&lt;Int&gt;?&gt; = ArrayList() // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_ii.rb<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subset_sum_ii}\n</code></pre> subset_sum_ii.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumII}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 13-14 \u5c55\u793a\u4e86\u6570\u7ec4 \\([4, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u7684\u56de\u6eaf\u8fc7\u7a0b\uff0c\u5171\u5305\u542b\u56db\u79cd\u526a\u679d\u64cd\u4f5c\u3002\u8bf7\u4f60\u5c06\u56fe\u793a\u4e0e\u4ee3\u7801\u6ce8\u91ca\u76f8\u7ed3\u5408\uff0c\u7406\u89e3\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ee5\u53ca\u6bcf\u79cd\u526a\u679d\u64cd\u4f5c\u662f\u5982\u4f55\u5de5\u4f5c\u7684\u3002</p> <p></p> <p> \u56fe 13-14 \u00a0 \u5b50\u96c6\u548c II \u56de\u6eaf\u8fc7\u7a0b </p>"},{"location":"chapter_backtracking/summary/","title":"13.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_backtracking/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u662f\u7a77\u4e3e\u6cd5\uff0c\u901a\u8fc7\u5bf9\u89e3\u7a7a\u95f4\u8fdb\u884c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u6765\u5bfb\u627e\u7b26\u5408\u6761\u4ef6\u7684\u89e3\u3002\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u9047\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u5219\u8bb0\u5f55\uff0c\u76f4\u81f3\u627e\u5230\u6240\u6709\u89e3\u6216\u904d\u5386\u5b8c\u6210\u540e\u7ed3\u675f\u3002</li> <li>\u56de\u6eaf\u7b97\u6cd5\u7684\u641c\u7d22\u8fc7\u7a0b\u5305\u62ec\u5c1d\u8bd5\u4e0e\u56de\u9000\u4e24\u4e2a\u90e8\u5206\u3002\u5b83\u901a\u8fc7\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u6765\u5c1d\u8bd5\u5404\u79cd\u9009\u62e9\uff0c\u5f53\u9047\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u60c5\u51b5\u65f6\uff0c\u5219\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002\u5c1d\u8bd5\u4e0e\u56de\u9000\u662f\u4e24\u4e2a\u65b9\u5411\u76f8\u53cd\u7684\u64cd\u4f5c\u3002</li> <li>\u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u5b83\u4eec\u53ef\u7528\u4e8e\u5b9e\u73b0\u526a\u679d\u64cd\u4f5c\u3002\u526a\u679d\u53ef\u4ee5\u63d0\u524d\u7ed3\u675f\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\uff0c\u5927\u5e45\u63d0\u5347\u641c\u7d22\u6548\u7387\u3002</li> <li>\u56de\u6eaf\u7b97\u6cd5\u4e3b\u8981\u53ef\u7528\u4e8e\u89e3\u51b3\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u3002\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u867d\u7136\u53ef\u4ee5\u7528\u56de\u6eaf\u7b97\u6cd5\u89e3\u51b3\uff0c\u4f46\u5f80\u5f80\u5b58\u5728\u6548\u7387\u66f4\u9ad8\u6216\u6548\u679c\u66f4\u597d\u7684\u89e3\u6cd5\u3002</li> <li>\u5168\u6392\u5217\u95ee\u9898\u65e8\u5728\u641c\u7d22\u7ed9\u5b9a\u96c6\u5408\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e2a\u5143\u7d20\u662f\u5426\u88ab\u9009\u62e9\uff0c\u526a\u6389\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\u7684\u641c\u7d22\u5206\u652f\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002</li> <li>\u5728\u5168\u6392\u5217\u95ee\u9898\u4e2d\uff0c\u5982\u679c\u96c6\u5408\u4e2d\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff0c\u5219\u6700\u7ec8\u7ed3\u679c\u4f1a\u51fa\u73b0\u91cd\u590d\u6392\u5217\u3002\u6211\u4eec\u9700\u8981\u7ea6\u675f\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u8fd9\u901a\u5e38\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002</li> <li>\u5b50\u96c6\u548c\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u7ed9\u5b9a\u96c6\u5408\u4e2d\u627e\u5230\u548c\u4e3a\u76ee\u6807\u503c\u7684\u6240\u6709\u5b50\u96c6\u3002\u96c6\u5408\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\uff0c\u800c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u8f93\u51fa\u6240\u6709\u987a\u5e8f\u7684\u7ed3\u679c\uff0c\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002\u6211\u4eec\u5728\u56de\u6eaf\u524d\u5c06\u6570\u636e\u8fdb\u884c\u6392\u5e8f\uff0c\u5e76\u8bbe\u7f6e\u4e00\u4e2a\u53d8\u91cf\u6765\u6307\u793a\u6bcf\u4e00\u8f6e\u7684\u904d\u5386\u8d77\u59cb\u70b9\uff0c\u4ece\u800c\u5c06\u751f\u6210\u91cd\u590d\u5b50\u96c6\u7684\u641c\u7d22\u5206\u652f\u8fdb\u884c\u526a\u679d\u3002</li> <li>\u5bf9\u4e8e\u5b50\u96c6\u548c\u95ee\u9898\uff0c\u6570\u7ec4\u4e2d\u7684\u76f8\u7b49\u5143\u7d20\u4f1a\u4ea7\u751f\u91cd\u590d\u96c6\u5408\u3002\u6211\u4eec\u5229\u7528\u6570\u7ec4\u5df2\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\uff0c\u901a\u8fc7\u5224\u65ad\u76f8\u90bb\u5143\u7d20\u662f\u5426\u76f8\u7b49\u5b9e\u73b0\u526a\u679d\uff0c\u4ece\u800c\u786e\u4fdd\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u4e2d\u4e00\u6b21\u3002</li> <li>\\(n\\) \u7687\u540e\u95ee\u9898\u65e8\u5728\u5bfb\u627e\u5c06 \\(n\\) \u4e2a\u7687\u540e\u653e\u7f6e\u5230 \\(n \\times n\\) \u5c3a\u5bf8\u68cb\u76d8\u4e0a\u7684\u65b9\u6848\uff0c\u8981\u6c42\u6240\u6709\u7687\u540e\u4e24\u4e24\u4e4b\u95f4\u65e0\u6cd5\u653b\u51fb\u5bf9\u65b9\u3002\u8be5\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6\u6709\u884c\u7ea6\u675f\u3001\u5217\u7ea6\u675f\u3001\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002\u4e3a\u6ee1\u8db3\u884c\u7ea6\u675f\uff0c\u6211\u4eec\u91c7\u7528\u6309\u884c\u653e\u7f6e\u7684\u7b56\u7565\uff0c\u4fdd\u8bc1\u6bcf\u4e00\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002</li> <li>\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u5904\u7406\u65b9\u5f0f\u7c7b\u4f3c\u3002\u5bf9\u4e8e\u5217\u7ea6\u675f\uff0c\u6211\u4eec\u5229\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\uff0c\u4ece\u800c\u6307\u793a\u9009\u4e2d\u7684\u683c\u5b50\u662f\u5426\u5408\u6cd5\u3002\u5bf9\u4e8e\u5bf9\u89d2\u7ebf\u7ea6\u675f\uff0c\u6211\u4eec\u501f\u52a9\u4e24\u4e2a\u6570\u7ec4\u6765\u5206\u522b\u8bb0\u5f55\u8be5\u4e3b\u3001\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u5b58\u5728\u7687\u540e\uff1b\u96be\u70b9\u5728\u4e8e\u627e\u5904\u5728\u5230\u540c\u4e00\u4e3b\uff08\u526f\uff09\u5bf9\u89d2\u7ebf\u4e0a\u683c\u5b50\u6ee1\u8db3\u7684\u884c\u5217\u7d22\u5f15\u89c4\u5f8b\u3002</li> </ul>"},{"location":"chapter_backtracking/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u600e\u4e48\u7406\u89e3\u56de\u6eaf\u548c\u9012\u5f52\u7684\u5173\u7cfb\uff1f</p> <p>\u603b\u7684\u6765\u770b\uff0c\u56de\u6eaf\u662f\u4e00\u79cd\u201c\u7b97\u6cd5\u7b56\u7565\u201d\uff0c\u800c\u9012\u5f52\u66f4\u50cf\u662f\u4e00\u4e2a\u201c\u5de5\u5177\u201d\u3002</p> <ul> <li>\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\u3002\u7136\u800c\uff0c\u56de\u6eaf\u662f\u9012\u5f52\u7684\u5e94\u7528\u573a\u666f\u4e4b\u4e00\uff0c\u662f\u9012\u5f52\u5728\u641c\u7d22\u95ee\u9898\u4e2d\u7684\u5e94\u7528\u3002</li> <li>\u9012\u5f52\u7684\u7ed3\u6784\u4f53\u73b0\u4e86\u201c\u5b50\u95ee\u9898\u5206\u89e3\u201d\u7684\u89e3\u9898\u8303\u5f0f\uff0c\u5e38\u7528\u4e8e\u89e3\u51b3\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\uff08\u8bb0\u5fc6\u5316\u9012\u5f52\uff09\u7b49\u95ee\u9898\u3002</li> </ul>"},{"location":"chapter_computational_complexity/","title":"\u7b2c 2 \u7ae0 \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>Abstract</p> <p>\u590d\u6742\u5ea6\u5206\u6790\u72b9\u5982\u6d69\u701a\u7684\u7b97\u6cd5\u5b87\u5b99\u4e2d\u7684\u65f6\u7a7a\u5411\u5bfc\u3002</p> <p>\u5b83\u5e26\u9886\u6211\u4eec\u5728\u65f6\u95f4\u4e0e\u7a7a\u95f4\u8fd9\u4e24\u4e2a\u7ef4\u5ea6\u4e0a\u6df1\u5165\u63a2\u7d22\uff0c\u5bfb\u627e\u66f4\u4f18\u96c5\u7684\u89e3\u51b3\u65b9\u6848\u3002</p>"},{"location":"chapter_computational_complexity/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>2.1 \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30</li> <li>2.2 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52</li> <li>2.3 \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6</li> <li>2.4 \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6</li> <li>2.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/","title":"2.2 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52","text":"<p>\u5728\u7b97\u6cd5\u4e2d\uff0c\u91cd\u590d\u6267\u884c\u67d0\u4e2a\u4efb\u52a1\u662f\u5f88\u5e38\u89c1\u7684\uff0c\u5b83\u4e0e\u590d\u6742\u5ea6\u5206\u6790\u606f\u606f\u76f8\u5173\u3002\u56e0\u6b64\uff0c\u5728\u4ecb\u7ecd\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e4b\u524d\uff0c\u6211\u4eec\u5148\u6765\u4e86\u89e3\u5982\u4f55\u5728\u7a0b\u5e8f\u4e2d\u5b9e\u73b0\u91cd\u590d\u6267\u884c\u4efb\u52a1\uff0c\u5373\u4e24\u79cd\u57fa\u672c\u7684\u7a0b\u5e8f\u63a7\u5236\u7ed3\u6784\uff1a\u8fed\u4ee3\u3001\u9012\u5f52\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#221","title":"2.2.1 \u00a0 \u8fed\u4ee3","text":"<p>\u8fed\u4ee3\uff08iteration\uff09\u662f\u4e00\u79cd\u91cd\u590d\u6267\u884c\u67d0\u4e2a\u4efb\u52a1\u7684\u63a7\u5236\u7ed3\u6784\u3002\u5728\u8fed\u4ee3\u4e2d\uff0c\u7a0b\u5e8f\u4f1a\u5728\u6ee1\u8db3\u4e00\u5b9a\u7684\u6761\u4ef6\u4e0b\u91cd\u590d\u6267\u884c\u67d0\u6bb5\u4ee3\u7801\uff0c\u76f4\u5230\u8fd9\u4e2a\u6761\u4ef6\u4e0d\u518d\u6ee1\u8db3\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for","title":"1. \u00a0 for \u5faa\u73af","text":"<p><code>for</code> \u5faa\u73af\u662f\u6700\u5e38\u89c1\u7684\u8fed\u4ee3\u5f62\u5f0f\u4e4b\u4e00\uff0c\u9002\u5408\u5728\u9884\u5148\u77e5\u9053\u8fed\u4ee3\u6b21\u6570\u65f6\u4f7f\u7528\u3002</p> <p>\u4ee5\u4e0b\u51fd\u6570\u57fa\u4e8e <code>for</code> \u5faa\u73af\u5b9e\u73b0\u4e86\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff0c\u6c42\u548c\u7ed3\u679c\u4f7f\u7528\u53d8\u91cf <code>res</code> \u8bb0\u5f55\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0cPython \u4e2d <code>range(a, b)</code> \u5bf9\u5e94\u7684\u533a\u95f4\u662f\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u7684\uff0c\u5bf9\u5e94\u7684\u904d\u5386\u8303\u56f4\u4e3a \\(a, a + 1, \\dots, b-1\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def for_loop(n: int) -&gt; int:\n    \"\"\"for \u5faa\u73af\"\"\"\n    res = 0\n    # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        res += i\n    return res\n</code></pre> iteration.cpp<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* for \u5faa\u73af */\nint ForLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* for \u5faa\u73af */\nfunc forLoop(n int) int {\n    res := 0\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* for \u5faa\u73af */\nfunc forLoop(n: Int) -&gt; Int {\n    var res = 0\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* for \u5faa\u73af */\nfunction forLoop(n) {\n    let res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* for \u5faa\u73af */\nfunction forLoop(n: number): number {\n    let res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n  int res = 0;\n  // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    res += i;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* for \u5faa\u73af */\nfn for_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i in 1..=n {\n        res += i;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* for \u5faa\u73af */\nfun forLoop(n: Int): Int {\n    var res = 0\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (i in 1..n) {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{for_loop}\n</code></pre> iteration.zig<pre><code>// for \u5faa\u73af\nfn forLoop(n: usize) i32 {\n    var res: i32 = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (1..n+1) |i| {\n        res = res + @as(i32, @intCast(i));\n    }\n    return res;\n} \n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 2-1 \u662f\u8be5\u6c42\u548c\u51fd\u6570\u7684\u6d41\u7a0b\u6846\u56fe\u3002</p> <p></p> <p> \u56fe 2-1 \u00a0 \u6c42\u548c\u51fd\u6570\u7684\u6d41\u7a0b\u6846\u56fe </p> <p>\u6b64\u6c42\u548c\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8bf4\u6210\u201c\u7ebf\u6027\u5173\u7cfb\u201d\u3002\u5b9e\u9645\u4e0a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u63cf\u8ff0\u7684\u5c31\u662f\u8fd9\u4e2a\u201c\u7ebf\u6027\u5173\u7cfb\u201d\u3002\u76f8\u5173\u5185\u5bb9\u5c06\u4f1a\u5728\u4e0b\u4e00\u8282\u4e2d\u8be6\u7ec6\u4ecb\u7ecd\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while","title":"2. \u00a0 while \u5faa\u73af","text":"<p>\u4e0e <code>for</code> \u5faa\u73af\u7c7b\u4f3c\uff0c<code>while</code> \u5faa\u73af\u4e5f\u662f\u4e00\u79cd\u5b9e\u73b0\u8fed\u4ee3\u7684\u65b9\u6cd5\u3002\u5728 <code>while</code> \u5faa\u73af\u4e2d\uff0c\u7a0b\u5e8f\u6bcf\u8f6e\u90fd\u4f1a\u5148\u68c0\u67e5\u6761\u4ef6\uff0c\u5982\u679c\u6761\u4ef6\u4e3a\u771f\uff0c\u5219\u7ee7\u7eed\u6267\u884c\uff0c\u5426\u5219\u5c31\u7ed3\u675f\u5faa\u73af\u3002</p> <p>\u4e0b\u9762\u6211\u4eec\u7528 <code>while</code> \u5faa\u73af\u6765\u5b9e\u73b0\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def while_loop(n: int) -&gt; int:\n    \"\"\"while \u5faa\u73af\"\"\"\n    res = 0\n    i = 1  # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n:\n        res += i\n        i += 1  # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    return res\n</code></pre> iteration.cpp<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u5faa\u73af */\nint WhileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while \u5faa\u73af */\nfunc whileLoop(n int) int {\n    res := 0\n    // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    i := 1\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while \u5faa\u73af */\nfunc whileLoop(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i\n        i += 1 // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while \u5faa\u73af */\nfunction whileLoop(n) {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u5faa\u73af */\nfunction whileLoop(n: number): number {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n  int res = 0;\n  int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n  // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n  while (i &lt;= n) {\n    res += i;\n    i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u5faa\u73af */\nfn while_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i;\n        i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while \u5faa\u73af */\nfun whileLoop(n: Int): Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i\n        i++ // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{while_loop}\n</code></pre> iteration.zig<pre><code>// while \u5faa\u73af\nfn whileLoop(n: i32) i32 {\n    var res: i32 = 0;\n    var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += @intCast(i);\n        i += 1;\n    }\n    return res;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p><code>while</code> \u5faa\u73af\u6bd4 <code>for</code> \u5faa\u73af\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\u3002\u5728 <code>while</code> \u5faa\u73af\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u81ea\u7531\u5730\u8bbe\u8ba1\u6761\u4ef6\u53d8\u91cf\u7684\u521d\u59cb\u5316\u548c\u66f4\u65b0\u6b65\u9aa4\u3002</p> <p>\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u6761\u4ef6\u53d8\u91cf \\(i\\) \u6bcf\u8f6e\u8fdb\u884c\u4e24\u6b21\u66f4\u65b0\uff0c\u8fd9\u79cd\u60c5\u51b5\u5c31\u4e0d\u592a\u65b9\u4fbf\u7528 <code>for</code> \u5faa\u73af\u5b9e\u73b0\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def while_loop_ii(n: int) -&gt; int:\n    \"\"\"while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\"\"\"\n    res = 0\n    i = 1  # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    # \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n:\n        res += i\n        # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1\n        i *= 2\n    return res\n</code></pre> iteration.cpp<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint WhileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1; \n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n int) int {\n    res := 0\n    // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    i := 1\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    for i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n) {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n: number): number {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n  int res = 0;\n  int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n  // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n  while (i &lt;= n) {\n    res += i;\n    // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    i++;\n    i *= 2;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfn while_loop_ii(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1;\n        i *= 2;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfun whileLoopII(n: Int): Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{while_loop_ii}\n</code></pre> iteration.zig<pre><code>//  while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\nfn whileLoopII(n: i32) i32 {\n    var res: i32 = 0;\n    var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += @intCast(i);\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u603b\u7684\u6765\u8bf4\uff0c<code>for</code> \u5faa\u73af\u7684\u4ee3\u7801\u66f4\u52a0\u7d27\u51d1\uff0c<code>while</code> \u5faa\u73af\u66f4\u52a0\u7075\u6d3b\uff0c\u4e24\u8005\u90fd\u53ef\u4ee5\u5b9e\u73b0\u8fed\u4ee3\u7ed3\u6784\u3002\u9009\u62e9\u4f7f\u7528\u54ea\u4e00\u4e2a\u5e94\u8be5\u6839\u636e\u7279\u5b9a\u95ee\u9898\u7684\u9700\u6c42\u6765\u51b3\u5b9a\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3","title":"3. \u00a0 \u5d4c\u5957\u5faa\u73af","text":"<p>\u6211\u4eec\u53ef\u4ee5\u5728\u4e00\u4e2a\u5faa\u73af\u7ed3\u6784\u5185\u5d4c\u5957\u53e6\u4e00\u4e2a\u5faa\u73af\u7ed3\u6784\uff0c\u4e0b\u9762\u4ee5 <code>for</code> \u5faa\u73af\u4e3a\u4f8b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig iteration.py<pre><code>def nested_for_loop(n: int) -&gt; str:\n    \"\"\"\u53cc\u5c42 for \u5faa\u73af\"\"\"\n    res = \"\"\n    # \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        # \u5faa\u73af j = 1, 2, ..., n-1, n\n        for j in range(1, n + 1):\n            res += f\"({i}, {j}), \"\n    return res\n</code></pre> iteration.cpp<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nstring nestedForLoop(int n) {\n    ostringstream res;\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; ++j) {\n            res &lt;&lt; \"(\" &lt;&lt; i &lt;&lt; \", \" &lt;&lt; j &lt;&lt; \"), \";\n        }\n    }\n    return res.str();\n}\n</code></pre> iteration.java<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n    StringBuilder res = new StringBuilder();\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.append(\"(\" + i + \", \" + j + \"), \");\n        }\n    }\n    return res.toString();\n}\n</code></pre> iteration.cs<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nstring NestedForLoop(int n) {\n    StringBuilder res = new();\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.Append($\"({i}, {j}), \");\n        }\n    }\n    return res.ToString();\n}\n</code></pre> iteration.go<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n int) string {\n    res := \"\"\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= n; j++ {\n            // \u5faa\u73af j = 1, 2, ..., n-1, n\n            res += fmt.Sprintf(\"(%d, %d), \", i, j)\n        }\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n: Int) -&gt; String {\n    var res = \"\"\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for j in 1 ... n {\n            res.append(\"(\\(i), \\(j)), \")\n        }\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n) {\n    let res = '';\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n: number): string {\n    let res = '';\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n  String res = \"\";\n  // \u5faa\u73af i = 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    // \u5faa\u73af j = 1, 2, ..., n-1, n\n    for (int j = 1; j &lt;= n; j++) {\n      res += \"($i, $j), \";\n    }\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfn nested_for_loop(n: i32) -&gt; String {\n    let mut res = vec![];\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i in 1..=n {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for j in 1..=n {\n            res.push(format!(\"({}, {}), \", i, j));\n        }\n    }\n    res.join(\"\")\n}\n</code></pre> iteration.c<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nchar *nestedForLoop(int n) {\n    // n * n \u4e3a\u5bf9\u5e94\u70b9\u6570\u91cf\uff0c\"(i, j), \" \u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u6700\u5927\u4e3a 6+10*2\uff0c\u52a0\u4e0a\u6700\u540e\u4e00\u4e2a\u7a7a\u5b57\u7b26 \\0 \u7684\u989d\u5916\u7a7a\u95f4\n    int size = n * n * 26 + 1;\n    char *res = malloc(size * sizeof(char));\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            char tmp[26];\n            snprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\n            strncat(res, tmp, size - strlen(res) - 1);\n        }\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfun nestedForLoop(n: Int): String {\n    val res = StringBuilder()\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (i in 1..n) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (j in 1..n) {\n            res.append(\" ($i, $j), \")\n        }\n    }\n    return res.toString()\n}\n</code></pre> iteration.rb<pre><code>[class]{}-[func]{nested_for_loop}\n</code></pre> iteration.zig<pre><code>// \u53cc\u5c42 for \u5faa\u73af\nfn nestedForLoop(allocator: Allocator, n: usize) ![]const u8 {\n    var res = std.ArrayList(u8).init(allocator);\n    defer res.deinit();\n    var buffer: [20]u8 = undefined;\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (1..n+1) |i| {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (1..n+1) |j| {\n            var _str = try std.fmt.bufPrint(&amp;buffer, \"({d}, {d}), \", .{i, j});\n            try res.appendSlice(_str);\n        }\n    }\n    return res.toOwnedSlice();\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 2-2 \u662f\u8be5\u5d4c\u5957\u5faa\u73af\u7684\u6d41\u7a0b\u6846\u56fe\u3002</p> <p></p> <p> \u56fe 2-2 \u00a0 \u5d4c\u5957\u5faa\u73af\u7684\u6d41\u7a0b\u6846\u56fe </p> <p>\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e \\(n^2\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8bf4\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u548c\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u6210\u201c\u5e73\u65b9\u5173\u7cfb\u201d\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u7ee7\u7eed\u6dfb\u52a0\u5d4c\u5957\u5faa\u73af\uff0c\u6bcf\u4e00\u6b21\u5d4c\u5957\u90fd\u662f\u4e00\u6b21\u201c\u5347\u7ef4\u201d\uff0c\u5c06\u4f1a\u4f7f\u65f6\u95f4\u590d\u6742\u5ea6\u63d0\u9ad8\u81f3\u201c\u7acb\u65b9\u5173\u7cfb\u201d\u201c\u56db\u6b21\u65b9\u5173\u7cfb\u201d\uff0c\u4ee5\u6b64\u7c7b\u63a8\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#222","title":"2.2.2 \u00a0 \u9012\u5f52","text":"<p>\u9012\u5f52\uff08recursion\uff09\u662f\u4e00\u79cd\u7b97\u6cd5\u7b56\u7565\uff0c\u901a\u8fc7\u51fd\u6570\u8c03\u7528\u81ea\u8eab\u6765\u89e3\u51b3\u95ee\u9898\u3002\u5b83\u4e3b\u8981\u5305\u542b\u4e24\u4e2a\u9636\u6bb5\u3002</p> <ol> <li>\u9012\uff1a\u7a0b\u5e8f\u4e0d\u65ad\u6df1\u5165\u5730\u8c03\u7528\u81ea\u8eab\uff0c\u901a\u5e38\u4f20\u5165\u66f4\u5c0f\u6216\u66f4\u7b80\u5316\u7684\u53c2\u6570\uff0c\u76f4\u5230\u8fbe\u5230\u201c\u7ec8\u6b62\u6761\u4ef6\u201d\u3002</li> <li>\u5f52\uff1a\u89e6\u53d1\u201c\u7ec8\u6b62\u6761\u4ef6\u201d\u540e\uff0c\u7a0b\u5e8f\u4ece\u6700\u6df1\u5c42\u7684\u9012\u5f52\u51fd\u6570\u5f00\u59cb\u9010\u5c42\u8fd4\u56de\uff0c\u6c47\u805a\u6bcf\u4e00\u5c42\u7684\u7ed3\u679c\u3002</li> </ol> <p>\u800c\u4ece\u5b9e\u73b0\u7684\u89d2\u5ea6\u770b\uff0c\u9012\u5f52\u4ee3\u7801\u4e3b\u8981\u5305\u542b\u4e09\u4e2a\u8981\u7d20\u3002</p> <ol> <li>\u7ec8\u6b62\u6761\u4ef6\uff1a\u7528\u4e8e\u51b3\u5b9a\u4ec0\u4e48\u65f6\u5019\u7531\u201c\u9012\u201d\u8f6c\u201c\u5f52\u201d\u3002</li> <li>\u9012\u5f52\u8c03\u7528\uff1a\u5bf9\u5e94\u201c\u9012\u201d\uff0c\u51fd\u6570\u8c03\u7528\u81ea\u8eab\uff0c\u901a\u5e38\u8f93\u5165\u66f4\u5c0f\u6216\u66f4\u7b80\u5316\u7684\u53c2\u6570\u3002</li> <li>\u8fd4\u56de\u7ed3\u679c\uff1a\u5bf9\u5e94\u201c\u5f52\u201d\uff0c\u5c06\u5f53\u524d\u9012\u5f52\u5c42\u7ea7\u7684\u7ed3\u679c\u8fd4\u56de\u81f3\u4e0a\u4e00\u5c42\u3002</li> </ol> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6211\u4eec\u53ea\u9700\u8c03\u7528\u51fd\u6570 <code>recur(n)</code>  \uff0c\u5c31\u53ef\u4ee5\u5b8c\u6210 \\(1 + 2 + \\dots + n\\) \u7684\u8ba1\u7b97\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def recur(n: int) -&gt; int:\n    \"\"\"\u9012\u5f52\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6\n    if n == 1:\n        return 1\n    # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    res = recur(n - 1)\n    # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res\n</code></pre> recursion.cpp<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.java<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.cs<pre><code>/* \u9012\u5f52 */\nint Recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = Recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.go<pre><code>/* \u9012\u5f52 */\nfunc recur(n int) int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 1 {\n        return 1\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    res := recur(n - 1)\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res\n}\n</code></pre> recursion.swift<pre><code>/* \u9012\u5f52 */\nfunc recur(n: Int) -&gt; Int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 1 {\n        return 1\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    let res = recur(n: n - 1)\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res\n}\n</code></pre> recursion.js<pre><code>/* \u9012\u5f52 */\nfunction recur(n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 1) return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    const res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.ts<pre><code>/* \u9012\u5f52 */\nfunction recur(n: number): number {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 1) return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    const res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.dart<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n  // \u7ec8\u6b62\u6761\u4ef6\n  if (n == 1) return 1;\n  // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n  int res = recur(n - 1);\n  // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n  return n + res;\n}\n</code></pre> recursion.rs<pre><code>/* \u9012\u5f52 */\nfn recur(n: i32) -&gt; i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 1 {\n        return 1;\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    let res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    n + res\n}\n</code></pre> recursion.c<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.kt<pre><code>/* \u9012\u5f52 */\nfun recur(n: Int): Int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1\n    // \u9012: \u9012\u5f52\u8c03\u7528\n    val res = recur(n - 1)\n    // \u5f52: \u8fd4\u56de\u7ed3\u679c\n    return n + res\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{recur}\n</code></pre> recursion.zig<pre><code>// \u9012\u5f52\u51fd\u6570\nfn recur(n: i32) i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1) {\n        return 1;\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    var res: i32 = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 2-3 \u5c55\u793a\u4e86\u8be5\u51fd\u6570\u7684\u9012\u5f52\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 2-3 \u00a0 \u6c42\u548c\u51fd\u6570\u7684\u9012\u5f52\u8fc7\u7a0b </p> <p>\u867d\u7136\u4ece\u8ba1\u7b97\u89d2\u5ea6\u770b\uff0c\u8fed\u4ee3\u4e0e\u9012\u5f52\u53ef\u4ee5\u5f97\u5230\u76f8\u540c\u7684\u7ed3\u679c\uff0c\u4f46\u5b83\u4eec\u4ee3\u8868\u4e86\u4e24\u79cd\u5b8c\u5168\u4e0d\u540c\u7684\u601d\u8003\u548c\u89e3\u51b3\u95ee\u9898\u7684\u8303\u5f0f\u3002</p> <ul> <li>\u8fed\u4ee3\uff1a\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u5730\u89e3\u51b3\u95ee\u9898\u3002\u4ece\u6700\u57fa\u7840\u7684\u6b65\u9aa4\u5f00\u59cb\uff0c\u7136\u540e\u4e0d\u65ad\u91cd\u590d\u6216\u7d2f\u52a0\u8fd9\u4e9b\u6b65\u9aa4\uff0c\u76f4\u5230\u4efb\u52a1\u5b8c\u6210\u3002</li> <li>\u9012\u5f52\uff1a\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u5730\u89e3\u51b3\u95ee\u9898\u3002\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u8fd9\u4e9b\u5b50\u95ee\u9898\u548c\u539f\u95ee\u9898\u5177\u6709\u76f8\u540c\u7684\u5f62\u5f0f\u3002\u63a5\u4e0b\u6765\u5c06\u5b50\u95ee\u9898\u7ee7\u7eed\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u5230\u57fa\u672c\u60c5\u51b5\u65f6\u505c\u6b62\uff08\u57fa\u672c\u60c5\u51b5\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff09\u3002</li> </ul> <p>\u4ee5\u4e0a\u8ff0\u6c42\u548c\u51fd\u6570\u4e3a\u4f8b\uff0c\u8bbe\u95ee\u9898 \\(f(n) = 1 + 2 + \\dots + n\\) \u3002</p> <ul> <li>\u8fed\u4ee3\uff1a\u5728\u5faa\u73af\u4e2d\u6a21\u62df\u6c42\u548c\u8fc7\u7a0b\uff0c\u4ece \\(1\\) \u904d\u5386\u5230 \\(n\\) \uff0c\u6bcf\u8f6e\u6267\u884c\u6c42\u548c\u64cd\u4f5c\uff0c\u5373\u53ef\u6c42\u5f97 \\(f(n)\\) \u3002</li> <li>\u9012\u5f52\uff1a\u5c06\u95ee\u9898\u5206\u89e3\u4e3a\u5b50\u95ee\u9898 \\(f(n) = n + f(n-1)\\) \uff0c\u4e0d\u65ad\uff08\u9012\u5f52\u5730\uff09\u5206\u89e3\u4e0b\u53bb\uff0c\u76f4\u81f3\u57fa\u672c\u60c5\u51b5 \\(f(1) = 1\\) \u65f6\u7ec8\u6b62\u3002</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1","title":"1. \u00a0 \u8c03\u7528\u6808","text":"<p>\u9012\u5f52\u51fd\u6570\u6bcf\u6b21\u8c03\u7528\u81ea\u8eab\u65f6\uff0c\u7cfb\u7edf\u90fd\u4f1a\u4e3a\u65b0\u5f00\u542f\u7684\u51fd\u6570\u5206\u914d\u5185\u5b58\uff0c\u4ee5\u5b58\u50a8\u5c40\u90e8\u53d8\u91cf\u3001\u8c03\u7528\u5730\u5740\u548c\u5176\u4ed6\u4fe1\u606f\u7b49\u3002\u8fd9\u5c06\u5bfc\u81f4\u4e24\u65b9\u9762\u7684\u7ed3\u679c\u3002</p> <ul> <li>\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u90fd\u5b58\u50a8\u5728\u79f0\u4e3a\u201c\u6808\u5e27\u7a7a\u95f4\u201d\u7684\u5185\u5b58\u533a\u57df\u4e2d\uff0c\u76f4\u81f3\u51fd\u6570\u8fd4\u56de\u540e\u624d\u4f1a\u88ab\u91ca\u653e\u3002\u56e0\u6b64\uff0c\u9012\u5f52\u901a\u5e38\u6bd4\u8fed\u4ee3\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u9012\u5f52\u8c03\u7528\u51fd\u6570\u4f1a\u4ea7\u751f\u989d\u5916\u7684\u5f00\u9500\u3002\u56e0\u6b64\u9012\u5f52\u901a\u5e38\u6bd4\u5faa\u73af\u7684\u65f6\u95f4\u6548\u7387\u66f4\u4f4e\u3002</li> </ul> <p>\u5982\u56fe 2-4 \u6240\u793a\uff0c\u5728\u89e6\u53d1\u7ec8\u6b62\u6761\u4ef6\u524d\uff0c\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684\u9012\u5f52\u51fd\u6570\uff0c\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \u3002</p> <p></p> <p> \u56fe 2-4 \u00a0 \u9012\u5f52\u8c03\u7528\u6df1\u5ea6 </p> <p>\u5728\u5b9e\u9645\u4e2d\uff0c\u7f16\u7a0b\u8bed\u8a00\u5141\u8bb8\u7684\u9012\u5f52\u6df1\u5ea6\u901a\u5e38\u662f\u6709\u9650\u7684\uff0c\u8fc7\u6df1\u7684\u9012\u5f52\u53ef\u80fd\u5bfc\u81f4\u6808\u6ea2\u51fa\u9519\u8bef\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2","title":"2. \u00a0 \u5c3e\u9012\u5f52","text":"<p>\u6709\u8da3\u7684\u662f\uff0c\u5982\u679c\u51fd\u6570\u5728\u8fd4\u56de\u524d\u7684\u6700\u540e\u4e00\u6b65\u624d\u8fdb\u884c\u9012\u5f52\u8c03\u7528\uff0c\u5219\u8be5\u51fd\u6570\u53ef\u4ee5\u88ab\u7f16\u8bd1\u5668\u6216\u89e3\u91ca\u5668\u4f18\u5316\uff0c\u4f7f\u5176\u5728\u7a7a\u95f4\u6548\u7387\u4e0a\u4e0e\u8fed\u4ee3\u76f8\u5f53\u3002\u8fd9\u79cd\u60c5\u51b5\u88ab\u79f0\u4e3a\u5c3e\u9012\u5f52\uff08tail recursion\uff09\u3002</p> <ul> <li>\u666e\u901a\u9012\u5f52\uff1a\u5f53\u51fd\u6570\u8fd4\u56de\u5230\u4e0a\u4e00\u5c42\u7ea7\u7684\u51fd\u6570\u540e\uff0c\u9700\u8981\u7ee7\u7eed\u6267\u884c\u4ee3\u7801\uff0c\u56e0\u6b64\u7cfb\u7edf\u9700\u8981\u4fdd\u5b58\u4e0a\u4e00\u5c42\u8c03\u7528\u7684\u4e0a\u4e0b\u6587\u3002</li> <li>\u5c3e\u9012\u5f52\uff1a\u9012\u5f52\u8c03\u7528\u662f\u51fd\u6570\u8fd4\u56de\u524d\u7684\u6700\u540e\u4e00\u4e2a\u64cd\u4f5c\uff0c\u8fd9\u610f\u5473\u7740\u51fd\u6570\u8fd4\u56de\u5230\u4e0a\u4e00\u5c42\u7ea7\u540e\uff0c\u65e0\u987b\u7ee7\u7eed\u6267\u884c\u5176\u4ed6\u64cd\u4f5c\uff0c\u56e0\u6b64\u7cfb\u7edf\u65e0\u987b\u4fdd\u5b58\u4e0a\u4e00\u5c42\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u3002</li> </ul> <p>\u4ee5\u8ba1\u7b97 \\(1 + 2 + \\dots + n\\) \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u7ed3\u679c\u53d8\u91cf <code>res</code> \u8bbe\u4e3a\u51fd\u6570\u53c2\u6570\uff0c\u4ece\u800c\u5b9e\u73b0\u5c3e\u9012\u5f52\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def tail_recur(n, res):\n    \"\"\"\u5c3e\u9012\u5f52\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6\n    if n == 0:\n        return res\n    # \u5c3e\u9012\u5f52\u8c03\u7528\n    return tail_recur(n - 1, res + n)\n</code></pre> recursion.cpp<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.java<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.cs<pre><code>/* \u5c3e\u9012\u5f52 */\nint TailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return TailRecur(n - 1, res + n);\n}\n</code></pre> recursion.go<pre><code>/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n int, res int) int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 0 {\n        return res\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n-1, res+n)\n}\n</code></pre> recursion.swift<pre><code>/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n: Int, res: Int) -&gt; Int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 0 {\n        return res\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n: n - 1, res: res + n)\n}\n</code></pre> recursion.js<pre><code>/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n, res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 0) return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.ts<pre><code>/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n: number, res: number): number {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 0) return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.dart<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n  // \u7ec8\u6b62\u6761\u4ef6\n  if (n == 0) return res;\n  // \u5c3e\u9012\u5f52\u8c03\u7528\n  return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.rs<pre><code>/* \u5c3e\u9012\u5f52 */\nfn tail_recur(n: i32, res: i32) -&gt; i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 0 {\n        return res;\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    tail_recur(n - 1, res + n)\n}\n</code></pre> recursion.c<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.kt<pre><code>/* \u5c3e\u9012\u5f52 */\ntailrec fun tailRecur(n: Int, res: Int): Int {\n    // \u6dfb\u52a0 tailrec \u5173\u952e\u8bcd\uff0c\u4ee5\u5f00\u542f\u5c3e\u9012\u5f52\u4f18\u5316\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n)\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{tail_recur}\n</code></pre> recursion.zig<pre><code>// \u5c3e\u9012\u5f52\u51fd\u6570\nfn tailRecur(n: i32, res: i32) i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0) {\n        return res;\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5c3e\u9012\u5f52\u7684\u6267\u884c\u8fc7\u7a0b\u5982\u56fe 2-5 \u6240\u793a\u3002\u5bf9\u6bd4\u666e\u901a\u9012\u5f52\u548c\u5c3e\u9012\u5f52\uff0c\u4e24\u8005\u7684\u6c42\u548c\u64cd\u4f5c\u7684\u6267\u884c\u70b9\u662f\u4e0d\u540c\u7684\u3002</p> <ul> <li>\u666e\u901a\u9012\u5f52\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u4e2d\u6267\u884c\u7684\uff0c\u6bcf\u5c42\u8fd4\u56de\u540e\u90fd\u8981\u518d\u6267\u884c\u4e00\u6b21\u6c42\u548c\u64cd\u4f5c\u3002</li> <li>\u5c3e\u9012\u5f52\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u9012\u201d\u7684\u8fc7\u7a0b\u4e2d\u6267\u884c\u7684\uff0c\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u53ea\u9700\u5c42\u5c42\u8fd4\u56de\u3002</li> </ul> <p></p> <p> \u56fe 2-5 \u00a0 \u5c3e\u9012\u5f52\u8fc7\u7a0b </p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u8bb8\u591a\u7f16\u8bd1\u5668\u6216\u89e3\u91ca\u5668\u5e76\u4e0d\u652f\u6301\u5c3e\u9012\u5f52\u4f18\u5316\u3002\u4f8b\u5982\uff0cPython \u9ed8\u8ba4\u4e0d\u652f\u6301\u5c3e\u9012\u5f52\u4f18\u5316\uff0c\u56e0\u6b64\u5373\u4f7f\u51fd\u6570\u662f\u5c3e\u9012\u5f52\u5f62\u5f0f\uff0c\u4ecd\u7136\u53ef\u80fd\u4f1a\u9047\u5230\u6808\u6ea2\u51fa\u95ee\u9898\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3_1","title":"3. \u00a0 \u9012\u5f52\u6811","text":"<p>\u5f53\u5904\u7406\u4e0e\u201c\u5206\u6cbb\u201d\u76f8\u5173\u7684\u7b97\u6cd5\u95ee\u9898\u65f6\uff0c\u9012\u5f52\u5f80\u5f80\u6bd4\u8fed\u4ee3\u7684\u601d\u8def\u66f4\u52a0\u76f4\u89c2\u3001\u4ee3\u7801\u66f4\u52a0\u6613\u8bfb\u3002\u4ee5\u201c\u6590\u6ce2\u90a3\u5951\u6570\u5217\u201d\u4e3a\u4f8b\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6590\u6ce2\u90a3\u5951\u6570\u5217 \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\) \uff0c\u6c42\u8be5\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\u3002</p> <p>\u8bbe\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\u4e3a \\(f(n)\\) \uff0c\u6613\u5f97\u4e24\u4e2a\u7ed3\u8bba\u3002</p> <ul> <li>\u6570\u5217\u7684\u524d\u4e24\u4e2a\u6570\u5b57\u4e3a \\(f(1) = 0\\) \u548c \\(f(2) = 1\\) \u3002</li> <li>\u6570\u5217\u4e2d\u7684\u6bcf\u4e2a\u6570\u5b57\u662f\u524d\u4e24\u4e2a\u6570\u5b57\u7684\u548c\uff0c\u5373 \\(f(n) = f(n - 1) + f(n - 2)\\) \u3002</li> </ul> <p>\u6309\u7167\u9012\u63a8\u5173\u7cfb\u8fdb\u884c\u9012\u5f52\u8c03\u7528\uff0c\u5c06\u524d\u4e24\u4e2a\u6570\u5b57\u4f5c\u4e3a\u7ec8\u6b62\u6761\u4ef6\uff0c\u4fbf\u53ef\u5199\u51fa\u9012\u5f52\u4ee3\u7801\u3002\u8c03\u7528 <code>fib(n)</code> \u5373\u53ef\u5f97\u5230\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def fib(n: int) -&gt; int:\n    \"\"\"\u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 or n == 2:\n        return n - 1\n    # \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    res = fib(n - 1) + fib(n - 2)\n    # \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n</code></pre> recursion.cpp<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint Fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = Fib(n - 1) + Fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n int) int {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    res := fib(n-1) + fib(n-2)\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n: Int) -&gt; Int {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    let res = fib(n: n - 1) + fib(n: n - 2)\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n}\n</code></pre> recursion.js<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n: number): number {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n  // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n  if (n == 1 || n == 2) return n - 1;\n  // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n  int res = fib(n - 1) + fib(n - 2);\n  // \u8fd4\u56de\u7ed3\u679c f(n)\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfn fib(n: i32) -&gt; i32 {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1;\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    let res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c\n    res\n}\n</code></pre> recursion.c<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfun fib(n: Int): Int {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    val res = fib(n - 1) + fib(n - 2)\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{fib}\n</code></pre> recursion.zig<pre><code>// \u6590\u6ce2\u90a3\u5951\u6570\u5217\nfn fib(n: i32) i32 {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 or n == 2) {\n        return n - 1;\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    var res: i32 = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u6211\u4eec\u5728\u51fd\u6570\u5185\u9012\u5f52\u8c03\u7528\u4e86\u4e24\u4e2a\u51fd\u6570\uff0c\u8fd9\u610f\u5473\u7740\u4ece\u4e00\u4e2a\u8c03\u7528\u4ea7\u751f\u4e86\u4e24\u4e2a\u8c03\u7528\u5206\u652f\u3002\u5982\u56fe 2-6 \u6240\u793a\uff0c\u8fd9\u6837\u4e0d\u65ad\u9012\u5f52\u8c03\u7528\u4e0b\u53bb\uff0c\u6700\u7ec8\u5c06\u4ea7\u751f\u4e00\u68f5\u5c42\u6570\u4e3a \\(n\\) \u7684\u9012\u5f52\u6811\uff08recursion tree\uff09\u3002</p> <p></p> <p> \u56fe 2-6 \u00a0 \u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u9012\u5f52\u6811 </p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9012\u5f52\u4f53\u73b0\u4e86\u201c\u5c06\u95ee\u9898\u5206\u89e3\u4e3a\u66f4\u5c0f\u5b50\u95ee\u9898\u201d\u7684\u601d\u7ef4\u8303\u5f0f\uff0c\u8fd9\u79cd\u5206\u6cbb\u7b56\u7565\u81f3\u5173\u91cd\u8981\u3002</p> <ul> <li>\u4ece\u7b97\u6cd5\u89d2\u5ea6\u770b\uff0c\u641c\u7d22\u3001\u6392\u5e8f\u3001\u56de\u6eaf\u3001\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u7b49\u8bb8\u591a\u91cd\u8981\u7b97\u6cd5\u7b56\u7565\u76f4\u63a5\u6216\u95f4\u63a5\u5730\u5e94\u7528\u4e86\u8fd9\u79cd\u601d\u7ef4\u65b9\u5f0f\u3002</li> <li>\u4ece\u6570\u636e\u7ed3\u6784\u89d2\u5ea6\u770b\uff0c\u9012\u5f52\u5929\u7136\u9002\u5408\u5904\u7406\u94fe\u8868\u3001\u6811\u548c\u56fe\u7684\u76f8\u5173\u95ee\u9898\uff0c\u56e0\u4e3a\u5b83\u4eec\u975e\u5e38\u9002\u5408\u7528\u5206\u6cbb\u601d\u60f3\u8fdb\u884c\u5206\u6790\u3002</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#223","title":"2.2.3 \u00a0 \u4e24\u8005\u5bf9\u6bd4","text":"<p>\u603b\u7ed3\u4ee5\u4e0a\u5185\u5bb9\uff0c\u5982\u8868 2-1 \u6240\u793a\uff0c\u8fed\u4ee3\u548c\u9012\u5f52\u5728\u5b9e\u73b0\u3001\u6027\u80fd\u548c\u9002\u7528\u6027\u4e0a\u6709\u6240\u4e0d\u540c\u3002</p> <p> \u8868 2-1 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52\u7279\u70b9\u5bf9\u6bd4 </p> \u8fed\u4ee3 \u9012\u5f52 \u5b9e\u73b0\u65b9\u5f0f \u5faa\u73af\u7ed3\u6784 \u51fd\u6570\u8c03\u7528\u81ea\u8eab \u65f6\u95f4\u6548\u7387 \u6548\u7387\u901a\u5e38\u8f83\u9ad8\uff0c\u65e0\u51fd\u6570\u8c03\u7528\u5f00\u9500 \u6bcf\u6b21\u51fd\u6570\u8c03\u7528\u90fd\u4f1a\u4ea7\u751f\u5f00\u9500 \u5185\u5b58\u4f7f\u7528 \u901a\u5e38\u4f7f\u7528\u56fa\u5b9a\u5927\u5c0f\u7684\u5185\u5b58\u7a7a\u95f4 \u7d2f\u79ef\u51fd\u6570\u8c03\u7528\u53ef\u80fd\u4f7f\u7528\u5927\u91cf\u7684\u6808\u5e27\u7a7a\u95f4 \u9002\u7528\u95ee\u9898 \u9002\u7528\u4e8e\u7b80\u5355\u5faa\u73af\u4efb\u52a1\uff0c\u4ee3\u7801\u76f4\u89c2\u3001\u53ef\u8bfb\u6027\u597d \u9002\u7528\u4e8e\u5b50\u95ee\u9898\u5206\u89e3\uff0c\u5982\u6811\u3001\u56fe\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u7b49\uff0c\u4ee3\u7801\u7ed3\u6784\u7b80\u6d01\u3001\u6e05\u6670 <p>Tip</p> <p>\u5982\u679c\u611f\u89c9\u4ee5\u4e0b\u5185\u5bb9\u7406\u89e3\u56f0\u96be\uff0c\u53ef\u4ee5\u5728\u8bfb\u5b8c\u201c\u6808\u201d\u7ae0\u8282\u540e\u518d\u6765\u590d\u4e60\u3002</p> <p>\u90a3\u4e48\uff0c\u8fed\u4ee3\u548c\u9012\u5f52\u5177\u6709\u4ec0\u4e48\u5185\u5728\u8054\u7cfb\u5462\uff1f\u4ee5\u4e0a\u8ff0\u9012\u5f52\u51fd\u6570\u4e3a\u4f8b\uff0c\u6c42\u548c\u64cd\u4f5c\u5728\u9012\u5f52\u7684\u201c\u5f52\u201d\u9636\u6bb5\u8fdb\u884c\u3002\u8fd9\u610f\u5473\u7740\u6700\u521d\u88ab\u8c03\u7528\u7684\u51fd\u6570\u5b9e\u9645\u4e0a\u662f\u6700\u540e\u5b8c\u6210\u5176\u6c42\u548c\u64cd\u4f5c\u7684\uff0c\u8fd9\u79cd\u5de5\u4f5c\u673a\u5236\u4e0e\u6808\u7684\u201c\u5148\u5165\u540e\u51fa\u201d\u539f\u5219\u5f02\u66f2\u540c\u5de5\u3002</p> <p>\u4e8b\u5b9e\u4e0a\uff0c\u201c\u8c03\u7528\u6808\u201d\u548c\u201c\u6808\u5e27\u7a7a\u95f4\u201d\u8fd9\u7c7b\u9012\u5f52\u672f\u8bed\u5df2\u7ecf\u6697\u793a\u4e86\u9012\u5f52\u4e0e\u6808\u4e4b\u95f4\u7684\u5bc6\u5207\u5173\u7cfb\u3002</p> <ol> <li>\u9012\uff1a\u5f53\u51fd\u6570\u88ab\u8c03\u7528\u65f6\uff0c\u7cfb\u7edf\u4f1a\u5728\u201c\u8c03\u7528\u6808\u201d\u4e0a\u4e3a\u8be5\u51fd\u6570\u5206\u914d\u65b0\u7684\u6808\u5e27\uff0c\u7528\u4e8e\u5b58\u50a8\u51fd\u6570\u7684\u5c40\u90e8\u53d8\u91cf\u3001\u53c2\u6570\u3001\u8fd4\u56de\u5730\u5740\u7b49\u6570\u636e\u3002</li> <li>\u5f52\uff1a\u5f53\u51fd\u6570\u5b8c\u6210\u6267\u884c\u5e76\u8fd4\u56de\u65f6\uff0c\u5bf9\u5e94\u7684\u6808\u5e27\u4f1a\u88ab\u4ece\u201c\u8c03\u7528\u6808\u201d\u4e0a\u79fb\u9664\uff0c\u6062\u590d\u4e4b\u524d\u51fd\u6570\u7684\u6267\u884c\u73af\u5883\u3002</li> </ol> <p>\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u8c03\u7528\u6808\u7684\u884c\u4e3a\uff0c\u4ece\u800c\u5c06\u9012\u5f52\u8f6c\u5316\u4e3a\u8fed\u4ee3\u5f62\u5f0f\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig recursion.py<pre><code>def for_loop_recur(n: int) -&gt; int:\n    \"\"\"\u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\"\"\"\n    # \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    stack = []\n    res = 0\n    # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i in range(n, 0, -1):\n        # \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.append(i)\n    # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while stack:\n        # \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop()\n    # res = 1+2+3+...+n\n    return res\n</code></pre> recursion.cpp<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    stack&lt;int&gt; stack;\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (!stack.empty()) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.top();\n        stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    Stack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (!stack.isEmpty()) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint ForLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    Stack&lt;int&gt; stack = new();\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.Push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (stack.Count &gt; 0) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.Pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n int) int {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    stack := list.New()\n    res := 0\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i := n; i &gt; 0; i-- {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.PushBack(i)\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    for stack.Len() != 0 {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.Back().Value.(int)\n        stack.Remove(stack.Back())\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n: Int) -&gt; Int {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    var stack: [Int] = []\n    var res = 0\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i in (1 ... n).reversed() {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.append(i)\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while !stack.isEmpty {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.removeLast()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.js<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    const stack = [];\n    let res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (let i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (stack.length) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n: number): number {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808 \n    const stack: number[] = [];\n    let res: number = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (let i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (stack.length) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n  // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n  List&lt;int&gt; stack = [];\n  int res = 0;\n  // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n  for (int i = n; i &gt; 0; i--) {\n    // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n    stack.add(i);\n  }\n  // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n  while (!stack.isEmpty) {\n    // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n    res += stack.removeLast();\n  }\n  // res = 1+2+3+...+n\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfn for_loop_recur(n: i32) -&gt; i32 {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    let mut stack = Vec::new();\n    let mut res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i in (1..=n).rev() {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while !stack.is_empty() {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop().unwrap();\n    }\n    // res = 1+2+3+...+n\n    res\n}\n</code></pre> recursion.c<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n    int stack[1000]; // \u501f\u52a9\u4e00\u4e2a\u5927\u6570\u7ec4\u6765\u6a21\u62df\u6808\n    int top = -1;    // \u6808\u9876\u7d22\u5f15\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack[1 + top++] = i;\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (top &gt;= 0) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack[top--];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfun forLoopRecur(n: Int): Int {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    val stack = Stack&lt;Int&gt;()\n    var res = 0\n    // \u9012: \u9012\u5f52\u8c03\u7528\n    for (i in n downTo 0) {\n        stack.push(i)\n    }\n    // \u5f52: \u8fd4\u56de\u7ed3\u679c\n    while (stack.isNotEmpty()) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.rb<pre><code>[class]{}-[func]{for_loop_recur}\n</code></pre> recursion.zig<pre><code>// \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\nfn forLoopRecur(comptime n: i32) i32 {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    var stack: [n]i32 = undefined;\n    var res: i32 = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    var i: usize = n;\n    while (i &gt; 0) {\n        stack[i - 1] = @intCast(i);\n        i -= 1;\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    var index: usize = n;\n    while (index &gt; 0) {\n        index -= 1;\n        res += stack[index];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u5f53\u9012\u5f52\u8f6c\u5316\u4e3a\u8fed\u4ee3\u540e\uff0c\u4ee3\u7801\u53d8\u5f97\u66f4\u52a0\u590d\u6742\u4e86\u3002\u5c3d\u7ba1\u8fed\u4ee3\u548c\u9012\u5f52\u5728\u5f88\u591a\u60c5\u51b5\u4e0b\u53ef\u4ee5\u4e92\u76f8\u8f6c\u5316\uff0c\u4f46\u4e0d\u4e00\u5b9a\u503c\u5f97\u8fd9\u6837\u505a\uff0c\u6709\u4ee5\u4e0b\u4e24\u70b9\u539f\u56e0\u3002</p> <ul> <li>\u8f6c\u5316\u540e\u7684\u4ee3\u7801\u53ef\u80fd\u66f4\u52a0\u96be\u4ee5\u7406\u89e3\uff0c\u53ef\u8bfb\u6027\u66f4\u5dee\u3002</li> <li>\u5bf9\u4e8e\u67d0\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\u7684\u884c\u4e3a\u53ef\u80fd\u975e\u5e38\u56f0\u96be\u3002</li> </ul> <p>\u603b\u4e4b\uff0c\u9009\u62e9\u8fed\u4ee3\u8fd8\u662f\u9012\u5f52\u53d6\u51b3\u4e8e\u7279\u5b9a\u95ee\u9898\u7684\u6027\u8d28\u3002\u5728\u7f16\u7a0b\u5b9e\u8df5\u4e2d\uff0c\u6743\u8861\u4e24\u8005\u7684\u4f18\u52a3\u5e76\u6839\u636e\u60c5\u5883\u9009\u62e9\u5408\u9002\u7684\u65b9\u6cd5\u81f3\u5173\u91cd\u8981\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1 \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30","text":"<p>\u5728\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\uff0c\u6211\u4eec\u5148\u540e\u8ffd\u6c42\u4ee5\u4e0b\u4e24\u4e2a\u5c42\u9762\u7684\u76ee\u6807\u3002</p> <ol> <li>\u627e\u5230\u95ee\u9898\u89e3\u6cd5\uff1a\u7b97\u6cd5\u9700\u8981\u5728\u89c4\u5b9a\u7684\u8f93\u5165\u8303\u56f4\u5185\u53ef\u9760\u5730\u6c42\u5f97\u95ee\u9898\u7684\u6b63\u786e\u89e3\u3002</li> <li>\u5bfb\u6c42\u6700\u4f18\u89e3\u6cd5\uff1a\u540c\u4e00\u4e2a\u95ee\u9898\u53ef\u80fd\u5b58\u5728\u591a\u79cd\u89e3\u6cd5\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u5c3d\u53ef\u80fd\u9ad8\u6548\u7684\u7b97\u6cd5\u3002</li> </ol> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5728\u80fd\u591f\u89e3\u51b3\u95ee\u9898\u7684\u524d\u63d0\u4e0b\uff0c\u7b97\u6cd5\u6548\u7387\u5df2\u6210\u4e3a\u8861\u91cf\u7b97\u6cd5\u4f18\u52a3\u7684\u4e3b\u8981\u8bc4\u4ef7\u6307\u6807\uff0c\u5b83\u5305\u62ec\u4ee5\u4e0b\u4e24\u4e2a\u7ef4\u5ea6\u3002</p> <ul> <li>\u65f6\u95f4\u6548\u7387\uff1a\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u5feb\u6162\u3002</li> <li>\u7a7a\u95f4\u6548\u7387\uff1a\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u7684\u5927\u5c0f\u3002</li> </ul> <p>\u7b80\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u8bbe\u8ba1\u201c\u65e2\u5feb\u53c8\u7701\u201d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u3002\u800c\u6709\u6548\u5730\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u81f3\u5173\u91cd\u8981\uff0c\u56e0\u4e3a\u53ea\u6709\u8fd9\u6837\uff0c\u6211\u4eec\u624d\u80fd\u5c06\u5404\u79cd\u7b97\u6cd5\u8fdb\u884c\u5bf9\u6bd4\uff0c\u8fdb\u800c\u6307\u5bfc\u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u4f18\u5316\u8fc7\u7a0b\u3002</p> <p>\u6548\u7387\u8bc4\u4f30\u65b9\u6cd5\u4e3b\u8981\u5206\u4e3a\u4e24\u79cd\uff1a\u5b9e\u9645\u6d4b\u8bd5\u3001\u7406\u8bba\u4f30\u7b97\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#211","title":"2.1.1 \u00a0 \u5b9e\u9645\u6d4b\u8bd5","text":"<p>\u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u7b97\u6cd5 <code>A</code> \u548c\u7b97\u6cd5 <code>B</code> \uff0c\u5b83\u4eec\u90fd\u80fd\u89e3\u51b3\u540c\u4e00\u95ee\u9898\uff0c\u73b0\u5728\u9700\u8981\u5bf9\u6bd4\u8fd9\u4e24\u4e2a\u7b97\u6cd5\u7684\u6548\u7387\u3002\u6700\u76f4\u63a5\u7684\u65b9\u6cd5\u662f\u627e\u4e00\u53f0\u8ba1\u7b97\u673a\uff0c\u8fd0\u884c\u8fd9\u4e24\u4e2a\u7b97\u6cd5\uff0c\u5e76\u76d1\u63a7\u8bb0\u5f55\u5b83\u4eec\u7684\u8fd0\u884c\u65f6\u95f4\u548c\u5185\u5b58\u5360\u7528\u60c5\u51b5\u3002\u8fd9\u79cd\u8bc4\u4f30\u65b9\u5f0f\u80fd\u591f\u53cd\u6620\u771f\u5b9e\u60c5\u51b5\uff0c\u4f46\u4e5f\u5b58\u5728\u8f83\u5927\u7684\u5c40\u9650\u6027\u3002</p> <p>\u4e00\u65b9\u9762\uff0c\u96be\u4ee5\u6392\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5e72\u6270\u56e0\u7d20\u3002\u786c\u4ef6\u914d\u7f6e\u4f1a\u5f71\u54cd\u7b97\u6cd5\u7684\u6027\u80fd\u3002\u6bd4\u5982\u5728\u67d0\u53f0\u8ba1\u7b97\u673a\u4e2d\uff0c\u7b97\u6cd5 <code>A</code> \u7684\u8fd0\u884c\u65f6\u95f4\u6bd4\u7b97\u6cd5 <code>B</code> \u77ed\uff1b\u4f46\u5728\u53e6\u4e00\u53f0\u914d\u7f6e\u4e0d\u540c\u7684\u8ba1\u7b97\u673a\u4e2d\uff0c\u53ef\u80fd\u5f97\u5230\u76f8\u53cd\u7684\u6d4b\u8bd5\u7ed3\u679c\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u9700\u8981\u5728\u5404\u79cd\u673a\u5668\u4e0a\u8fdb\u884c\u6d4b\u8bd5\uff0c\u7edf\u8ba1\u5e73\u5747\u6548\u7387\uff0c\u800c\u8fd9\u662f\u4e0d\u73b0\u5b9e\u7684\u3002</p> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5c55\u5f00\u5b8c\u6574\u6d4b\u8bd5\u975e\u5e38\u8017\u8d39\u8d44\u6e90\u3002\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u7684\u53d8\u5316\uff0c\u7b97\u6cd5\u4f1a\u8868\u73b0\u51fa\u4e0d\u540c\u7684\u6548\u7387\u3002\u4f8b\u5982\uff0c\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 <code>A</code> \u7684\u8fd0\u884c\u65f6\u95f4\u6bd4\u7b97\u6cd5 <code>B</code> \u77ed\uff1b\u800c\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5927\u65f6\uff0c\u6d4b\u8bd5\u7ed3\u679c\u53ef\u80fd\u6070\u6070\u76f8\u53cd\u3002\u56e0\u6b64\uff0c\u4e3a\u4e86\u5f97\u5230\u6709\u8bf4\u670d\u529b\u7684\u7ed3\u8bba\uff0c\u6211\u4eec\u9700\u8981\u6d4b\u8bd5\u5404\u79cd\u89c4\u6a21\u7684\u8f93\u5165\u6570\u636e\uff0c\u800c\u8fd9\u9700\u8981\u8017\u8d39\u5927\u91cf\u7684\u8ba1\u7b97\u8d44\u6e90\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#212","title":"2.1.2 \u00a0 \u7406\u8bba\u4f30\u7b97","text":"<p>\u7531\u4e8e\u5b9e\u9645\u6d4b\u8bd5\u5177\u6709\u8f83\u5927\u7684\u5c40\u9650\u6027\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u4ec5\u901a\u8fc7\u4e00\u4e9b\u8ba1\u7b97\u6765\u8bc4\u4f30\u7b97\u6cd5\u7684\u6548\u7387\u3002\u8fd9\u79cd\u4f30\u7b97\u65b9\u6cd5\u88ab\u79f0\u4e3a\u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790\uff08asymptotic complexity analysis\uff09\uff0c\u7b80\u79f0\u590d\u6742\u5ea6\u5206\u6790\u3002</p> <p>\u590d\u6742\u5ea6\u5206\u6790\u80fd\u591f\u4f53\u73b0\u7b97\u6cd5\u8fd0\u884c\u6240\u9700\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u5b83\u63cf\u8ff0\u4e86\u968f\u7740\u8f93\u5165\u6570\u636e\u5927\u5c0f\u7684\u589e\u52a0\uff0c\u7b97\u6cd5\u6267\u884c\u6240\u9700\u65f6\u95f4\u548c\u7a7a\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u5b9a\u4e49\u6709\u4e9b\u62d7\u53e3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u5206\u4e3a\u4e09\u4e2a\u91cd\u70b9\u6765\u7406\u89e3\u3002</p> <ul> <li>\u201c\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\u201d\u5206\u522b\u5bf9\u5e94\u65f6\u95f4\u590d\u6742\u5ea6\uff08time complexity\uff09\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\uff08space complexity\uff09\u3002</li> <li>\u201c\u968f\u7740\u8f93\u5165\u6570\u636e\u5927\u5c0f\u7684\u589e\u52a0\u201d\u610f\u5473\u7740\u590d\u6742\u5ea6\u53cd\u6620\u4e86\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u4e0e\u8f93\u5165\u6570\u636e\u4f53\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u3002</li> <li>\u201c\u65f6\u95f4\u548c\u7a7a\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u201d\u8868\u793a\u590d\u6742\u5ea6\u5206\u6790\u5173\u6ce8\u7684\u4e0d\u662f\u8fd0\u884c\u65f6\u95f4\u6216\u5360\u7528\u7a7a\u95f4\u7684\u5177\u4f53\u503c\uff0c\u800c\u662f\u65f6\u95f4\u6216\u7a7a\u95f4\u589e\u957f\u7684\u201c\u5feb\u6162\u201d\u3002</li> </ul> <p>\u590d\u6742\u5ea6\u5206\u6790\u514b\u670d\u4e86\u5b9e\u9645\u6d4b\u8bd5\u65b9\u6cd5\u7684\u5f0a\u7aef\uff0c\u4f53\u73b0\u5728\u4ee5\u4e0b\u4e24\u4e2a\u65b9\u9762\u3002</p> <ul> <li>\u5b83\u72ec\u7acb\u4e8e\u6d4b\u8bd5\u73af\u5883\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\u3002</li> <li>\u5b83\u53ef\u4ee5\u4f53\u73b0\u4e0d\u540c\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6548\u7387\uff0c\u5c24\u5176\u662f\u5728\u5927\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6027\u80fd\u3002</li> </ul> <p>Tip</p> <p>\u5982\u679c\u4f60\u4ecd\u5bf9\u590d\u6742\u5ea6\u7684\u6982\u5ff5\u611f\u5230\u56f0\u60d1\uff0c\u65e0\u987b\u62c5\u5fc3\uff0c\u6211\u4eec\u4f1a\u5728\u540e\u7eed\u7ae0\u8282\u4e2d\u8be6\u7ec6\u4ecb\u7ecd\u3002</p> <p>\u590d\u6742\u5ea6\u5206\u6790\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u4e00\u628a\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u7684\u201c\u6807\u5c3a\u201d\uff0c\u4f7f\u6211\u4eec\u53ef\u4ee5\u8861\u91cf\u6267\u884c\u67d0\u4e2a\u7b97\u6cd5\u6240\u9700\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\uff0c\u5bf9\u6bd4\u4e0d\u540c\u7b97\u6cd5\u4e4b\u95f4\u7684\u6548\u7387\u3002</p> <p>\u590d\u6742\u5ea6\u662f\u4e2a\u6570\u5b66\u6982\u5ff5\uff0c\u5bf9\u4e8e\u521d\u5b66\u8005\u53ef\u80fd\u6bd4\u8f83\u62bd\u8c61\uff0c\u5b66\u4e60\u96be\u5ea6\u76f8\u5bf9\u8f83\u9ad8\u3002\u4ece\u8fd9\u4e2a\u89d2\u5ea6\u770b\uff0c\u590d\u6742\u5ea6\u5206\u6790\u53ef\u80fd\u4e0d\u592a\u9002\u5408\u4f5c\u4e3a\u6700\u5148\u4ecb\u7ecd\u7684\u5185\u5bb9\u3002\u7136\u800c\uff0c\u5f53\u6211\u4eec\u8ba8\u8bba\u67d0\u4e2a\u6570\u636e\u7ed3\u6784\u6216\u7b97\u6cd5\u7684\u7279\u70b9\u65f6\uff0c\u96be\u4ee5\u907f\u514d\u8981\u5206\u6790\u5176\u8fd0\u884c\u901f\u5ea6\u548c\u7a7a\u95f4\u4f7f\u7528\u60c5\u51b5\u3002</p> <p>\u7efc\u4e0a\u6240\u8ff0\uff0c\u5efa\u8bae\u4f60\u5728\u6df1\u5165\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u524d\uff0c\u5148\u5bf9\u590d\u6742\u5ea6\u5206\u6790\u5efa\u7acb\u521d\u6b65\u7684\u4e86\u89e3\uff0c\u4ee5\u4fbf\u80fd\u591f\u5b8c\u6210\u7b80\u5355\u7b97\u6cd5\u7684\u590d\u6742\u5ea6\u5206\u6790\u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/","title":"2.4 \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6","text":"<p>\u7a7a\u95f4\u590d\u6742\u5ea6\uff08space complexity\uff09\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u6982\u5ff5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u5e38\u7c7b\u4f3c\uff0c\u53ea\u9700\u5c06\u201c\u8fd0\u884c\u65f6\u95f4\u201d\u66ff\u6362\u4e3a\u201c\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u201d\u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/#241","title":"2.4.1 \u00a0 \u7b97\u6cd5\u76f8\u5173\u7a7a\u95f4","text":"<p>\u7b97\u6cd5\u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f7f\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u3002</p> <ul> <li>\u8f93\u5165\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u5165\u6570\u636e\u3002</li> <li>\u6682\u5b58\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u53d8\u91cf\u3001\u5bf9\u8c61\u3001\u51fd\u6570\u4e0a\u4e0b\u6587\u7b49\u6570\u636e\u3002</li> <li>\u8f93\u51fa\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u51fa\u6570\u636e\u3002</li> </ul> <p>\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u7edf\u8ba1\u8303\u56f4\u662f\u201c\u6682\u5b58\u7a7a\u95f4\u201d\u52a0\u4e0a\u201c\u8f93\u51fa\u7a7a\u95f4\u201d\u3002</p> <p>\u6682\u5b58\u7a7a\u95f4\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\u3002</p> <ul> <li>\u6682\u5b58\u6570\u636e\uff1a\u7528\u4e8e\u4fdd\u5b58\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u5404\u79cd\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u7b49\u3002</li> <li>\u6808\u5e27\u7a7a\u95f4\uff1a\u7528\u4e8e\u4fdd\u5b58\u8c03\u7528\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u3002\u7cfb\u7edf\u5728\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\u90fd\u4f1a\u5728\u6808\u9876\u90e8\u521b\u5efa\u4e00\u4e2a\u6808\u5e27\uff0c\u51fd\u6570\u8fd4\u56de\u540e\uff0c\u6808\u5e27\u7a7a\u95f4\u4f1a\u88ab\u91ca\u653e\u3002</li> <li>\u6307\u4ee4\u7a7a\u95f4\uff1a\u7528\u4e8e\u4fdd\u5b58\u7f16\u8bd1\u540e\u7684\u7a0b\u5e8f\u6307\u4ee4\uff0c\u5728\u5b9e\u9645\u7edf\u8ba1\u4e2d\u901a\u5e38\u5ffd\u7565\u4e0d\u8ba1\u3002</li> </ul> <p>\u5728\u5206\u6790\u4e00\u6bb5\u7a0b\u5e8f\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u7edf\u8ba1\u6682\u5b58\u6570\u636e\u3001\u6808\u5e27\u7a7a\u95f4\u548c\u8f93\u51fa\u6570\u636e\u4e09\u90e8\u5206\uff0c\u5982\u56fe 2-15 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 2-15 \u00a0 \u7b97\u6cd5\u4f7f\u7528\u7684\u76f8\u5173\u7a7a\u95f4 </p> <p>\u76f8\u5173\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class Node:\n    \"\"\"\u7c7b\"\"\"\n    def __init__(self, x: int):\n        self.val: int = x              # \u8282\u70b9\u503c\n        self.next: Node | None = None  # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n\ndef function() -&gt; int:\n    \"\"\"\u51fd\u6570\"\"\"\n    # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n\ndef algorithm(n) -&gt; int:  # \u8f93\u5165\u6570\u636e\n    A = 0                 # \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff0c\u4e00\u822c\u7528\u5927\u5199\u5b57\u6bcd\u8868\u793a\uff09\n    b = 0                 # \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    node = Node(0)        # \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    c = function()        # \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return A + b + c      # \u8f93\u51fa\u6570\u636e\n</code></pre> <pre><code>/* \u7ed3\u6784\u4f53 */\nstruct Node {\n    int val;\n    Node *next;\n    Node(int x) : val(x), next(nullptr) {}\n};\n\n/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\n    const int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    int b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    Node* node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    int c = func();           // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\n    int val;\n    Node next;\n    Node(int x) { val = x; }\n}\n\n/* \u51fd\u6570 */\nint function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\n    final int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    int b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    Node node = new Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    int c = function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node(int x) {\n    int val = x;\n    Node next;\n}\n\n/* \u51fd\u6570 */\nint Function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint Algorithm(int n) {        // \u8f93\u5165\u6570\u636e\n    const int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    int b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    Node node = new(0);       // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    int c = Function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7ed3\u6784\u4f53 */\ntype node struct {\n    val  int\n    next *node\n}\n\n/* \u521b\u5efa node \u7ed3\u6784\u4f53  */\nfunc newNode(val int) *node {\n    return &amp;node{val: val}\n}\n\n/* \u51fd\u6570 */\nfunc function() int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\nfunc algorithm(n int) int { // \u8f93\u5165\u6570\u636e\n    const a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    b := 0                  // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    newNode(0)              // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    c := function()         // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c        // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\n    var val: Int\n    var next: Node?\n\n    init(x: Int) {\n        val = x\n    }\n}\n\n/* \u51fd\u6570 */\nfunc function() -&gt; Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\nfunc algorithm(n: Int) -&gt; Int { // \u8f93\u5165\u6570\u636e\n    let a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    var b = 0             // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    let node = Node(x: 0) // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    let c = function()    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c      // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\n    val;\n    next;\n    constructor(val) {\n        this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n        this.next = null;                       // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n\n/* \u51fd\u6570 */\nfunction constFunc() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\nfunction algorithm(n) {       // \u8f93\u5165\u6570\u636e\n    const a = 0;              // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    let b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    const node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    const c = constFunc();    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\n    val: number;\n    next: Node | null;\n    constructor(val?: number) {\n        this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n        this.next = null;                       // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n\n/* \u51fd\u6570 */\nfunction constFunc(): number {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\nfunction algorithm(n: number): number { // \u8f93\u5165\u6570\u636e\n    const a = 0;                        // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    let b = 0;                          // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    const node = new Node(0);           // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    const c = constFunc();              // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;                   // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\n  int val;\n  Node next;\n  Node(this.val, [this.next]);\n}\n\n/* \u51fd\u6570 */\nint function() {\n  // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n  return 0;\n}\n\nint algorithm(int n) {  // \u8f93\u5165\u6570\u636e\n  const int a = 0;      // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n  int b = 0;            // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n  Node node = Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n  int c = function();   // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n  return a + b + c;     // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u7ed3\u6784\u4f53 */\nstruct Node {\n    val: i32,\n    next: Option&lt;Rc&lt;RefCell&lt;Node&gt;&gt;&gt;,\n}\n\n/* \u521b\u5efa Node \u7ed3\u6784\u4f53 */\nimpl Node {\n    fn new(val: i32) -&gt; Self {\n        Self { val: val, next: None }\n    }\n}\n\n/* \u51fd\u6570 */\nfn function() -&gt; i32 {      \n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nfn algorithm(n: i32) -&gt; i32 {       // \u8f93\u5165\u6570\u636e\n    const a: i32 = 0;               // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    let mut b = 0;                  // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    let node = Node::new(0);        // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    let c = function();             // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;               // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint algorithm(int n) { // \u8f93\u5165\u6570\u636e\n    const int a = 0;   // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    int b = 0;         // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    int c = func();    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;  // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node(var _val: Int) {\n    var next: Node? = null\n}\n\n/* \u51fd\u6570 */\nfun function(): Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\nfun algorithm(n: Int): Int { // \u8f93\u5165\u6570\u636e\n    val a = 0                // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    var b = 0                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    val node = Node(0)       // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    val c = function()       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre>"},{"location":"chapter_computational_complexity/space_complexity/#242","title":"2.4.2 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"<p>\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u5927\u81f4\u76f8\u540c\uff0c\u53ea\u9700\u5c06\u7edf\u8ba1\u5bf9\u8c61\u4ece\u201c\u64cd\u4f5c\u6570\u91cf\u201d\u8f6c\u4e3a\u201c\u4f7f\u7528\u7a7a\u95f4\u5927\u5c0f\u201d\u3002</p> <p>\u800c\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u7684\u662f\uff0c\u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u3002\u8fd9\u662f\u56e0\u4e3a\u5185\u5b58\u7a7a\u95f4\u662f\u4e00\u9879\u786c\u6027\u8981\u6c42\uff0c\u6211\u4eec\u5fc5\u987b\u786e\u4fdd\u5728\u6240\u6709\u8f93\u5165\u6570\u636e\u4e0b\u90fd\u6709\u8db3\u591f\u7684\u5185\u5b58\u7a7a\u95f4\u9884\u7559\u3002</p> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u201c\u6700\u5dee\u201d\u6709\u4e24\u5c42\u542b\u4e49\u3002</p> <ol> <li>\u4ee5\u6700\u5dee\u8f93\u5165\u6570\u636e\u4e3a\u51c6\uff1a\u5f53 \\(n &lt; 10\\) \u65f6\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff1b\u4f46\u5f53 \\(n &gt; 10\\) \u65f6\uff0c\u521d\u59cb\u5316\u7684\u6570\u7ec4 <code>nums</code> \u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</li> <li>\u4ee5\u7b97\u6cd5\u8fd0\u884c\u4e2d\u7684\u5cf0\u503c\u5185\u5b58\u4e3a\u51c6\uff1a\u4f8b\u5982\uff0c\u7a0b\u5e8f\u5728\u6267\u884c\u6700\u540e\u4e00\u884c\u4e4b\u524d\uff0c\u5360\u7528 \\(O(1)\\) \u7a7a\u95f4\uff1b\u5f53\u521d\u59cb\u5316\u6570\u7ec4 <code>nums</code> \u65f6\uff0c\u7a0b\u5e8f\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</li> </ol> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def algorithm(n: int):\n    a = 0               # O(1)\n    b = [0] * 10000     # O(1)\n    if n &gt; 10:\n        nums = [0] * n  # O(n)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    vector&lt;int&gt; b(10000);    // O(1)\n    if (n &gt; 10)\n        vector&lt;int&gt; nums(n); // O(n)\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10)\n        int[] nums = new int[n]; // O(n)\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10) {\n        int[] nums = new int[n]; // O(n)\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 0                      // O(1)\n    b := make([]int, 10000)     // O(1)\n    var nums []int\n    if n &gt; 10 {\n        nums := make([]int, n)  // O(n)\n    }\n    fmt.Println(a, b, nums)\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    let a = 0 // O(1)\n    let b = Array(repeating: 0, count: 10000) // O(1)\n    if n &gt; 10 {\n        let nums = Array(repeating: 0, count: n) // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 0;                            // O(1)\n  List&lt;int&gt; b = List.filled(10000, 0);  // O(1)\n  if (n &gt; 10) {\n    List&lt;int&gt; nums = List.filled(n, 0); // O(n)\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let a = 0;                              // O(1)\n    let b = [0; 10000];                     // O(1)\n    if n &gt; 10 {\n        let nums = vec![0; n as usize];     // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    int b[10000];            // O(1)\n    if (n &gt; 10)\n        int nums[n] = {0};   // O(n)\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    val a = 0                    // O(1)\n    val b = IntArray(10000)      // O(1)\n    if (n &gt; 10) {\n        val nums = IntArray(n)   // O(n)\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u7edf\u8ba1\u6808\u5e27\u7a7a\u95f4\u3002\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def function() -&gt; int:\n    # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n\ndef loop(n: int):\n    \"\"\"\u5faa\u73af\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(1)\"\"\"\n    for _ in range(n):\n        function()\n\ndef recur(n: int):\n    \"\"\"\u9012\u5f52\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\"\"\"\n    if n == 1:\n        return\n    return recur(n - 1)\n</code></pre> <pre><code>int func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int Function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid Loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n/* \u9012\u5f52 O(n) */\nint Recur(int n) {\n    if (n == 1) return 1;\n    return Recur(n - 1);\n}\n</code></pre> <pre><code>func function() int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5faa\u73af O(1) */\nfunc loop(n int) {\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n}\n\n/* \u9012\u5f52 O(n) */\nfunc recur(n int) {\n    if n == 1 {\n        return\n    }\n    recur(n - 1)\n}\n</code></pre> <pre><code>@discardableResult\nfunc function() -&gt; Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5faa\u73af O(1) */\nfunc loop(n: Int) {\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n\n/* \u9012\u5f52 O(n) */\nfunc recur(n: Int) {\n    if n == 1 {\n        return\n    }\n    recur(n: n - 1)\n}\n</code></pre> <pre><code>function constFunc() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n) {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n) {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>function constFunc(): number {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n: number): void {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n  // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n  return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n  if (n == 1) return;\n  return recur(n - 1);\n}\n</code></pre> <pre><code>fn function() -&gt; i32 {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nfn loop(n: i32) {\n    for i in 0..n {\n        function();\n    }\n}\n/* \u9012\u5f52 O(n) */\nfn recur(n: i32) {\n    if n == 1 {\n        return;\n    }\n    recur(n - 1);\n}\n</code></pre> <pre><code>int func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>fun function(): Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n/* \u5faa\u73af O(1) */\nfun loop(n: Int) {\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n/* \u9012\u5f52 O(n) */\nfun recur(n: Int) {\n    if (n == 1) return\n    return recur(n - 1)\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>\u51fd\u6570 <code>loop()</code> \u548c <code>recur()</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n)\\) \uff0c\u4f46\u7a7a\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u3002</p> <ul> <li>\u51fd\u6570 <code>loop()</code> \u5728\u5faa\u73af\u4e2d\u8c03\u7528\u4e86 \\(n\\) \u6b21 <code>function()</code> \uff0c\u6bcf\u8f6e\u4e2d\u7684 <code>function()</code> \u90fd\u8fd4\u56de\u5e76\u91ca\u653e\u4e86\u6808\u5e27\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002</li> <li>\u9012\u5f52\u51fd\u6570 <code>recur()</code> \u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f1a\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 <code>recur()</code> \uff0c\u4ece\u800c\u5360\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_computational_complexity/space_complexity/#243","title":"2.4.3 \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"<p>\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u56fe 2-16 \u5c55\u793a\u4e86\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\uff08\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\uff09\u3002</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \\newline \\text{\u5e38\u6570\u9636} &lt; \\text{\u5bf9\u6570\u9636} &lt; \\text{\u7ebf\u6027\u9636} &lt; \\text{\u5e73\u65b9\u9636} &lt; \\text{\u6307\u6570\u9636} \\end{aligned} \\] <p></p> <p> \u56fe 2-16 \u00a0 \u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b </p>"},{"location":"chapter_computational_complexity/space_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6570\u9636 \\(O(1)\\)","text":"<p>\u5e38\u6570\u9636\u5e38\u89c1\u4e8e\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u7684\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u5faa\u73af\u4e2d\u521d\u59cb\u5316\u53d8\u91cf\u6216\u8c03\u7528\u51fd\u6570\u800c\u5360\u7528\u7684\u5185\u5b58\uff0c\u5728\u8fdb\u5165\u4e0b\u4e00\u5faa\u73af\u540e\u5c31\u4f1a\u88ab\u91ca\u653e\uff0c\u56e0\u6b64\u4e0d\u4f1a\u7d2f\u79ef\u5360\u7528\u7a7a\u95f4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def function() -&gt; int:\n    \"\"\"\u51fd\u6570\"\"\"\n    # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n\ndef constant(n: int):\n    \"\"\"\u5e38\u6570\u9636\"\"\"\n    # \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    a = 0\n    nums = [0] * 10000\n    node = ListNode(0)\n    # \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in range(n):\n        c = 0\n    # \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in range(n):\n        function()\n</code></pre> space_complexity.cpp<pre><code>/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const int a = 0;\n    int b = 0;\n    vector&lt;int&gt; nums(10000);\n    ListNode node(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u51fd\u6570 */\nint function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    final int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new ListNode(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u51fd\u6570 */\nint Function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid Constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u51fd\u6570 */\nfunc function() int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc spaceConstant(n int) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a = 0\n    b := 0\n    nums := make([]int, 10000)\n    node := newNode(0)\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    var c int\n    for i := 0; i &lt; n; i++ {\n        c = 0\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n    b += 0\n    c += 0\n    nums[0] = 0\n    node.val = 0\n}\n</code></pre> space_complexity.swift<pre><code>/* \u51fd\u6570 */\n@discardableResult\nfunc function() -&gt; Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    let a = 0\n    var b = 0\n    let nums = Array(repeating: 0, count: 10000)\n    let node = ListNode(x: 0)\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in 0 ..&lt; n {\n        let c = 0\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n</code></pre> space_complexity.js<pre><code>/* \u51fd\u6570 */\nfunction constFunc() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u51fd\u6570 */\nfunction constFunc(): number {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n: number): void {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u51fd\u6570 */\nint function() {\n  // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n  return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n  // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n  final int a = 0;\n  int b = 0;\n  List&lt;int&gt; nums = List.filled(10000, 0);\n  ListNode node = ListNode(0);\n  // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n  for (var i = 0; i &lt; n; i++) {\n    int c = 0;\n  }\n  // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n  for (var i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u51fd\u6570 */\nfn function() -&gt; i32 {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\n#[allow(unused)]\nfn constant(n: i32) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const A: i32 = 0;\n    let b = 0;\n    let nums = vec![0; 10000];\n    let node = ListNode::new(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for i in 0..n {\n        let c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for i in 0..n {\n        function();\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const int a = 0;\n    int b = 0;\n    int nums[1000];\n    ListNode *node = newListNode(0);\n    free(node);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* \u51fd\u6570 */\nfun function(): Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfun constant(n: Int) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    val a = 0\n    var b = 0\n    val nums = Array(10000) { 0 }\n    val node = ListNode(0)\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (i in 0..&lt;n) {\n        val c = 0\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{function}\n\n[class]{}-[func]{constant}\n</code></pre> space_complexity.zig<pre><code>// \u51fd\u6570\nfn function() i32 {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n// \u5e38\u6570\u9636\nfn constant(n: i32) void {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a: i32 = 0;\n    var b: i32 = 0;\n    var nums = [_]i32{0}**10000;\n    var node = inc.ListNode(i32){.val = 0};\n    var i: i32 = 0;\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    while (i &lt; n) : (i += 1) {\n        var c: i32 = 0;\n        _ = c;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    i = 0;\n    while (i &lt; n) : (i += 1) {\n        _ = function();\n    }\n    _ = a;\n    _ = b;\n    _ = nums;\n    _ = node;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_computational_complexity/space_complexity/#2-on","title":"2. \u00a0 \u7ebf\u6027\u9636 \\(O(n)\\)","text":"<p>\u7ebf\u6027\u9636\u5e38\u89c1\u4e8e\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u6b63\u6bd4\u7684\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u7b49\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def linear(n: int):\n    \"\"\"\u7ebf\u6027\u9636\"\"\"\n    # \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    nums = [0] * n\n    # \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    hmap = dict[int, str]()\n    for i in range(n):\n        hmap[i] = str(i)\n</code></pre> space_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    vector&lt;int&gt; nums(n);\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    vector&lt;ListNode&gt; nodes;\n    for (int i = 0; i &lt; n; i++) {\n        nodes.push_back(ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    unordered_map&lt;int, string&gt; map;\n    for (int i = 0; i &lt; n; i++) {\n        map[i] = to_string(i);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    int[] nums = new int[n];\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    List&lt;ListNode&gt; nodes = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        nodes.add(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    Map&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        map.put(i, String.valueOf(i));\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7ebf\u6027\u9636 */\nvoid Linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    int[] nums = new int[n];\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    List&lt;ListNode&gt; nodes = [];\n    for (int i = 0; i &lt; n; i++) {\n        nodes.Add(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    Dictionary&lt;int, string&gt; map = [];\n    for (int i = 0; i &lt; n; i++) {\n        map.Add(i, i.ToString());\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u7ebf\u6027\u9636 */\nfunc spaceLinear(n int) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    _ = make([]int, n)\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    var nodes []*node\n    for i := 0; i &lt; n; i++ {\n        nodes = append(nodes, newNode(i))\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    m := make(map[int]string, n)\n    for i := 0; i &lt; n; i++ {\n        m[i] = strconv.Itoa(i)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    let nums = Array(repeating: 0, count: n)\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let nodes = (0 ..&lt; n).map { ListNode(x: $0) }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let map = Dictionary(uniqueKeysWithValues: (0 ..&lt; n).map { ($0, \"\\($0)\") })\n}\n</code></pre> space_complexity.js<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    const nums = new Array(n);\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const nodes = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): void {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    const nums = new Array(n);\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const nodes: ListNode[] = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n  // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n  List&lt;int&gt; nums = List.filled(n, 0);\n  // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n  List&lt;ListNode&gt; nodes = [];\n  for (var i = 0; i &lt; n; i++) {\n    nodes.add(ListNode(i));\n  }\n  // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n  Map&lt;int, String&gt; map = HashMap();\n  for (var i = 0; i &lt; n; i++) {\n    map.putIfAbsent(i, () =&gt; i.toString());\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7ebf\u6027\u9636 */\n#[allow(unused)]\nfn linear(n: i32) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    let mut nums = vec![0; n as usize];\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let mut nodes = Vec::new();\n    for i in 0..n {\n        nodes.push(ListNode::new(i))\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let mut map = HashMap::new();\n    for i in 0..n {\n        map.insert(i, i.to_string());\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u54c8\u5e0c\u8868 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n\n/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    int *nums = malloc(sizeof(int) * n);\n    free(nums);\n\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    ListNode **nodes = malloc(sizeof(ListNode *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        nodes[i] = newListNode(i);\n    }\n    // \u5185\u5b58\u91ca\u653e\n    for (int i = 0; i &lt; n; i++) {\n        free(nodes[i]);\n    }\n    free(nodes);\n\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    HashTable *h = NULL;\n    for (int i = 0; i &lt; n; i++) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = i;\n        tmp-&gt;val = i;\n        HASH_ADD_INT(h, key, tmp);\n    }\n\n    // \u5185\u5b58\u91ca\u653e\n    HashTable *curr, *tmp;\n    HASH_ITER(hh, h, curr, tmp) {\n        HASH_DEL(h, curr);\n        free(curr);\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* \u7ebf\u6027\u9636 */\nfun linear(n: Int) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    val nums = Array(n) { 0 }\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    val nodes = mutableListOf&lt;ListNode&gt;()\n    for (i in 0..&lt;n) {\n        nodes.add(ListNode(i))\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    val map = mutableMapOf&lt;Int, String&gt;()\n    for (i in 0..&lt;n) {\n        map[i] = i.toString()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{linear}\n</code></pre> space_complexity.zig<pre><code>// \u7ebf\u6027\u9636\nfn linear(comptime n: i32) !void {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    var nums = [_]i32{0}**n;\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    var nodes = std.ArrayList(i32).init(std.heap.page_allocator);\n    defer nodes.deinit();\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        try nodes.append(i);\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    var map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\n    defer map.deinit();\n    var j: i32 = 0;\n    while (j &lt; n) : (j += 1) {\n        const string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\n        defer std.heap.page_allocator.free(string);\n        try map.put(i, string);\n    }\n    _ = nums;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 2-17 \u6240\u793a\uff0c\u6b64\u51fd\u6570\u7684\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u5373\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 <code>linear_recur()</code> \u51fd\u6570\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def linear_recur(n: int):\n    \"\"\"\u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    print(\"\u9012\u5f52 n =\", n)\n    if n == 1:\n        return\n    linear_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n    cout &lt;&lt; \"\u9012\u5f52 n = \" &lt;&lt; n &lt;&lt; endl;\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n    System.out.println(\"\u9012\u5f52 n = \" + n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid LinearRecur(int n) {\n    Console.WriteLine(\"\u9012\u5f52 n = \" + n);\n    if (n == 1) return;\n    LinearRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceLinearRecur(n int) {\n    fmt.Println(\"\u9012\u5f52 n =\", n)\n    if n == 1 {\n        return\n    }\n    spaceLinearRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc linearRecur(n: Int) {\n    print(\"\u9012\u5f52 n = \\(n)\")\n    if n == 1 {\n        return\n    }\n    linearRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n) {\n    console.log(`\u9012\u5f52 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n: number): void {\n    console.log(`\u9012\u5f52 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n  print('\u9012\u5f52 n = $n');\n  if (n == 1) return;\n  linearRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn linear_recur(n: i32) {\n    println!(\"\u9012\u5f52 n = {}\", n);\n    if n == 1 {\n        return;\n    };\n    linear_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n    printf(\"\u9012\u5f52 n = %d\\r\\n\", n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.kt<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun linearRecur(n: Int) {\n    println(\"\u9012\u5f52 n = $n\")\n    if (n == 1)\n        return\n    linearRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{linear_recur}\n</code></pre> space_complexity.zig<pre><code>// \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn linearRecur(comptime n: i32) void {\n    std.debug.print(\"\u9012\u5f52 n = {}\\n\", .{n});\n    if (n == 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 2-17 \u00a0 \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u7ebf\u6027\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"<p>\u5e73\u65b9\u9636\u5e38\u89c1\u4e8e\u77e9\u9635\u548c\u56fe\uff0c\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u5e73\u65b9\u5173\u7cfb\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def quadratic(n: int):\n    \"\"\"\u5e73\u65b9\u9636\"\"\"\n    # \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    num_matrix = [[0] * n for _ in range(n)]\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    vector&lt;vector&lt;int&gt;&gt; numMatrix;\n    for (int i = 0; i &lt; n; i++) {\n        vector&lt;int&gt; tmp;\n        for (int j = 0; j &lt; n; j++) {\n            tmp.push_back(0);\n        }\n        numMatrix.push_back(tmp);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    int[][] numMatrix = new int[n][n];\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    List&lt;List&lt;Integer&gt;&gt; numList = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            tmp.add(0);\n        }\n        numList.add(tmp);\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u9636 */\nvoid Quadratic(int n) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    int[,] numMatrix = new int[n, n];\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    List&lt;List&lt;int&gt;&gt; numList = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;int&gt; tmp = [];\n        for (int j = 0; j &lt; n; j++) {\n            tmp.Add(0);\n        }\n        numList.Add(tmp);\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u9636 */\nfunc spaceQuadratic(n int) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    numMatrix := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        numMatrix[i] = make([]int, n)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    let numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): void {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n  // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n  List&lt;List&lt;int&gt;&gt; numMatrix = List.generate(n, (_) =&gt; List.filled(n, 0));\n  // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n  List&lt;List&lt;int&gt;&gt; numList = [];\n  for (var i = 0; i &lt; n; i++) {\n    List&lt;int&gt; tmp = [];\n    for (int j = 0; j &lt; n; j++) {\n      tmp.add(0);\n    }\n    numList.add(tmp);\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u9636 */\n#[allow(unused)]\nfn quadratic(n: i32) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    let num_matrix = vec![vec![0; n as usize]; n as usize];\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    let mut num_list = Vec::new();\n    for i in 0..n {\n        let mut tmp = Vec::new();\n        for j in 0..n {\n            tmp.push(0);\n        }\n        num_list.push(tmp);\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    int **numMatrix = malloc(sizeof(int *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        int *tmp = malloc(sizeof(int) * n);\n        for (int j = 0; j &lt; n; j++) {\n            tmp[j] = 0;\n        }\n        numMatrix[i] = tmp;\n    }\n\n    // \u5185\u5b58\u91ca\u653e\n    for (int i = 0; i &lt; n; i++) {\n        free(numMatrix[i]);\n    }\n    free(numMatrix);\n}\n</code></pre> space_complexity.kt<pre><code>/* \u5e73\u65b9\u9636 */\nfun quadratic(n: Int) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    val numMatrix: Array&lt;Array&lt;Int&gt;?&gt; = arrayOfNulls(n)\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    val numList: MutableList&lt;MutableList&lt;Int&gt;&gt; = arrayListOf()\n    for (i in 0..&lt;n) {\n        val tmp = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            tmp.add(0)\n        }\n        numList.add(tmp)\n    }\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{quadratic}\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u9636\nfn quadratic(n: i32) !void {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    var nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\n    defer nodes.deinit();\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        var tmp = std.ArrayList(i32).init(std.heap.page_allocator);\n        defer tmp.deinit();\n        var j: i32 = 0;\n        while (j &lt; n) : (j += 1) {\n            try tmp.append(0);\n        }\n        try nodes.append(tmp);\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 2-18 \u6240\u793a\uff0c\u8be5\u51fd\u6570\u7684\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u5728\u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u4e2d\u90fd\u521d\u59cb\u5316\u4e86\u4e00\u4e2a\u6570\u7ec4\uff0c\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u5e73\u5747\u957f\u5ea6\u4e3a \\(n / 2\\) \uff0c\u56e0\u6b64\u603b\u4f53\u5360\u7528 \\(O(n^2)\\) \u7a7a\u95f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def quadratic_recur(n: int) -&gt; int:\n    \"\"\"\u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n &lt;= 0:\n        return 0\n    # \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    nums = [0] * n\n    return quadratic_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    vector&lt;int&gt; nums(n);\n    cout &lt;&lt; \"\u9012\u5f52 n = \" &lt;&lt; n &lt;&lt; \" \u4e2d\u7684 nums \u957f\u5ea6 = \" &lt;&lt; nums.size() &lt;&lt; endl;\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    int[] nums = new int[n];\n    System.out.println(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.length);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint QuadraticRecur(int n) {\n    if (n &lt;= 0) return 0;\n    int[] nums = new int[n];\n    Console.WriteLine(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.Length);\n    return QuadraticRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceQuadraticRecur(n int) int {\n    if n &lt;= 0 {\n        return 0\n    }\n    nums := make([]int, n)\n    fmt.Printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d \\n\", n, len(nums))\n    return spaceQuadraticRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -&gt; Int {\n    if n &lt;= 0 {\n        return 0\n    }\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    let nums = Array(repeating: 0, count: n)\n    print(\"\u9012\u5f52 n = \\(n) \u4e2d\u7684 nums \u957f\u5ea6 = \\(nums.count)\")\n    return quadraticRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n) {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n: number): number {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n  if (n &lt;= 0) return 0;\n  List&lt;int&gt; nums = List.filled(n, 0);\n  print('\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}');\n  return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn quadratic_recur(n: i32) -&gt; i32 {\n    if n &lt;= 0 {\n        return 0;\n    };\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    let nums = vec![0; n as usize];\n    println!(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\", n, nums.len());\n    return quadratic_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    int *nums = malloc(sizeof(int) * n);\n    printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d\\r\\n\", n, n);\n    int res = quadraticRecur(n - 1);\n    free(nums);\n    return res;\n}\n</code></pre> space_complexity.kt<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\ntailrec fun quadraticRecur(n: Int): Int {\n    if (n &lt;= 0)\n        return 0\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    val nums = Array(n) { 0 }\n    println(\"\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.size}\")\n    return quadraticRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{quadratic_recur}\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn quadraticRecur(comptime n: i32) i32 {\n    if (n &lt;= 0) return 0;\n    var nums = [_]i32{0}**n;\n    std.debug.print(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\\n\", .{n, nums.len});\n    return quadraticRecur(n - 1);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 2-18 \u00a0 \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u5e73\u65b9\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6570\u9636 \\(O(2^n)\\)","text":"<p>\u6307\u6570\u9636\u5e38\u89c1\u4e8e\u4e8c\u53c9\u6811\u3002\u89c2\u5bdf\u56fe 2-19 \uff0c\u5c42\u6570\u4e3a \\(n\\) \u7684\u201c\u6ee1\u4e8c\u53c9\u6811\u201d\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(2^n - 1\\) \uff0c\u5360\u7528 \\(O(2^n)\\) \u7a7a\u95f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig space_complexity.py<pre><code>def build_tree(n: int) -&gt; TreeNode | None:\n    \"\"\"\u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\"\"\"\n    if n == 0:\n        return None\n    root = TreeNode(0)\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n    return root\n</code></pre> space_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return nullptr;\n    TreeNode *root = new TreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode buildTree(int n) {\n    if (n == 0)\n        return null;\n    TreeNode root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? BuildTree(int n) {\n    if (n == 0) return null;\n    TreeNode root = new(0) {\n        left = BuildTree(n - 1),\n        right = BuildTree(n - 1)\n    };\n    return root;\n}\n</code></pre> space_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n int) *TreeNode {\n    if n == 0 {\n        return nil\n    }\n    root := NewTreeNode(0)\n    root.Left = buildTree(n - 1)\n    root.Right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n: Int) -&gt; TreeNode? {\n    if n == 0 {\n        return nil\n    }\n    let root = TreeNode(x: 0)\n    root.left = buildTree(n: n - 1)\n    root.right = buildTree(n: n - 1)\n    return root\n}\n</code></pre> space_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n) {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n: number): TreeNode | null {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n) {\n  if (n == 0) return null;\n  TreeNode root = TreeNode(0);\n  root.left = buildTree(n - 1);\n  root.right = buildTree(n - 1);\n  return root;\n}\n</code></pre> space_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfn build_tree(n: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    if n == 0 {\n        return None;\n    };\n    let root = TreeNode::new(0);\n    root.borrow_mut().left = build_tree(n - 1);\n    root.borrow_mut().right = build_tree(n - 1);\n    return Some(root);\n}\n</code></pre> space_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return NULL;\n    TreeNode *root = newTreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.kt<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfun buildTree(n: Int): TreeNode? {\n    if (n == 0)\n        return null\n    val root = TreeNode(0)\n    root.left = buildTree(n - 1)\n    root.right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.rb<pre><code>[class]{}-[func]{build_tree}\n</code></pre> space_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\n    if (n == 0) return null;\n    const root = try mem_allocator.create(inc.TreeNode(i32));\n    root.init(0);\n    root.left = try buildTree(mem_allocator, n - 1);\n    root.right = try buildTree(mem_allocator, n - 1);\n    return root;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 2-19 \u00a0 \u6ee1\u4e8c\u53c9\u6811\u4ea7\u751f\u7684\u6307\u6570\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#5-olog-n","title":"5. \u00a0 \u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"<p>\u5bf9\u6570\u9636\u5e38\u89c1\u4e8e\u5206\u6cbb\u7b97\u6cd5\u3002\u4f8b\u5982\u5f52\u5e76\u6392\u5e8f\uff0c\u8f93\u5165\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u6bcf\u8f6e\u9012\u5f52\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5212\u5206\u4e3a\u4e24\u534a\uff0c\u5f62\u6210\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u6808\u5e27\u7a7a\u95f4\u3002</p> <p>\u518d\u4f8b\u5982\u5c06\u6570\u5b57\u8f6c\u5316\u4e3a\u5b57\u7b26\u4e32\uff0c\u8f93\u5165\u4e00\u4e2a\u6b63\u6574\u6570 \\(n\\) \uff0c\u5b83\u7684\u4f4d\u6570\u4e3a \\(\\lfloor \\log_{10} n \\rfloor + 1\\) \uff0c\u5373\u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u5ea6\u4e3a \\(\\lfloor \\log_{10} n \\rfloor + 1\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_{10} n + 1) = O(\\log n)\\) \u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/#244","title":"2.4.4 \u00a0 \u6743\u8861\u65f6\u95f4\u4e0e\u7a7a\u95f4","text":"<p>\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u80fd\u8fbe\u5230\u6700\u4f18\u3002\u7136\u800c\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u540c\u65f6\u4f18\u5316\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u975e\u5e38\u56f0\u96be\u3002</p> <p>\u964d\u4f4e\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u9700\u8981\u4ee5\u63d0\u5347\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a\u4ee3\u4ef7\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u6211\u4eec\u5c06\u727a\u7272\u5185\u5b58\u7a7a\u95f4\u6765\u63d0\u5347\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u601d\u8def\u79f0\u4e3a\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff1b\u53cd\u4e4b\uff0c\u5219\u79f0\u4e3a\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002</p> <p>\u9009\u62e9\u54ea\u79cd\u601d\u8def\u53d6\u51b3\u4e8e\u6211\u4eec\u66f4\u770b\u91cd\u54ea\u4e2a\u65b9\u9762\u3002\u5728\u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u65f6\u95f4\u6bd4\u7a7a\u95f4\u66f4\u5b9d\u8d35\uff0c\u56e0\u6b64\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u901a\u5e38\u662f\u66f4\u5e38\u7528\u7684\u7b56\u7565\u3002\u5f53\u7136\uff0c\u5728\u6570\u636e\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\uff0c\u63a7\u5236\u7a7a\u95f4\u590d\u6742\u5ea6\u4e5f\u975e\u5e38\u91cd\u8981\u3002</p>"},{"location":"chapter_computational_complexity/summary/","title":"2.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_computational_complexity/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<p>\u7b97\u6cd5\u6548\u7387\u8bc4\u4f30</p> <ul> <li>\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u8861\u91cf\u7b97\u6cd5\u4f18\u52a3\u7684\u4e24\u4e2a\u4e3b\u8981\u8bc4\u4ef7\u6307\u6807\u3002</li> <li>\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5b9e\u9645\u6d4b\u8bd5\u6765\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u96be\u4ee5\u6d88\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5f71\u54cd\uff0c\u4e14\u4f1a\u8017\u8d39\u5927\u91cf\u8ba1\u7b97\u8d44\u6e90\u3002</li> <li>\u590d\u6742\u5ea6\u5206\u6790\u53ef\u4ee5\u6d88\u9664\u5b9e\u9645\u6d4b\u8bd5\u7684\u5f0a\u7aef\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\uff0c\u5e76\u4e14\u80fd\u591f\u63ed\u793a\u7b97\u6cd5\u5728\u4e0d\u540c\u6570\u636e\u89c4\u6a21\u4e0b\u7684\u6548\u7387\u3002</li> </ul> <p>\u65f6\u95f4\u590d\u6742\u5ea6</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\uff0c\u53ef\u4ee5\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u53ef\u80fd\u5931\u6548\uff0c\u5982\u5728\u8f93\u5165\u7684\u6570\u636e\u91cf\u8f83\u5c0f\u6216\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\u65f6\uff0c\u65e0\u6cd5\u7cbe\u786e\u5bf9\u6bd4\u7b97\u6cd5\u6548\u7387\u7684\u4f18\u52a3\u3002</li> <li>\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f7f\u7528\u5927 \\(O\\) \u7b26\u53f7\u8868\u793a\uff0c\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u53cd\u6620\u5f53 \\(n\\) \u8d8b\u5411\u6b63\u65e0\u7a77\u65f6\uff0c\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u7684\u589e\u957f\u7ea7\u522b\u3002</li> <li>\u63a8\u7b97\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u4e24\u6b65\uff0c\u9996\u5148\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002</li> <li>\u5e38\u89c1\u65f6\u95f4\u590d\u6742\u5ea6\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n \\log n)\\)\u3001\\(O(n^2)\\)\u3001\\(O(2^n)\\) \u548c \\(O(n!)\\) \u7b49\u3002</li> <li>\u67d0\u4e9b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u56fa\u5b9a\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u51e0\u4e4e\u4e0d\u7528\uff0c\u56e0\u4e3a\u8f93\u5165\u6570\u636e\u4e00\u822c\u9700\u8981\u6ee1\u8db3\u4e25\u683c\u6761\u4ef6\u624d\u80fd\u8fbe\u5230\u6700\u4f73\u60c5\u51b5\u3002</li> <li>\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53cd\u6620\u7b97\u6cd5\u5728\u968f\u673a\u6570\u636e\u8f93\u5165\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u6700\u63a5\u8fd1\u5b9e\u9645\u5e94\u7528\u4e2d\u7684\u7b97\u6cd5\u6027\u80fd\u3002\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u9700\u8981\u7edf\u8ba1\u8f93\u5165\u6570\u636e\u5206\u5e03\u4ee5\u53ca\u7efc\u5408\u540e\u7684\u6570\u5b66\u671f\u671b\u3002</li> </ul> <p>\u7a7a\u95f4\u590d\u6742\u5ea6</p> <ul> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u4f5c\u7528\u7c7b\u4f3c\u4e8e\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\u3002</li> <li>\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u76f8\u5173\u5185\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u8f93\u5165\u7a7a\u95f4\u3001\u6682\u5b58\u7a7a\u95f4\u3001\u8f93\u51fa\u7a7a\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u8f93\u5165\u7a7a\u95f4\u4e0d\u7eb3\u5165\u7a7a\u95f4\u590d\u6742\u5ea6\u8ba1\u7b97\u3002\u6682\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u6682\u5b58\u6570\u636e\u3001\u6808\u5e27\u7a7a\u95f4\u548c\u6307\u4ee4\u7a7a\u95f4\uff0c\u5176\u4e2d\u6808\u5e27\u7a7a\u95f4\u901a\u5e38\u4ec5\u5728\u9012\u5f52\u51fd\u6570\u4e2d\u5f71\u54cd\u7a7a\u95f4\u590d\u6742\u5ea6\u3002</li> <li>\u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5373\u7edf\u8ba1\u7b97\u6cd5\u5728\u6700\u5dee\u8f93\u5165\u6570\u636e\u548c\u6700\u5dee\u8fd0\u884c\u65f6\u523b\u4e0b\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u3002</li> <li>\u5e38\u89c1\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n^2)\\) \u548c \\(O(2^n)\\) \u7b49\u3002</li> </ul>"},{"location":"chapter_computational_complexity/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u5c3e\u9012\u5f52\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u5417\uff1f</p> <p>\u7406\u8bba\u4e0a\uff0c\u5c3e\u9012\u5f52\u51fd\u6570\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \u3002\u4e0d\u8fc7\u7edd\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java\u3001Python\u3001C++\u3001Go\u3001C# \u7b49\uff09\u4e0d\u652f\u6301\u81ea\u52a8\u4f18\u5316\u5c3e\u9012\u5f52\uff0c\u56e0\u6b64\u901a\u5e38\u8ba4\u4e3a\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(n)\\) \u3002</p> <p>Q\uff1a\u51fd\u6570\u548c\u65b9\u6cd5\u8fd9\u4e24\u4e2a\u672f\u8bed\u7684\u533a\u522b\u662f\u4ec0\u4e48\uff1f</p> <p>\u51fd\u6570\uff08function\uff09\u53ef\u4ee5\u88ab\u72ec\u7acb\u6267\u884c\uff0c\u6240\u6709\u53c2\u6570\u90fd\u4ee5\u663e\u5f0f\u4f20\u9012\u3002\u65b9\u6cd5\uff08method\uff09\u4e0e\u4e00\u4e2a\u5bf9\u8c61\u5173\u8054\uff0c\u88ab\u9690\u5f0f\u4f20\u9012\u7ed9\u8c03\u7528\u5b83\u7684\u5bf9\u8c61\uff0c\u80fd\u591f\u5bf9\u7c7b\u7684\u5b9e\u4f8b\u4e2d\u5305\u542b\u7684\u6570\u636e\u8fdb\u884c\u64cd\u4f5c\u3002</p> <p>\u4e0b\u9762\u4ee5\u51e0\u79cd\u5e38\u89c1\u7684\u7f16\u7a0b\u8bed\u8a00\u4e3a\u4f8b\u6765\u8bf4\u660e\u3002</p> <ul> <li>C \u8bed\u8a00\u662f\u8fc7\u7a0b\u5f0f\u7f16\u7a0b\u8bed\u8a00\uff0c\u6ca1\u6709\u9762\u5411\u5bf9\u8c61\u7684\u6982\u5ff5\uff0c\u6240\u4ee5\u53ea\u6709\u51fd\u6570\u3002\u4f46\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u521b\u5efa\u7ed3\u6784\u4f53\uff08struct\uff09\u6765\u6a21\u62df\u9762\u5411\u5bf9\u8c61\u7f16\u7a0b\uff0c\u4e0e\u7ed3\u6784\u4f53\u76f8\u5173\u8054\u7684\u51fd\u6570\u5c31\u76f8\u5f53\u4e8e\u5176\u4ed6\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u65b9\u6cd5\u3002</li> <li>Java \u548c C# \u662f\u9762\u5411\u5bf9\u8c61\u7684\u7f16\u7a0b\u8bed\u8a00\uff0c\u4ee3\u7801\u5757\uff08\u65b9\u6cd5\uff09\u901a\u5e38\u4f5c\u4e3a\u67d0\u4e2a\u7c7b\u7684\u4e00\u90e8\u5206\u3002\u9759\u6001\u65b9\u6cd5\u7684\u884c\u4e3a\u7c7b\u4f3c\u4e8e\u51fd\u6570\uff0c\u56e0\u4e3a\u5b83\u88ab\u7ed1\u5b9a\u5728\u7c7b\u4e0a\uff0c\u4e0d\u80fd\u8bbf\u95ee\u7279\u5b9a\u7684\u5b9e\u4f8b\u53d8\u91cf\u3002</li> <li>C++ \u548c Python \u65e2\u652f\u6301\u8fc7\u7a0b\u5f0f\u7f16\u7a0b\uff08\u51fd\u6570\uff09\uff0c\u4e5f\u652f\u6301\u9762\u5411\u5bf9\u8c61\u7f16\u7a0b\uff08\u65b9\u6cd5\uff09\u3002</li> </ul> <p>Q\uff1a\u56fe\u89e3\u201c\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u201d\u53cd\u6620\u7684\u662f\u5426\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\uff1f</p> <p>\u4e0d\u662f\uff0c\u8be5\u56fe\u5c55\u793a\u7684\u662f\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5176\u53cd\u6620\u7684\u662f\u589e\u957f\u8d8b\u52bf\uff0c\u800c\u4e0d\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\u3002</p> <p>\u5047\u8bbe\u53d6 \\(n = 8\\) \uff0c\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\u6bcf\u6761\u66f2\u7ebf\u7684\u503c\u4e0e\u51fd\u6570\u5bf9\u5e94\u4e0d\u4e0a\u3002\u8fd9\u662f\u56e0\u4e3a\u6bcf\u6761\u66f2\u7ebf\u90fd\u5305\u542b\u4e00\u4e2a\u5e38\u6570\u9879\uff0c\u7528\u4e8e\u5c06\u53d6\u503c\u8303\u56f4\u538b\u7f29\u5230\u4e00\u4e2a\u89c6\u89c9\u8212\u9002\u7684\u8303\u56f4\u5185\u3002</p> <p>\u5728\u5b9e\u9645\u4e2d\uff0c\u56e0\u4e3a\u6211\u4eec\u901a\u5e38\u4e0d\u77e5\u9053\u6bcf\u4e2a\u65b9\u6cd5\u7684\u201c\u5e38\u6570\u9879\u201d\u590d\u6742\u5ea6\u662f\u591a\u5c11\uff0c\u6240\u4ee5\u4e00\u822c\u65e0\u6cd5\u4ec5\u51ed\u590d\u6742\u5ea6\u6765\u9009\u62e9 \\(n = 8\\) \u4e4b\u4e0b\u7684\u6700\u4f18\u89e3\u6cd5\u3002\u4f46\u5bf9\u4e8e \\(n = 8^5\\) \u5c31\u5f88\u597d\u9009\u4e86\uff0c\u8fd9\u65f6\u589e\u957f\u8d8b\u52bf\u5df2\u7ecf\u5360\u4e3b\u5bfc\u4e86\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/","title":"2.3 \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6","text":"<p>\u8fd0\u884c\u65f6\u95f4\u53ef\u4ee5\u76f4\u89c2\u4e14\u51c6\u786e\u5730\u53cd\u6620\u7b97\u6cd5\u7684\u6548\u7387\u3002\u5982\u679c\u6211\u4eec\u60f3\u51c6\u786e\u9884\u4f30\u4e00\u6bb5\u4ee3\u7801\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u5e94\u8be5\u5982\u4f55\u64cd\u4f5c\u5462\uff1f</p> <ol> <li>\u786e\u5b9a\u8fd0\u884c\u5e73\u53f0\uff0c\u5305\u62ec\u786c\u4ef6\u914d\u7f6e\u3001\u7f16\u7a0b\u8bed\u8a00\u3001\u7cfb\u7edf\u73af\u5883\u7b49\uff0c\u8fd9\u4e9b\u56e0\u7d20\u90fd\u4f1a\u5f71\u54cd\u4ee3\u7801\u7684\u8fd0\u884c\u6548\u7387\u3002</li> <li>\u8bc4\u4f30\u5404\u79cd\u8ba1\u7b97\u64cd\u4f5c\u6240\u9700\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u4f8b\u5982\u52a0\u6cd5\u64cd\u4f5c <code>+</code> \u9700\u8981 1 ns \uff0c\u4e58\u6cd5\u64cd\u4f5c <code>*</code> \u9700\u8981 10 ns \uff0c\u6253\u5370\u64cd\u4f5c <code>print()</code> \u9700\u8981 5 ns \u7b49\u3002</li> <li>\u7edf\u8ba1\u4ee3\u7801\u4e2d\u6240\u6709\u7684\u8ba1\u7b97\u64cd\u4f5c\uff0c\u5e76\u5c06\u6240\u6709\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u6c42\u548c\uff0c\u4ece\u800c\u5f97\u5230\u8fd0\u884c\u65f6\u95f4\u3002</li> </ol> <p>\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\ndef algorithm(n: int):\n    a = 2      # 1 ns\n    a = a + 1  # 1 ns\n    a = a * 2  # 10 ns\n    # \u5faa\u73af n \u6b21\n    for _ in range(n):  # 1 ns\n        print(0)        # 5 ns\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        cout &lt;&lt; 0 &lt;&lt; endl;         // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        System.out.println(0);     // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid Algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        Console.WriteLine(0);      // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n int) {\n    a := 2     // 1 ns\n    a = a + 1  // 1 ns\n    a = a * 2  // 10 ns\n    // \u5faa\u73af n \u6b21\n    for i := 0; i &lt; n; i++ {  // 1 ns\n        fmt.Println(a)        // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // \u5faa\u73af n \u6b21\n    for _ in 0 ..&lt; n { // 1 ns\n        print(0) // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n) {\n    var a = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // \u5faa\u73af n \u6b21\n    for(let i = 0; i &lt; n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n: number): void {\n    var a: number = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // \u5faa\u73af n \u6b21\n    for(let i = 0; i &lt; n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n  int a = 2; // 1 ns\n  a = a + 1; // 1 ns\n  a = a * 2; // 10 ns\n  // \u5faa\u73af n \u6b21\n  for (int i = 0; i &lt; n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n    print(0); // 5 ns\n  }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfn algorithm(n: i32) {\n    let mut a = 2;      // 1 ns\n    a = a + 1;          // 1 ns\n    a = a * 2;          // 10 ns\n    // \u5faa\u73af n \u6b21\n    for _ in 0..n {     // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        println!(\"{}\", 0);  // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {   // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        printf(\"%d\", 0);            // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfun algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (i in 0..&lt;n) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        println(0)      // 5 ns\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfn algorithm(n: usize) void {\n    var a: i32 = 2; // 1 ns\n    a += 1; // 1 ns\n    a *= 2; // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (0..n) |_| { // 1 ns\n        std.debug.print(\"{}\\n\", .{0}); // 5 ns\n    }\n}\n</code></pre> <p>\u6839\u636e\u4ee5\u4e0a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5f97\u5230\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u4e3a \\((6n + 12)\\) ns \uff1a</p> \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] <p>\u4f46\u5b9e\u9645\u4e0a\uff0c\u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u65e2\u4e0d\u5408\u7406\u4e5f\u4e0d\u73b0\u5b9e\u3002\u9996\u5148\uff0c\u6211\u4eec\u4e0d\u5e0c\u671b\u5c06\u9884\u4f30\u65f6\u95f4\u548c\u8fd0\u884c\u5e73\u53f0\u7ed1\u5b9a\uff0c\u56e0\u4e3a\u7b97\u6cd5\u9700\u8981\u5728\u5404\u79cd\u4e0d\u540c\u7684\u5e73\u53f0\u4e0a\u8fd0\u884c\u3002\u5176\u6b21\uff0c\u6211\u4eec\u5f88\u96be\u83b7\u77e5\u6bcf\u79cd\u64cd\u4f5c\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u8fd9\u7ed9\u9884\u4f30\u8fc7\u7a0b\u5e26\u6765\u4e86\u6781\u5927\u7684\u96be\u5ea6\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#231","title":"2.3.1 \u00a0 \u7edf\u8ba1\u65f6\u95f4\u589e\u957f\u8d8b\u52bf","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u7edf\u8ba1\u7684\u4e0d\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\uff0c\u800c\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002</p> <p>\u201c\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u201d\u8fd9\u4e2a\u6982\u5ff5\u6bd4\u8f83\u62bd\u8c61\uff0c\u6211\u4eec\u901a\u8fc7\u4e00\u4e2a\u4f8b\u5b50\u6765\u52a0\u4ee5\u7406\u89e3\u3002\u5047\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7ed9\u5b9a\u4e09\u4e2a\u7b97\u6cd5 <code>A</code>\u3001<code>B</code> \u548c <code>C</code> \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_A(n: int):\n    print(0)\n# \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\ndef algorithm_B(n: int):\n    for _ in range(n):\n        print(0)\n# \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_C(n: int):\n    for _ in range(1000000):\n        print(0)\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\n    cout &lt;&lt; 0 &lt;&lt; endl;\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\n    System.out.println(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        System.out.println(0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        System.out.println(0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid AlgorithmA(int n) {\n    Console.WriteLine(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid AlgorithmB(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Console.WriteLine(0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid AlgorithmC(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        Console.WriteLine(0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_A(n int) {\n    fmt.Println(0)\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithm_B(n int) {\n    for i := 0; i &lt; n; i++ {\n        fmt.Println(0)\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_C(n int) {\n    for i := 0; i &lt; 1000000; i++ {\n        fmt.Println(0)\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmA(n: Int) {\n    print(0)\n}\n\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithmB(n: Int) {\n    for _ in 0 ..&lt; n {\n        print(0)\n    }\n}\n\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmC(n: Int) {\n    for _ in 0 ..&lt; 1_000_000 {\n        print(0)\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n) {\n    console.log(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n) {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n) {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n: number): void {\n    console.log(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n: number): void {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmA(int n) {\n  print(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithmB(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    print(0);\n  }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmC(int n) {\n  for (int i = 0; i &lt; 1000000; i++) {\n    print(0);\n  }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_A(n: i32) {\n    println!(\"{}\", 0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfn algorithm_B(n: i32) {\n    for _ in 0..n {\n        println!(\"{}\", 0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_C(n: i32) {\n    for _ in 0..1000000 {\n        println!(\"{}\", 0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\n    printf(\"%d\", 0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        printf(\"%d\", 0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        printf(\"%d\", 0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfun algoritm_A(n: Int) {\n    println(0)\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfun algorithm_B(n: Int) {\n    for (i in 0..&lt;n){\n        println(0)\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfun algorithm_C(n: Int) {\n    for (i in 0..&lt;1000000) {\n        println(0)\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_A(n: usize) void {\n    _ = n;\n    std.debug.print(\"{}\\n\", .{0});\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfn algorithm_B(n: i32) void {\n    for (0..n) |_| {\n        std.debug.print(\"{}\\n\", .{0});\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_C(n: i32) void {\n    _ = n;\n    for (0..1000000) |_| { \n        std.debug.print(\"{}\\n\", .{0});\n    }\n}\n</code></pre> <p>\u56fe 2-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u4e09\u4e2a\u7b97\u6cd5\u51fd\u6570\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <ul> <li>\u7b97\u6cd5 <code>A</code> \u53ea\u6709 \\(1\\) \u4e2a\u6253\u5370\u64cd\u4f5c\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u4e0d\u968f\u7740 \\(n\\) \u589e\u5927\u800c\u589e\u957f\u3002\u6211\u4eec\u79f0\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u201c\u5e38\u6570\u9636\u201d\u3002</li> <li>\u7b97\u6cd5 <code>B</code> \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(n\\) \u6b21\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740 \\(n\\) \u589e\u5927\u5448\u7ebf\u6027\u589e\u957f\u3002\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u88ab\u79f0\u4e3a\u201c\u7ebf\u6027\u9636\u201d\u3002</li> <li>\u7b97\u6cd5 <code>C</code> \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(1000000\\) \u6b21\uff0c\u867d\u7136\u8fd0\u884c\u65f6\u95f4\u5f88\u957f\uff0c\u4f46\u5b83\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u3002\u56e0\u6b64 <code>C</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c <code>A</code> \u76f8\u540c\uff0c\u4ecd\u4e3a\u201c\u5e38\u6570\u9636\u201d\u3002</li> </ul> <p></p> <p> \u56fe 2-7 \u00a0 \u7b97\u6cd5 A\u3001B \u548c C \u7684\u65f6\u95f4\u589e\u957f\u8d8b\u52bf </p> <p>\u76f8\u8f83\u4e8e\u76f4\u63a5\u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u6709\u54ea\u4e9b\u7279\u70b9\u5462\uff1f</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u80fd\u591f\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u3002\u4f8b\u5982\uff0c\u7b97\u6cd5 <code>B</code> \u7684\u8fd0\u884c\u65f6\u95f4\u5448\u7ebf\u6027\u589e\u957f\uff0c\u5728 \\(n &gt; 1\\) \u65f6\u6bd4\u7b97\u6cd5 <code>A</code> \u66f4\u6162\uff0c\u5728 \\(n &gt; 1000000\\) \u65f6\u6bd4\u7b97\u6cd5 <code>C</code> \u66f4\u6162\u3002\u4e8b\u5b9e\u4e0a\uff0c\u53ea\u8981\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8db3\u591f\u5927\uff0c\u590d\u6742\u5ea6\u4e3a\u201c\u5e38\u6570\u9636\u201d\u7684\u7b97\u6cd5\u4e00\u5b9a\u4f18\u4e8e\u201c\u7ebf\u6027\u9636\u201d\u7684\u7b97\u6cd5\uff0c\u8fd9\u6b63\u662f\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u7684\u542b\u4e49\u3002</li> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u66f4\u7b80\u4fbf\u3002\u663e\u7136\uff0c\u8fd0\u884c\u5e73\u53f0\u548c\u8ba1\u7b97\u64cd\u4f5c\u7c7b\u578b\u90fd\u4e0e\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u65e0\u5173\u3002\u56e0\u6b64\u5728\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u5c06\u6240\u6709\u8ba1\u7b97\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u89c6\u4e3a\u76f8\u540c\u7684\u201c\u5355\u4f4d\u65f6\u95f4\u201d\uff0c\u4ece\u800c\u5c06\u201c\u8ba1\u7b97\u64cd\u4f5c\u8fd0\u884c\u65f6\u95f4\u7edf\u8ba1\u201d\u7b80\u5316\u4e3a\u201c\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u7edf\u8ba1\u201d\uff0c\u8fd9\u6837\u4e00\u6765\u4f30\u7b97\u96be\u5ea6\u5c31\u5927\u5927\u964d\u4f4e\u4e86\u3002</li> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u5b58\u5728\u4e00\u5b9a\u7684\u5c40\u9650\u6027\u3002\u4f8b\u5982\uff0c\u5c3d\u7ba1\u7b97\u6cd5 <code>A</code> \u548c <code>C</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\uff0c\u4f46\u5b9e\u9645\u8fd0\u884c\u65f6\u95f4\u5dee\u522b\u5f88\u5927\u3002\u540c\u6837\uff0c\u5c3d\u7ba1\u7b97\u6cd5 <code>B</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6bd4 <code>C</code> \u9ad8\uff0c\u4f46\u5728\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 <code>B</code> \u660e\u663e\u4f18\u4e8e\u7b97\u6cd5 <code>C</code> \u3002\u5728\u8fd9\u4e9b\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5f88\u96be\u4ec5\u51ed\u65f6\u95f4\u590d\u6742\u5ea6\u5224\u65ad\u7b97\u6cd5\u6548\u7387\u7684\u9ad8\u4f4e\u3002\u5f53\u7136\uff0c\u5c3d\u7ba1\u5b58\u5728\u4e0a\u8ff0\u95ee\u9898\uff0c\u590d\u6742\u5ea6\u5206\u6790\u4ecd\u7136\u662f\u8bc4\u5224\u7b97\u6cd5\u6548\u7387\u6700\u6709\u6548\u4e14\u5e38\u7528\u7684\u65b9\u6cd5\u3002</li> </ul>"},{"location":"chapter_computational_complexity/time_complexity/#232","title":"2.3.2 \u00a0 \u51fd\u6570\u6e10\u8fd1\u4e0a\u754c","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u8f93\u5165\u5927\u5c0f\u4e3a \\(n\\) \u7684\u51fd\u6570\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def algorithm(n: int):\n    a = 1      # +1\n    a = a + 1  # +1\n    a = a * 2  # +1\n    # \u5faa\u73af n \u6b21\n    for i in range(n):  # +1\n        print(0)        # +1\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        cout &lt;&lt; 0 &lt;&lt; endl;    // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        System.out.println(0);    // +1\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {   // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        Console.WriteLine(0);   // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1      // +1\n    a = a + 1   // +1\n    a = a * 2   // +1\n    // \u5faa\u73af n \u6b21\n    for i := 0; i &lt; n; i++ {   // +1\n        fmt.Println(a)         // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // \u5faa\u73af n \u6b21\n    for _ in 0 ..&lt; n { // +1\n        print(0) // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    var a = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // \u5faa\u73af n \u6b21\n    for(let i = 0; i &lt; n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void{\n    var a: number = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // \u5faa\u73af n \u6b21\n    for(let i = 0; i &lt; n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +1\n  a = a + 1; // +1\n  a = a * 2; // +1\n  // \u5faa\u73af n \u6b21\n  for (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n    print(0); // +1\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;   // +1\n    a = a + 1;      // +1\n    a = a * 2;      // +1\n\n    // \u5faa\u73af n \u6b21\n    for _ in 0..n { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        println!(\"{}\", 0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {   // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        printf(\"%d\", 0);            // +1\n    }\n}  \n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // \u5faa\u73af n \u6b21\n    for (i in 0..&lt;n) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        println(0) // +1\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>fn algorithm(n: usize) void {\n    var a: i32 = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // \u5faa\u73af n \u6b21\n    for (0..n) |_| { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        std.debug.print(\"{}\\n\", .{0}); // +1\n    }\n}\n</code></pre> <p>\u8bbe\u7b97\u6cd5\u7684\u64cd\u4f5c\u6570\u91cf\u662f\u4e00\u4e2a\u5173\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u7684\u51fd\u6570\uff0c\u8bb0\u4e3a \\(T(n)\\) \uff0c\u5219\u4ee5\u4e0a\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e3a\uff1a</p> \\[ T(n) = 3 + 2n \\] <p>\\(T(n)\\) \u662f\u4e00\u6b21\u51fd\u6570\uff0c\u8bf4\u660e\u5176\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u662f\u7ebf\u6027\u7684\uff0c\u56e0\u6b64\u5b83\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u7ebf\u6027\u9636\u3002</p> <p>\u6211\u4eec\u5c06\u7ebf\u6027\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u8bb0\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e2a\u6570\u5b66\u7b26\u53f7\u79f0\u4e3a\u5927\uff08\\(O\\) \u8bb0\u53f7 big-\\(O\\) notation\uff09\uff0c\u8868\u793a\u51fd\u6570 \\(T(n)\\) \u7684\u6e10\u8fd1\u4e0a\u754c\uff08asymptotic upper bound\uff09\u3002</p> <p>\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u672c\u8d28\u4e0a\u662f\u8ba1\u7b97\u201c\u64cd\u4f5c\u6570\u91cf \\(T(n)\\)\u201d\u7684\u6e10\u8fd1\u4e0a\u754c\uff0c\u5b83\u5177\u6709\u660e\u786e\u7684\u6570\u5b66\u5b9a\u4e49\u3002</p> <p>\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c</p> <p>\u82e5\u5b58\u5728\u6b63\u5b9e\u6570 \\(c\\) \u548c\u5b9e\u6570 \\(n_0\\) \uff0c\u4f7f\u5f97\u5bf9\u4e8e\u6240\u6709\u7684 \\(n &gt; n_0\\) \uff0c\u5747\u6709 \\(T(n) \\leq c \\cdot f(n)\\) \uff0c\u5219\u53ef\u8ba4\u4e3a \\(f(n)\\) \u7ed9\u51fa\u4e86 \\(T(n)\\) \u7684\u4e00\u4e2a\u6e10\u8fd1\u4e0a\u754c\uff0c\u8bb0\u4e3a \\(T(n) = O(f(n))\\) \u3002</p> <p>\u5982\u56fe 2-8 \u6240\u793a\uff0c\u8ba1\u7b97\u6e10\u8fd1\u4e0a\u754c\u5c31\u662f\u5bfb\u627e\u4e00\u4e2a\u51fd\u6570 \\(f(n)\\) \uff0c\u4f7f\u5f97\u5f53 \\(n\\) \u8d8b\u5411\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\\(T(n)\\) \u548c \\(f(n)\\) \u5904\u4e8e\u76f8\u540c\u7684\u589e\u957f\u7ea7\u522b\uff0c\u4ec5\u76f8\u5dee\u4e00\u4e2a\u5e38\u6570\u9879 \\(c\\) \u7684\u500d\u6570\u3002</p> <p></p> <p> \u56fe 2-8 \u00a0 \u51fd\u6570\u7684\u6e10\u8fd1\u4e0a\u754c </p>"},{"location":"chapter_computational_complexity/time_complexity/#233","title":"2.3.3 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"<p>\u6e10\u8fd1\u4e0a\u754c\u7684\u6570\u5b66\u5473\u513f\u6709\u70b9\u91cd\uff0c\u5982\u679c\u4f60\u611f\u89c9\u6ca1\u6709\u5b8c\u5168\u7406\u89e3\uff0c\u4e5f\u65e0\u987b\u62c5\u5fc3\u3002\u6211\u4eec\u53ef\u4ee5\u5148\u638c\u63e1\u63a8\u7b97\u65b9\u6cd5\uff0c\u5728\u4e0d\u65ad\u7684\u5b9e\u8df5\u4e2d\uff0c\u5c31\u53ef\u4ee5\u9010\u6e10\u9886\u609f\u5176\u6570\u5b66\u610f\u4e49\u3002</p> <p>\u6839\u636e\u5b9a\u4e49\uff0c\u786e\u5b9a \\(f(n)\\) \u4e4b\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \u3002\u90a3\u4e48\u5982\u4f55\u786e\u5b9a\u6e10\u8fd1\u4e0a\u754c \\(f(n)\\) \u5462\uff1f\u603b\u4f53\u5206\u4e3a\u4e24\u6b65\uff1a\u9996\u5148\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#1","title":"1. \u00a0 \u7b2c\u4e00\u6b65\uff1a\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf","text":"<p>\u9488\u5bf9\u4ee3\u7801\uff0c\u9010\u884c\u4ece\u4e0a\u5230\u4e0b\u8ba1\u7b97\u5373\u53ef\u3002\u7136\u800c\uff0c\u7531\u4e8e\u4e0a\u8ff0 \\(c \\cdot f(n)\\) \u4e2d\u7684\u5e38\u6570\u9879 \\(c\\) \u53ef\u4ee5\u53d6\u4efb\u610f\u5927\u5c0f\uff0c\u56e0\u6b64\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u4e2d\u7684\u5404\u79cd\u7cfb\u6570\u3001\u5e38\u6570\u9879\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002\u6839\u636e\u6b64\u539f\u5219\uff0c\u53ef\u4ee5\u603b\u7ed3\u51fa\u4ee5\u4e0b\u8ba1\u6570\u7b80\u5316\u6280\u5de7\u3002</p> <ol> <li>\u5ffd\u7565 \\(T(n)\\) \u4e2d\u7684\u5e38\u6570\u9879\u3002\u56e0\u4e3a\u5b83\u4eec\u90fd\u4e0e \\(n\\) \u65e0\u5173\uff0c\u6240\u4ee5\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4ea7\u751f\u5f71\u54cd\u3002</li> <li>\u7701\u7565\u6240\u6709\u7cfb\u6570\u3002\u4f8b\u5982\uff0c\u5faa\u73af \\(2n\\) \u6b21\u3001\\(5n + 1\\) \u6b21\u7b49\uff0c\u90fd\u53ef\u4ee5\u7b80\u5316\u8bb0\u4e3a \\(n\\) \u6b21\uff0c\u56e0\u4e3a \\(n\\) \u524d\u9762\u7684\u7cfb\u6570\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u6ca1\u6709\u5f71\u54cd\u3002</li> <li>\u5faa\u73af\u5d4c\u5957\u65f6\u4f7f\u7528\u4e58\u6cd5\u3002\u603b\u64cd\u4f5c\u6570\u91cf\u7b49\u4e8e\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u64cd\u4f5c\u6570\u91cf\u4e4b\u79ef\uff0c\u6bcf\u4e00\u5c42\u5faa\u73af\u4f9d\u7136\u53ef\u4ee5\u5206\u522b\u5957\u7528\u7b2c <code>1.</code> \u70b9\u548c\u7b2c <code>2.</code> \u70b9\u7684\u6280\u5de7\u3002</li> </ol> <p>\u7ed9\u5b9a\u4e00\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u4e0a\u8ff0\u6280\u5de7\u6765\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>def algorithm(n: int):\n    a = 1      # +0\uff08\u6280\u5de7 1\uff09\n    a = a + n  # +0\uff08\u6280\u5de7 1\uff09\n    # +n\uff08\u6280\u5de7 2\uff09\n    for i in range(5 * n + 1):\n        print(0)\n    # +n*n\uff08\u6280\u5de7 3\uff09\n    for i in range(2 * n):\n        for j in range(n + 1):\n            print(0)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            cout &lt;&lt; 0 &lt;&lt; endl;\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        System.out.println(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            System.out.println(0);\n        }\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        Console.WriteLine(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            Console.WriteLine(0);\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1     // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for i := 0; i &lt; 5 * n + 1; i++ {\n        fmt.Println(0)\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for i := 0; i &lt; 2 * n; i++ {\n        for j := 0; j &lt; n + 1; j++ {\n            fmt.Println(0)\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for _ in 0 ..&lt; (5 * n + 1) {\n        print(0)\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for _ in 0 ..&lt; (2 * n) {\n        for _ in 0 ..&lt; (n + 1) {\n            print(0)\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    let a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    let a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +0\uff08\u6280\u5de7 1\uff09\n  a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n  // +n\uff08\u6280\u5de7 2\uff09\n  for (int i = 0; i &lt; 5 * n + 1; i++) {\n    print(0);\n  }\n  // +n*n\uff08\u6280\u5de7 3\uff09\n  for (int i = 0; i &lt; 2 * n; i++) {\n    for (int j = 0; j &lt; n + 1; j++) {\n      print(0);\n    }\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;     // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;        // +0\uff08\u6280\u5de7 1\uff09\n\n    // +n\uff08\u6280\u5de7 2\uff09\n    for i in 0..(5 * n + 1) {\n        println!(\"{}\", 0);\n    }\n\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for i in 0..(2 * n) {\n        for j in 0..(n + 1) {\n            println!(\"{}\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        printf(\"%d\", 0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            printf(\"%d\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1   // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n   // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (i in 0..&lt;5 * n + 1) {\n        println(0)\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (i in 0..&lt;2 * n) {\n        for (j in 0..&lt;n + 1) {\n            println(0)\n        }\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>fn algorithm(n: usize) void {\n    var a: i32 = 1;     // +0\uff08\u6280\u5de7 1\uff09\n    a = a + @as(i32, @intCast(n));        // +0\uff08\u6280\u5de7 1\uff09\n\n    // +n\uff08\u6280\u5de7 2\uff09\n    for(0..(5 * n + 1)) |_| {\n        std.debug.print(\"{}\\n\", .{0}); \n    }\n\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for(0..(2 * n)) |_| {\n        for(0..(n + 1)) |_| {\n            std.debug.print(\"{}\\n\", .{0}); \n        }\n    }\n}\n</code></pre> <p>\u4ee5\u4e0b\u516c\u5f0f\u5c55\u793a\u4e86\u4f7f\u7528\u4e0a\u8ff0\u6280\u5de7\u524d\u540e\u7684\u7edf\u8ba1\u7ed3\u679c\uff0c\u4e24\u8005\u63a8\u7b97\u51fa\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \u3002</p> \\[ \\begin{aligned} T(n) &amp; = 2n(n + 1) + (5n + 1) + 2 &amp; \\text{\u5b8c\u6574\u7edf\u8ba1 (-.-|||)} \\newline &amp; = 2n^2 + 7n + 3 \\newline T(n) &amp; = n^2 + n &amp; \\text{\u5077\u61d2\u7edf\u8ba1 (o.O)} \\end{aligned} \\]"},{"location":"chapter_computational_complexity/time_complexity/#2","title":"2. \u00a0 \u7b2c\u4e8c\u6b65\uff1a\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\u7531 \\(T(n)\\) \u4e2d\u6700\u9ad8\u9636\u7684\u9879\u6765\u51b3\u5b9a\u3002\u8fd9\u662f\u56e0\u4e3a\u5728 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u6700\u9ad8\u9636\u7684\u9879\u5c06\u53d1\u6325\u4e3b\u5bfc\u4f5c\u7528\uff0c\u5176\u4ed6\u9879\u7684\u5f71\u54cd\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002</p> <p>\u8868 2-2 \u5c55\u793a\u4e86\u4e00\u4e9b\u4f8b\u5b50\uff0c\u5176\u4e2d\u4e00\u4e9b\u5938\u5f20\u7684\u503c\u662f\u4e3a\u4e86\u5f3a\u8c03\u201c\u7cfb\u6570\u65e0\u6cd5\u64bc\u52a8\u9636\u6570\u201d\u8fd9\u4e00\u7ed3\u8bba\u3002\u5f53 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u8fd9\u4e9b\u5e38\u6570\u53d8\u5f97\u65e0\u8db3\u8f7b\u91cd\u3002</p> <p> \u8868 2-2 \u00a0 \u4e0d\u540c\u64cd\u4f5c\u6570\u91cf\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> \u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#234","title":"2.3.4 \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"<p>\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u5982\u56fe 2-9 \u6240\u793a\uff08\u6309\u7167\u4ece\u4f4e\u5230\u9ad8\u7684\u987a\u5e8f\u6392\u5217\uff09\u3002</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n \\log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!) \\newline \\text{\u5e38\u6570\u9636} &lt; \\text{\u5bf9\u6570\u9636} &lt; \\text{\u7ebf\u6027\u9636} &lt; \\text{\u7ebf\u6027\u5bf9\u6570\u9636} &lt; \\text{\u5e73\u65b9\u9636} &lt; \\text{\u6307\u6570\u9636} &lt; \\text{\u9636\u4e58\u9636} \\end{aligned} \\] <p></p> <p> \u56fe 2-9 \u00a0 \u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b </p>"},{"location":"chapter_computational_complexity/time_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6570\u9636 \\(O(1)\\)","text":"<p>\u5e38\u6570\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u5373\u4e0d\u968f\u7740 \\(n\\) \u7684\u53d8\u5316\u800c\u53d8\u5316\u3002</p> <p>\u5728\u4ee5\u4e0b\u51fd\u6570\u4e2d\uff0c\u5c3d\u7ba1\u64cd\u4f5c\u6570\u91cf <code>size</code> \u53ef\u80fd\u5f88\u5927\uff0c\u4f46\u7531\u4e8e\u5176\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def constant(n: int) -&gt; int:\n    \"\"\"\u5e38\u6570\u9636\"\"\"\n    count = 0\n    size = 100000\n    for _ in range(size):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e38\u6570\u9636 */\nint Constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e38\u6570\u9636 */\nfunc constant(n int) int {\n    count := 0\n    size := 100000\n    for i := 0; i &lt; size; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) -&gt; Int {\n    var count = 0\n    let size = 100_000\n    for _ in 0 ..&lt; size {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e38\u6570\u9636 */\nfunction constant(n: number): number {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n  int count = 0;\n  int size = 100000;\n  for (var i = 0; i &lt; size; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e38\u6570\u9636 */\nfn constant(n: i32) -&gt; i32 {\n    _ = n;\n    let mut count = 0;\n    let size = 100_000;\n    for _ in 0..size {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    int i = 0;\n    for (int i = 0; i &lt; size; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e38\u6570\u9636 */\nfun constant(n: Int): Int {\n    var count = 0\n    val size = 10_0000\n    for (i in 0..&lt;size)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{constant}\n</code></pre> time_complexity.zig<pre><code>// \u5e38\u6570\u9636\nfn constant(n: i32) i32 {\n    _ = n;\n    var count: i32 = 0;\n    const size: i32 = 100_000;\n    var i: i32 = 0;\n    while(i&lt;size) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_computational_complexity/time_complexity/#2-on","title":"2. \u00a0 \u7ebf\u6027\u9636 \\(O(n)\\)","text":"<p>\u7ebf\u6027\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u4ee5\u7ebf\u6027\u7ea7\u522b\u589e\u957f\u3002\u7ebf\u6027\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5355\u5c42\u5faa\u73af\u4e2d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def linear(n: int) -&gt; int:\n    \"\"\"\u7ebf\u6027\u9636\"\"\"\n    count = 0\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u9636 */\nint Linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n int) int {\n    count := 0\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) -&gt; Int {\n    var count = 0\n    for _ in 0 ..&lt; n {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): number {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n  int count = 0;\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u9636 */\nfn linear(n: i32) -&gt; i32 {\n    let mut count = 0;\n    for _ in 0..n {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7ebf\u6027\u9636 */\nfun linear(n: Int): Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (i in 0..&lt;n)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{linear}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u9636\nfn linear(n: i32) i32 {\n    var count: i32 = 0;\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u904d\u5386\u6570\u7ec4\u548c\u904d\u5386\u94fe\u8868\u7b49\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u7684\u957f\u5ea6\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def array_traversal(nums: list[int]) -&gt; int:\n    \"\"\"\u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\"\"\"\n    count = 0\n    # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for num in nums:\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(vector&lt;int&gt; &amp;nums) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint ArrayTraversal(int[] nums) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    foreach (int num in nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums []int) int {\n    count := 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for range nums {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums: [Int]) -&gt; Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for _ in nums {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums) {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums: number[]): number {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(List&lt;int&gt; nums) {\n  int count = 0;\n  // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n  for (var _num in nums) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfn array_traversal(nums: &amp;[i32]) -&gt; i32 {\n    let mut count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for _ in nums {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int *nums, int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfun arrayTraversal(nums: IntArray): Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (num in nums) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{array_traversal}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\nfn arrayTraversal(nums: []i32) i32 {\n    var count: i32 = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (nums) |_| {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u9700\u6839\u636e\u8f93\u5165\u6570\u636e\u7684\u7c7b\u578b\u6765\u5177\u4f53\u786e\u5b9a\u3002\u6bd4\u5982\u5728\u7b2c\u4e00\u4e2a\u793a\u4f8b\u4e2d\uff0c\u53d8\u91cf \\(n\\) \u4e3a\u8f93\u5165\u6570\u636e\u5927\u5c0f\uff1b\u5728\u7b2c\u4e8c\u4e2a\u793a\u4f8b\u4e2d\uff0c\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u4e3a\u6570\u636e\u5927\u5c0f\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"<p>\u5e73\u65b9\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u4ee5\u5e73\u65b9\u7ea7\u522b\u589e\u957f\u3002\u5e73\u65b9\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n)\\) \uff0c\u56e0\u6b64\u603b\u4f53\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def quadratic(n: int) -&gt; int:\n    \"\"\"\u5e73\u65b9\u9636\"\"\"\n    count = 0\n    # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for i in range(n):\n        for j in range(n):\n            count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u9636 */\nint Quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n int) int {\n    count := 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; n; j++ {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) -&gt; Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; n {\n            count += 1\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): number {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n  int count = 0;\n  // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n  for (int i = 0; i &lt; n; i++) {\n    for (int j = 0; j &lt; n; j++) {\n      count++;\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u9636 */\nfn quadratic(n: i32) -&gt; i32 {\n    let mut count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for _ in 0..n {\n        for _ in 0..n {\n            count += 1;\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e73\u65b9\u9636 */\nfun quadratic(n: Int): Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;n) {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{quadratic}\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u9636\nfn quadratic(n: i32) i32 {\n    var count: i32 = 0;\n    var i: i32 = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u636e\u5927\u5c0f n \u6210\u5e73\u65b9\u5173\u7cfb\n    while (i &lt; n) : (i += 1) {\n        var j: i32 = 0;\n        while (j &lt; n) : (j += 1) {\n            count += 1;\n        }\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 2-10 \u5bf9\u6bd4\u4e86\u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u548c\u5e73\u65b9\u9636\u4e09\u79cd\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <p></p> <p> \u56fe 2-10 \u00a0 \u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u548c\u5e73\u65b9\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u4ee5\u5192\u6ce1\u6392\u5e8f\u4e3a\u4f8b\uff0c\u5916\u5c42\u5faa\u73af\u6267\u884c \\(n - 1\\) \u6b21\uff0c\u5185\u5c42\u5faa\u73af\u6267\u884c \\(n-1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u5e73\u5747\u4e3a \\(n / 2\\) \u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O((n - 1) n / 2) = O(n^2)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def bubble_sort(nums: list[int]) -&gt; int:\n    \"\"\"\u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\"\"\"\n    count = 0  # \u8ba1\u6570\u5668\n    # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in range(len(nums) - 1, 0, -1):\n        # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                tmp: int = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3  # \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(vector&lt;int&gt; &amp;nums) {\n    int count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\n    int count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint BubbleSort(int[] nums) {\n    int count = 0;  // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                count += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\n    count := 0 // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                tmp := nums[j]\n                nums[j] = nums[j+1]\n                nums[j+1] = tmp\n                count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -&gt; Int {\n    var count = 0 // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\n    let count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\n    let count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(List&lt;int&gt; nums) {\n  int count = 0; // \u8ba1\u6570\u5668\n  // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n  for (var i = nums.length - 1; i &gt; 0; i--) {\n    // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n    for (var j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n      }\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfn bubble_sort(nums: &amp;mut [i32]) -&gt; i32 {\n    let mut count = 0; // \u8ba1\u6570\u5668\n\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in (1..nums.len()).rev() {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int *nums, int n) {\n    int count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = n - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfun bubbleSort(nums: IntArray): Int {\n    var count = 0\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j] = nums[j + 1].also { nums[j + 1] = nums[j] }\n                count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{bubble_sort}\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\n    var count: i32 = 0;  // \u8ba1\u6570\u5668 \n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    var i: i32 = @as(i32, @intCast(nums.len)) - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var j: usize = 0;\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_computational_complexity/time_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6570\u9636 \\(O(2^n)\\)","text":"<p>\u751f\u7269\u5b66\u7684\u201c\u7ec6\u80de\u5206\u88c2\u201d\u662f\u6307\u6570\u9636\u589e\u957f\u7684\u5178\u578b\u4f8b\u5b50\uff1a\u521d\u59cb\u72b6\u6001\u4e3a \\(1\\) \u4e2a\u7ec6\u80de\uff0c\u5206\u88c2\u4e00\u8f6e\u540e\u53d8\u4e3a \\(2\\) \u4e2a\uff0c\u5206\u88c2\u4e24\u8f6e\u540e\u53d8\u4e3a \\(4\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5206\u88c2 \\(n\\) \u8f6e\u540e\u6709 \\(2^n\\) \u4e2a\u7ec6\u80de\u3002</p> <p>\u56fe 2-11 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6a21\u62df\u4e86\u7ec6\u80de\u5206\u88c2\u7684\u8fc7\u7a0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def exponential(n: int) -&gt; int:\n    \"\"\"\u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n    count = 0\n    base = 1\n    # \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in range(n):\n        for _ in range(base):\n            count += 1\n        base *= 2\n    # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Exponential(int n) {\n    int count = 0, bas = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc exponential(n int) int {\n    count, base := 0, 1\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; base; j++ {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc exponential(n: Int) -&gt; Int {\n    var count = 0\n    var base = 1\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; base {\n            count += 1\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n) {\n    let count = 0,\n        base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n: number): number {\n    let count = 0,\n        base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n  int count = 0, base = 1;\n  // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n  for (var i = 0; i &lt; n; i++) {\n    for (var j = 0; j &lt; base; j++) {\n      count++;\n    }\n    base *= 2;\n  }\n  // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn exponential(n: i32) -&gt; i32 {\n    let mut count = 0;\n    let mut base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0..n {\n        for _ in 0..base {\n            count += 1\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n    int count = 0;\n    int bas = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfun exponential(n: Int): Int {\n    var count = 0\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    var base = 1\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;base) {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{exponential}\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn exponential(n: i32) i32 {\n    var count: i32 = 0;\n    var bas: i32 = 1;\n    var i: i32 = 0;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    while (i &lt; n) : (i += 1) {\n        var j: i32 = 0;\n        while (j &lt; bas) : (j += 1) {\n            count += 1;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 2-11 \u00a0 \u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u5728\u5b9e\u9645\u7b97\u6cd5\u4e2d\uff0c\u6307\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u4e2d\u3002\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u5176\u9012\u5f52\u5730\u4e00\u5206\u4e3a\u4e8c\uff0c\u7ecf\u8fc7 \\(n\\) \u6b21\u5206\u88c2\u540e\u505c\u6b62\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def exp_recur(n: int) -&gt; int:\n    \"\"\"\u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n == 1:\n        return 1\n    return exp_recur(n - 1) + exp_recur(n - 1) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint ExpRecur(int n) {\n    if (n == 1) return 1;\n    return ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc expRecur(n int) int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n-1) + expRecur(n-1) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc expRecur(n: Int) -&gt; Int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n) {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n: number): number {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n  if (n == 1) return 1;\n  return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn exp_recur(n: i32) -&gt; i32 {\n    if n == 1 {\n        return 1;\n    }\n    exp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun expRecur(n: Int): Int {\n    if (n == 1) {\n        return 1\n    }\n    return expRecur(n - 1) + expRecur(n - 1) + 1\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{exp_recur}\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn expRecur(n: i32) i32 {\n    if (n == 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u6307\u6570\u9636\u589e\u957f\u975e\u5e38\u8fc5\u901f\uff0c\u5728\u7a77\u4e3e\u6cd5\uff08\u66b4\u529b\u641c\u7d22\u3001\u56de\u6eaf\u7b49\uff09\u4e2d\u6bd4\u8f83\u5e38\u89c1\u3002\u5bf9\u4e8e\u6570\u636e\u89c4\u6a21\u8f83\u5927\u7684\u95ee\u9898\uff0c\u6307\u6570\u9636\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\uff0c\u901a\u5e38\u9700\u8981\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u6216\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u6765\u89e3\u51b3\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#5-olog-n","title":"5. \u00a0 \u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"<p>\u4e0e\u6307\u6570\u9636\u76f8\u53cd\uff0c\u5bf9\u6570\u9636\u53cd\u6620\u4e86\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u201d\u7684\u60c5\u51b5\u3002\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7531\u4e8e\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\uff0c\u56e0\u6b64\u5faa\u73af\u6b21\u6570\u662f \\(\\log_2 n\\) \uff0c\u5373 \\(2^n\\) \u7684\u53cd\u51fd\u6570\u3002</p> <p>\u56fe 2-12 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6a21\u62df\u4e86\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u201d\u7684\u8fc7\u7a0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_2 n)\\) \uff0c\u7b80\u8bb0\u4e3a \\(O(\\log n)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def logarithmic(n: int) -&gt; int:\n    \"\"\"\u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n    count = 0\n    while n &gt; 1:\n        n = n / 2\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n /= 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc logarithmic(n int) int {\n    count := 0\n    for n &gt; 1 {\n        n = n / 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc logarithmic(n: Int) -&gt; Int {\n    var count = 0\n    var n = n\n    while n &gt; 1 {\n        n = n / 2\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n) {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n: number): number {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n  int count = 0;\n  while (n &gt; 1) {\n    n = n ~/ 2;\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn logarithmic(mut n: i32) -&gt; i32 {\n    let mut count = 0;\n    while n &gt; 1 {\n        n = n / 2;\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfun logarithmic(n: Int): Int {\n    var n1 = n\n    var count = 0\n    while (n1 &gt; 1) {\n        n1 /= 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{logarithmic}\n</code></pre> time_complexity.zig<pre><code>// \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn logarithmic(n: i32) i32 {\n    var count: i32 = 0;\n    var n_var = n;\n    while (n_var &gt; 1)\n    {\n        n_var = n_var / 2;\n        count +=1;\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 2-12 \u00a0 \u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u4e0e\u6307\u6570\u9636\u7c7b\u4f3c\uff0c\u5bf9\u6570\u9636\u4e5f\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u4e2d\u3002\u4ee5\u4e0b\u4ee3\u7801\u5f62\u6210\u4e86\u4e00\u68f5\u9ad8\u5ea6\u4e3a \\(\\log_2 n\\) \u7684\u9012\u5f52\u6811\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def log_recur(n: int) -&gt; int:\n    \"\"\"\u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n &lt;= 1:\n        return 0\n    return log_recur(n / 2) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint LogRecur(int n) {\n    if (n &lt;= 1) return 0;\n    return LogRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc logRecur(n int) int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n/2) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc logRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n: n / 2) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n) {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n: number): number {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n  if (n &lt;= 1) return 0;\n  return logRecur(n ~/ 2) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 0;\n    }\n    log_recur(n / 2) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun logRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 0\n    return logRecur(n / 2) + 1\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{log_recur}\n</code></pre> time_complexity.zig<pre><code>// \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn logRecur(n: i32) i32 {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u7b97\u6cd5\u4e2d\uff0c\u4f53\u73b0\u4e86\u201c\u4e00\u5206\u4e3a\u591a\u201d\u548c\u201c\u5316\u7e41\u4e3a\u7b80\u201d\u7684\u7b97\u6cd5\u601d\u60f3\u3002\u5b83\u589e\u957f\u7f13\u6162\uff0c\u662f\u4ec5\u6b21\u4e8e\u5e38\u6570\u9636\u7684\u7406\u60f3\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <p>\\(O(\\log n)\\) \u7684\u5e95\u6570\u662f\u591a\u5c11\uff1f</p> <p>\u51c6\u786e\u6765\u8bf4\uff0c\u201c\u4e00\u5206\u4e3a \\(m\\)\u201d\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(\\log_m n)\\) \u3002\u800c\u901a\u8fc7\u5bf9\u6570\u6362\u5e95\u516c\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5177\u6709\u4e0d\u540c\u5e95\u6570\u3001\u76f8\u7b49\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a</p> \\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\] <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5e95\u6570 \\(m\\) \u53ef\u4ee5\u5728\u4e0d\u5f71\u54cd\u590d\u6742\u5ea6\u7684\u524d\u63d0\u4e0b\u8f6c\u6362\u3002\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f1a\u7701\u7565\u5e95\u6570 \\(m\\) \uff0c\u5c06\u5bf9\u6570\u9636\u76f4\u63a5\u8bb0\u4e3a \\(O(\\log n)\\) \u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#6-on-log-n","title":"6. \u00a0 \u7ebf\u6027\u5bf9\u6570\u9636 \\(O(n \\log n)\\)","text":"<p>\u7ebf\u6027\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u4e24\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u522b\u4e3a \\(O(\\log n)\\) \u548c \\(O(n)\\) \u3002\u76f8\u5173\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def linear_log_recur(n: int) -&gt; int:\n    \"\"\"\u7ebf\u6027\u5bf9\u6570\u9636\"\"\"\n    if n &lt;= 1:\n        return 1\n    count: int = linear_log_recur(n // 2) + linear_log_recur(n // 2)\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint LinearLogRecur(int n) {\n    if (n &lt;= 1) return 1;\n    int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n int) int {\n    if n &lt;= 1 {\n        return 1\n    }\n    count := linearLogRecur(n/2) + linearLogRecur(n/2)\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 1\n    }\n    var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\n    for _ in stride(from: 0, to: n, by: 1) {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n) {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n: number): number {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n  if (n &lt;= 1) return 1;\n  int count = linearLogRecur(n ~/ 2) + linearLogRecur(n ~/ 2);\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfn linear_log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 1;\n    }\n    let mut count = linear_log_recur(n / 2) + linear_log_recur(n / 2);\n    for _ in 0..n as i32 {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfun linearLogRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 1\n    var count = linearLogRecur(n / 2) + linearLogRecur(n / 2)\n    for (i in 0..&lt;n.toInt()) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{linear_log_recur}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u5bf9\u6570\u9636\nfn linearLogRecur(n: i32) i32 {\n    if (n &lt;= 1) return 1;\n    var count: i32 = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 2-13 \u5c55\u793a\u4e86\u7ebf\u6027\u5bf9\u6570\u9636\u7684\u751f\u6210\u65b9\u5f0f\u3002\u4e8c\u53c9\u6811\u7684\u6bcf\u4e00\u5c42\u7684\u64cd\u4f5c\u603b\u6570\u90fd\u4e3a \\(n\\) \uff0c\u6811\u5171\u6709 \\(\\log_2 n + 1\\) \u5c42\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p> <p></p> <p> \u56fe 2-13 \u00a0 \u7ebf\u6027\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u7b49\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#7-on","title":"7. \u00a0 \u9636\u4e58\u9636 \\(O(n!)\\)","text":"<p>\u9636\u4e58\u9636\u5bf9\u5e94\u6570\u5b66\u4e0a\u7684\u201c\u5168\u6392\u5217\u201d\u95ee\u9898\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u4e92\u4e0d\u91cd\u590d\u7684\u5143\u7d20\uff0c\u6c42\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u65b9\u6848\uff0c\u65b9\u6848\u6570\u91cf\u4e3a\uff1a</p> \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\] <p>\u9636\u4e58\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u5b9e\u73b0\u3002\u5982\u56fe 2-14 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u7b2c\u4e00\u5c42\u5206\u88c2\u51fa \\(n\\) \u4e2a\uff0c\u7b2c\u4e8c\u5c42\u5206\u88c2\u51fa \\(n - 1\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u7b2c \\(n\\) \u5c42\u65f6\u505c\u6b62\u5206\u88c2\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig time_complexity.py<pre><code>def factorial_recur(n: int) -&gt; int:\n    \"\"\"\u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n == 0:\n        return 1\n    count = 0\n    # \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for _ in range(n):\n        count += factorial_recur(n - 1)\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint FactorialRecur(int n) {\n    if (n == 0) return 1;\n    int count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (int i = 0; i &lt; n; i++) {\n        count += FactorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n int) int {\n    if n == 0 {\n        return 1\n    }\n    count := 0\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for i := 0; i &lt; n; i++ {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n: Int) -&gt; Int {\n    if n == 0 {\n        return 1\n    }\n    var count = 0\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for _ in 0 ..&lt; n {\n        count += factorialRecur(n: n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n) {\n    if (n === 0) return 1;\n    let count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n: number): number {\n    if (n === 0) return 1;\n    let count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n  if (n == 0) return 1;\n  int count = 0;\n  // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n  for (var i = 0; i &lt; n; i++) {\n    count += factorialRecur(n - 1);\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn factorial_recur(n: i32) -&gt; i32 {\n    if n == 0 {\n        return 1;\n    }\n    let mut count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for _ in 0..n {\n        count += factorial_recur(n - 1);\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfun factorialRecur(n: Int): Int {\n    if (n == 0)\n        return 1\n    var count = 0\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (i in 0..&lt;n) {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>[class]{}-[func]{factorial_recur}\n</code></pre> time_complexity.zig<pre><code>// \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn factorialRecur(n: i32) i32 {\n    if (n == 0) return 1;\n    var count: i32 = 0;\n    var i: i32 = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    while (i &lt; n) : (i += 1) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 2-14 \u00a0 \u9636\u4e58\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u8bf7\u6ce8\u610f\uff0c\u56e0\u4e3a\u5f53 \\(n \\geq 4\\) \u65f6\u6052\u6709 \\(n! &gt; 2^n\\) \uff0c\u6240\u4ee5\u9636\u4e58\u9636\u6bd4\u6307\u6570\u9636\u589e\u957f\u5f97\u66f4\u5feb\uff0c\u5728 \\(n\\) \u8f83\u5927\u65f6\u4e5f\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#235","title":"2.3.5 \u00a0 \u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6","text":"<p>\u7b97\u6cd5\u7684\u65f6\u95f4\u6548\u7387\u5f80\u5f80\u4e0d\u662f\u56fa\u5b9a\u7684\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u5047\u8bbe\u8f93\u5165\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d <code>nums</code> \u7531\u4ece \\(1\\) \u81f3 \\(n\\) \u7684\u6570\u5b57\u7ec4\u6210\uff0c\u6bcf\u4e2a\u6570\u5b57\u53ea\u51fa\u73b0\u4e00\u6b21\uff1b\u4f46\u5143\u7d20\u987a\u5e8f\u662f\u968f\u673a\u6253\u4e71\u7684\uff0c\u4efb\u52a1\u76ee\u6807\u662f\u8fd4\u56de\u5143\u7d20 \\(1\\) \u7684\u7d22\u5f15\u3002\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\u3002</p> <ul> <li>\u5f53 <code>nums = [?, ?, ..., 1]</code> \uff0c\u5373\u5f53\u672b\u5c3e\u5143\u7d20\u662f \\(1\\) \u65f6\uff0c\u9700\u8981\u5b8c\u6574\u904d\u5386\u6570\u7ec4\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002</li> <li>\u5f53 <code>nums = [1, ?, ?, ...]</code> \uff0c\u5373\u5f53\u9996\u4e2a\u5143\u7d20\u4e3a \\(1\\) \u65f6\uff0c\u65e0\u8bba\u6570\u7ec4\u591a\u957f\u90fd\u4e0d\u9700\u8981\u7ee7\u7eed\u904d\u5386\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(\\Omega(1)\\) \u3002</li> </ul> <p>\u201c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u201d\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u4f7f\u7528\u5927 \\(O\\) \u8bb0\u53f7\u8868\u793a\u3002\u76f8\u5e94\u5730\uff0c\u201c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u201d\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0b\u754c\uff0c\u7528 \\(\\Omega\\) \u8bb0\u53f7\u8868\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig worst_best_time_complexity.py<pre><code>def random_numbers(n: int) -&gt; list[int]:\n    \"\"\"\u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a: 1, 2, ..., n \uff0c\u987a\u5e8f\u88ab\u6253\u4e71\"\"\"\n    # \u751f\u6210\u6570\u7ec4 nums =: 1, 2, 3, ..., n\n    nums = [i for i in range(1, n + 1)]\n    # \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    random.shuffle(nums)\n    return nums\n\ndef find_one(nums: list[int]) -&gt; int:\n    \"\"\"\u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\"\"\"\n    for i in range(len(nums)):\n        # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1:\n            return i\n    return -1\n</code></pre> worst_best_time_complexity.cpp<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nvector&lt;int&gt; randomNumbers(int n) {\n    vector&lt;int&gt; nums(n);\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u4f7f\u7528\u7cfb\u7edf\u65f6\u95f4\u751f\u6210\u968f\u673a\u79cd\u5b50\n    unsigned seed = chrono::system_clock::now().time_since_epoch().count();\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    shuffle(nums.begin(), nums.end(), default_random_engine(seed));\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector&lt;int&gt; &amp;nums) {\n    for (int i = 0; i &lt; nums.size(); i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.java<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\n    Integer[] nums = new Integer[n];\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    Collections.shuffle(Arrays.asList(nums));\n    // Integer[] -&gt; int[]\n    int[] res = new int[n];\n    for (int i = 0; i &lt; n; i++) {\n        res[i] = nums[i];\n    }\n    return res;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.cs<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] RandomNumbers(int n) {\n    int[] nums = new int[n];\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (int i = 0; i &lt; nums.Length; i++) {\n        int index = new Random().Next(i, nums.Length);\n        (nums[i], nums[index]) = (nums[index], nums[i]);\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint FindOne(int[] nums) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.go<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n int) []int {\n    nums := make([]int, n)\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for i := 0; i &lt; n; i++ {\n        nums[i] = i + 1\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    rand.Shuffle(len(nums), func(i, j int) {\n        nums[i], nums[j] = nums[j], nums[i]\n    })\n    return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\n    for i := 0; i &lt; len(nums); i++ {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.swift<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n: Int) -&gt; [Int] {\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    var nums = Array(1 ... n)\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    nums.shuffle()\n    return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -&gt; Int {\n    for i in nums.indices {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.js<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n) {\n    const nums = Array(n);\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.ts<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n: number): number[] {\n    const nums = Array(n);\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.dart<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nList&lt;int&gt; randomNumbers(int n) {\n  final nums = List.filled(n, 0);\n  // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n  for (var i = 0; i &lt; n; i++) {\n    nums[i] = i + 1;\n  }\n  // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n  nums.shuffle();\n\n  return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(List&lt;int&gt; nums) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n    // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n    if (nums[i] == 1) return i;\n  }\n\n  return -1;\n}\n</code></pre> worst_best_time_complexity.rs<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfn random_numbers(n: i32) -&gt; Vec&lt;i32&gt; {\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    let mut nums = (1..=n).collect::&lt;Vec&lt;i32&gt;&gt;();\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    nums.shuffle(&amp;mut thread_rng());\n    nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfn find_one(nums: &amp;[i32]) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1 {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> worst_best_time_complexity.c<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint *randomNumbers(int n) {\n    // \u5206\u914d\u5806\u533a\u5185\u5b58\uff08\u521b\u5efa\u4e00\u7ef4\u53ef\u53d8\u957f\u6570\u7ec4\uff1a\u6570\u7ec4\u4e2d\u5143\u7d20\u6570\u91cf\u4e3a n \uff0c\u5143\u7d20\u7c7b\u578b\u4e3a int \uff09\n    int *nums = (int *)malloc(n * sizeof(int));\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (int i = n - 1; i &gt; 0; i--) {\n        int j = rand() % (i + 1);\n        int temp = nums[i];\n        nums[i] = nums[j];\n        nums[j] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int *nums, int n) {\n    for (int i = 0; i &lt; n; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.kt<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfun randomNumbers(n: Int): Array&lt;Int?&gt; {\n    val nums = IntArray(n)\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (i in 0..&lt;n) {\n        nums[i] = i + 1\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    val mutableList = nums.toMutableList()\n    mutableList.shuffle()\n    // Integer[] -&gt; int[]\n    val res = arrayOfNulls&lt;Int&gt;(n)\n    for (i in 0..&lt;n) {\n        res[i] = mutableList[i]\n    }\n    return res\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfun findOne(nums: Array&lt;Int?&gt;): Int {\n    for (i in nums.indices) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.rb<pre><code>[class]{}-[func]{random_numbers}\n\n[class]{}-[func]{find_one}\n</code></pre> worst_best_time_complexity.zig<pre><code>// \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71\nfn randomNumbers(comptime n: usize) [n]i32 {\n    var nums: [n]i32 = undefined;\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (&amp;nums, 0..) |*num, i| {\n        num.* = @as(i32, @intCast(i)) + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    const rand = std.crypto.random;\n    rand.shuffle(i32, &amp;nums);\n    return nums;\n}\n\n// \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\nfn findOne(nums: []i32) i32 {\n    for (nums, 0..) |num, i| {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (num == 1) return @intCast(i);\n    }\n    return -1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6211\u4eec\u5728\u5b9e\u9645\u4e2d\u5f88\u5c11\u4f7f\u7528\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u56e0\u4e3a\u901a\u5e38\u53ea\u6709\u5728\u5f88\u5c0f\u6982\u7387\u4e0b\u624d\u80fd\u8fbe\u5230\uff0c\u53ef\u80fd\u4f1a\u5e26\u6765\u4e00\u5b9a\u7684\u8bef\u5bfc\u6027\u3002\u800c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u66f4\u4e3a\u5b9e\u7528\uff0c\u56e0\u4e3a\u5b83\u7ed9\u51fa\u4e86\u4e00\u4e2a\u6548\u7387\u5b89\u5168\u503c\uff0c\u8ba9\u6211\u4eec\u53ef\u4ee5\u653e\u5fc3\u5730\u4f7f\u7528\u7b97\u6cd5\u3002</p> <p>\u4ece\u4e0a\u8ff0\u793a\u4f8b\u53ef\u4ee5\u770b\u51fa\uff0c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ea\u51fa\u73b0\u4e8e\u201c\u7279\u6b8a\u7684\u6570\u636e\u5206\u5e03\u201d\uff0c\u8fd9\u4e9b\u60c5\u51b5\u7684\u51fa\u73b0\u6982\u7387\u53ef\u80fd\u5f88\u5c0f\uff0c\u5e76\u4e0d\u80fd\u771f\u5b9e\u5730\u53cd\u6620\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f53\u73b0\u7b97\u6cd5\u5728\u968f\u673a\u8f93\u5165\u6570\u636e\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u7528 \\(\\Theta\\) \u8bb0\u53f7\u6765\u8868\u793a\u3002</p> <p>\u5bf9\u4e8e\u90e8\u5206\u7b97\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u63a8\u7b97\u51fa\u968f\u673a\u6570\u636e\u5206\u5e03\u4e0b\u7684\u5e73\u5747\u60c5\u51b5\u3002\u6bd4\u5982\u4e0a\u8ff0\u793a\u4f8b\uff0c\u7531\u4e8e\u8f93\u5165\u6570\u7ec4\u662f\u88ab\u6253\u4e71\u7684\uff0c\u56e0\u6b64\u5143\u7d20 \\(1\\) \u51fa\u73b0\u5728\u4efb\u610f\u7d22\u5f15\u7684\u6982\u7387\u90fd\u662f\u76f8\u7b49\u7684\uff0c\u90a3\u4e48\u7b97\u6cd5\u7684\u5e73\u5747\u5faa\u73af\u6b21\u6570\u5c31\u662f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a \\(n / 2\\) \uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(\\Theta(n / 2) = \\Theta(n)\\) \u3002</p> <p>\u4f46\u5bf9\u4e8e\u8f83\u4e3a\u590d\u6742\u7684\u7b97\u6cd5\uff0c\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5f80\u5f80\u6bd4\u8f83\u56f0\u96be\uff0c\u56e0\u4e3a\u5f88\u96be\u5206\u6790\u51fa\u5728\u6570\u636e\u5206\u5e03\u4e0b\u7684\u6574\u4f53\u6570\u5b66\u671f\u671b\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f5c\u4e3a\u7b97\u6cd5\u6548\u7387\u7684\u8bc4\u5224\u6807\u51c6\u3002</p> <p>\u4e3a\u4ec0\u4e48\u5f88\u5c11\u770b\u5230 \\(\\Theta\\) \u7b26\u53f7\uff1f</p> <p>\u53ef\u80fd\u7531\u4e8e \\(O\\) \u7b26\u53f7\u8fc7\u4e8e\u6717\u6717\u4e0a\u53e3\uff0c\u56e0\u6b64\u6211\u4eec\u5e38\u5e38\u4f7f\u7528\u5b83\u6765\u8868\u793a\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u4f46\u4ece\u4e25\u683c\u610f\u4e49\u4e0a\u8bb2\uff0c\u8fd9\u79cd\u505a\u6cd5\u5e76\u4e0d\u89c4\u8303\u3002\u5728\u672c\u4e66\u548c\u5176\u4ed6\u8d44\u6599\u4e2d\uff0c\u82e5\u9047\u5230\u7c7b\u4f3c\u201c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u201d\u7684\u8868\u8ff0\uff0c\u8bf7\u5c06\u5176\u76f4\u63a5\u7406\u89e3\u4e3a \\(\\Theta(n)\\) \u3002</p>"},{"location":"chapter_data_structure/","title":"\u7b2c 3 \u7ae0 \u00a0 \u6570\u636e\u7ed3\u6784","text":"<p>Abstract</p> <p>\u6570\u636e\u7ed3\u6784\u5982\u540c\u4e00\u526f\u7a33\u56fa\u800c\u591a\u6837\u7684\u6846\u67b6\u3002</p> <p>\u5b83\u4e3a\u6570\u636e\u7684\u6709\u5e8f\u7ec4\u7ec7\u63d0\u4f9b\u4e86\u84dd\u56fe\uff0c\u7b97\u6cd5\u5f97\u4ee5\u5728\u6b64\u57fa\u7840\u4e0a\u751f\u52a8\u8d77\u6765\u3002</p>"},{"location":"chapter_data_structure/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>3.1 \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b</li> <li>3.2 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b</li> <li>3.3 \u00a0 \u6570\u5b57\u7f16\u7801 *</li> <li>3.4 \u00a0 \u5b57\u7b26\u7f16\u7801 *</li> <li>3.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_data_structure/basic_data_types/","title":"3.2 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b","text":"<p>\u5f53\u8c08\u53ca\u8ba1\u7b97\u673a\u4e2d\u7684\u6570\u636e\u65f6\uff0c\u6211\u4eec\u4f1a\u60f3\u5230\u6587\u672c\u3001\u56fe\u7247\u3001\u89c6\u9891\u3001\u8bed\u97f3\u30013D \u6a21\u578b\u7b49\u5404\u79cd\u5f62\u5f0f\u3002\u5c3d\u7ba1\u8fd9\u4e9b\u6570\u636e\u7684\u7ec4\u7ec7\u5f62\u5f0f\u5404\u5f02\uff0c\u4f46\u5b83\u4eec\u90fd\u7531\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6784\u6210\u3002</p> <p>\u57fa\u672c\u6570\u636e\u7c7b\u578b\u662f CPU \u53ef\u4ee5\u76f4\u63a5\u8fdb\u884c\u8fd0\u7b97\u7684\u7c7b\u578b\uff0c\u5728\u7b97\u6cd5\u4e2d\u76f4\u63a5\u88ab\u4f7f\u7528\uff0c\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u3002</p> <ul> <li>\u6574\u6570\u7c7b\u578b <code>byte</code>\u3001<code>short</code>\u3001<code>int</code>\u3001<code>long</code> \u3002</li> <li>\u6d6e\u70b9\u6570\u7c7b\u578b <code>float</code>\u3001<code>double</code> \uff0c\u7528\u4e8e\u8868\u793a\u5c0f\u6570\u3002</li> <li>\u5b57\u7b26\u7c7b\u578b <code>char</code> \uff0c\u7528\u4e8e\u8868\u793a\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u6bcd\u3001\u6807\u70b9\u7b26\u53f7\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u7b49\u3002</li> <li>\u5e03\u5c14\u7c7b\u578b <code>bool</code> \uff0c\u7528\u4e8e\u8868\u793a\u201c\u662f\u201d\u4e0e\u201c\u5426\u201d\u5224\u65ad\u3002</li> </ul> <p>\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4ee5\u4e8c\u8fdb\u5236\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u3002\u4e00\u4e2a\u4e8c\u8fdb\u5236\u4f4d\u5373\u4e3a \\(1\\) \u6bd4\u7279\u3002\u5728\u7edd\u5927\u591a\u6570\u73b0\u4ee3\u64cd\u4f5c\u7cfb\u7edf\u4e2d\uff0c\\(1\\) \u5b57\u8282\uff08byte\uff09\u7531 \\(8\\) \u6bd4\u7279\uff08bit\uff09\u7ec4\u6210\u3002</p> <p>\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5176\u5360\u7528\u7684\u7a7a\u95f4\u5927\u5c0f\u3002\u4e0b\u9762\u4ee5 Java \u4e3a\u4f8b\u3002</p> <ul> <li>\u6574\u6570\u7c7b\u578b <code>byte</code> \u5360\u7528 \\(1\\) \u5b57\u8282 = \\(8\\) \u6bd4\u7279 \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{8}\\) \u4e2a\u6570\u5b57\u3002</li> <li>\u6574\u6570\u7c7b\u578b <code>int</code> \u5360\u7528 \\(4\\) \u5b57\u8282 = \\(32\\) \u6bd4\u7279 \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{32}\\) \u4e2a\u6570\u5b57\u3002</li> </ul> <p>\u8868 3-1 \u5217\u4e3e\u4e86 Java \u4e2d\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5360\u7528\u7a7a\u95f4\u3001\u53d6\u503c\u8303\u56f4\u548c\u9ed8\u8ba4\u503c\u3002\u6b64\u8868\u683c\u65e0\u987b\u6b7b\u8bb0\u786c\u80cc\uff0c\u5927\u81f4\u7406\u89e3\u5373\u53ef\uff0c\u9700\u8981\u65f6\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u6765\u56de\u5fc6\u3002</p> <p> \u8868 3-1 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5360\u7528\u7a7a\u95f4\u548c\u53d6\u503c\u8303\u56f4 </p> \u7c7b\u578b \u7b26\u53f7 \u5360\u7528\u7a7a\u95f4 \u6700\u5c0f\u503c \u6700\u5927\u503c \u9ed8\u8ba4\u503c \u6574\u6570 <code>byte</code> 1 \u5b57\u8282 \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) \\(0\\) <code>short</code> 2 \u5b57\u8282 \\(-2^{15}\\) \\(2^{15} - 1\\) \\(0\\) <code>int</code> 4 \u5b57\u8282 \\(-2^{31}\\) \\(2^{31} - 1\\) \\(0\\) <code>long</code> 8 \u5b57\u8282 \\(-2^{63}\\) \\(2^{63} - 1\\) \\(0\\) \u6d6e\u70b9\u6570 <code>float</code> 4 \u5b57\u8282 \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0\\text{f}\\) <code>double</code> 8 \u5b57\u8282 \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) \\(0.0\\) \u5b57\u7b26 <code>char</code> 2 \u5b57\u8282 \\(0\\) \\(2^{16} - 1\\) \\(0\\) \u5e03\u5c14 <code>bool</code> 1 \u5b57\u8282 \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\) <p>\u8bf7\u6ce8\u610f\uff0c\u8868 3-1 \u9488\u5bf9\u7684\u662f Java \u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u60c5\u51b5\u3002\u6bcf\u79cd\u7f16\u7a0b\u8bed\u8a00\u90fd\u6709\u5404\u81ea\u7684\u6570\u636e\u7c7b\u578b\u5b9a\u4e49\uff0c\u5b83\u4eec\u7684\u5360\u7528\u7a7a\u95f4\u3001\u53d6\u503c\u8303\u56f4\u548c\u9ed8\u8ba4\u503c\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002</p> <ul> <li>\u5728 Python \u4e2d\uff0c\u6574\u6570\u7c7b\u578b <code>int</code> \u53ef\u4ee5\u662f\u4efb\u610f\u5927\u5c0f\uff0c\u53ea\u53d7\u9650\u4e8e\u53ef\u7528\u5185\u5b58\uff1b\u6d6e\u70b9\u6570 <code>float</code> \u662f\u53cc\u7cbe\u5ea6 64 \u4f4d\uff1b\u6ca1\u6709 <code>char</code> \u7c7b\u578b\uff0c\u5355\u4e2a\u5b57\u7b26\u5b9e\u9645\u4e0a\u662f\u957f\u5ea6\u4e3a 1 \u7684\u5b57\u7b26\u4e32 <code>str</code> \u3002</li> <li>C \u548c C++ \u672a\u660e\u786e\u89c4\u5b9a\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5927\u5c0f\uff0c\u800c\u56e0\u5b9e\u73b0\u548c\u5e73\u53f0\u5404\u5f02\u3002\u8868 3-1 \u9075\u5faa LP64 \u6570\u636e\u6a21\u578b\uff0c\u5176\u7528\u4e8e\u5305\u62ec Linux \u548c macOS \u5728\u5185\u7684 Unix 64 \u4f4d\u64cd\u4f5c\u7cfb\u7edf\u3002</li> <li>\u5b57\u7b26 <code>char</code> \u7684\u5927\u5c0f\u5728 C \u548c C++ \u4e2d\u4e3a 1 \u5b57\u8282\uff0c\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\u53d6\u51b3\u4e8e\u7279\u5b9a\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\uff0c\u8be6\u89c1\u201c\u5b57\u7b26\u7f16\u7801\u201d\u7ae0\u8282\u3002</li> <li>\u5373\u4f7f\u8868\u793a\u5e03\u5c14\u91cf\u4ec5\u9700 1 \u4f4d\uff08\\(0\\) \u6216 \\(1\\)\uff09\uff0c\u5b83\u5728\u5185\u5b58\u4e2d\u901a\u5e38\u4e5f\u5b58\u50a8\u4e3a 1 \u5b57\u8282\u3002\u8fd9\u662f\u56e0\u4e3a\u73b0\u4ee3\u8ba1\u7b97\u673a CPU \u901a\u5e38\u5c06 1 \u5b57\u8282\u4f5c\u4e3a\u6700\u5c0f\u5bfb\u5740\u5185\u5b58\u5355\u5143\u3002</li> </ul> <p>\u90a3\u4e48\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4e0e\u6570\u636e\u7ed3\u6784\u4e4b\u95f4\u6709\u4ec0\u4e48\u8054\u7cfb\u5462\uff1f\u6211\u4eec\u77e5\u9053\uff0c\u6570\u636e\u7ed3\u6784\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u4e0e\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002\u8fd9\u53e5\u8bdd\u7684\u4e3b\u8bed\u662f\u201c\u7ed3\u6784\u201d\u800c\u975e\u201c\u6570\u636e\u201d\u3002</p> <p>\u5982\u679c\u60f3\u8868\u793a\u201c\u4e00\u6392\u6570\u5b57\u201d\uff0c\u6211\u4eec\u81ea\u7136\u4f1a\u60f3\u5230\u4f7f\u7528\u6570\u7ec4\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u7684\u7ebf\u6027\u7ed3\u6784\u53ef\u4ee5\u8868\u793a\u6570\u5b57\u7684\u76f8\u90bb\u5173\u7cfb\u548c\u987a\u5e8f\u5173\u7cfb\uff0c\u4f46\u81f3\u4e8e\u5b58\u50a8\u7684\u5185\u5bb9\u662f\u6574\u6570 <code>int</code>\u3001\u5c0f\u6570 <code>float</code> \u8fd8\u662f\u5b57\u7b26 <code>char</code> \uff0c\u5219\u4e0e\u201c\u6570\u636e\u7ed3\u6784\u201d\u65e0\u5173\u3002</p> <p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u5185\u5bb9\u7c7b\u578b\u201d\uff0c\u800c\u6570\u636e\u7ed3\u6784\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u7ec4\u7ec7\u65b9\u5f0f\u201d\u3002\u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6211\u4eec\u7528\u76f8\u540c\u7684\u6570\u636e\u7ed3\u6784\uff08\u6570\u7ec4\uff09\u6765\u5b58\u50a8\u4e0e\u8868\u793a\u4e0d\u540c\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u5305\u62ec <code>int</code>\u3001<code>float</code>\u3001<code>char</code>\u3001<code>bool</code> \u7b49\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nnumbers: list[int] = [0] * 5\ndecimals: list[float] = [0.0] * 5\n# Python \u7684\u5b57\u7b26\u5b9e\u9645\u4e0a\u662f\u957f\u5ea6\u4e3a 1 \u7684\u5b57\u7b26\u4e32\ncharacters: list[str] = ['0'] * 5\nbools: list[bool] = [False] * 5\n# Python \u7684\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61\u5f15\u7528\ndata = [0, 0.0, 'a', False, ListNode(0)]\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool bools[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] bools = new boolean[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] bools = new bool[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar bools = [5]bool{}\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nlet numbers = Array(repeating: 0, count: 5)\nlet decimals = Array(repeating: 0.0, count: 5)\nlet characters: [Character] = Array(repeating: \"a\", count: 5)\nlet bools = Array(repeating: false, count: 5)\n</code></pre> <pre><code>// JavaScript \u7684\u6570\u7ec4\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61\nconst array = [0, 0.0, 'a', false];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst bools: boolean[] = [];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nList&lt;int&gt; numbers = List.filled(5, 0);\nList&lt;double&gt; decimals = List.filled(5, 0.0);\nList&lt;String&gt; characters = List.filled(5, 'a');\nList&lt;bool&gt; bools = List.filled(5, false);\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nlet numbers: Vec&lt;i32&gt; = vec![0; 5];\nlet decimals: Vec&lt;f32&gt; = vec![0.0; 5];\nlet characters: Vec&lt;char&gt; = vec!['0'; 5];\nlet bools: Vec&lt;bool&gt; = vec![false; 5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool bools[10];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nval numbers = IntArray(5)\nval decinals = FloatArray(5)\nval characters = CharArray(5)\nval bools = BooleanArray(5)\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_data_structure/character_encoding/","title":"3.4 \u00a0 \u5b57\u7b26\u7f16\u7801 *","text":"<p>\u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u6240\u6709\u6570\u636e\u90fd\u662f\u4ee5\u4e8c\u8fdb\u5236\u6570\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\uff0c\u5b57\u7b26 <code>char</code> \u4e5f\u4e0d\u4f8b\u5916\u3002\u4e3a\u4e86\u8868\u793a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u5efa\u7acb\u4e00\u5957\u201c\u5b57\u7b26\u96c6\u201d\uff0c\u89c4\u5b9a\u6bcf\u4e2a\u5b57\u7b26\u548c\u4e8c\u8fdb\u5236\u6570\u4e4b\u95f4\u7684\u4e00\u4e00\u5bf9\u5e94\u5173\u7cfb\u3002\u6709\u4e86\u5b57\u7b26\u96c6\u4e4b\u540e\uff0c\u8ba1\u7b97\u673a\u5c31\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u5b8c\u6210\u4e8c\u8fdb\u5236\u6570\u5230\u5b57\u7b26\u7684\u8f6c\u6362\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#341-ascii","title":"3.4.1 \u00a0 ASCII \u5b57\u7b26\u96c6","text":"<p>ASCII\uff08\u7801\uff09\u662f\u6700\u65e9\u51fa\u73b0\u7684\u5b57\u7b26\u96c6\uff0c\u5176\u5168\u79f0\u4e3a American Standard Code for Information Interchange\uff08\u7f8e\u56fd\u6807\u51c6\u4fe1\u606f\u4ea4\u6362\u4ee3\u7801\uff09\u3002\u5b83\u4f7f\u7528 7 \u4f4d\u4e8c\u8fdb\u5236\u6570\uff08\u4e00\u4e2a\u5b57\u8282\u7684\u4f4e 7 \u4f4d\uff09\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6700\u591a\u80fd\u591f\u8868\u793a 128 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002\u5982\u56fe 3-6 \u6240\u793a\uff0cASCII \u7801\u5305\u62ec\u82f1\u6587\u5b57\u6bcd\u7684\u5927\u5c0f\u5199\u3001\u6570\u5b57 0 ~ 9\u3001\u4e00\u4e9b\u6807\u70b9\u7b26\u53f7\uff0c\u4ee5\u53ca\u4e00\u4e9b\u63a7\u5236\u5b57\u7b26\uff08\u5982\u6362\u884c\u7b26\u548c\u5236\u8868\u7b26\uff09\u3002</p> <p></p> <p> \u56fe 3-6 \u00a0 ASCII \u7801 </p> <p>\u7136\u800c\uff0cASCII \u7801\u4ec5\u80fd\u591f\u8868\u793a\u82f1\u6587\u3002\u968f\u7740\u8ba1\u7b97\u673a\u7684\u5168\u7403\u5316\uff0c\u8bde\u751f\u4e86\u4e00\u79cd\u80fd\u591f\u8868\u793a\u66f4\u591a\u8bed\u8a00\u7684EASCII\u5b57\u7b26\u96c6\u3002\u5b83\u5728 ASCII \u7684 7 \u4f4d\u57fa\u7840\u4e0a\u6269\u5c55\u5230 8 \u4f4d\uff0c\u80fd\u591f\u8868\u793a 256 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002</p> <p>\u5728\u4e16\u754c\u8303\u56f4\u5185\uff0c\u9646\u7eed\u51fa\u73b0\u4e86\u4e00\u6279\u9002\u7528\u4e8e\u4e0d\u540c\u5730\u533a\u7684 EASCII \u5b57\u7b26\u96c6\u3002\u8fd9\u4e9b\u5b57\u7b26\u96c6\u7684\u524d 128 \u4e2a\u5b57\u7b26\u7edf\u4e00\u4e3a ASCII \u7801\uff0c\u540e 128 \u4e2a\u5b57\u7b26\u5b9a\u4e49\u4e0d\u540c\uff0c\u4ee5\u9002\u5e94\u4e0d\u540c\u8bed\u8a00\u7684\u9700\u6c42\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#342-gbk","title":"3.4.2 \u00a0 GBK \u5b57\u7b26\u96c6","text":"<p>\u540e\u6765\u4eba\u4eec\u53d1\u73b0\uff0cEASCII \u7801\u4ecd\u7136\u65e0\u6cd5\u6ee1\u8db3\u8bb8\u591a\u8bed\u8a00\u7684\u5b57\u7b26\u6570\u91cf\u8981\u6c42\u3002\u6bd4\u5982\u6c49\u5b57\u6709\u8fd1\u5341\u4e07\u4e2a\uff0c\u5149\u65e5\u5e38\u4f7f\u7528\u7684\u5c31\u6709\u51e0\u5343\u4e2a\u3002\u4e2d\u56fd\u56fd\u5bb6\u6807\u51c6\u603b\u5c40\u4e8e 1980 \u5e74\u53d1\u5e03\u4e86GB2312\u5b57\u7b26\u96c6\uff0c\u5176\u6536\u5f55\u4e86 6763 \u4e2a\u6c49\u5b57\uff0c\u57fa\u672c\u6ee1\u8db3\u4e86\u6c49\u5b57\u7684\u8ba1\u7b97\u673a\u5904\u7406\u9700\u8981\u3002</p> <p>\u7136\u800c\uff0cGB2312 \u65e0\u6cd5\u5904\u7406\u90e8\u5206\u7f55\u89c1\u5b57\u548c\u7e41\u4f53\u5b57\u3002GBK\u5b57\u7b26\u96c6\u662f\u5728 GB2312 \u7684\u57fa\u7840\u4e0a\u6269\u5c55\u5f97\u5230\u7684\uff0c\u5b83\u5171\u6536\u5f55\u4e86 21886 \u4e2a\u6c49\u5b57\u3002\u5728 GBK \u7684\u7f16\u7801\u65b9\u6848\u4e2d\uff0cASCII \u5b57\u7b26\u4f7f\u7528\u4e00\u4e2a\u5b57\u8282\u8868\u793a\uff0c\u6c49\u5b57\u4f7f\u7528\u4e24\u4e2a\u5b57\u8282\u8868\u793a\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#343-unicode","title":"3.4.3 \u00a0 Unicode \u5b57\u7b26\u96c6","text":"<p>\u968f\u7740\u8ba1\u7b97\u673a\u6280\u672f\u7684\u84ec\u52c3\u53d1\u5c55\uff0c\u5b57\u7b26\u96c6\u4e0e\u7f16\u7801\u6807\u51c6\u767e\u82b1\u9f50\u653e\uff0c\u800c\u8fd9\u5e26\u6765\u4e86\u8bb8\u591a\u95ee\u9898\u3002\u4e00\u65b9\u9762\uff0c\u8fd9\u4e9b\u5b57\u7b26\u96c6\u4e00\u822c\u53ea\u5b9a\u4e49\u4e86\u7279\u5b9a\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u65e0\u6cd5\u5728\u591a\u8bed\u8a00\u73af\u5883\u4e0b\u6b63\u5e38\u5de5\u4f5c\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u540c\u4e00\u79cd\u8bed\u8a00\u5b58\u5728\u591a\u79cd\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u5982\u679c\u4e24\u53f0\u8ba1\u7b97\u673a\u4f7f\u7528\u7684\u662f\u4e0d\u540c\u7684\u7f16\u7801\u6807\u51c6\uff0c\u5219\u5728\u4fe1\u606f\u4f20\u9012\u65f6\u5c31\u4f1a\u51fa\u73b0\u4e71\u7801\u3002</p> <p>\u90a3\u4e2a\u65f6\u4ee3\u7684\u7814\u7a76\u4eba\u5458\u5c31\u5728\u60f3\uff1a\u5982\u679c\u63a8\u51fa\u4e00\u4e2a\u8db3\u591f\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\uff0c\u5c06\u4e16\u754c\u8303\u56f4\u5185\u7684\u6240\u6709\u8bed\u8a00\u548c\u7b26\u53f7\u90fd\u6536\u5f55\u5176\u4e2d\uff0c\u4e0d\u5c31\u53ef\u4ee5\u89e3\u51b3\u8de8\u8bed\u8a00\u73af\u5883\u548c\u4e71\u7801\u95ee\u9898\u4e86\u5417\uff1f\u5728\u8fd9\u79cd\u60f3\u6cd5\u7684\u9a71\u52a8\u4e0b\uff0c\u4e00\u4e2a\u5927\u800c\u5168\u7684\u5b57\u7b26\u96c6 Unicode \u5e94\u8fd0\u800c\u751f\u3002</p> <p>Unicode\u7684\u4e2d\u6587\u540d\u79f0\u4e3a\u201c\u7edf\u4e00\u7801\u201d\uff0c\u7406\u8bba\u4e0a\u80fd\u5bb9\u7eb3 100 \u591a\u4e07\u4e2a\u5b57\u7b26\u3002\u5b83\u81f4\u529b\u4e8e\u5c06\u5168\u7403\u8303\u56f4\u5185\u7684\u5b57\u7b26\u7eb3\u5165\u7edf\u4e00\u7684\u5b57\u7b26\u96c6\u4e4b\u4e2d\uff0c\u63d0\u4f9b\u4e00\u79cd\u901a\u7528\u7684\u5b57\u7b26\u96c6\u6765\u5904\u7406\u548c\u663e\u793a\u5404\u79cd\u8bed\u8a00\u6587\u5b57\uff0c\u51cf\u5c11\u56e0\u4e3a\u7f16\u7801\u6807\u51c6\u4e0d\u540c\u800c\u4ea7\u751f\u7684\u4e71\u7801\u95ee\u9898\u3002</p> <p>\u81ea 1991 \u5e74\u53d1\u5e03\u4ee5\u6765\uff0cUnicode \u4e0d\u65ad\u6269\u5145\u65b0\u7684\u8bed\u8a00\u4e0e\u5b57\u7b26\u3002\u622a\u81f3 2022 \u5e74 9 \u6708\uff0cUnicode \u5df2\u7ecf\u5305\u542b 149186 \u4e2a\u5b57\u7b26\uff0c\u5305\u62ec\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\u3001\u7b26\u53f7\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u7b49\u3002\u5728\u5e9e\u5927\u7684 Unicode \u5b57\u7b26\u96c6\u4e2d\uff0c\u5e38\u7528\u7684\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\uff0c\u6709\u4e9b\u751f\u50fb\u7684\u5b57\u7b26\u5360\u7528 3 \u5b57\u8282\u751a\u81f3 4 \u5b57\u8282\u3002</p> <p>Unicode \u662f\u4e00\u79cd\u901a\u7528\u5b57\u7b26\u96c6\uff0c\u672c\u8d28\u4e0a\u662f\u7ed9\u6bcf\u4e2a\u5b57\u7b26\u5206\u914d\u4e00\u4e2a\u7f16\u53f7\uff08\u79f0\u4e3a\u201c\u7801\u70b9\u201d\uff09\uff0c\u4f46\u5b83\u5e76\u6ca1\u6709\u89c4\u5b9a\u5728\u8ba1\u7b97\u673a\u4e2d\u5982\u4f55\u5b58\u50a8\u8fd9\u4e9b\u5b57\u7b26\u7801\u70b9\u3002\u6211\u4eec\u4e0d\u7981\u4f1a\u95ee\uff1a\u5f53\u591a\u79cd\u957f\u5ea6\u7684 Unicode \u7801\u70b9\u540c\u65f6\u51fa\u73b0\u5728\u4e00\u4e2a\u6587\u672c\u4e2d\u65f6\uff0c\u7cfb\u7edf\u5982\u4f55\u89e3\u6790\u5b57\u7b26\uff1f\u4f8b\u5982\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a 2 \u5b57\u8282\u7684\u7f16\u7801\uff0c\u7cfb\u7edf\u5982\u4f55\u786e\u8ba4\u5b83\u662f\u4e00\u4e2a 2 \u5b57\u8282\u7684\u5b57\u7b26\u8fd8\u662f\u4e24\u4e2a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff1f</p> <p>\u5bf9\u4e8e\u4ee5\u4e0a\u95ee\u9898\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u89e3\u51b3\u65b9\u6848\u662f\u5c06\u6240\u6709\u5b57\u7b26\u5b58\u50a8\u4e3a\u7b49\u957f\u7684\u7f16\u7801\u3002\u5982\u56fe 3-7 \u6240\u793a\uff0c\u201cHello\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u5b57\u8282\uff0c\u201c\u7b97\u6cd5\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9ad8\u4f4d\u586b 0 \u5c06\u201cHello \u7b97\u6cd5\u201d\u4e2d\u7684\u6240\u6709\u5b57\u7b26\u90fd\u7f16\u7801\u4e3a 2 \u5b57\u8282\u957f\u5ea6\u3002\u8fd9\u6837\u7cfb\u7edf\u5c31\u53ef\u4ee5\u6bcf\u9694 2 \u5b57\u8282\u89e3\u6790\u4e00\u4e2a\u5b57\u7b26\uff0c\u6062\u590d\u8fd9\u4e2a\u77ed\u8bed\u7684\u5185\u5bb9\u4e86\u3002</p> <p></p> <p> \u56fe 3-7 \u00a0 Unicode \u7f16\u7801\u793a\u4f8b </p> <p>\u7136\u800c ASCII \u7801\u5df2\u7ecf\u5411\u6211\u4eec\u8bc1\u660e\uff0c\u7f16\u7801\u82f1\u6587\u53ea\u9700 1 \u5b57\u8282\u3002\u82e5\u91c7\u7528\u4e0a\u8ff0\u65b9\u6848\uff0c\u82f1\u6587\u6587\u672c\u5360\u7528\u7a7a\u95f4\u7684\u5927\u5c0f\u5c06\u4f1a\u662f ASCII \u7f16\u7801\u4e0b\u7684\u4e24\u500d\uff0c\u975e\u5e38\u6d6a\u8d39\u5185\u5b58\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u66f4\u52a0\u9ad8\u6548\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#344-utf-8","title":"3.4.4 \u00a0 UTF-8 \u7f16\u7801","text":"<p>\u76ee\u524d\uff0cUTF-8 \u5df2\u6210\u4e3a\u56fd\u9645\u4e0a\u4f7f\u7528\u6700\u5e7f\u6cdb\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002\u5b83\u662f\u4e00\u79cd\u53ef\u53d8\u957f\u5ea6\u7684\u7f16\u7801\uff0c\u4f7f\u7528 1 \u5230 4 \u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6839\u636e\u5b57\u7b26\u7684\u590d\u6742\u6027\u800c\u53d8\u3002ASCII \u5b57\u7b26\u53ea\u9700 1 \u5b57\u8282\uff0c\u62c9\u4e01\u5b57\u6bcd\u548c\u5e0c\u814a\u5b57\u6bcd\u9700\u8981 2 \u5b57\u8282\uff0c\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u9700\u8981 3 \u5b57\u8282\uff0c\u5176\u4ed6\u7684\u4e00\u4e9b\u751f\u50fb\u5b57\u7b26\u9700\u8981 4 \u5b57\u8282\u3002</p> <p>UTF-8 \u7684\u7f16\u7801\u89c4\u5219\u5e76\u4e0d\u590d\u6742\uff0c\u5206\u4e3a\u4ee5\u4e0b\u4e24\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u5bf9\u4e8e\u957f\u5ea6\u4e3a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff0c\u5c06\u6700\u9ad8\u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \uff0c\u5176\u4f59 7 \u4f4d\u8bbe\u7f6e\u4e3a Unicode \u7801\u70b9\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0cASCII \u5b57\u7b26\u5728 Unicode \u5b57\u7b26\u96c6\u4e2d\u5360\u636e\u4e86\u524d 128 \u4e2a\u7801\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0cUTF-8 \u7f16\u7801\u53ef\u4ee5\u5411\u4e0b\u517c\u5bb9 ASCII \u7801\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528 UTF-8 \u6765\u89e3\u6790\u5e74\u4ee3\u4e45\u8fdc\u7684 ASCII \u7801\u6587\u672c\u3002</li> <li>\u5bf9\u4e8e\u957f\u5ea6\u4e3a \\(n\\) \u5b57\u8282\u7684\u5b57\u7b26\uff08\u5176\u4e2d \\(n &gt; 1\\)\uff09\uff0c\u5c06\u9996\u4e2a\u5b57\u8282\u7684\u9ad8 \\(n\\) \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(1\\) \uff0c\u7b2c \\(n + 1\\) \u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \uff1b\u4ece\u7b2c\u4e8c\u4e2a\u5b57\u8282\u5f00\u59cb\uff0c\u5c06\u6bcf\u4e2a\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \uff1b\u5176\u4f59\u6240\u6709\u4f4d\u7528\u4e8e\u586b\u5145\u5b57\u7b26\u7684 Unicode \u7801\u70b9\u3002</li> </ul> <p>\u56fe 3-8 \u5c55\u793a\u4e86\u201cHello\u7b97\u6cd5\u201d\u5bf9\u5e94\u7684 UTF-8 \u7f16\u7801\u3002\u89c2\u5bdf\u53d1\u73b0\uff0c\u7531\u4e8e\u6700\u9ad8 \\(n\\) \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(1\\) \uff0c\u56e0\u6b64\u7cfb\u7edf\u53ef\u4ee5\u901a\u8fc7\u8bfb\u53d6\u6700\u9ad8\u4f4d \\(1\\) \u7684\u4e2a\u6570\u6765\u89e3\u6790\u51fa\u5b57\u7b26\u7684\u957f\u5ea6\u4e3a \\(n\\) \u3002</p> <p>\u4f46\u4e3a\u4ec0\u4e48\u8981\u5c06\u5176\u4f59\u6240\u6709\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \u5462\uff1f\u5b9e\u9645\u4e0a\uff0c\u8fd9\u4e2a \\(10\\) \u80fd\u591f\u8d77\u5230\u6821\u9a8c\u7b26\u7684\u4f5c\u7528\u3002\u5047\u8bbe\u7cfb\u7edf\u4ece\u4e00\u4e2a\u9519\u8bef\u7684\u5b57\u8282\u5f00\u59cb\u89e3\u6790\u6587\u672c\uff0c\u5b57\u8282\u5934\u90e8\u7684 \\(10\\) \u80fd\u591f\u5e2e\u52a9\u7cfb\u7edf\u5feb\u901f\u5224\u65ad\u51fa\u5f02\u5e38\u3002</p> <p>\u4e4b\u6240\u4ee5\u5c06 \\(10\\) \u5f53\u4f5c\u6821\u9a8c\u7b26\uff0c\u662f\u56e0\u4e3a\u5728 UTF-8 \u7f16\u7801\u89c4\u5219\u4e0b\uff0c\u4e0d\u53ef\u80fd\u6709\u5b57\u7b26\u7684\u6700\u9ad8\u4e24\u4f4d\u662f \\(10\\) \u3002\u8fd9\u4e2a\u7ed3\u8bba\u53ef\u4ee5\u7528\u53cd\u8bc1\u6cd5\u6765\u8bc1\u660e\uff1a\u5047\u8bbe\u4e00\u4e2a\u5b57\u7b26\u7684\u6700\u9ad8\u4e24\u4f4d\u662f \\(10\\) \uff0c\u8bf4\u660e\u8be5\u5b57\u7b26\u7684\u957f\u5ea6\u4e3a \\(1\\) \uff0c\u5bf9\u5e94 ASCII \u7801\u3002\u800c ASCII \u7801\u7684\u6700\u9ad8\u4f4d\u5e94\u8be5\u662f \\(0\\) \uff0c\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</p> <p></p> <p> \u56fe 3-8 \u00a0 UTF-8 \u7f16\u7801\u793a\u4f8b </p> <p>\u9664\u4e86 UTF-8 \u4e4b\u5916\uff0c\u5e38\u89c1\u7684\u7f16\u7801\u65b9\u5f0f\u8fd8\u5305\u62ec\u4ee5\u4e0b\u4e24\u79cd\u3002</p> <ul> <li>UTF-16 \u7f16\u7801\uff1a\u4f7f\u7528 2 \u6216 4 \u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\u3002\u6240\u6709\u7684 ASCII \u5b57\u7b26\u548c\u5e38\u7528\u7684\u975e\u82f1\u6587\u5b57\u7b26\uff0c\u90fd\u7528 2 \u5b57\u8282\u8868\u793a\uff1b\u5c11\u6570\u5b57\u7b26\u9700\u8981\u7528\u5230 4 \u5b57\u8282\u8868\u793a\u3002\u5bf9\u4e8e 2 \u5b57\u8282\u7684\u5b57\u7b26\uff0cUTF-16 \u7f16\u7801\u4e0e Unicode \u7801\u70b9\u76f8\u7b49\u3002</li> <li>UTF-32 \u7f16\u7801\uff1a\u6bcf\u4e2a\u5b57\u7b26\u90fd\u4f7f\u7528 4 \u5b57\u8282\u3002\u8fd9\u610f\u5473\u7740 UTF-32 \u6bd4 UTF-8 \u548c UTF-16 \u66f4\u5360\u7528\u7a7a\u95f4\uff0c\u7279\u522b\u662f\u5bf9\u4e8e ASCII \u5b57\u7b26\u5360\u6bd4\u8f83\u9ad8\u7684\u6587\u672c\u3002</li> </ul> <p>\u4ece\u5b58\u50a8\u7a7a\u95f4\u5360\u7528\u7684\u89d2\u5ea6\u770b\uff0c\u4f7f\u7528 UTF-8 \u8868\u793a\u82f1\u6587\u5b57\u7b26\u975e\u5e38\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u4ec5\u9700 1 \u5b57\u8282\uff1b\u4f7f\u7528 UTF-16 \u7f16\u7801\u67d0\u4e9b\u975e\u82f1\u6587\u5b57\u7b26\uff08\u4f8b\u5982\u4e2d\u6587\uff09\u4f1a\u66f4\u52a0\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u4ec5\u9700 2 \u5b57\u8282\uff0c\u800c UTF-8 \u53ef\u80fd\u9700\u8981 3 \u5b57\u8282\u3002</p> <p>\u4ece\u517c\u5bb9\u6027\u7684\u89d2\u5ea6\u770b\uff0cUTF-8 \u7684\u901a\u7528\u6027\u6700\u4f73\uff0c\u8bb8\u591a\u5de5\u5177\u548c\u5e93\u4f18\u5148\u652f\u6301 UTF-8 \u3002</p>"},{"location":"chapter_data_structure/character_encoding/#345","title":"3.4.5 \u00a0 \u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801","text":"<p>\u5bf9\u4e8e\u4ee5\u5f80\u7684\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff0c\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u7684\u5b57\u7b26\u4e32\u90fd\u91c7\u7528 UTF-16 \u6216 UTF-32 \u8fd9\u7c7b\u7b49\u957f\u7f16\u7801\u3002\u5728\u7b49\u957f\u7f16\u7801\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b57\u7b26\u4e32\u770b\u4f5c\u6570\u7ec4\u6765\u5904\u7406\uff0c\u8fd9\u79cd\u505a\u6cd5\u5177\u6709\u4ee5\u4e0b\u4f18\u70b9\u3002</p> <ul> <li>\u968f\u673a\u8bbf\u95ee\uff1aUTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u8fdb\u884c\u968f\u673a\u8bbf\u95ee\u3002UTF-8 \u662f\u4e00\u79cd\u53d8\u957f\u7f16\u7801\uff0c\u8981\u60f3\u627e\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u4ece\u5b57\u7b26\u4e32\u7684\u5f00\u59cb\u5904\u904d\u5386\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u8fd9\u9700\u8981 \\(O(n)\\) \u7684\u65f6\u95f4\u3002</li> <li>\u5b57\u7b26\u8ba1\u6570\uff1a\u4e0e\u968f\u673a\u8bbf\u95ee\u7c7b\u4f3c\uff0c\u8ba1\u7b97 UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u4e5f\u662f \\(O(1)\\) \u7684\u64cd\u4f5c\u3002\u4f46\u662f\uff0c\u8ba1\u7b97 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u9700\u8981\u904d\u5386\u6574\u4e2a\u5b57\u7b26\u4e32\u3002</li> <li>\u5b57\u7b26\u4e32\u64cd\u4f5c\uff1a\u5728 UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e0a\uff0c\u5f88\u591a\u5b57\u7b26\u4e32\u64cd\u4f5c\uff08\u5982\u5206\u5272\u3001\u8fde\u63a5\u3001\u63d2\u5165\u3001\u5220\u9664\u7b49\uff09\u66f4\u5bb9\u6613\u8fdb\u884c\u3002\u5728 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e0a\uff0c\u8fdb\u884c\u8fd9\u4e9b\u64cd\u4f5c\u901a\u5e38\u9700\u8981\u989d\u5916\u7684\u8ba1\u7b97\uff0c\u4ee5\u786e\u4fdd\u4e0d\u4f1a\u4ea7\u751f\u65e0\u6548\u7684 UTF-8 \u7f16\u7801\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6848\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5f88\u6709\u8da3\u7684\u8bdd\u9898\uff0c\u6d89\u53ca\u8bb8\u591a\u56e0\u7d20\u3002</p> <ul> <li>Java \u7684 <code>String</code> \u7c7b\u578b\u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u8fd9\u662f\u56e0\u4e3a Java \u8bed\u8a00\u8bbe\u8ba1\u4e4b\u521d\uff0c\u4eba\u4eec\u8ba4\u4e3a 16 \u4f4d\u8db3\u4ee5\u8868\u793a\u6240\u6709\u53ef\u80fd\u7684\u5b57\u7b26\u3002\u7136\u800c\uff0c\u8fd9\u662f\u4e00\u4e2a\u4e0d\u6b63\u786e\u7684\u5224\u65ad\u3002\u540e\u6765 Unicode \u89c4\u8303\u6269\u5c55\u5230\u4e86\u8d85\u8fc7 16 \u4f4d\uff0c\u6240\u4ee5 Java \u4e2d\u7684\u5b57\u7b26\u73b0\u5728\u53ef\u80fd\u7531\u4e00\u5bf9 16 \u4f4d\u7684\u503c\uff08\u79f0\u4e3a\u201c\u4ee3\u7406\u5bf9\u201d\uff09\u8868\u793a\u3002</li> <li>JavaScript \u548c TypeScript \u7684\u5b57\u7b26\u4e32\u4f7f\u7528 UTF-16 \u7f16\u7801\u7684\u539f\u56e0\u4e0e Java \u7c7b\u4f3c\u3002\u5f53 1995 \u5e74 Netscape \u516c\u53f8\u9996\u6b21\u63a8\u51fa JavaScript \u8bed\u8a00\u65f6\uff0cUnicode \u8fd8\u5904\u4e8e\u53d1\u5c55\u65e9\u671f\uff0c\u90a3\u65f6\u5019\u4f7f\u7528 16 \u4f4d\u7684\u7f16\u7801\u5c31\u8db3\u4ee5\u8868\u793a\u6240\u6709\u7684 Unicode \u5b57\u7b26\u4e86\u3002</li> <li>C# \u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u4e3b\u8981\u662f\u56e0\u4e3a .NET \u5e73\u53f0\u662f\u7531 Microsoft \u8bbe\u8ba1\u7684\uff0c\u800c Microsoft \u7684\u5f88\u591a\u6280\u672f\uff08\u5305\u62ec Windows \u64cd\u4f5c\u7cfb\u7edf\uff09\u90fd\u5e7f\u6cdb\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002</li> </ul> <p>\u7531\u4e8e\u4ee5\u4e0a\u7f16\u7a0b\u8bed\u8a00\u5bf9\u5b57\u7b26\u6570\u91cf\u7684\u4f4e\u4f30\uff0c\u5b83\u4eec\u4e0d\u5f97\u4e0d\u91c7\u53d6\u201c\u4ee3\u7406\u5bf9\u201d\u7684\u65b9\u5f0f\u6765\u8868\u793a\u8d85\u8fc7 16 \u4f4d\u957f\u5ea6\u7684 Unicode \u5b57\u7b26\u3002\u8fd9\u662f\u4e00\u4e2a\u4e0d\u5f97\u5df2\u4e3a\u4e4b\u7684\u65e0\u5948\u4e4b\u4e3e\u3002\u4e00\u65b9\u9762\uff0c\u5305\u542b\u4ee3\u7406\u5bf9\u7684\u5b57\u7b26\u4e32\u4e2d\uff0c\u4e00\u4e2a\u5b57\u7b26\u53ef\u80fd\u5360\u7528 2 \u5b57\u8282\u6216 4 \u5b57\u8282\uff0c\u4ece\u800c\u4e27\u5931\u4e86\u7b49\u957f\u7f16\u7801\u7684\u4f18\u52bf\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5904\u7406\u4ee3\u7406\u5bf9\u9700\u8981\u989d\u5916\u589e\u52a0\u4ee3\u7801\uff0c\u8fd9\u63d0\u9ad8\u4e86\u7f16\u7a0b\u7684\u590d\u6742\u6027\u548c\u8c03\u8bd5\u96be\u5ea6\u3002</p> <p>\u51fa\u4e8e\u4ee5\u4e0a\u539f\u56e0\uff0c\u90e8\u5206\u7f16\u7a0b\u8bed\u8a00\u63d0\u51fa\u4e86\u4e00\u4e9b\u4e0d\u540c\u7684\u7f16\u7801\u65b9\u6848\u3002</p> <ul> <li>Python \u4e2d\u7684 <code>str</code> \u4f7f\u7528 Unicode \u7f16\u7801\uff0c\u5e76\u91c7\u7528\u4e00\u79cd\u7075\u6d3b\u7684\u5b57\u7b26\u4e32\u8868\u793a\uff0c\u5b58\u50a8\u7684\u5b57\u7b26\u957f\u5ea6\u53d6\u51b3\u4e8e\u5b57\u7b26\u4e32\u4e2d\u6700\u5927\u7684 Unicode \u7801\u70b9\u3002\u82e5\u5b57\u7b26\u4e32\u4e2d\u5168\u90e8\u662f ASCII \u5b57\u7b26\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u5b57\u8282\uff1b\u5982\u679c\u6709\u5b57\u7b26\u8d85\u51fa\u4e86 ASCII \u8303\u56f4\uff0c\u4f46\u5168\u90e8\u5728\u57fa\u672c\u591a\u8bed\u8a00\u5e73\u9762\uff08BMP\uff09\u5185\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\uff1b\u5982\u679c\u6709\u8d85\u51fa BMP \u7684\u5b57\u7b26\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 4 \u5b57\u8282\u3002</li> <li>Go \u8bed\u8a00\u7684 <code>string</code> \u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Go \u8bed\u8a00\u8fd8\u63d0\u4f9b\u4e86 <code>rune</code> \u7c7b\u578b\uff0c\u5b83\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002</li> <li>Rust \u8bed\u8a00\u7684 <code>str</code> \u548c <code>String</code> \u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Rust \u4e5f\u63d0\u4f9b\u4e86 <code>char</code> \u7c7b\u578b\uff0c\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002</li> </ul> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ee5\u4e0a\u8ba8\u8bba\u7684\u90fd\u662f\u5b57\u7b26\u4e32\u5728\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u8fd9\u548c\u5b57\u7b26\u4e32\u5982\u4f55\u5728\u6587\u4ef6\u4e2d\u5b58\u50a8\u6216\u5728\u7f51\u7edc\u4e2d\u4f20\u8f93\u662f\u4e0d\u540c\u7684\u95ee\u9898\u3002\u5728\u6587\u4ef6\u5b58\u50a8\u6216\u7f51\u7edc\u4f20\u8f93\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u5b57\u7b26\u4e32\u7f16\u7801\u4e3a UTF-8 \u683c\u5f0f\uff0c\u4ee5\u8fbe\u5230\u6700\u4f18\u7684\u517c\u5bb9\u6027\u548c\u7a7a\u95f4\u6548\u7387\u3002</p>"},{"location":"chapter_data_structure/classification_of_data_structure/","title":"3.1 \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b","text":"<p>\u5e38\u89c1\u7684\u6570\u636e\u7ed3\u6784\u5305\u62ec\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\uff0c\u5b83\u4eec\u53ef\u4ee5\u4ece\u201c\u903b\u8f91\u7ed3\u6784\u201d\u548c\u201c\u7269\u7406\u7ed3\u6784\u201d\u4e24\u4e2a\u7ef4\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002</p>"},{"location":"chapter_data_structure/classification_of_data_structure/#311","title":"3.1.1 \u00a0 \u903b\u8f91\u7ed3\u6784\uff1a\u7ebf\u6027\u4e0e\u975e\u7ebf\u6027","text":"<p>\u903b\u8f91\u7ed3\u6784\u63ed\u793a\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\u3002\u5728\u6570\u7ec4\u548c\u94fe\u8868\u4e2d\uff0c\u6570\u636e\u6309\u7167\u4e00\u5b9a\u987a\u5e8f\u6392\u5217\uff0c\u4f53\u73b0\u4e86\u6570\u636e\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\uff1b\u800c\u5728\u6811\u4e2d\uff0c\u6570\u636e\u4ece\u9876\u90e8\u5411\u4e0b\u6309\u5c42\u6b21\u6392\u5217\uff0c\u8868\u73b0\u51fa\u201c\u7956\u5148\u201d\u4e0e\u201c\u540e\u4ee3\u201d\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff1b\u56fe\u5219\u7531\u8282\u70b9\u548c\u8fb9\u6784\u6210\uff0c\u53cd\u6620\u4e86\u590d\u6742\u7684\u7f51\u7edc\u5173\u7cfb\u3002</p> <p>\u5982\u56fe 3-1 \u6240\u793a\uff0c\u903b\u8f91\u7ed3\u6784\u53ef\u5206\u4e3a\u201c\u7ebf\u6027\u201d\u548c\u201c\u975e\u7ebf\u6027\u201d\u4e24\u5927\u7c7b\u3002\u7ebf\u6027\u7ed3\u6784\u6bd4\u8f83\u76f4\u89c2\uff0c\u6307\u6570\u636e\u5728\u903b\u8f91\u5173\u7cfb\u4e0a\u5448\u7ebf\u6027\u6392\u5217\uff1b\u975e\u7ebf\u6027\u7ed3\u6784\u5219\u76f8\u53cd\uff0c\u5448\u975e\u7ebf\u6027\u6392\u5217\u3002</p> <ul> <li>\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u4e00\u5bf9\u4e00\u7684\u987a\u5e8f\u5173\u7cfb\u3002</li> <li>\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6811\u3001\u5806\u3001\u56fe\u3001\u54c8\u5e0c\u8868\u3002</li> </ul> <p>\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a\u6811\u5f62\u7ed3\u6784\u548c\u7f51\u72b6\u7ed3\u6784\u3002</p> <ul> <li>\u6811\u5f62\u7ed3\u6784\uff1a\u6811\u3001\u5806\u3001\u54c8\u5e0c\u8868\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u4e00\u5bf9\u591a\u7684\u5173\u7cfb\u3002</li> <li>\u7f51\u72b6\u7ed3\u6784\uff1a\u56fe\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u591a\u5bf9\u591a\u7684\u5173\u7cfb\u3002</li> </ul> <p></p> <p> \u56fe 3-1 \u00a0 \u7ebf\u6027\u6570\u636e\u7ed3\u6784\u4e0e\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784 </p>"},{"location":"chapter_data_structure/classification_of_data_structure/#312","title":"3.1.2 \u00a0 \u7269\u7406\u7ed3\u6784\uff1a\u8fde\u7eed\u4e0e\u5206\u6563","text":"<p>\u5f53\u7b97\u6cd5\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6b63\u5728\u5904\u7406\u7684\u6570\u636e\u4e3b\u8981\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u56fe 3-2 \u5c55\u793a\u4e86\u4e00\u4e2a\u8ba1\u7b97\u673a\u5185\u5b58\u6761\uff0c\u5176\u4e2d\u6bcf\u4e2a\u9ed1\u8272\u65b9\u5757\u90fd\u5305\u542b\u4e00\u5757\u5185\u5b58\u7a7a\u95f4\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u5185\u5b58\u60f3\u8c61\u6210\u4e00\u4e2a\u5de8\u5927\u7684 Excel \u8868\u683c\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5355\u5143\u683c\u90fd\u53ef\u4ee5\u5b58\u50a8\u4e00\u5b9a\u5927\u5c0f\u7684\u6570\u636e\u3002</p> <p>\u7cfb\u7edf\u901a\u8fc7\u5185\u5b58\u5730\u5740\u6765\u8bbf\u95ee\u76ee\u6807\u4f4d\u7f6e\u7684\u6570\u636e\u3002\u5982\u56fe 3-2 \u6240\u793a\uff0c\u8ba1\u7b97\u673a\u6839\u636e\u7279\u5b9a\u89c4\u5219\u4e3a\u8868\u683c\u4e2d\u7684\u6bcf\u4e2a\u5355\u5143\u683c\u5206\u914d\u7f16\u53f7\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u6709\u552f\u4e00\u7684\u5185\u5b58\u5730\u5740\u3002\u6709\u4e86\u8fd9\u4e9b\u5730\u5740\uff0c\u7a0b\u5e8f\u4fbf\u53ef\u4ee5\u8bbf\u95ee\u5185\u5b58\u4e2d\u7684\u6570\u636e\u3002</p> <p></p> <p> \u56fe 3-2 \u00a0 \u5185\u5b58\u6761\u3001\u5185\u5b58\u7a7a\u95f4\u3001\u5185\u5b58\u5730\u5740 </p> <p>Tip</p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5c06\u5185\u5b58\u6bd4\u4f5c Excel \u8868\u683c\u662f\u4e00\u4e2a\u7b80\u5316\u7684\u7c7b\u6bd4\uff0c\u5b9e\u9645\u5185\u5b58\u7684\u5de5\u4f5c\u673a\u5236\u6bd4\u8f83\u590d\u6742\uff0c\u6d89\u53ca\u5730\u5740\u7a7a\u95f4\u3001\u5185\u5b58\u7ba1\u7406\u3001\u7f13\u5b58\u673a\u5236\u3001\u865a\u62df\u5185\u5b58\u548c\u7269\u7406\u5185\u5b58\u7b49\u6982\u5ff5\u3002</p> <p>\u5185\u5b58\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u5171\u4eab\u8d44\u6e90\uff0c\u5f53\u67d0\u5757\u5185\u5b58\u88ab\u67d0\u4e2a\u7a0b\u5e8f\u5360\u7528\u65f6\uff0c\u5219\u65e0\u6cd5\u88ab\u5176\u4ed6\u7a0b\u5e8f\u540c\u65f6\u4f7f\u7528\u4e86\u3002\u56e0\u6b64\u5728\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8bbe\u8ba1\u4e2d\uff0c\u5185\u5b58\u8d44\u6e90\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u8003\u8651\u56e0\u7d20\u3002\u6bd4\u5982\uff0c\u7b97\u6cd5\u6240\u5360\u7528\u7684\u5185\u5b58\u5cf0\u503c\u4e0d\u5e94\u8d85\u8fc7\u7cfb\u7edf\u5269\u4f59\u7a7a\u95f2\u5185\u5b58\uff1b\u5982\u679c\u7f3a\u5c11\u8fde\u7eed\u5927\u5757\u7684\u5185\u5b58\u7a7a\u95f4\uff0c\u90a3\u4e48\u6240\u9009\u7528\u7684\u6570\u636e\u7ed3\u6784\u5fc5\u987b\u80fd\u591f\u5b58\u50a8\u5728\u5206\u6563\u7684\u5185\u5b58\u7a7a\u95f4\u5185\u3002</p> <p>\u5982\u56fe 3-3 \u6240\u793a\uff0c\u7269\u7406\u7ed3\u6784\u53cd\u6620\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u53ef\u5206\u4e3a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\u3002\u7269\u7406\u7ed3\u6784\u4ece\u5e95\u5c42\u51b3\u5b9a\u4e86\u6570\u636e\u7684\u8bbf\u95ee\u3001\u66f4\u65b0\u3001\u589e\u5220\u7b49\u64cd\u4f5c\u65b9\u6cd5\uff0c\u4e24\u79cd\u7269\u7406\u7ed3\u6784\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u65b9\u9762\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u70b9\u3002</p> <p></p> <p> \u56fe 3-3 \u00a0 \u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u4e0e\u5206\u6563\u7a7a\u95f4\u5b58\u50a8 </p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u57fa\u4e8e\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e8c\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002\u4f8b\u5982\uff0c\u6808\u548c\u961f\u5217\u65e2\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff1b\u800c\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u6570\u7ec4\u548c\u94fe\u8868\u3002</p> <ul> <li>\u57fa\u4e8e\u6570\u7ec4\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u3001\u77e9\u9635\u3001\u5f20\u91cf\uff08\u7ef4\u5ea6 \\(\\geq 3\\) \u7684\u6570\u7ec4\uff09\u7b49\u3002</li> <li>\u57fa\u4e8e\u94fe\u8868\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u3002</li> </ul> <p>\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u79f0\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d\uff0c\u8fd9\u610f\u5473\u7740\u6b64\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u76f8\u5bf9\u5e94\u5730\uff0c\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u79f0\u201c\u52a8\u6001\u6570\u636e\u7ed3\u6784\u201d\uff0c\u8fd9\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\uff0c\u4ecd\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u5bf9\u5176\u957f\u5ea6\u8fdb\u884c\u8c03\u6574\u3002</p> <p>Tip</p> <p>\u5982\u679c\u4f60\u611f\u89c9\u7269\u7406\u7ed3\u6784\u7406\u89e3\u8d77\u6765\u6709\u56f0\u96be\uff0c\u5efa\u8bae\u5148\u9605\u8bfb\u4e0b\u4e00\u7ae0\uff0c\u7136\u540e\u518d\u56de\u987e\u672c\u8282\u5185\u5bb9\u3002</p>"},{"location":"chapter_data_structure/number_encoding/","title":"3.3 \u00a0 \u6570\u5b57\u7f16\u7801 *","text":"<p>Note</p> <p>\u5728\u672c\u4e66\u4e2d\uff0c\u6807\u9898\u5e26\u6709 * \u7b26\u53f7\u7684\u662f\u9009\u8bfb\u7ae0\u8282\u3002\u5982\u679c\u4f60\u65f6\u95f4\u6709\u9650\u6216\u611f\u5230\u7406\u89e3\u56f0\u96be\uff0c\u53ef\u4ee5\u5148\u8df3\u8fc7\uff0c\u7b49\u5b66\u5b8c\u5fc5\u8bfb\u7ae0\u8282\u540e\u518d\u5355\u72ec\u653b\u514b\u3002</p>"},{"location":"chapter_data_structure/number_encoding/#331","title":"3.3.1 \u00a0 \u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801","text":"<p>\u5728\u4e0a\u4e00\u8282\u7684\u8868\u683c\u4e2d\u6211\u4eec\u53d1\u73b0\uff0c\u6240\u6709\u6574\u6570\u7c7b\u578b\u80fd\u591f\u8868\u793a\u7684\u8d1f\u6570\u90fd\u6bd4\u6b63\u6570\u591a\u4e00\u4e2a\uff0c\u4f8b\u5982 <code>byte</code> \u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \u3002\u8fd9\u4e2a\u73b0\u8c61\u6bd4\u8f83\u53cd\u76f4\u89c9\uff0c\u5b83\u7684\u5185\u5728\u539f\u56e0\u6d89\u53ca\u539f\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u7684\u76f8\u5173\u77e5\u8bc6\u3002</p> <p>\u9996\u5148\u9700\u8981\u6307\u51fa\uff0c\u6570\u5b57\u662f\u4ee5\u201c\u8865\u7801\u201d\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u7684\u3002\u5728\u5206\u6790\u8fd9\u6837\u505a\u7684\u539f\u56e0\u4e4b\u524d\uff0c\u9996\u5148\u7ed9\u51fa\u4e09\u8005\u7684\u5b9a\u4e49\u3002</p> <ul> <li>\u539f\u7801\uff1a\u6211\u4eec\u5c06\u6570\u5b57\u7684\u4e8c\u8fdb\u5236\u8868\u793a\u7684\u6700\u9ad8\u4f4d\u89c6\u4e3a\u7b26\u53f7\u4f4d\uff0c\u5176\u4e2d \\(0\\) \u8868\u793a\u6b63\u6570\uff0c\\(1\\) \u8868\u793a\u8d1f\u6570\uff0c\u5176\u4f59\u4f4d\u8868\u793a\u6570\u5b57\u7684\u503c\u3002</li> <li>\u53cd\u7801\uff1a\u6b63\u6570\u7684\u53cd\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u53cd\u7801\u662f\u5bf9\u5176\u539f\u7801\u9664\u7b26\u53f7\u4f4d\u5916\u7684\u6240\u6709\u4f4d\u53d6\u53cd\u3002</li> <li>\u8865\u7801\uff1a\u6b63\u6570\u7684\u8865\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u8865\u7801\u662f\u5728\u5176\u53cd\u7801\u7684\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u3002</li> </ul> <p>\u56fe 3-4 \u5c55\u793a\u4e86\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\u4e4b\u95f4\u7684\u8f6c\u6362\u65b9\u6cd5\u3002</p> <p></p> <p> \u56fe 3-4 \u00a0 \u539f\u7801\u3001\u53cd\u7801\u4e0e\u8865\u7801\u4e4b\u95f4\u7684\u76f8\u4e92\u8f6c\u6362 </p> <p>\u539f\u7801\uff08sign-magnitude\uff09\u867d\u7136\u6700\u76f4\u89c2\uff0c\u4f46\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\u3002\u4e00\u65b9\u9762\uff0c\u8d1f\u6570\u7684\u539f\u7801\u4e0d\u80fd\u76f4\u63a5\u7528\u4e8e\u8fd0\u7b97\u3002\u4f8b\u5982\u5728\u539f\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u5f97\u5230\u7684\u7ed3\u679c\u662f \\(-3\\) \uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 + 1000 \\; 0010 \\newline &amp; = 1000 \\; 0011 \\newline &amp; \\rightarrow -3 \\end{aligned} \\] <p>\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u8ba1\u7b97\u673a\u5f15\u5165\u4e86\u53cd\u7801\uff081's complement\uff09\u3002\u5982\u679c\u6211\u4eec\u5148\u5c06\u539f\u7801\u8f6c\u6362\u4e3a\u53cd\u7801\uff0c\u5e76\u5728\u53cd\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u6700\u540e\u5c06\u7ed3\u679c\u4ece\u53cd\u7801\u8f6c\u6362\u56de\u539f\u7801\uff0c\u5219\u53ef\u5f97\u5230\u6b63\u786e\u7ed3\u679c \\(-1\\) \u3002</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 \\; \\text{(\u539f\u7801)} + 1000 \\; 0010 \\; \\text{(\u539f\u7801)} \\newline &amp; = 0000 \\; 0001 \\; \\text{(\u53cd\u7801)} + 1111  \\; 1101 \\; \\text{(\u53cd\u7801)} \\newline &amp; = 1111 \\; 1110 \\; \\text{(\u53cd\u7801)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(\u539f\u7801)} \\newline &amp; \\rightarrow -1 \\end{aligned} \\] <p>\u53e6\u4e00\u65b9\u9762\uff0c\u6570\u5b57\u96f6\u7684\u539f\u7801\u6709 \\(+0\\) \u548c \\(-0\\) \u4e24\u79cd\u8868\u793a\u65b9\u5f0f\u3002\u8fd9\u610f\u5473\u7740\u6570\u5b57\u96f6\u5bf9\u5e94\u4e24\u4e2a\u4e0d\u540c\u7684\u4e8c\u8fdb\u5236\u7f16\u7801\uff0c\u8fd9\u53ef\u80fd\u4f1a\u5e26\u6765\u6b67\u4e49\u3002\u6bd4\u5982\u5728\u6761\u4ef6\u5224\u65ad\u4e2d\uff0c\u5982\u679c\u6ca1\u6709\u533a\u5206\u6b63\u96f6\u548c\u8d1f\u96f6\uff0c\u5219\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5224\u65ad\u7ed3\u679c\u51fa\u9519\u3002\u800c\u5982\u679c\u6211\u4eec\u60f3\u5904\u7406\u6b63\u96f6\u548c\u8d1f\u96f6\u6b67\u4e49\uff0c\u5219\u9700\u8981\u5f15\u5165\u989d\u5916\u7684\u5224\u65ad\u64cd\u4f5c\uff0c\u8fd9\u53ef\u80fd\u4f1a\u964d\u4f4e\u8ba1\u7b97\u673a\u7684\u8fd0\u7b97\u6548\u7387\u3002</p> \\[ \\begin{aligned} +0 &amp; \\rightarrow 0000 \\; 0000 \\newline -0 &amp; \\rightarrow 1000 \\; 0000 \\end{aligned} \\] <p>\u4e0e\u539f\u7801\u4e00\u6837\uff0c\u53cd\u7801\u4e5f\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u95ee\u9898\uff0c\u56e0\u6b64\u8ba1\u7b97\u673a\u8fdb\u4e00\u6b65\u5f15\u5165\u4e86\u8865\u7801\uff082's complement\uff09\u3002\u6211\u4eec\u5148\u6765\u89c2\u5bdf\u4e00\u4e0b\u8d1f\u96f6\u7684\u539f\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u7684\u8f6c\u6362\u8fc7\u7a0b\uff1a</p> \\[ \\begin{aligned} -0 \\rightarrow \\; &amp; 1000 \\; 0000 \\; \\text{(\u539f\u7801)} \\newline = \\; &amp; 1111 \\; 1111 \\; \\text{(\u53cd\u7801)} \\newline = 1 \\; &amp; 0000 \\; 0000 \\; \\text{(\u8865\u7801)} \\newline \\end{aligned} \\] <p>\u5728\u8d1f\u96f6\u7684\u53cd\u7801\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u4f1a\u4ea7\u751f\u8fdb\u4f4d\uff0c\u4f46 <code>byte</code> \u7c7b\u578b\u7684\u957f\u5ea6\u53ea\u6709 8 \u4f4d\uff0c\u56e0\u6b64\u6ea2\u51fa\u5230\u7b2c 9 \u4f4d\u7684 \\(1\\) \u4f1a\u88ab\u820d\u5f03\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u8d1f\u96f6\u7684\u8865\u7801\u4e3a \\(0000 \\; 0000\\) \uff0c\u4e0e\u6b63\u96f6\u7684\u8865\u7801\u76f8\u540c\u3002\u8fd9\u610f\u5473\u7740\u5728\u8865\u7801\u8868\u793a\u4e2d\u53ea\u5b58\u5728\u4e00\u4e2a\u96f6\uff0c\u6b63\u8d1f\u96f6\u6b67\u4e49\u4ece\u800c\u5f97\u5230\u89e3\u51b3\u3002</p> <p>\u8fd8\u5269\u6700\u540e\u4e00\u4e2a\u7591\u60d1\uff1a<code>byte</code> \u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \uff0c\u591a\u51fa\u6765\u7684\u4e00\u4e2a\u8d1f\u6570 \\(-128\\) \u662f\u5982\u4f55\u5f97\u5230\u7684\u5462\uff1f\u6211\u4eec\u6ce8\u610f\u5230\uff0c\u533a\u95f4 \\([-127, +127]\\) \u5185\u7684\u6240\u6709\u6574\u6570\u90fd\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\uff0c\u5e76\u4e14\u539f\u7801\u548c\u8865\u7801\u4e4b\u95f4\u53ef\u4ee5\u4e92\u76f8\u8f6c\u6362\u3002</p> <p>\u7136\u800c\uff0c\u8865\u7801 \\(1000 \\; 0000\\) \u662f\u4e00\u4e2a\u4f8b\u5916\uff0c\u5b83\u5e76\u6ca1\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3002\u6839\u636e\u8f6c\u6362\u65b9\u6cd5\uff0c\u6211\u4eec\u5f97\u5230\u8be5\u8865\u7801\u7684\u539f\u7801\u4e3a \\(0000 \\; 0000\\) \u3002\u8fd9\u663e\u7136\u662f\u77db\u76fe\u7684\uff0c\u56e0\u4e3a\u8be5\u539f\u7801\u8868\u793a\u6570\u5b57 \\(0\\) \uff0c\u5b83\u7684\u8865\u7801\u5e94\u8be5\u662f\u81ea\u8eab\u3002\u8ba1\u7b97\u673a\u89c4\u5b9a\u8fd9\u4e2a\u7279\u6b8a\u7684\u8865\u7801 \\(1000 \\; 0000\\) \u4ee3\u8868 \\(-128\\) \u3002\u5b9e\u9645\u4e0a\uff0c\\((-1) + (-127)\\) \u5728\u8865\u7801\u4e0b\u7684\u8ba1\u7b97\u7ed3\u679c\u5c31\u662f \\(-128\\) \u3002</p> \\[ \\begin{aligned} &amp; (-127) + (-1) \\newline &amp; \\rightarrow 1111 \\; 1111 \\; \\text{(\u539f\u7801)} + 1000 \\; 0001 \\; \\text{(\u539f\u7801)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(\u53cd\u7801)} + 1111  \\; 1110 \\; \\text{(\u53cd\u7801)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(\u8865\u7801)} + 1111  \\; 1111 \\; \\text{(\u8865\u7801)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(\u8865\u7801)} \\newline &amp; \\rightarrow -128 \\end{aligned} \\] <p>\u4f60\u53ef\u80fd\u5df2\u7ecf\u53d1\u73b0\u4e86\uff0c\u4e0a\u8ff0\u6240\u6709\u8ba1\u7b97\u90fd\u662f\u52a0\u6cd5\u8fd0\u7b97\u3002\u8fd9\u6697\u793a\u7740\u4e00\u4e2a\u91cd\u8981\u4e8b\u5b9e\uff1a\u8ba1\u7b97\u673a\u5185\u90e8\u7684\u786c\u4ef6\u7535\u8def\u4e3b\u8981\u662f\u57fa\u4e8e\u52a0\u6cd5\u8fd0\u7b97\u8bbe\u8ba1\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u52a0\u6cd5\u8fd0\u7b97\u76f8\u5bf9\u4e8e\u5176\u4ed6\u8fd0\u7b97\uff08\u6bd4\u5982\u4e58\u6cd5\u3001\u9664\u6cd5\u548c\u51cf\u6cd5\uff09\u6765\u8bf4\uff0c\u786c\u4ef6\u5b9e\u73b0\u8d77\u6765\u66f4\u7b80\u5355\uff0c\u66f4\u5bb9\u6613\u8fdb\u884c\u5e76\u884c\u5316\u5904\u7406\uff0c\u8fd0\u7b97\u901f\u5ea6\u66f4\u5feb\u3002</p> <p>\u8bf7\u6ce8\u610f\uff0c\u8fd9\u5e76\u4e0d\u610f\u5473\u7740\u8ba1\u7b97\u673a\u53ea\u80fd\u505a\u52a0\u6cd5\u3002\u901a\u8fc7\u5c06\u52a0\u6cd5\u4e0e\u4e00\u4e9b\u57fa\u672c\u903b\u8f91\u8fd0\u7b97\u7ed3\u5408\uff0c\u8ba1\u7b97\u673a\u80fd\u591f\u5b9e\u73b0\u5404\u79cd\u5176\u4ed6\u7684\u6570\u5b66\u8fd0\u7b97\u3002\u4f8b\u5982\uff0c\u8ba1\u7b97\u51cf\u6cd5 \\(a - b\\) \u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u52a0\u6cd5 \\(a + (-b)\\) \uff1b\u8ba1\u7b97\u4e58\u6cd5\u548c\u9664\u6cd5\u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u591a\u6b21\u52a0\u6cd5\u6216\u51cf\u6cd5\u3002</p> <p>\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u8ba1\u7b97\u673a\u4f7f\u7528\u8865\u7801\u7684\u539f\u56e0\uff1a\u57fa\u4e8e\u8865\u7801\u8868\u793a\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u7528\u540c\u6837\u7684\u7535\u8def\u548c\u64cd\u4f5c\u6765\u5904\u7406\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\uff0c\u4e0d\u9700\u8981\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\u6765\u5904\u7406\u51cf\u6cd5\uff0c\u5e76\u4e14\u65e0\u987b\u7279\u522b\u5904\u7406\u6b63\u8d1f\u96f6\u7684\u6b67\u4e49\u95ee\u9898\u3002\u8fd9\u5927\u5927\u7b80\u5316\u4e86\u786c\u4ef6\u8bbe\u8ba1\uff0c\u63d0\u9ad8\u4e86\u8fd0\u7b97\u6548\u7387\u3002</p> <p>\u8865\u7801\u7684\u8bbe\u8ba1\u975e\u5e38\u7cbe\u5999\uff0c\u56e0\u7bc7\u5e45\u5173\u7cfb\u6211\u4eec\u5c31\u5148\u4ecb\u7ecd\u5230\u8fd9\u91cc\uff0c\u5efa\u8bae\u6709\u5174\u8da3\u7684\u8bfb\u8005\u8fdb\u4e00\u6b65\u6df1\u5165\u4e86\u89e3\u3002</p>"},{"location":"chapter_data_structure/number_encoding/#332","title":"3.3.2 \u00a0 \u6d6e\u70b9\u6570\u7f16\u7801","text":"<p>\u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\uff1a<code>int</code> \u548c <code>float</code> \u957f\u5ea6\u76f8\u540c\uff0c\u90fd\u662f 4 \u5b57\u8282 \uff0c\u4f46\u4e3a\u4ec0\u4e48 <code>float</code> \u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e <code>int</code> \uff1f\u8fd9\u975e\u5e38\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u6309\u7406\u8bf4 <code>float</code> \u9700\u8981\u8868\u793a\u5c0f\u6570\uff0c\u53d6\u503c\u8303\u56f4\u5e94\u8be5\u53d8\u5c0f\u624d\u5bf9\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8fd9\u662f\u56e0\u4e3a\u6d6e\u70b9\u6570 <code>float</code> \u91c7\u7528\u4e86\u4e0d\u540c\u7684\u8868\u793a\u65b9\u5f0f\u3002\u8bb0\u4e00\u4e2a 32 \u6bd4\u7279\u957f\u5ea6\u7684\u4e8c\u8fdb\u5236\u6570\u4e3a\uff1a</p> \\[ b_{31} b_{30} b_{29} \\ldots b_2 b_1 b_0 \\] <p>\u6839\u636e IEEE 754 \u6807\u51c6\uff0c32-bit \u957f\u5ea6\u7684 <code>float</code> \u7531\u4ee5\u4e0b\u4e09\u4e2a\u90e8\u5206\u6784\u6210\u3002</p> <ul> <li>\u7b26\u53f7\u4f4d \\(\\mathrm{S}\\) \uff1a\u5360 1 \u4f4d \uff0c\u5bf9\u5e94 \\(b_{31}\\) \u3002</li> <li>\u6307\u6570\u4f4d \\(\\mathrm{E}\\) \uff1a\u5360 8 \u4f4d \uff0c\u5bf9\u5e94 \\(b_{30} b_{29} \\ldots b_{23}\\) \u3002</li> <li>\u5206\u6570\u4f4d \\(\\mathrm{N}\\) \uff1a\u5360 23 \u4f4d \uff0c\u5bf9\u5e94 \\(b_{22} b_{21} \\ldots b_0\\) \u3002</li> </ul> <p>\u4e8c\u8fdb\u5236\u6570 <code>float</code> \u5bf9\u5e94\u503c\u7684\u8ba1\u7b97\u65b9\u6cd5\u4e3a\uff1a</p> \\[ \\text {val} = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\] <p>\u8f6c\u5316\u5230\u5341\u8fdb\u5236\u4e0b\u7684\u8ba1\u7b97\u516c\u5f0f\u4e3a\uff1a</p> \\[ \\text {val}=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\] <p>\u5176\u4e2d\u5404\u9879\u7684\u53d6\u503c\u8303\u56f4\u4e3a\uff1a</p> \\[ \\begin{aligned} \\mathrm{S} \\in &amp; \\{ 0, 1\\}, \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = &amp; (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\] <p></p> <p> \u56fe 3-5 \u00a0 IEEE 754 \u6807\u51c6\u4e0b\u7684 float \u7684\u8ba1\u7b97\u793a\u4f8b </p> <p>\u89c2\u5bdf\u56fe 3-5 \uff0c\u7ed9\u5b9a\u4e00\u4e2a\u793a\u4f8b\u6570\u636e \\(\\mathrm{S} = 0\\) \uff0c \\(\\mathrm{E} = 124\\) \uff0c\\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\) \uff0c\u5219\u6709\uff1a</p> \\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\] <p>\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u56de\u7b54\u6700\u521d\u7684\u95ee\u9898\uff1a<code>float</code> \u7684\u8868\u793a\u65b9\u5f0f\u5305\u542b\u6307\u6570\u4f4d\uff0c\u5bfc\u81f4\u5176\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e <code>int</code> \u3002\u6839\u636e\u4ee5\u4e0a\u8ba1\u7b97\uff0c<code>float</code> \u53ef\u8868\u793a\u7684\u6700\u5927\u6b63\u6570\u4e3a \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\) \uff0c\u5207\u6362\u7b26\u53f7\u4f4d\u4fbf\u53ef\u5f97\u5230\u6700\u5c0f\u8d1f\u6570\u3002</p> <p>\u5c3d\u7ba1\u6d6e\u70b9\u6570 <code>float</code> \u6269\u5c55\u4e86\u53d6\u503c\u8303\u56f4\uff0c\u4f46\u5176\u526f\u4f5c\u7528\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002\u6574\u6570\u7c7b\u578b <code>int</code> \u5c06\u5168\u90e8 32 \u6bd4\u7279\u7528\u4e8e\u8868\u793a\u6570\u5b57\uff0c\u6570\u5b57\u662f\u5747\u5300\u5206\u5e03\u7684\uff1b\u800c\u7531\u4e8e\u6307\u6570\u4f4d\u7684\u5b58\u5728\uff0c\u6d6e\u70b9\u6570 <code>float</code> \u7684\u6570\u503c\u8d8a\u5927\uff0c\u76f8\u90bb\u4e24\u4e2a\u6570\u5b57\u4e4b\u95f4\u7684\u5dee\u503c\u5c31\u4f1a\u8d8b\u5411\u8d8a\u5927\u3002</p> <p>\u5982\u8868 3-2 \u6240\u793a\uff0c\u6307\u6570\u4f4d \\(E = 0\\) \u548c \\(E = 255\\) \u5177\u6709\u7279\u6b8a\u542b\u4e49\uff0c\u7528\u4e8e\u8868\u793a\u96f6\u3001\u65e0\u7a77\u5927\u3001\\(\\mathrm{NaN}\\) \u7b49\u3002</p> <p> \u8868 3-2 \u00a0 \u6307\u6570\u4f4d\u542b\u4e49 </p> \u6307\u6570\u4f4d E \u5206\u6570\u4f4d \\(\\mathrm{N} = 0\\) \u5206\u6570\u4f4d \\(\\mathrm{N} \\ne 0\\) \u8ba1\u7b97\u516c\u5f0f \\(0\\) \\(\\pm 0\\) \u6b21\u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) \u6b63\u89c4\u6570 \u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\) <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6b21\u6b63\u89c4\u6570\u663e\u8457\u63d0\u5347\u4e86\u6d6e\u70b9\u6570\u7684\u7cbe\u5ea6\u3002\u6700\u5c0f\u6b63\u6b63\u89c4\u6570\u4e3a \\(2^{-126}\\) \uff0c\u6700\u5c0f\u6b63\u6b21\u6b63\u89c4\u6570\u4e3a \\(2^{-126} \\times 2^{-23}\\) \u3002</p> <p>\u53cc\u7cbe\u5ea6 <code>double</code> \u4e5f\u91c7\u7528\u7c7b\u4f3c\u4e8e <code>float</code> \u7684\u8868\u793a\u65b9\u6cd5\uff0c\u5728\u6b64\u4e0d\u505a\u8d58\u8ff0\u3002</p>"},{"location":"chapter_data_structure/summary/","title":"3.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_data_structure/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u4ece\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\u4e24\u4e2a\u89d2\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002\u903b\u8f91\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\uff0c\u800c\u7269\u7406\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\u3002</li> <li>\u5e38\u89c1\u7684\u903b\u8f91\u7ed3\u6784\u5305\u62ec\u7ebf\u6027\u3001\u6811\u72b6\u548c\u7f51\u72b6\u7b49\u3002\u901a\u5e38\u6211\u4eec\u6839\u636e\u903b\u8f91\u7ed3\u6784\u5c06\u6570\u636e\u7ed3\u6784\u5206\u4e3a\u7ebf\u6027\uff08\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\uff09\u548c\u975e\u7ebf\u6027\uff08\u6811\u3001\u56fe\u3001\u5806\uff09\u4e24\u79cd\u3002\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u5f53\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u88ab\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u3002\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u62e5\u6709\u5bf9\u5e94\u7684\u5185\u5b58\u5730\u5740\uff0c\u7a0b\u5e8f\u901a\u8fc7\u8fd9\u4e9b\u5185\u5b58\u5730\u5740\u8bbf\u95ee\u6570\u636e\u3002</li> <li>\u7269\u7406\u7ed3\u6784\u4e3b\u8981\u5206\u4e3a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\u3002\u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u7531\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e24\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002</li> <li>\u8ba1\u7b97\u673a\u4e2d\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\u5305\u62ec\u6574\u6570 <code>byte</code>\u3001<code>short</code>\u3001<code>int</code>\u3001<code>long</code> \uff0c\u6d6e\u70b9\u6570 <code>float</code>\u3001<code>double</code> \uff0c\u5b57\u7b26 <code>char</code> \u548c\u5e03\u5c14 <code>bool</code> \u3002\u5b83\u4eec\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5360\u7528\u7a7a\u95f4\u5927\u5c0f\u548c\u8868\u793a\u65b9\u5f0f\u3002</li> <li>\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7f16\u7801\u6570\u5b57\u7684\u4e09\u79cd\u65b9\u6cd5\uff0c\u5b83\u4eec\u4e4b\u95f4\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\u3002\u6574\u6570\u7684\u539f\u7801\u7684\u6700\u9ad8\u4f4d\u662f\u7b26\u53f7\u4f4d\uff0c\u5176\u4f59\u4f4d\u662f\u6570\u5b57\u7684\u503c\u3002</li> <li>\u6574\u6570\u5728\u8ba1\u7b97\u673a\u4e2d\u662f\u4ee5\u8865\u7801\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\u3002\u5728\u8865\u7801\u8868\u793a\u4e0b\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u5bf9\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\u4e00\u89c6\u540c\u4ec1\uff0c\u4e0d\u9700\u8981\u4e3a\u51cf\u6cd5\u64cd\u4f5c\u5355\u72ec\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\uff0c\u5e76\u4e14\u4e0d\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u7684\u95ee\u9898\u3002</li> <li>\u6d6e\u70b9\u6570\u7684\u7f16\u7801\u7531 1 \u4f4d\u7b26\u53f7\u4f4d\u30018 \u4f4d\u6307\u6570\u4f4d\u548c 23 \u4f4d\u5206\u6570\u4f4d\u6784\u6210\u3002\u7531\u4e8e\u5b58\u5728\u6307\u6570\u4f4d\uff0c\u56e0\u6b64\u6d6e\u70b9\u6570\u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e\u6574\u6570\uff0c\u4ee3\u4ef7\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002</li> <li>ASCII \u7801\u662f\u6700\u65e9\u51fa\u73b0\u7684\u82f1\u6587\u5b57\u7b26\u96c6\uff0c\u957f\u5ea6\u4e3a 1 \u5b57\u8282\uff0c\u5171\u6536\u5f55 127 \u4e2a\u5b57\u7b26\u3002GBK \u5b57\u7b26\u96c6\u662f\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u96c6\uff0c\u5171\u6536\u5f55\u4e24\u4e07\u591a\u4e2a\u6c49\u5b57\u3002Unicode \u81f4\u529b\u4e8e\u63d0\u4f9b\u4e00\u4e2a\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u6536\u5f55\u4e16\u754c\u4e0a\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u4ece\u800c\u89e3\u51b3\u7531\u4e8e\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\u4e0d\u4e00\u81f4\u800c\u5bfc\u81f4\u7684\u4e71\u7801\u95ee\u9898\u3002</li> <li>UTF-8 \u662f\u6700\u53d7\u6b22\u8fce\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\uff0c\u901a\u7528\u6027\u975e\u5e38\u597d\u3002\u5b83\u662f\u4e00\u79cd\u53d8\u957f\u7684\u7f16\u7801\u65b9\u6cd5\uff0c\u5177\u6709\u5f88\u597d\u7684\u6269\u5c55\u6027\uff0c\u6709\u6548\u63d0\u5347\u4e86\u5b58\u50a8\u7a7a\u95f4\u7684\u4f7f\u7528\u6548\u7387\u3002UTF-16 \u548c UTF-32 \u662f\u7b49\u957f\u7684\u7f16\u7801\u65b9\u6cd5\u3002\u5728\u7f16\u7801\u4e2d\u6587\u65f6\uff0cUTF-16 \u5360\u7528\u7684\u7a7a\u95f4\u6bd4 UTF-8 \u66f4\u5c0f\u3002Java \u548c C# \u7b49\u7f16\u7a0b\u8bed\u8a00\u9ed8\u8ba4\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002</li> </ul>"},{"location":"chapter_data_structure/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1f</p> <p>\u54c8\u5e0c\u8868\u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u800c\u4e3a\u4e86\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\uff0c\u6211\u4eec\u53ef\u80fd\u4f1a\u4f7f\u7528\u201c\u94fe\u5f0f\u5730\u5740\u201d\uff08\u540e\u7eed\u201c\u54c8\u5e0c\u51b2\u7a81\u201d\u7ae0\u8282\u4f1a\u8bb2\uff09\uff1a\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u6876\u6307\u5411\u4e00\u4e2a\u94fe\u8868\uff0c\u5f53\u94fe\u8868\u957f\u5ea6\u8d85\u8fc7\u4e00\u5b9a\u9608\u503c\u65f6\uff0c\u53c8\u53ef\u80fd\u88ab\u8f6c\u5316\u4e3a\u6811\uff08\u901a\u5e38\u4e3a\u7ea2\u9ed1\u6811\uff09\u3002</p> <p>\u4ece\u5b58\u50a8\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u54c8\u5e0c\u8868\u7684\u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u5176\u4e2d\u6bcf\u4e00\u4e2a\u6876\u69fd\u4f4d\u53ef\u80fd\u5305\u542b\u4e00\u4e2a\u503c\uff0c\u4e5f\u53ef\u80fd\u5305\u542b\u4e00\u4e2a\u94fe\u8868\u6216\u4e00\u68f5\u6811\u3002\u56e0\u6b64\uff0c\u54c8\u5e0c\u8868\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff08\u6570\u7ec4\u3001\u94fe\u8868\uff09\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff08\u6811\uff09\u3002</p> <p>Q\uff1a<code>char</code> \u7c7b\u578b\u7684\u957f\u5ea6\u662f 1 \u5b57\u8282\u5417\uff1f</p> <p><code>char</code> \u7c7b\u578b\u7684\u957f\u5ea6\u7531\u7f16\u7a0b\u8bed\u8a00\u91c7\u7528\u7684\u7f16\u7801\u65b9\u6cd5\u51b3\u5b9a\u3002\u4f8b\u5982\uff0cJava\u3001JavaScript\u3001TypeScript\u3001C# \u90fd\u91c7\u7528 UTF-16 \u7f16\u7801\uff08\u4fdd\u5b58 Unicode \u7801\u70b9\uff09\uff0c\u56e0\u6b64 <code>char</code> \u7c7b\u578b\u7684\u957f\u5ea6\u4e3a 2 \u5b57\u8282\u3002</p> <p>Q\uff1a\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u79f0\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d \u662f\u5426\u6709\u6b67\u4e49\uff1f\u6808\u4e5f\u53ef\u4ee5\u8fdb\u884c\u51fa\u6808\u548c\u5165\u6808\u7b49\u64cd\u4f5c\uff0c\u8fd9\u4e9b\u64cd\u4f5c\u90fd\u662f\u201c\u52a8\u6001\u201d\u7684\u3002</p> <p>\u6808\u786e\u5b9e\u53ef\u4ee5\u5b9e\u73b0\u52a8\u6001\u7684\u6570\u636e\u64cd\u4f5c\uff0c\u4f46\u6570\u636e\u7ed3\u6784\u4ecd\u7136\u662f\u201c\u9759\u6001\u201d\uff08\u957f\u5ea6\u4e0d\u53ef\u53d8\uff09\u7684\u3002\u5c3d\u7ba1\u57fa\u4e8e\u6570\u7ec4\u7684\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u52a8\u6001\u5730\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\uff0c\u4f46\u5b83\u4eec\u7684\u5bb9\u91cf\u662f\u56fa\u5b9a\u7684\u3002\u5982\u679c\u6570\u636e\u91cf\u8d85\u51fa\u4e86\u9884\u5206\u914d\u7684\u5927\u5c0f\uff0c\u5c31\u9700\u8981\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u65e7\u6570\u7ec4\u7684\u5185\u5bb9\u590d\u5236\u5230\u65b0\u6570\u7ec4\u4e2d\u3002</p> <p>Q\uff1a\u5728\u6784\u5efa\u6808\uff08\u961f\u5217\uff09\u7684\u65f6\u5019\uff0c\u672a\u6307\u5b9a\u5b83\u7684\u5927\u5c0f\uff0c\u4e3a\u4ec0\u4e48\u5b83\u4eec\u662f\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d\u5462\uff1f</p> <p>\u5728\u9ad8\u7ea7\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u65e0\u987b\u4eba\u5de5\u6307\u5b9a\u6808\uff08\u961f\u5217\uff09\u7684\u521d\u59cb\u5bb9\u91cf\uff0c\u8fd9\u4e2a\u5de5\u4f5c\u7531\u7c7b\u5185\u90e8\u81ea\u52a8\u5b8c\u6210\u3002\u4f8b\u5982\uff0cJava \u7684 <code>ArrayList</code> \u7684\u521d\u59cb\u5bb9\u91cf\u901a\u5e38\u4e3a 10\u3002\u53e6\u5916\uff0c\u6269\u5bb9\u64cd\u4f5c\u4e5f\u662f\u81ea\u52a8\u5b9e\u73b0\u7684\u3002\u8be6\u89c1\u540e\u7eed\u7684\u201c\u5217\u8868\u201d\u7ae0\u8282\u3002</p>"},{"location":"chapter_divide_and_conquer/","title":"\u7b2c 12 \u7ae0 \u00a0 \u5206\u6cbb","text":"<p>Abstract</p> <p>\u96be\u9898\u88ab\u9010\u5c42\u62c6\u89e3\uff0c\u6bcf\u4e00\u6b21\u7684\u62c6\u89e3\u90fd\u4f7f\u5b83\u53d8\u5f97\u66f4\u4e3a\u7b80\u5355\u3002</p> <p>\u5206\u800c\u6cbb\u4e4b\u63ed\u793a\u4e86\u4e00\u4e2a\u91cd\u8981\u7684\u4e8b\u5b9e\uff1a\u4ece\u7b80\u5355\u505a\u8d77\uff0c\u4e00\u5207\u90fd\u4e0d\u518d\u590d\u6742\u3002</p>"},{"location":"chapter_divide_and_conquer/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>12.1 \u00a0 \u5206\u6cbb\u7b97\u6cd5</li> <li>12.2 \u00a0 \u5206\u6cbb\u641c\u7d22\u7b56\u7565</li> <li>12.3 \u00a0 \u6784\u5efa\u6811\u95ee\u9898</li> <li>12.4 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898</li> <li>12.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_divide_and_conquer/binary_search_recur/","title":"12.2 \u00a0 \u5206\u6cbb\u641c\u7d22\u7b56\u7565","text":"<p>\u6211\u4eec\u5df2\u7ecf\u5b66\u8fc7\uff0c\u641c\u7d22\u7b97\u6cd5\u5206\u4e3a\u4e24\u5927\u7c7b\u3002</p> <ul> <li>\u66b4\u529b\u641c\u7d22\uff1a\u5b83\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</li> <li>\u81ea\u9002\u5e94\u641c\u7d22\uff1a\u5b83\u5229\u7528\u7279\u6709\u7684\u6570\u636e\u7ec4\u7ec7\u5f62\u5f0f\u6216\u5148\u9a8c\u4fe1\u606f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u7684\u641c\u7d22\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u5b9e\u73b0\u7684\uff0c\u4f8b\u5982\u4e8c\u5206\u67e5\u627e\u548c\u6811\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\u7684\u6bcf\u4e00\u6b65\u90fd\u5c06\u95ee\u9898\uff08\u5728\u6570\u7ec4\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff09\u5206\u89e3\u4e3a\u4e00\u4e2a\u5c0f\u95ee\u9898\uff08\u5728\u6570\u7ec4\u7684\u4e00\u534a\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff09\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u4e00\u76f4\u6301\u7eed\u5230\u6570\u7ec4\u4e3a\u7a7a\u6216\u627e\u5230\u76ee\u6807\u5143\u7d20\u4e3a\u6b62\u3002</li> <li>\u6811\u662f\u5206\u6cbb\u601d\u60f3\u7684\u4ee3\u8868\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u3001AVL \u6811\u3001\u5806\u7b49\u6570\u636e\u7ed3\u6784\u4e2d\uff0c\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7686\u4e3a \\(O(\\log n)\\) \u3002</li> </ul> <p>\u4e8c\u5206\u67e5\u627e\u7684\u5206\u6cbb\u7b56\u7565\u5982\u4e0b\u6240\u793a\u3002</p> <ul> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u4e8c\u5206\u67e5\u627e\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\uff08\u5728\u6570\u7ec4\u4e2d\u8fdb\u884c\u67e5\u627e\uff09\u5206\u89e3\u4e3a\u5b50\u95ee\u9898\uff08\u5728\u6570\u7ec4\u7684\u4e00\u534a\u4e2d\u8fdb\u884c\u67e5\u627e\uff09\uff0c\u8fd9\u662f\u901a\u8fc7\u6bd4\u8f83\u4e2d\u95f4\u5143\u7d20\u548c\u76ee\u6807\u5143\u7d20\u6765\u5b9e\u73b0\u7684\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u5904\u7406\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u5b83\u4e0d\u53d7\u5176\u4ed6\u5b50\u95ee\u9898\u7684\u5f71\u54cd\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u65e0\u987b\u5408\u5e76\uff1a\u4e8c\u5206\u67e5\u627e\u65e8\u5728\u67e5\u627e\u4e00\u4e2a\u7279\u5b9a\u5143\u7d20\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\u3002\u5f53\u5b50\u95ee\u9898\u5f97\u5230\u89e3\u51b3\u65f6\uff0c\u539f\u95ee\u9898\u4e5f\u4f1a\u540c\u65f6\u5f97\u5230\u89e3\u51b3\u3002</li> </ul> <p>\u5206\u6cbb\u80fd\u591f\u63d0\u5347\u641c\u7d22\u6548\u7387\uff0c\u672c\u8d28\u4e0a\u662f\u56e0\u4e3a\u66b4\u529b\u641c\u7d22\u6bcf\u8f6e\u53ea\u80fd\u6392\u9664\u4e00\u4e2a\u9009\u9879\uff0c\u800c\u5206\u6cbb\u641c\u7d22\u6bcf\u8f6e\u53ef\u4ee5\u6392\u9664\u4e00\u534a\u9009\u9879\u3002</p>"},{"location":"chapter_divide_and_conquer/binary_search_recur/#1","title":"1. \u00a0 \u57fa\u4e8e\u5206\u6cbb\u5b9e\u73b0\u4e8c\u5206\u67e5\u627e","text":"<p>\u5728\u4e4b\u524d\u7684\u7ae0\u8282\u4e2d\uff0c\u4e8c\u5206\u67e5\u627e\u662f\u57fa\u4e8e\u9012\u63a8\uff08\u8fed\u4ee3\uff09\u5b9e\u73b0\u7684\u3002\u73b0\u5728\u6211\u4eec\u57fa\u4e8e\u5206\u6cbb\uff08\u9012\u5f52\uff09\u6765\u5b9e\u73b0\u5b83\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u662f\u552f\u4e00\u7684\uff0c\u8bf7\u67e5\u627e\u5143\u7d20 <code>target</code> \u3002</p> <p>\u4ece\u5206\u6cbb\u89d2\u5ea6\uff0c\u6211\u4eec\u5c06\u641c\u7d22\u533a\u95f4 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u8bb0\u4e3a \\(f(i, j)\\) \u3002</p> <p>\u4ee5\u539f\u95ee\u9898 \\(f(0, n-1)\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u901a\u8fc7\u4ee5\u4e0b\u6b65\u9aa4\u8fdb\u884c\u4e8c\u5206\u67e5\u627e\u3002</p> <ol> <li>\u8ba1\u7b97\u641c\u7d22\u533a\u95f4 \\([i, j]\\) \u7684\u4e2d\u70b9 \\(m\\) \uff0c\u6839\u636e\u5b83\u6392\u9664\u4e00\u534a\u641c\u7d22\u533a\u95f4\u3002</li> <li>\u9012\u5f52\u6c42\u89e3\u89c4\u6a21\u51cf\u5c0f\u4e00\u534a\u7684\u5b50\u95ee\u9898\uff0c\u53ef\u80fd\u4e3a \\(f(i, m-1)\\) \u6216 \\(f(m+1, j)\\) \u3002</li> <li>\u5faa\u73af\u7b2c <code>1.</code> \u6b65\u548c\u7b2c <code>2.</code> \u6b65\uff0c\u76f4\u81f3\u627e\u5230 <code>target</code> \u6216\u533a\u95f4\u4e3a\u7a7a\u65f6\u8fd4\u56de\u3002</li> </ol> <p>\u56fe 12-4 \u5c55\u793a\u4e86\u5728\u6570\u7ec4\u4e2d\u4e8c\u5206\u67e5\u627e\u5143\u7d20 \\(6\\) \u7684\u5206\u6cbb\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 12-4 \u00a0 \u4e8c\u5206\u67e5\u627e\u7684\u5206\u6cbb\u8fc7\u7a0b </p> <p>\u5728\u5b9e\u73b0\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u9012\u5f52\u51fd\u6570 <code>dfs()</code> \u6765\u6c42\u89e3\u95ee\u9898 \\(f(i, j)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_recur.py<pre><code>def dfs(nums: list[int], target: int, i: int, j: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j)\"\"\"\n    # \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if i &gt; j:\n        return -1\n    # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    m = (i + j) // 2\n    if nums[m] &lt; target:\n        # \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j)\n    elif nums[m] &gt; target:\n        # \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1)\n    else:\n        # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m\n\ndef binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\"\"\"\n    n = len(nums)\n    # \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1)\n</code></pre> binary_search_recur.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(vector&lt;int&gt; &amp;nums, int target, int i, int j) {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    int n = nums.size();\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(int[] nums, int target, int i, int j) {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(int[] nums, int target) {\n    int n = nums.length;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint DFS(int[] nums, int target, int i, int j) {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return DFS(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return DFS(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint BinarySearch(int[] nums, int target) {\n    int n = nums.Length;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return DFS(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunc dfs(nums []int, target, i, j int) int {\n    // \u5982\u679c\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u6ca1\u6709\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if i &gt; j {\n        return -1\n    }\n    //    \u8ba1\u7b97\u7d22\u5f15\u4e2d\u70b9\n    m := i + ((j - i) &gt;&gt; 1)\n    //\u5224\u65ad\u4e2d\u70b9\u4e0e\u76ee\u6807\u5143\u7d20\u5927\u5c0f\n    if nums[m] &lt; target {\n        // \u5c0f\u4e8e\u5219\u9012\u5f52\u53f3\u534a\u6570\u7ec4\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m+1, j)\n    } else if nums[m] &gt; target {\n        // \u5c0f\u4e8e\u5219\u9012\u5f52\u5de6\u534a\u6570\u7ec4\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m-1)\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunc binarySearch(nums []int, target int) int {\n    n := len(nums)\n    return dfs(nums, target, 0, n-1)\n}\n</code></pre> binary_search_recur.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunc dfs(nums: [Int], target: Int, i: Int, j: Int) -&gt; Int {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if i &gt; j {\n        return -1\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    let m = (i + j) / 2\n    if nums[m] &lt; target {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums: nums, target: target, i: m + 1, j: j)\n    } else if nums[m] &gt; target {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums: nums, target: target, i: i, j: m - 1)\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    dfs(nums: nums, target: target, i: nums.startIndex, j: nums.endIndex - 1)\n}\n</code></pre> binary_search_recur.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunction dfs(nums, target, i, j) {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunction binarySearch(nums, target) {\n    const n = nums.length;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunction dfs(nums: number[], target: number, i: number, j: number): number {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunction binarySearch(nums: number[], target: number): number {\n    const n = nums.length;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(List&lt;int&gt; nums, int target, int i, int j) {\n  // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n  if (i &gt; j) {\n    return -1;\n  }\n  // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n  int m = (i + j) ~/ 2;\n  if (nums[m] &lt; target) {\n    // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n    return dfs(nums, target, m + 1, j);\n  } else if (nums[m] &gt; target) {\n    // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n    return dfs(nums, target, i, m - 1);\n  } else {\n    // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return m;\n  }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  int n = nums.length;\n  // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n  return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfn dfs(nums: &amp;[i32], target: i32, i: i32, j: i32) -&gt; i32 {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if i &gt; j {\n        return -1;\n    }\n    let m: i32 = (i + j) / 2;\n    if nums[m as usize] &lt; target {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if nums[m as usize] &gt; target {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let n = nums.len() as i32;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(int nums[], int target, int i, int j) {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(int nums[], int target, int numsSize) {\n    int n = numsSize;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.kt<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfun dfs(\n    nums: IntArray,\n    target: Int,\n    i: Int,\n    j: Int\n): Int {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    val m = (i + j) / 2\n    return if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        dfs(nums, target, m + 1, j)\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        dfs(nums, target, i, m - 1)\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        m\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    val n = nums.size\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.rb<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{binary_search}\n</code></pre> binary_search_recur.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{binarySearch}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/","title":"12.3 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u68f5\u4e8c\u53c9\u6811\u7684\u524d\u5e8f\u904d\u5386 <code>preorder</code> \u548c\u4e2d\u5e8f\u904d\u5386 <code>inorder</code> \uff0c\u8bf7\u4ece\u4e2d\u6784\u5efa\u4e8c\u53c9\u6811\uff0c\u8fd4\u56de\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\u3002\u5047\u8bbe\u4e8c\u53c9\u6811\u4e2d\u6ca1\u6709\u503c\u91cd\u590d\u7684\u8282\u70b9\uff08\u5982\u56fe 12-5 \u6240\u793a\uff09\u3002</p> <p></p> <p> \u56fe 12-5 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#1","title":"1. \u00a0 \u5224\u65ad\u662f\u5426\u4e3a\u5206\u6cbb\u95ee\u9898","text":"<p>\u539f\u95ee\u9898\u5b9a\u4e49\u4e3a\u4ece <code>preorder</code> \u548c <code>inorder</code> \u6784\u5efa\u4e8c\u53c9\u6811\uff0c\u662f\u4e00\u4e2a\u5178\u578b\u7684\u5206\u6cbb\u95ee\u9898\u3002</p> <ul> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u4ece\u5206\u6cbb\u7684\u89d2\u5ea6\u5207\u5165\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u539f\u95ee\u9898\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\u3001\u6784\u5efa\u53f3\u5b50\u6811\uff0c\u52a0\u4e0a\u4e00\u6b65\u64cd\u4f5c\uff1a\u521d\u59cb\u5316\u6839\u8282\u70b9\u3002\u800c\u5bf9\u4e8e\u6bcf\u68f5\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u590d\u7528\u4ee5\u4e0a\u5212\u5206\u65b9\u6cd5\uff0c\u5c06\u5176\u5212\u5206\u4e3a\u66f4\u5c0f\u7684\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u76f4\u81f3\u8fbe\u5230\u6700\u5c0f\u5b50\u95ee\u9898\uff08\u7a7a\u5b50\u6811\uff09\u65f6\u7ec8\u6b62\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u4ea4\u96c6\u3002\u5728\u6784\u5efa\u5de6\u5b50\u6811\u65f6\uff0c\u6211\u4eec\u53ea\u9700\u5173\u6ce8\u4e2d\u5e8f\u904d\u5386\u548c\u524d\u5e8f\u904d\u5386\u4e2d\u4e0e\u5de6\u5b50\u6811\u5bf9\u5e94\u7684\u90e8\u5206\u3002\u53f3\u5b50\u6811\u540c\u7406\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u4e00\u65e6\u5f97\u5230\u4e86\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u5b83\u4eec\u94fe\u63a5\u5230\u6839\u8282\u70b9\u4e0a\uff0c\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002</li> </ul>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#2","title":"2. \u00a0 \u5982\u4f55\u5212\u5206\u5b50\u6811","text":"<p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u8fd9\u9053\u9898\u53ef\u4ee5\u4f7f\u7528\u5206\u6cbb\u6765\u6c42\u89e3\uff0c\u4f46\u5982\u4f55\u901a\u8fc7\u524d\u5e8f\u904d\u5386 <code>preorder</code> \u548c\u4e2d\u5e8f\u904d\u5386 <code>inorder</code> \u6765\u5212\u5206\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u5462\uff1f</p> <p>\u6839\u636e\u5b9a\u4e49\uff0c<code>preorder</code> \u548c <code>inorder</code> \u90fd\u53ef\u4ee5\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\u3002</p> <ul> <li>\u524d\u5e8f\u904d\u5386\uff1a<code>[ \u6839\u8282\u70b9 | \u5de6\u5b50\u6811 | \u53f3\u5b50\u6811 ]</code> \uff0c\u4f8b\u5982\u56fe 12-5 \u7684\u6811\u5bf9\u5e94 <code>[ 3 | 9 | 2 1 7 ]</code> \u3002</li> <li>\u4e2d\u5e8f\u904d\u5386\uff1a<code>[ \u5de6\u5b50\u6811 | \u6839\u8282\u70b9 \uff5c \u53f3\u5b50\u6811 ]</code> \uff0c\u4f8b\u5982\u56fe 12-5 \u7684\u6811\u5bf9\u5e94 <code>[ 9 | 3 | 1 2 7 ]</code> \u3002</li> </ul> <p>\u4ee5\u4e0a\u56fe\u6570\u636e\u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u56fe 12-6 \u6240\u793a\u7684\u6b65\u9aa4\u5f97\u5230\u5212\u5206\u7ed3\u679c\u3002</p> <ol> <li>\u524d\u5e8f\u904d\u5386\u7684\u9996\u5143\u7d20 3 \u662f\u6839\u8282\u70b9\u7684\u503c\u3002</li> <li>\u67e5\u627e\u6839\u8282\u70b9 3 \u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\uff0c\u5229\u7528\u8be5\u7d22\u5f15\u53ef\u5c06 <code>inorder</code> \u5212\u5206\u4e3a <code>[ 9 | 3 \uff5c 1 2 7 ]</code> \u3002</li> <li>\u6839\u636e <code>inorder</code> \u7684\u5212\u5206\u7ed3\u679c\uff0c\u6613\u5f97\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u8282\u70b9\u6570\u91cf\u5206\u522b\u4e3a 1 \u548c 3 \uff0c\u4ece\u800c\u53ef\u5c06 <code>preorder</code> \u5212\u5206\u4e3a <code>[ 3 | 9 | 2 1 7 ]</code> \u3002</li> </ol> <p></p> <p> \u56fe 12-6 \u00a0 \u5728\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u4e2d\u5212\u5206\u5b50\u6811 </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#3","title":"3. \u00a0 \u57fa\u4e8e\u53d8\u91cf\u63cf\u8ff0\u5b50\u6811\u533a\u95f4","text":"<p>\u6839\u636e\u4ee5\u4e0a\u5212\u5206\u65b9\u6cd5\uff0c\u6211\u4eec\u5df2\u7ecf\u5f97\u5230\u6839\u8282\u70b9\u3001\u5de6\u5b50\u6811\u3001\u53f3\u5b50\u6811\u5728 <code>preorder</code> \u548c <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u3002\u800c\u4e3a\u4e86\u63cf\u8ff0\u8fd9\u4e9b\u7d22\u5f15\u533a\u95f4\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u51e0\u4e2a\u6307\u9488\u53d8\u91cf\u3002</p> <ul> <li>\u5c06\u5f53\u524d\u6811\u7684\u6839\u8282\u70b9\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15\u8bb0\u4e3a \\(i\\) \u3002</li> <li>\u5c06\u5f53\u524d\u6811\u7684\u6839\u8282\u70b9\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u8bb0\u4e3a \\(m\\) \u3002</li> <li>\u5c06\u5f53\u524d\u6811\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u8bb0\u4e3a \\([l, r]\\) \u3002</li> </ul> <p>\u5982\u8868 12-1 \u6240\u793a\uff0c\u901a\u8fc7\u4ee5\u4e0a\u53d8\u91cf\u5373\u53ef\u8868\u793a\u6839\u8282\u70b9\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15\uff0c\u4ee5\u53ca\u5b50\u6811\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u3002</p> <p> \u8868 12-1 \u00a0 \u6839\u8282\u70b9\u548c\u5b50\u6811\u5728\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u7d22\u5f15 </p> \u6839\u8282\u70b9\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15 \u5b50\u6811\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4 \u5f53\u524d\u6811 \\(i\\) \\([l, r]\\) \u5de6\u5b50\u6811 \\(i + 1\\) \\([l, m-1]\\) \u53f3\u5b50\u6811 \\(i + 1 + (m - l)\\) \\([m+1, r]\\) <p>\u8bf7\u6ce8\u610f\uff0c\u53f3\u5b50\u6811\u6839\u8282\u70b9\u7d22\u5f15\u4e2d\u7684 \\((m-l)\\) \u7684\u542b\u4e49\u662f\u201c\u5de6\u5b50\u6811\u7684\u8282\u70b9\u6570\u91cf\u201d\uff0c\u5efa\u8bae\u7ed3\u5408\u56fe 12-7 \u7406\u89e3\u3002</p> <p></p> <p> \u56fe 12-7 \u00a0 \u6839\u8282\u70b9\u548c\u5de6\u53f3\u5b50\u6811\u7684\u7d22\u5f15\u533a\u95f4\u8868\u793a </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#4","title":"4. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u4e3a\u4e86\u63d0\u5347\u67e5\u8be2 \\(m\\) \u7684\u6548\u7387\uff0c\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>hmap</code> \u6765\u5b58\u50a8\u6570\u7ec4 <code>inorder</code> \u4e2d\u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig build_tree.py<pre><code>def dfs(\n    preorder: list[int],\n    inorder_map: dict[int, int],\n    i: int,\n    l: int,\n    r: int,\n) -&gt; TreeNode | None:\n    \"\"\"\u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb\"\"\"\n    # \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if r - l &lt; 0:\n        return None\n    # \u521d\u59cb\u5316\u6839\u8282\u70b9\n    root = TreeNode(preorder[i])\n    # \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    m = inorder_map[preorder[i]]\n    # \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n    # \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n    # \u8fd4\u56de\u6839\u8282\u70b9\n    return root\n\ndef build_tree(preorder: list[int], inorder: list[int]) -&gt; TreeNode | None:\n    \"\"\"\u6784\u5efa\u4e8c\u53c9\u6811\"\"\"\n    # \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    inorder_map = {val: i for i, val in enumerate(inorder)}\n    root = dfs(preorder, inorder_map, 0, 0, len(inorder) - 1)\n    return root\n</code></pre> build_tree.cpp<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode *dfs(vector&lt;int&gt; &amp;preorder, unordered_map&lt;int, int&gt; &amp;inorderMap, int i, int l, int r) {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0)\n        return NULL;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    TreeNode *root = new TreeNode(preorder[i]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    int m = inorderMap[preorder[i]];\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode *buildTree(vector&lt;int&gt; &amp;preorder, vector&lt;int&gt; &amp;inorder) {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    unordered_map&lt;int, int&gt; inorderMap;\n    for (int i = 0; i &lt; inorder.size(); i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorder.size() - 1);\n    return root;\n}\n</code></pre> build_tree.java<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode dfs(int[] preorder, Map&lt;Integer, Integer&gt; inorderMap, int i, int l, int r) {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0)\n        return null;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    TreeNode root = new TreeNode(preorder[i]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    int m = inorderMap.get(preorder[i]);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode buildTree(int[] preorder, int[] inorder) {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    Map&lt;Integer, Integer&gt; inorderMap = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; inorder.length; i++) {\n        inorderMap.put(inorder[i], i);\n    }\n    TreeNode root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.cs<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode? DFS(int[] preorder, Dictionary&lt;int, int&gt; inorderMap, int i, int l, int r) {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0)\n        return null;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    TreeNode root = new(preorder[i]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    int m = inorderMap[preorder[i]];\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = DFS(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = DFS(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode? BuildTree(int[] preorder, int[] inorder) {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    Dictionary&lt;int, int&gt; inorderMap = [];\n    for (int i = 0; i &lt; inorder.Length; i++) {\n        inorderMap.TryAdd(inorder[i], i);\n    }\n    TreeNode? root = DFS(preorder, inorderMap, 0, 0, inorder.Length - 1);\n    return root;\n}\n</code></pre> build_tree.go<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunc dfsBuildTree(preorder []int, inorderMap map[int]int, i, l, r int) *TreeNode {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if r-l &lt; 0 {\n        return nil\n    }\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    root := NewTreeNode(preorder[i])\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    m := inorderMap[preorder[i]]\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.Left = dfsBuildTree(preorder, inorderMap, i+1, l, m-1)\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.Right = dfsBuildTree(preorder, inorderMap, i+1+m-l, m+1, r)\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunc buildTree(preorder, inorder []int) *TreeNode {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    inorderMap := make(map[int]int, len(inorder))\n    for i := 0; i &lt; len(inorder); i++ {\n        inorderMap[inorder[i]] = i\n    }\n\n    root := dfsBuildTree(preorder, inorderMap, 0, 0, len(inorder)-1)\n    return root\n}\n</code></pre> build_tree.swift<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunc dfs(preorder: [Int], inorderMap: [Int: Int], i: Int, l: Int, r: Int) -&gt; TreeNode? {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if r - l &lt; 0 {\n        return nil\n    }\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    let root = TreeNode(x: preorder[i])\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    let m = inorderMap[preorder[i]]!\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1, l: l, r: m - 1)\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1 + m - l, l: m + 1, r: r)\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunc buildTree(preorder: [Int], inorder: [Int]) -&gt; TreeNode? {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    let inorderMap = inorder.enumerated().reduce(into: [:]) { $0[$1.element] = $1.offset }\n    return dfs(preorder: preorder, inorderMap: inorderMap, i: inorder.startIndex, l: inorder.startIndex, r: inorder.endIndex - 1)\n}\n</code></pre> build_tree.js<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunction dfs(preorder, inorderMap, i, l, r) {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0) return null;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    const root = new TreeNode(preorder[i]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    const m = inorderMap.get(preorder[i]);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunction buildTree(preorder, inorder) {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    let inorderMap = new Map();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.ts<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunction dfs(\n    preorder: number[],\n    inorderMap: Map&lt;number, number&gt;,\n    i: number,\n    l: number,\n    r: number\n): TreeNode | null {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0) return null;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    const root: TreeNode = new TreeNode(preorder[i]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    const m = inorderMap.get(preorder[i]);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunction buildTree(preorder: number[], inorder: number[]): TreeNode | null {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    let inorderMap = new Map&lt;number, number&gt;();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.dart<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode? dfs(\n  List&lt;int&gt; preorder,\n  Map&lt;int, int&gt; inorderMap,\n  int i,\n  int l,\n  int r,\n) {\n  // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n  if (r - l &lt; 0) {\n    return null;\n  }\n  // \u521d\u59cb\u5316\u6839\u8282\u70b9\n  TreeNode? root = TreeNode(preorder[i]);\n  // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n  int m = inorderMap[preorder[i]]!;\n  // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n  root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n  // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n  root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n  // \u8fd4\u56de\u6839\u8282\u70b9\n  return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode? buildTree(List&lt;int&gt; preorder, List&lt;int&gt; inorder) {\n  // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n  Map&lt;int, int&gt; inorderMap = {};\n  for (int i = 0; i &lt; inorder.length; i++) {\n    inorderMap[inorder[i]] = i;\n  }\n  TreeNode? root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n  return root;\n}\n</code></pre> build_tree.rs<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfn dfs(\n    preorder: &amp;[i32],\n    inorder_map: &amp;HashMap&lt;i32, i32&gt;,\n    i: i32,\n    l: i32,\n    r: i32,\n) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if r - l &lt; 0 {\n        return None;\n    }\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    let root = TreeNode::new(preorder[i as usize]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    let m = inorder_map.get(&amp;preorder[i as usize]).unwrap();\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.borrow_mut().left = dfs(preorder, inorder_map, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.borrow_mut().right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    Some(root)\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfn build_tree(preorder: &amp;[i32], inorder: &amp;[i32]) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    let mut inorder_map: HashMap&lt;i32, i32&gt; = HashMap::new();\n    for i in 0..inorder.len() {\n        inorder_map.insert(inorder[i], i as i32);\n    }\n    let root = dfs(preorder, &amp;inorder_map, 0, 0, inorder.len() as i32 - 1);\n    root\n}\n</code></pre> build_tree.c<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode *dfs(int *preorder, int *inorderMap, int i, int l, int r, int size) {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0)\n        return NULL;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    TreeNode *root = (TreeNode *)malloc(sizeof(TreeNode));\n    root-&gt;val = preorder[i];\n    root-&gt;left = NULL;\n    root-&gt;right = NULL;\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    int m = inorderMap[preorder[i]];\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1, size);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r, size);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode *buildTree(int *preorder, int preorderSize, int *inorder, int inorderSize) {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    int *inorderMap = (int *)malloc(sizeof(int) * MAX_SIZE);\n    for (int i = 0; i &lt; inorderSize; i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorderSize - 1, inorderSize);\n    free(inorderMap);\n    return root;\n}\n</code></pre> build_tree.kt<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfun dfs(preorder: IntArray, inorderMap: Map&lt;Int?, Int?&gt;, i: Int, l: Int, r: Int): TreeNode? {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0) return null\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    val root = TreeNode(preorder[i])\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    val m = inorderMap[preorder[i]]!!\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1)\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r)\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfun buildTree(preorder: IntArray, inorder: IntArray): TreeNode? {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    val inorderMap: MutableMap&lt;Int?, Int?&gt; = HashMap()\n    for (i in inorder.indices) {\n        inorderMap[inorder[i]] = i\n    }\n    val root = dfs(preorder, inorderMap, 0, 0, inorder.size - 1)\n    return root\n}\n</code></pre> build_tree.rb<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{build_tree}\n</code></pre> build_tree.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{buildTree}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 12-8 \u5c55\u793a\u4e86\u6784\u5efa\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b\uff0c\u5404\u4e2a\u8282\u70b9\u662f\u5728\u5411\u4e0b\u201c\u9012\u201d\u7684\u8fc7\u7a0b\u4e2d\u5efa\u7acb\u7684\uff0c\u800c\u5404\u6761\u8fb9\uff08\u5f15\u7528\uff09\u662f\u5728\u5411\u4e0a\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u4e2d\u5efa\u7acb\u7684\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 12-8 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b </p> <p>\u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u5185\u7684\u524d\u5e8f\u904d\u5386 <code>preorder</code> \u548c\u4e2d\u5e8f\u904d\u5386 <code>inorder</code> \u7684\u5212\u5206\u7ed3\u679c\u5982\u56fe 12-9 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 12-9 \u00a0 \u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u4e2d\u7684\u5212\u5206\u7ed3\u679c </p> <p>\u8bbe\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u521d\u59cb\u5316\u6bcf\u4e00\u4e2a\u8282\u70b9\uff08\u6267\u884c\u4e00\u4e2a\u9012\u5f52\u51fd\u6570 <code>dfs()</code> \uff09\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u6b64\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> <p>\u54c8\u5e0c\u8868\u5b58\u50a8 <code>inorder</code> \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u9012\u5f52\u6df1\u5ea6\u8fbe\u5230 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002\u56e0\u6b64\u603b\u4f53\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/","title":"12.1 \u00a0 \u5206\u6cbb\u7b97\u6cd5","text":"<p>\u5206\u6cbb\uff08divide and conquer\uff09\uff0c\u5168\u79f0\u5206\u800c\u6cbb\u4e4b\uff0c\u662f\u4e00\u79cd\u975e\u5e38\u91cd\u8981\u4e14\u5e38\u89c1\u7684\u7b97\u6cd5\u7b56\u7565\u3002\u5206\u6cbb\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\uff0c\u5305\u62ec\u201c\u5206\u201d\u548c\u201c\u6cbb\u201d\u4e24\u4e2a\u6b65\u9aa4\u3002</p> <ol> <li>\u5206\uff08\u5212\u5206\u9636\u6bb5\uff09\uff1a\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u4e24\u4e2a\u6216\u591a\u4e2a\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u5230\u8fbe\u6700\u5c0f\u5b50\u95ee\u9898\u65f6\u7ec8\u6b62\u3002</li> <li>\u6cbb\uff08\u5408\u5e76\u9636\u6bb5\uff09\uff1a\u4ece\u5df2\u77e5\u89e3\u7684\u6700\u5c0f\u5b50\u95ee\u9898\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\uff0c\u4ece\u800c\u6784\u5efa\u51fa\u539f\u95ee\u9898\u7684\u89e3\u3002</li> </ol> <p>\u5982\u56fe 12-1 \u6240\u793a\uff0c\u201c\u5f52\u5e76\u6392\u5e8f\u201d\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u5e94\u7528\u4e4b\u4e00\u3002</p> <ol> <li>\u5206\uff1a\u9012\u5f52\u5730\u5c06\u539f\u6570\u7ec4\uff08\u539f\u95ee\u9898\uff09\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\uff09\uff0c\u76f4\u5230\u5b50\u6570\u7ec4\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\uff08\u6700\u5c0f\u5b50\u95ee\u9898\uff09\u3002</li> <li>\u6cbb\uff1a\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u6709\u5e8f\u7684\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\u8fdb\u884c\u5408\u5e76\uff0c\u4ece\u800c\u5f97\u5230\u6709\u5e8f\u7684\u539f\u6570\u7ec4\uff08\u539f\u95ee\u9898\u7684\u89e3\uff09\u3002</li> </ol> <p></p> <p> \u56fe 12-1 \u00a0 \u5f52\u5e76\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565 </p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1211","title":"12.1.1 \u00a0 \u5982\u4f55\u5224\u65ad\u5206\u6cbb\u95ee\u9898","text":"<p>\u4e00\u4e2a\u95ee\u9898\u662f\u5426\u9002\u5408\u4f7f\u7528\u5206\u6cbb\u89e3\u51b3\uff0c\u901a\u5e38\u53ef\u4ee5\u53c2\u8003\u4ee5\u4e0b\u51e0\u4e2a\u5224\u65ad\u4f9d\u636e\u3002</p> <ol> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u539f\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\u6210\u89c4\u6a21\u66f4\u5c0f\u3001\u7c7b\u4f3c\u7684\u5b50\u95ee\u9898\uff0c\u4ee5\u53ca\u80fd\u591f\u4ee5\u76f8\u540c\u65b9\u5f0f\u9012\u5f52\u5730\u8fdb\u884c\u5212\u5206\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5b50\u95ee\u9898\u4e4b\u95f4\u6ca1\u6709\u91cd\u53e0\uff0c\u4e92\u4e0d\u4f9d\u8d56\uff0c\u53ef\u4ee5\u72ec\u7acb\u89e3\u51b3\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u539f\u95ee\u9898\u7684\u89e3\u901a\u8fc7\u5408\u5e76\u5b50\u95ee\u9898\u7684\u89e3\u5f97\u6765\u3002</li> </ol> <p>\u663e\u7136\uff0c\u5f52\u5e76\u6392\u5e8f\u6ee1\u8db3\u4ee5\u4e0a\u4e09\u4e2a\u5224\u65ad\u4f9d\u636e\u3002</p> <ol> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u9012\u5f52\u5730\u5c06\u6570\u7ec4\uff08\u539f\u95ee\u9898\uff09\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\uff09\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u53ef\u4ee5\u72ec\u7acb\u5730\u8fdb\u884c\u6392\u5e8f\uff08\u5b50\u95ee\u9898\u53ef\u4ee5\u72ec\u7acb\u8fdb\u884c\u6c42\u89e3\uff09\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u4e24\u4e2a\u6709\u5e8f\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\u53ef\u4ee5\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\uff08\u539f\u95ee\u9898\u7684\u89e3\uff09\u3002</li> </ol>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1212","title":"12.1.2 \u00a0 \u901a\u8fc7\u5206\u6cbb\u63d0\u5347\u6548\u7387","text":"<p>\u5206\u6cbb\u4e0d\u4ec5\u53ef\u4ee5\u6709\u6548\u5730\u89e3\u51b3\u7b97\u6cd5\u95ee\u9898\uff0c\u5f80\u5f80\u8fd8\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002\u5728\u6392\u5e8f\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u76f8\u8f83\u4e8e\u9009\u62e9\u3001\u5192\u6ce1\u3001\u63d2\u5165\u6392\u5e8f\u66f4\u5feb\uff0c\u5c31\u662f\u56e0\u4e3a\u5b83\u4eec\u5e94\u7528\u4e86\u5206\u6cbb\u7b56\u7565\u3002</p> <p>\u90a3\u4e48\uff0c\u6211\u4eec\u4e0d\u7981\u53d1\u95ee\uff1a\u4e3a\u4ec0\u4e48\u5206\u6cbb\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\uff0c\u5176\u5e95\u5c42\u903b\u8f91\u662f\u4ec0\u4e48\uff1f\u6362\u53e5\u8bdd\u8bf4\uff0c\u5c06\u5927\u95ee\u9898\u5206\u89e3\u4e3a\u591a\u4e2a\u5b50\u95ee\u9898\u3001\u89e3\u51b3\u5b50\u95ee\u9898\u3001\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u5408\u5e76\u4e3a\u539f\u95ee\u9898\u7684\u89e3\uff0c\u8fd9\u51e0\u6b65\u7684\u6548\u7387\u4e3a\u4ec0\u4e48\u6bd4\u76f4\u63a5\u89e3\u51b3\u539f\u95ee\u9898\u7684\u6548\u7387\u66f4\u9ad8\uff1f\u8fd9\u4e2a\u95ee\u9898\u53ef\u4ee5\u4ece\u64cd\u4f5c\u6570\u91cf\u548c\u5e76\u884c\u8ba1\u7b97\u4e24\u65b9\u9762\u6765\u8ba8\u8bba\u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1","title":"1. \u00a0 \u64cd\u4f5c\u6570\u91cf\u4f18\u5316","text":"<p>\u4ee5\u201c\u5192\u6ce1\u6392\u5e8f\u201d\u4e3a\u4f8b\uff0c\u5176\u5904\u7406\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u9700\u8981 \\(O(n^2)\\) \u65f6\u95f4\u3002\u5047\u8bbe\u6211\u4eec\u6309\u7167\u56fe 12-2 \u6240\u793a\u7684\u65b9\u5f0f\uff0c\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u5219\u5212\u5206\u9700\u8981 \\(O(n)\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6bcf\u4e2a\u5b50\u6570\u7ec4\u9700\u8981 \\(O((n / 2)^2)\\) \u65f6\u95f4\uff0c\u5408\u5e76\u4e24\u4e2a\u5b50\u6570\u7ec4\u9700\u8981 \\(O(n)\\) \u65f6\u95f4\uff0c\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\uff1a</p> \\[ O(n + (\\frac{n}{2})^2 \\times 2 + n) = O(\\frac{n^2}{2} + 2n) \\] <p></p> <p> \u56fe 12-2 \u00a0 \u5212\u5206\u6570\u7ec4\u524d\u540e\u7684\u5192\u6ce1\u6392\u5e8f </p> <p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u8ba1\u7b97\u4ee5\u4e0b\u4e0d\u7b49\u5f0f\uff0c\u5176\u5de6\u8fb9\u548c\u53f3\u8fb9\u5206\u522b\u4e3a\u5212\u5206\u524d\u548c\u5212\u5206\u540e\u7684\u64cd\u4f5c\u603b\u6570\uff1a</p> \\[ \\begin{aligned} n^2 &amp; &gt; \\frac{n^2}{2} + 2n \\newline n^2 - \\frac{n^2}{2} - 2n &amp; &gt; 0 \\newline n(n - 4) &amp; &gt; 0 \\end{aligned} \\] <p>\u8fd9\u610f\u5473\u7740\u5f53 \\(n &gt; 4\\) \u65f6\uff0c\u5212\u5206\u540e\u7684\u64cd\u4f5c\u6570\u91cf\u66f4\u5c11\uff0c\u6392\u5e8f\u6548\u7387\u5e94\u8be5\u66f4\u9ad8\u3002\u8bf7\u6ce8\u610f\uff0c\u5212\u5206\u540e\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u7136\u662f\u5e73\u65b9\u9636 \\(O(n^2)\\) \uff0c\u53ea\u662f\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\u53d8\u5c0f\u4e86\u3002</p> <p>\u8fdb\u4e00\u6b65\u60f3\uff0c\u5982\u679c\u6211\u4eec\u628a\u5b50\u6570\u7ec4\u4e0d\u65ad\u5730\u518d\u4ece\u4e2d\u70b9\u5904\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\u65f6\u505c\u6b62\u5212\u5206\u5462\uff1f\u8fd9\u79cd\u601d\u8def\u5b9e\u9645\u4e0a\u5c31\u662f\u201c\u5f52\u5e76\u6392\u5e8f\u201d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p> <p>\u518d\u601d\u8003\uff0c\u5982\u679c\u6211\u4eec\u591a\u8bbe\u7f6e\u51e0\u4e2a\u5212\u5206\u70b9\uff0c\u5c06\u539f\u6570\u7ec4\u5e73\u5747\u5212\u5206\u4e3a \\(k\\) \u4e2a\u5b50\u6570\u7ec4\u5462\uff1f\u8fd9\u79cd\u60c5\u51b5\u4e0e\u201c\u6876\u6392\u5e8f\u201d\u975e\u5e38\u7c7b\u4f3c\uff0c\u5b83\u975e\u5e38\u9002\u5408\u6392\u5e8f\u6d77\u91cf\u6570\u636e\uff0c\u7406\u8bba\u4e0a\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230 \\(O(n + k)\\) \u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#2","title":"2. \u00a0 \u5e76\u884c\u8ba1\u7b97\u4f18\u5316","text":"<p>\u6211\u4eec\u77e5\u9053\uff0c\u5206\u6cbb\u751f\u6210\u7684\u5b50\u95ee\u9898\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u56e0\u6b64\u901a\u5e38\u53ef\u4ee5\u5e76\u884c\u89e3\u51b3\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5206\u6cbb\u4e0d\u4ec5\u53ef\u4ee5\u964d\u4f4e\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u8fd8\u6709\u5229\u4e8e\u64cd\u4f5c\u7cfb\u7edf\u7684\u5e76\u884c\u4f18\u5316\u3002</p> <p>\u5e76\u884c\u4f18\u5316\u5728\u591a\u6838\u6216\u591a\u5904\u7406\u5668\u7684\u73af\u5883\u4e2d\u5c24\u5176\u6709\u6548\uff0c\u56e0\u4e3a\u7cfb\u7edf\u53ef\u4ee5\u540c\u65f6\u5904\u7406\u591a\u4e2a\u5b50\u95ee\u9898\uff0c\u66f4\u52a0\u5145\u5206\u5730\u5229\u7528\u8ba1\u7b97\u8d44\u6e90\uff0c\u4ece\u800c\u663e\u8457\u51cf\u5c11\u603b\u4f53\u7684\u8fd0\u884c\u65f6\u95f4\u3002</p> <p>\u6bd4\u5982\u5728\u56fe 12-3 \u6240\u793a\u7684\u201c\u6876\u6392\u5e8f\u201d\u4e2d\uff0c\u6211\u4eec\u5c06\u6d77\u91cf\u7684\u6570\u636e\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u5219\u53ef\u6240\u6709\u6876\u7684\u6392\u5e8f\u4efb\u52a1\u5206\u6563\u5230\u5404\u4e2a\u8ba1\u7b97\u5355\u5143\uff0c\u5b8c\u6210\u540e\u518d\u5408\u5e76\u7ed3\u679c\u3002</p> <p></p> <p> \u56fe 12-3 \u00a0 \u6876\u6392\u5e8f\u7684\u5e76\u884c\u8ba1\u7b97 </p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1213","title":"12.1.3 \u00a0 \u5206\u6cbb\u5e38\u89c1\u5e94\u7528","text":"<p>\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u53ef\u4ee5\u7528\u6765\u89e3\u51b3\u8bb8\u591a\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\u3002</p> <ul> <li>\u5bfb\u627e\u6700\u8fd1\u70b9\u5bf9\uff1a\u8be5\u7b97\u6cd5\u9996\u5148\u5c06\u70b9\u96c6\u5206\u6210\u4e24\u90e8\u5206\uff0c\u7136\u540e\u5206\u522b\u627e\u51fa\u4e24\u90e8\u5206\u4e2d\u7684\u6700\u8fd1\u70b9\u5bf9\uff0c\u6700\u540e\u627e\u51fa\u8de8\u8d8a\u4e24\u90e8\u5206\u7684\u6700\u8fd1\u70b9\u5bf9\u3002</li> <li>\u5927\u6574\u6570\u4e58\u6cd5\uff1a\u4f8b\u5982 Karatsuba \u7b97\u6cd5\uff0c\u5b83\u5c06\u5927\u6574\u6570\u4e58\u6cd5\u5206\u89e3\u4e3a\u51e0\u4e2a\u8f83\u5c0f\u7684\u6574\u6570\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002</li> <li>\u77e9\u9635\u4e58\u6cd5\uff1a\u4f8b\u5982 Strassen \u7b97\u6cd5\uff0c\u5b83\u5c06\u5927\u77e9\u9635\u4e58\u6cd5\u5206\u89e3\u4e3a\u591a\u4e2a\u5c0f\u77e9\u9635\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002</li> <li>\u6c49\u8bfa\u5854\u95ee\u9898\uff1a\u6c49\u8bfa\u5854\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u9012\u5f52\u89e3\u51b3\uff0c\u8fd9\u662f\u5178\u578b\u7684\u5206\u6cbb\u7b56\u7565\u5e94\u7528\u3002</li> <li>\u6c42\u89e3\u9006\u5e8f\u5bf9\uff1a\u5728\u4e00\u4e2a\u5e8f\u5217\u4e2d\uff0c\u5982\u679c\u524d\u9762\u7684\u6570\u5b57\u5927\u4e8e\u540e\u9762\u7684\u6570\u5b57\uff0c\u90a3\u4e48\u8fd9\u4e24\u4e2a\u6570\u5b57\u6784\u6210\u4e00\u4e2a\u9006\u5e8f\u5bf9\u3002\u6c42\u89e3\u9006\u5e8f\u5bf9\u95ee\u9898\u53ef\u4ee5\u5229\u7528\u5206\u6cbb\u7684\u601d\u60f3\uff0c\u501f\u52a9\u5f52\u5e76\u6392\u5e8f\u8fdb\u884c\u6c42\u89e3\u3002</li> </ul> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u5728\u7b97\u6cd5\u548c\u6570\u636e\u7ed3\u6784\u7684\u8bbe\u8ba1\u4e2d\u5e94\u7528\u5f97\u975e\u5e38\u5e7f\u6cdb\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\uff1a\u4e8c\u5206\u67e5\u627e\u662f\u5c06\u6709\u5e8f\u6570\u7ec4\u4ece\u4e2d\u70b9\u7d22\u5f15\u5904\u5206\u4e3a\u4e24\u90e8\u5206\uff0c\u7136\u540e\u6839\u636e\u76ee\u6807\u503c\u4e0e\u4e2d\u95f4\u5143\u7d20\u503c\u6bd4\u8f83\u7ed3\u679c\uff0c\u51b3\u5b9a\u6392\u9664\u54ea\u4e00\u534a\u533a\u95f4\uff0c\u5e76\u5728\u5269\u4f59\u533a\u95f4\u6267\u884c\u76f8\u540c\u7684\u4e8c\u5206\u64cd\u4f5c\u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\uff1a\u672c\u8282\u5f00\u5934\u5df2\u4ecb\u7ecd\uff0c\u4e0d\u518d\u8d58\u8ff0\u3002</li> <li>\u5feb\u901f\u6392\u5e8f\uff1a\u5feb\u901f\u6392\u5e8f\u662f\u9009\u53d6\u4e00\u4e2a\u57fa\u51c6\u503c\uff0c\u7136\u540e\u628a\u6570\u7ec4\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u4e00\u4e2a\u5b50\u6570\u7ec4\u7684\u5143\u7d20\u6bd4\u57fa\u51c6\u503c\u5c0f\uff0c\u53e6\u4e00\u5b50\u6570\u7ec4\u7684\u5143\u7d20\u6bd4\u57fa\u51c6\u503c\u5927\uff0c\u518d\u5bf9\u8fd9\u4e24\u90e8\u5206\u8fdb\u884c\u76f8\u540c\u7684\u5212\u5206\u64cd\u4f5c\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u53ea\u5269\u4e0b\u4e00\u4e2a\u5143\u7d20\u3002</li> <li>\u6876\u6392\u5e8f\uff1a\u6876\u6392\u5e8f\u7684\u57fa\u672c\u601d\u60f3\u662f\u5c06\u6570\u636e\u5206\u6563\u5230\u591a\u4e2a\u6876\uff0c\u7136\u540e\u5bf9\u6bcf\u4e2a\u6876\u5185\u7684\u5143\u7d20\u8fdb\u884c\u6392\u5e8f\uff0c\u6700\u540e\u5c06\u5404\u4e2a\u6876\u7684\u5143\u7d20\u4f9d\u6b21\u53d6\u51fa\uff0c\u4ece\u800c\u5f97\u5230\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002</li> <li>\u6811\uff1a\u4f8b\u5982\u4e8c\u53c9\u641c\u7d22\u6811\u3001AVL \u6811\u3001\u7ea2\u9ed1\u6811\u3001B \u6811\u3001B+ \u6811\u7b49\uff0c\u5b83\u4eec\u7684\u67e5\u627e\u3001\u63d2\u5165\u548c\u5220\u9664\u7b49\u64cd\u4f5c\u90fd\u53ef\u4ee5\u89c6\u4e3a\u5206\u6cbb\u7b56\u7565\u7684\u5e94\u7528\u3002</li> <li>\u5806\uff1a\u5806\u662f\u4e00\u79cd\u7279\u6b8a\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u5176\u5404\u79cd\u64cd\u4f5c\uff0c\u5982\u63d2\u5165\u3001\u5220\u9664\u548c\u5806\u5316\uff0c\u5b9e\u9645\u4e0a\u90fd\u9690\u542b\u4e86\u5206\u6cbb\u7684\u601d\u60f3\u3002</li> <li>\u54c8\u5e0c\u8868\uff1a\u867d\u7136\u54c8\u5e0c\u8868\u5e76\u4e0d\u76f4\u63a5\u5e94\u7528\u5206\u6cbb\uff0c\u4f46\u67d0\u4e9b\u54c8\u5e0c\u51b2\u7a81\u89e3\u51b3\u65b9\u6848\u95f4\u63a5\u5e94\u7528\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u4f8b\u5982\uff0c\u94fe\u5f0f\u5730\u5740\u4e2d\u7684\u957f\u94fe\u8868\u4f1a\u88ab\u8f6c\u5316\u4e3a\u7ea2\u9ed1\u6811\uff0c\u4ee5\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002</li> </ul> <p>\u53ef\u4ee5\u770b\u51fa\uff0c\u5206\u6cbb\u662f\u4e00\u79cd\u201c\u6da6\u7269\u7ec6\u65e0\u58f0\u201d\u7684\u7b97\u6cd5\u601d\u60f3\uff0c\u9690\u542b\u5728\u5404\u79cd\u7b97\u6cd5\u4e0e\u6570\u636e\u7ed3\u6784\u4e4b\u4e2d\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/","title":"12.4 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898","text":"<p>\u5728\u5f52\u5e76\u6392\u5e8f\u548c\u6784\u5efa\u4e8c\u53c9\u6811\u4e2d\uff0c\u6211\u4eec\u90fd\u662f\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u4e24\u4e2a\u89c4\u6a21\u4e3a\u539f\u95ee\u9898\u4e00\u534a\u7684\u5b50\u95ee\u9898\u3002\u7136\u800c\u5bf9\u4e8e\u6c49\u8bfa\u5854\u95ee\u9898\uff0c\u6211\u4eec\u91c7\u7528\u4e0d\u540c\u7684\u5206\u89e3\u7b56\u7565\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e09\u6839\u67f1\u5b50\uff0c\u8bb0\u4e3a <code>A</code>\u3001<code>B</code> \u548c <code>C</code> \u3002\u8d77\u59cb\u72b6\u6001\u4e0b\uff0c\u67f1\u5b50 <code>A</code> \u4e0a\u5957\u7740 \\(n\\) \u4e2a\u5706\u76d8\uff0c\u5b83\u4eec\u4ece\u4e0a\u5230\u4e0b\u6309\u7167\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\u3002\u6211\u4eec\u7684\u4efb\u52a1\u662f\u8981\u628a\u8fd9 \\(n\\) \u4e2a\u5706\u76d8\u79fb\u5230\u67f1\u5b50 <code>C</code> \u4e0a\uff0c\u5e76\u4fdd\u6301\u5b83\u4eec\u7684\u539f\u6709\u987a\u5e8f\u4e0d\u53d8\uff08\u5982\u56fe 12-10 \u6240\u793a\uff09\u3002\u5728\u79fb\u52a8\u5706\u76d8\u7684\u8fc7\u7a0b\u4e2d\uff0c\u9700\u8981\u9075\u5b88\u4ee5\u4e0b\u89c4\u5219\u3002</p> <ol> <li>\u5706\u76d8\u53ea\u80fd\u4ece\u4e00\u6839\u67f1\u5b50\u9876\u90e8\u62ff\u51fa\uff0c\u4ece\u53e6\u4e00\u6839\u67f1\u5b50\u9876\u90e8\u653e\u5165\u3002</li> <li>\u6bcf\u6b21\u53ea\u80fd\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\u3002</li> <li>\u5c0f\u5706\u76d8\u5fc5\u987b\u65f6\u523b\u4f4d\u4e8e\u5927\u5706\u76d8\u4e4b\u4e0a\u3002</li> </ol> <p></p> <p> \u56fe 12-10 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898\u793a\u4f8b </p> <p>\u6211\u4eec\u5c06\u89c4\u6a21\u4e3a \\(i\\) \u7684\u6c49\u8bfa\u5854\u95ee\u9898\u8bb0\u4f5c \\(f(i)\\) \u3002\u4f8b\u5982 \\(f(3)\\) \u4ee3\u8868\u5c06 \\(3\\) \u4e2a\u5706\u76d8\u4ece <code>A</code> \u79fb\u52a8\u81f3 <code>C</code> \u7684\u6c49\u8bfa\u5854\u95ee\u9898\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#1","title":"1. \u00a0 \u8003\u8651\u57fa\u672c\u60c5\u51b5","text":"<p>\u5982\u56fe 12-11 \u6240\u793a\uff0c\u5bf9\u4e8e\u95ee\u9898 \\(f(1)\\) \uff0c\u5373\u5f53\u53ea\u6709\u4e00\u4e2a\u5706\u76d8\u65f6\uff0c\u6211\u4eec\u5c06\u5b83\u76f4\u63a5\u4ece <code>A</code> \u79fb\u52a8\u81f3 <code>C</code> \u5373\u53ef\u3002</p> &lt;1&gt;&lt;2&gt; <p></p> <p></p> <p> \u56fe 12-11 \u00a0 \u89c4\u6a21\u4e3a 1 \u7684\u95ee\u9898\u7684\u89e3 </p> <p>\u5982\u56fe 12-12 \u6240\u793a\uff0c\u5bf9\u4e8e\u95ee\u9898 \\(f(2)\\) \uff0c\u5373\u5f53\u6709\u4e24\u4e2a\u5706\u76d8\u65f6\uff0c\u7531\u4e8e\u8981\u65f6\u523b\u6ee1\u8db3\u5c0f\u5706\u76d8\u5728\u5927\u5706\u76d8\u4e4b\u4e0a\uff0c\u56e0\u6b64\u9700\u8981\u501f\u52a9 <code>B</code> \u6765\u5b8c\u6210\u79fb\u52a8\u3002</p> <ol> <li>\u5148\u5c06\u4e0a\u9762\u7684\u5c0f\u5706\u76d8\u4ece <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u518d\u5c06\u5927\u5706\u76d8\u4ece <code>A</code> \u79fb\u81f3 <code>C</code> \u3002</li> <li>\u6700\u540e\u5c06\u5c0f\u5706\u76d8\u4ece <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 12-12 \u00a0 \u89c4\u6a21\u4e3a 2 \u7684\u95ee\u9898\u7684\u89e3 </p> <p>\u89e3\u51b3\u95ee\u9898 \\(f(2)\\) \u7684\u8fc7\u7a0b\u53ef\u603b\u7ed3\u4e3a\uff1a\u5c06\u4e24\u4e2a\u5706\u76d8\u501f\u52a9 <code>B</code> \u4ece <code>A</code> \u79fb\u81f3 <code>C</code> \u3002\u5176\u4e2d\uff0c<code>C</code> \u79f0\u4e3a\u76ee\u6807\u67f1\u3001<code>B</code> \u79f0\u4e3a\u7f13\u51b2\u67f1\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#2","title":"2. \u00a0 \u5b50\u95ee\u9898\u5206\u89e3","text":"<p>\u5bf9\u4e8e\u95ee\u9898 \\(f(3)\\) \uff0c\u5373\u5f53\u6709\u4e09\u4e2a\u5706\u76d8\u65f6\uff0c\u60c5\u51b5\u53d8\u5f97\u7a0d\u5fae\u590d\u6742\u4e86\u4e00\u4e9b\u3002</p> <p>\u56e0\u4e3a\u5df2\u77e5 \\(f(1)\\) \u548c \\(f(2)\\) \u7684\u89e3\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ece\u5206\u6cbb\u89d2\u5ea6\u601d\u8003\uff0c\u5c06 <code>A</code> \u9876\u90e8\u7684\u4e24\u4e2a\u5706\u76d8\u770b\u4f5c\u4e00\u4e2a\u6574\u4f53\uff0c\u6267\u884c\u56fe 12-13 \u6240\u793a\u7684\u6b65\u9aa4\u3002\u8fd9\u6837\u4e09\u4e2a\u5706\u76d8\u5c31\u88ab\u987a\u5229\u5730\u4ece <code>A</code> \u79fb\u81f3 <code>C</code> \u4e86\u3002</p> <ol> <li>\u4ee4 <code>B</code> \u4e3a\u76ee\u6807\u67f1\u3001<code>C</code> \u4e3a\u7f13\u51b2\u67f1\uff0c\u5c06\u4e24\u4e2a\u5706\u76d8\u4ece <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u5c06 <code>A</code> \u4e2d\u5269\u4f59\u7684\u4e00\u4e2a\u5706\u76d8\u4ece <code>A</code> \u76f4\u63a5\u79fb\u52a8\u81f3 <code>C</code> \u3002</li> <li>\u4ee4 <code>C</code> \u4e3a\u76ee\u6807\u67f1\u3001<code>A</code> \u4e3a\u7f13\u51b2\u67f1\uff0c\u5c06\u4e24\u4e2a\u5706\u76d8\u4ece <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 12-13 \u00a0 \u89c4\u6a21\u4e3a 3 \u7684\u95ee\u9898\u7684\u89e3 </p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u6211\u4eec\u5c06\u95ee\u9898 \\(f(3)\\) \u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(2)\\) \u548c\u4e00\u4e2a\u5b50\u95ee\u9898 \\(f(1)\\) \u3002\u6309\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u4e4b\u540e\uff0c\u539f\u95ee\u9898\u968f\u4e4b\u5f97\u5230\u89e3\u51b3\u3002\u8fd9\u8bf4\u660e\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff0c\u800c\u4e14\u89e3\u53ef\u4ee5\u5408\u5e76\u3002</p> <p>\u81f3\u6b64\uff0c\u6211\u4eec\u53ef\u603b\u7ed3\u51fa\u56fe 12-14 \u6240\u793a\u7684\u89e3\u51b3\u6c49\u8bfa\u5854\u95ee\u9898\u7684\u5206\u6cbb\u7b56\u7565\uff1a\u5c06\u539f\u95ee\u9898 \\(f(n)\\) \u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(n-1)\\) \u548c\u4e00\u4e2a\u5b50\u95ee\u9898 \\(f(1)\\) \uff0c\u5e76\u6309\u7167\u4ee5\u4e0b\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u3002</p> <ol> <li>\u5c06 \\(n-1\\) \u4e2a\u5706\u76d8\u501f\u52a9 <code>C</code> \u4ece <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u5c06\u5269\u4f59 \\(1\\) \u4e2a\u5706\u76d8\u4ece <code>A</code> \u76f4\u63a5\u79fb\u81f3 <code>C</code> \u3002</li> <li>\u5c06 \\(n-1\\) \u4e2a\u5706\u76d8\u501f\u52a9 <code>A</code> \u4ece <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> <p>\u5bf9\u4e8e\u8fd9\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(n-1)\\) \uff0c\u53ef\u4ee5\u901a\u8fc7\u76f8\u540c\u7684\u65b9\u5f0f\u8fdb\u884c\u9012\u5f52\u5212\u5206\uff0c\u76f4\u81f3\u8fbe\u5230\u6700\u5c0f\u5b50\u95ee\u9898 \\(f(1)\\) \u3002\u800c \\(f(1)\\) \u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u53ea\u9700\u4e00\u6b21\u79fb\u52a8\u64cd\u4f5c\u5373\u53ef\u3002</p> <p></p> <p> \u56fe 12-14 \u00a0 \u89e3\u51b3\u6c49\u8bfa\u5854\u95ee\u9898\u7684\u5206\u6cbb\u7b56\u7565 </p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u9012\u5f52\u51fd\u6570 <code>dfs(i, src, buf, tar)</code> \uff0c\u5b83\u7684\u4f5c\u7528\u662f\u5c06\u67f1 <code>src</code> \u9876\u90e8\u7684 \\(i\\) \u4e2a\u5706\u76d8\u501f\u52a9\u7f13\u51b2\u67f1 <code>buf</code> \u79fb\u52a8\u81f3\u76ee\u6807\u67f1 <code>tar</code> \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hanota.py<pre><code>def move(src: list[int], tar: list[int]):\n    \"\"\"\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\"\"\"\n    # \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    pan = src.pop()\n    # \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.append(pan)\n\ndef dfs(i: int, src: list[int], buf: list[int], tar: list[int]):\n    \"\"\"\u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i)\"\"\"\n    # \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if i == 1:\n        move(src, tar)\n        return\n    # \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf)\n    # \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar)\n    # \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar)\n\ndef solve_hanota(A: list[int], B: list[int], C: list[int]):\n    \"\"\"\u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898\"\"\"\n    n = len(A)\n    # \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C)\n</code></pre> hanota.cpp<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;tar) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    int pan = src.back();\n    src.pop_back();\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.push_back(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;buf, vector&lt;int&gt; &amp;tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(vector&lt;int&gt; &amp;A, vector&lt;int&gt; &amp;B, vector&lt;int&gt; &amp;C) {\n    int n = A.size();\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.java<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(List&lt;Integer&gt; src, List&lt;Integer&gt; tar) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    Integer pan = src.remove(src.size() - 1);\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.add(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, List&lt;Integer&gt; src, List&lt;Integer&gt; buf, List&lt;Integer&gt; tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(List&lt;Integer&gt; A, List&lt;Integer&gt; B, List&lt;Integer&gt; C) {\n    int n = A.size();\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.cs<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid Move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    int pan = src[^1];\n    src.RemoveAt(src.Count - 1);\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.Add(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid DFS(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        Move(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    DFS(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    Move(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    DFS(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid SolveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n    int n = A.Count;\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    DFS(n, A, B, C);\n}\n</code></pre> hanota.go<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunc move(src, tar *list.List) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    pan := src.Back()\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.PushBack(pan.Value)\n    // \u79fb\u9664 src \u9876\u90e8\u5706\u76d8\n    src.Remove(pan)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunc dfsHanota(i int, src, buf, tar *list.List) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if i == 1 {\n        move(src, tar)\n        return\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfsHanota(i-1, src, tar, buf)\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar)\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfsHanota(i-1, buf, src, tar)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunc solveHanota(A, B, C *list.List) {\n    n := A.Len()\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfsHanota(n, A, B, C)\n}\n</code></pre> hanota.swift<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunc move(src: inout [Int], tar: inout [Int]) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    let pan = src.popLast()!\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.append(pan)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunc dfs(i: Int, src: inout [Int], buf: inout [Int], tar: inout [Int]) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if i == 1 {\n        move(src: &amp;src, tar: &amp;tar)\n        return\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i: i - 1, src: &amp;src, buf: &amp;tar, tar: &amp;buf)\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src: &amp;src, tar: &amp;tar)\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i: i - 1, src: &amp;buf, buf: &amp;src, tar: &amp;tar)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunc solveHanota(A: inout [Int], B: inout [Int], C: inout [Int]) {\n    let n = A.count\n    // \u5217\u8868\u5c3e\u90e8\u662f\u67f1\u5b50\u9876\u90e8\n    // \u5c06 src \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(i: n, src: &amp;A, buf: &amp;B, tar: &amp;C)\n}\n</code></pre> hanota.js<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunction move(src, tar) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    const pan = src.pop();\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunction dfs(i, src, buf, tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunction solveHanota(A, B, C) {\n    const n = A.length;\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.ts<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunction move(src: number[], tar: number[]): void {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    const pan = src.pop();\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunction dfs(i: number, src: number[], buf: number[], tar: number[]): void {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunction solveHanota(A: number[], B: number[], C: number[]): void {\n    const n = A.length;\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.dart<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n  // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n  int pan = src.removeLast();\n  // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n  tar.add(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n  // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n  if (i == 1) {\n    move(src, tar);\n    return;\n  }\n  // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n  dfs(i - 1, src, tar, buf);\n  // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n  move(src, tar);\n  // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n  dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n  int n = A.length;\n  // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n  dfs(n, A, B, C);\n}\n</code></pre> hanota.rs<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfn move_pan(src: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    let pan = src.remove(src.len() - 1);\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfn dfs(i: i32, src: &amp;mut Vec&lt;i32&gt;, buf: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if i == 1 {\n        move_pan(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move_pan(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfn solve_hanota(A: &amp;mut Vec&lt;i32&gt;, B: &amp;mut Vec&lt;i32&gt;, C: &amp;mut Vec&lt;i32&gt;) {\n    let n = A.len() as i32;\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.c<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(int *src, int *srcSize, int *tar, int *tarSize) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    int pan = src[*srcSize - 1];\n    src[*srcSize - 1] = 0;\n    (*srcSize)--;\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar[*tarSize] = pan;\n    (*tarSize)++;\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, int *src, int *srcSize, int *buf, int *bufSize, int *tar, int *tarSize) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, srcSize, tar, tarSize);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, srcSize, tar, tarSize, buf, bufSize);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, srcSize, tar, tarSize);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, bufSize, src, srcSize, tar, tarSize);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(int *A, int *ASize, int *B, int *BSize, int *C, int *CSize) {\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(*ASize, A, ASize, B, BSize, C, CSize);\n}\n</code></pre> hanota.kt<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfun move(src: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    val pan: Int = src.removeAt(src.size - 1)\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.add(pan)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfun dfs(i: Int, src: MutableList&lt;Int&gt;, buf: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, tar)\n        return\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf)\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar)\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfun solveHanota(A: MutableList&lt;Int&gt;, B: MutableList&lt;Int&gt;, C: MutableList&lt;Int&gt;) {\n    val n = A.size\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C)\n}\n</code></pre> hanota.rb<pre><code>[class]{}-[func]{move}\n\n[class]{}-[func]{dfs}\n\n[class]{}-[func]{solve_hanota}\n</code></pre> hanota.zig<pre><code>[class]{}-[func]{move}\n\n[class]{}-[func]{dfs}\n\n[class]{}-[func]{solveHanota}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 12-15 \u6240\u793a\uff0c\u6c49\u8bfa\u5854\u95ee\u9898\u5f62\u6210\u4e00\u68f5\u9ad8\u5ea6\u4e3a \\(n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u5bf9\u5e94\u4e00\u4e2a\u5f00\u542f\u7684 <code>dfs()</code> \u51fd\u6570\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> <p></p> <p> \u56fe 12-15 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898\u7684\u9012\u5f52\u6811 </p> <p>Quote</p> <p>\u6c49\u8bfa\u5854\u95ee\u9898\u6e90\u81ea\u4e00\u4e2a\u53e4\u8001\u7684\u4f20\u8bf4\u3002\u5728\u53e4\u5370\u5ea6\u7684\u4e00\u4e2a\u5bfa\u5e99\u91cc\uff0c\u50e7\u4fa3\u4eec\u6709\u4e09\u6839\u9ad8\u5927\u7684\u94bb\u77f3\u67f1\u5b50\uff0c\u4ee5\u53ca \\(64\\) \u4e2a\u5927\u5c0f\u4e0d\u4e00\u7684\u91d1\u5706\u76d8\u3002\u50e7\u4fa3\u4eec\u4e0d\u65ad\u5730\u79fb\u52a8\u5706\u76d8\uff0c\u4ed6\u4eec\u76f8\u4fe1\u5728\u6700\u540e\u4e00\u4e2a\u5706\u76d8\u88ab\u6b63\u786e\u653e\u7f6e\u7684\u90a3\u4e00\u523b\uff0c\u8fd9\u4e2a\u4e16\u754c\u5c31\u4f1a\u7ed3\u675f\u3002</p> <p>\u7136\u800c\uff0c\u5373\u4f7f\u50e7\u4fa3\u4eec\u6bcf\u79d2\u949f\u79fb\u52a8\u4e00\u6b21\uff0c\u603b\u5171\u9700\u8981\u5927\u7ea6 \\(2^{64} \\approx 1.84\u00d710^{19}\\) \u79d2\uff0c\u5408\u7ea6 \\(5850\\) \u4ebf\u5e74\uff0c\u8fdc\u8fdc\u8d85\u8fc7\u4e86\u73b0\u5728\u5bf9\u5b87\u5b99\u5e74\u9f84\u7684\u4f30\u8ba1\u3002\u6240\u4ee5\uff0c\u5018\u82e5\u8fd9\u4e2a\u4f20\u8bf4\u662f\u771f\u7684\uff0c\u6211\u4eec\u5e94\u8be5\u4e0d\u9700\u8981\u62c5\u5fc3\u4e16\u754c\u672b\u65e5\u7684\u5230\u6765\u3002</p>"},{"location":"chapter_divide_and_conquer/summary/","title":"12.5 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u5206\u6cbb\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u7b97\u6cd5\u8bbe\u8ba1\u7b56\u7565\uff0c\u5305\u62ec\u5206\uff08\u5212\u5206\uff09\u548c\u6cbb\uff08\u5408\u5e76\uff09\u4e24\u4e2a\u9636\u6bb5\uff0c\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\u3002</li> <li>\u5224\u65ad\u662f\u5426\u662f\u5206\u6cbb\u7b97\u6cd5\u95ee\u9898\u7684\u4f9d\u636e\u5305\u62ec\uff1a\u95ee\u9898\u80fd\u5426\u5206\u89e3\u3001\u5b50\u95ee\u9898\u662f\u5426\u72ec\u7acb\u3001\u5b50\u95ee\u9898\u80fd\u5426\u5408\u5e76\u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u5e94\u7528\uff0c\u5176\u9012\u5f52\u5730\u5c06\u6570\u7ec4\u5212\u5206\u4e3a\u7b49\u957f\u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u5230\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\u65f6\u5f00\u59cb\u9010\u5c42\u5408\u5e76\uff0c\u4ece\u800c\u5b8c\u6210\u6392\u5e8f\u3002</li> <li>\u5f15\u5165\u5206\u6cbb\u7b56\u7565\u5f80\u5f80\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u7b56\u7565\u51cf\u5c11\u4e86\u64cd\u4f5c\u6570\u91cf\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u540e\u6709\u5229\u4e8e\u7cfb\u7edf\u7684\u5e76\u884c\u4f18\u5316\u3002</li> <li>\u5206\u6cbb\u65e2\u53ef\u4ee5\u89e3\u51b3\u8bb8\u591a\u7b97\u6cd5\u95ee\u9898\uff0c\u4e5f\u5e7f\u6cdb\u5e94\u7528\u4e8e\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\uff0c\u5904\u5904\u53ef\u89c1\u5176\u8eab\u5f71\u3002</li> <li>\u76f8\u8f83\u4e8e\u66b4\u529b\u641c\u7d22\uff0c\u81ea\u9002\u5e94\u641c\u7d22\u6548\u7387\u66f4\u9ad8\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u7684\u641c\u7d22\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u5b9e\u73b0\u7684\u3002</li> <li>\u4e8c\u5206\u67e5\u627e\u662f\u5206\u6cbb\u7b56\u7565\u7684\u53e6\u4e00\u4e2a\u5178\u578b\u5e94\u7528\uff0c\u5b83\u4e0d\u5305\u542b\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\u7684\u6b65\u9aa4\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9012\u5f52\u5206\u6cbb\u5b9e\u73b0\u4e8c\u5206\u67e5\u627e\u3002</li> <li>\u5728\u6784\u5efa\u4e8c\u53c9\u6811\u7684\u95ee\u9898\u4e2d\uff0c\u6784\u5efa\u6811\uff08\u539f\u95ee\u9898\uff09\u53ef\u4ee5\u5212\u5206\u4e3a\u6784\u5efa\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u8fd9\u53ef\u4ee5\u901a\u8fc7\u5212\u5206\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u7684\u7d22\u5f15\u533a\u95f4\u6765\u5b9e\u73b0\u3002</li> <li>\u5728\u6c49\u8bfa\u5854\u95ee\u9898\u4e2d\uff0c\u4e00\u4e2a\u89c4\u6a21\u4e3a \\(n\\) \u7684\u95ee\u9898\u53ef\u4ee5\u5212\u5206\u4e3a\u4e24\u4e2a\u89c4\u6a21\u4e3a \\(n-1\\) \u7684\u5b50\u95ee\u9898\u548c\u4e00\u4e2a\u89c4\u6a21\u4e3a \\(1\\) \u7684\u5b50\u95ee\u9898\u3002\u6309\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u540e\uff0c\u539f\u95ee\u9898\u968f\u4e4b\u5f97\u5230\u89e3\u51b3\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/","title":"\u7b2c 14 \u7ae0 \u00a0 \u52a8\u6001\u89c4\u5212","text":"<p>Abstract</p> <p>\u5c0f\u6eaa\u6c47\u5165\u6cb3\u6d41\uff0c\u6c5f\u6cb3\u6c47\u5165\u5927\u6d77\u3002</p> <p>\u52a8\u6001\u89c4\u5212\u5c06\u5c0f\u95ee\u9898\u7684\u89e3\u6c47\u96c6\u6210\u5927\u95ee\u9898\u7684\u7b54\u6848\uff0c\u4e00\u6b65\u6b65\u5f15\u9886\u6211\u4eec\u8d70\u5411\u89e3\u51b3\u95ee\u9898\u7684\u5f7c\u5cb8\u3002</p>"},{"location":"chapter_dynamic_programming/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>14.1 \u00a0 \u521d\u63a2\u52a8\u6001\u89c4\u5212</li> <li>14.2 \u00a0 DP \u95ee\u9898\u7279\u6027</li> <li>14.3 \u00a0 DP \u89e3\u9898\u601d\u8def</li> <li>14.4 \u00a0 0-1 \u80cc\u5305\u95ee\u9898</li> <li>14.5 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898</li> <li>14.6 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898</li> <li>14.7 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_dynamic_programming/dp_problem_features/","title":"14.2 \u00a0 \u52a8\u6001\u89c4\u5212\u95ee\u9898\u7279\u6027","text":"<p>\u5728\u4e0a\u4e00\u8282\u4e2d\uff0c\u6211\u4eec\u5b66\u4e60\u4e86\u52a8\u6001\u89c4\u5212\u662f\u5982\u4f55\u901a\u8fc7\u5b50\u95ee\u9898\u5206\u89e3\u6765\u6c42\u89e3\u539f\u95ee\u9898\u7684\u3002\u5b9e\u9645\u4e0a\uff0c\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u901a\u7528\u7684\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u3001\u56de\u6eaf\u4e2d\u7684\u4fa7\u91cd\u70b9\u4e0d\u540c\u3002</p> <ul> <li>\u5206\u6cbb\u7b97\u6cd5\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\u5212\u5206\u4e3a\u591a\u4e2a\u76f8\u4e92\u72ec\u7acb\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u6700\u5c0f\u5b50\u95ee\u9898\uff0c\u5e76\u5728\u56de\u6eaf\u4e2d\u5408\u5e76\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u6700\u7ec8\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002</li> <li>\u52a8\u6001\u89c4\u5212\u4e5f\u5bf9\u95ee\u9898\u8fdb\u884c\u9012\u5f52\u5206\u89e3\uff0c\u4f46\u4e0e\u5206\u6cbb\u7b97\u6cd5\u7684\u4e3b\u8981\u533a\u522b\u662f\uff0c\u52a8\u6001\u89c4\u5212\u4e2d\u7684\u5b50\u95ee\u9898\u662f\u76f8\u4e92\u4f9d\u8d56\u7684\uff0c\u5728\u5206\u89e3\u8fc7\u7a0b\u4e2d\u4f1a\u51fa\u73b0\u8bb8\u591a\u91cd\u53e0\u5b50\u95ee\u9898\u3002</li> <li>\u56de\u6eaf\u7b97\u6cd5\u5728\u5c1d\u8bd5\u548c\u56de\u9000\u4e2d\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u7684\u89e3\uff0c\u5e76\u901a\u8fc7\u526a\u679d\u907f\u514d\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\u3002\u539f\u95ee\u9898\u7684\u89e3\u7531\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\u6784\u6210\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6bcf\u4e2a\u51b3\u7b56\u6b65\u9aa4\u4e4b\u524d\u7684\u5b50\u5e8f\u5217\u770b\u4f5c\u4e00\u4e2a\u5b50\u95ee\u9898\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u52a8\u6001\u89c4\u5212\u5e38\u7528\u6765\u6c42\u89e3\u6700\u4f18\u5316\u95ee\u9898\uff0c\u5b83\u4eec\u4e0d\u4ec5\u5305\u542b\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u8fd8\u5177\u6709\u53e6\u5916\u4e24\u5927\u7279\u6027\uff1a\u6700\u4f18\u5b50\u7ed3\u6784\u3001\u65e0\u540e\u6548\u6027\u3002</p>"},{"location":"chapter_dynamic_programming/dp_problem_features/#1421","title":"14.2.1 \u00a0 \u6700\u4f18\u5b50\u7ed3\u6784","text":"<p>\u6211\u4eec\u5bf9\u722c\u697c\u68af\u95ee\u9898\u7a0d\u4f5c\u6539\u52a8\uff0c\u4f7f\u4e4b\u66f4\u52a0\u9002\u5408\u5c55\u793a\u6700\u4f18\u5b50\u7ed3\u6784\u6982\u5ff5\u3002</p> <p>\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u6bcf\u4e00\u9636\u697c\u68af\u4e0a\u90fd\u8d34\u6709\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a\u4f60\u5728\u8be5\u53f0\u9636\u6240\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\u3002\u7ed9\u5b9a\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\u6570\u7ec4 \\(cost\\) \uff0c\u5176\u4e2d \\(cost[i]\\) \u8868\u793a\u5728\u7b2c \\(i\\) \u4e2a\u53f0\u9636\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\\(cost[0]\\) \u4e3a\u5730\u9762\uff08\u8d77\u59cb\u70b9\uff09\u3002\u8bf7\u8ba1\u7b97\u6700\u5c11\u9700\u8981\u4ed8\u51fa\u591a\u5c11\u4ee3\u4ef7\u624d\u80fd\u5230\u8fbe\u9876\u90e8\uff1f</p> <p>\u5982\u56fe 14-6 \u6240\u793a\uff0c\u82e5\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u9636\u7684\u4ee3\u4ef7\u5206\u522b\u4e3a \\(1\\)\u3001\\(10\\)\u3001\\(1\\) \uff0c\u5219\u4ece\u5730\u9762\u722c\u5230\u7b2c \\(3\\) \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7\u4e3a \\(2\\) \u3002</p> <p></p> <p> \u56fe 14-6 \u00a0 \u722c\u5230\u7b2c 3 \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7 </p> <p>\u8bbe \\(dp[i]\\) \u4e3a\u722c\u5230\u7b2c \\(i\\) \u9636\u7d2f\u8ba1\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\u7531\u4e8e\u7b2c \\(i\\) \u9636\u53ea\u53ef\u80fd\u4ece \\(i - 1\\) \u9636\u6216 \\(i - 2\\) \u9636\u8d70\u6765\uff0c\u56e0\u6b64 \\(dp[i]\\) \u53ea\u53ef\u80fd\u7b49\u4e8e \\(dp[i - 1] + cost[i]\\) \u6216 \\(dp[i - 2] + cost[i]\\) \u3002\u4e3a\u4e86\u5c3d\u53ef\u80fd\u51cf\u5c11\u4ee3\u4ef7\uff0c\u6211\u4eec\u5e94\u8be5\u9009\u62e9\u4e24\u8005\u4e2d\u8f83\u5c0f\u7684\u90a3\u4e00\u4e2a\uff1a</p> \\[ dp[i] = \\min(dp[i-1], dp[i-2]) + cost[i] \\] <p>\u8fd9\u4fbf\u53ef\u4ee5\u5f15\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u542b\u4e49\uff1a\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u662f\u4ece\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u5efa\u5f97\u6765\u7684\u3002</p> <p>\u672c\u9898\u663e\u7136\u5177\u6709\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u6211\u4eec\u4ece\u4e24\u4e2a\u5b50\u95ee\u9898\u6700\u4f18\u89e3 \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u4e2d\u6311\u9009\u51fa\u8f83\u4f18\u7684\u90a3\u4e00\u4e2a\uff0c\u5e76\u7528\u5b83\u6784\u5efa\u51fa\u539f\u95ee\u9898 \\(dp[i]\\) \u7684\u6700\u4f18\u89e3\u3002</p> <p>\u90a3\u4e48\uff0c\u4e0a\u4e00\u8282\u7684\u722c\u697c\u68af\u9898\u76ee\u6709\u6ca1\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u5462\uff1f\u5b83\u7684\u76ee\u6807\u662f\u6c42\u89e3\u65b9\u6848\u6570\u91cf\uff0c\u770b\u4f3c\u662f\u4e00\u4e2a\u8ba1\u6570\u95ee\u9898\uff0c\u4f46\u5982\u679c\u6362\u4e00\u79cd\u95ee\u6cd5\uff1a\u201c\u6c42\u89e3\u6700\u5927\u65b9\u6848\u6570\u91cf\u201d\u3002\u6211\u4eec\u610f\u5916\u5730\u53d1\u73b0\uff0c\u867d\u7136\u9898\u76ee\u4fee\u6539\u524d\u540e\u662f\u7b49\u4ef7\u7684\uff0c\u4f46\u6700\u4f18\u5b50\u7ed3\u6784\u6d6e\u73b0\u51fa\u6765\u4e86\uff1a\u7b2c \\(n\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u7b49\u4e8e\u7b2c \\(n-1\\) \u9636\u548c\u7b2c \\(n-2\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u4e4b\u548c\u3002\u6240\u4ee5\u8bf4\uff0c\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u89e3\u91ca\u65b9\u5f0f\u6bd4\u8f83\u7075\u6d3b\uff0c\u5728\u4e0d\u540c\u95ee\u9898\u4e2d\u4f1a\u6709\u4e0d\u540c\u7684\u542b\u4e49\u3002</p> <p>\u6839\u636e\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u4ee5\u53ca\u521d\u59cb\u72b6\u6001 \\(dp[1] = cost[1]\\) \u548c \\(dp[2] = cost[2]\\) \uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u52a8\u6001\u89c4\u5212\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp(cost: list[int]) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp = [0] * (n + 1)\n    # \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1], dp[2] = cost[1], cost[2]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in range(3, n + 1):\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    return dp[n]\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    vector&lt;int&gt; dp(n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint MinCostClimbingStairsDP(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.Min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDP(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp := make([]int, n+1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = min(dp[i-1], dp[i-2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDP(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = Array(repeating: 0, count: n + 1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in 3 ... n {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDP(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = new Array(n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDP(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = new Array(n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n  dp[1] = cost[1];\n  dp[2] = cost[2];\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n  }\n  return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfn min_cost_climbing_stairs_dp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    let mut dp = vec![-1; n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in 3..=n {\n        dp[i] = cmp::min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int *dp = calloc(n + 1, sizeof(int));\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = myMin(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    int res = dp[n];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfun minCostClimbingStairsDP(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    val dp = IntArray(n + 1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (i in 3..n) {\n        dp[i] = (min(dp[i - 1].toDouble(), dp[i - 2].toDouble()) + cost[i]).toInt()\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>[class]{}-[func]{min_cost_climbing_stairs_dp}\n</code></pre> min_cost_climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212\nfn minCostClimbingStairsDP(comptime cost: []i32) i32 {\n    comptime var n = cost.len - 1;\n    if (n == 1 or n == 2) {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = [_]i32{-1} ** (n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (3..n + 1) |i| {\n        dp[i] = @min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 14-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u4ee3\u7801\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 14-7 \u00a0 \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u672c\u9898\u4e5f\u53ef\u4ee5\u8fdb\u884c\u7a7a\u95f4\u4f18\u5316\uff0c\u5c06\u4e00\u7ef4\u538b\u7f29\u81f3\u96f6\u7ef4\uff0c\u4f7f\u5f97\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp_comp(cost: list[int]) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    a, b = cost[1], cost[2]\n    for i in range(3, n + 1):\n        a, b = b, min(a, b) + cost[i]\n    return b\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint MinCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.Min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDPComp(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    a, b := cost[1], cost[2]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i := 3; i &lt;= n; i++ {\n        tmp := b\n        b = min(a, tmp) + cost[i]\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDPComp(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    var (a, b) = (cost[1], cost[2])\n    for i in 3 ... n {\n        (a, b) = (b, min(a, b) + cost[i])\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDPComp(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDPComp(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  int a = cost[1], b = cost[2];\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = min(a, tmp) + cost[i];\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn min_cost_climbing_stairs_dp_comp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    };\n    let (mut a, mut b) = (cost[1], cost[2]);\n    for i in 3..=n {\n        let tmp = b;\n        b = cmp::min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    b\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = myMin(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun minCostClimbingStairsDPComp(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    var a = cost[1]\n    var b = cost[2]\n    for (i in 3..n) {\n        val tmp = b\n        b = (min(a.toDouble(), tmp.toDouble()) + cost[i]).toInt()\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>[class]{}-[func]{min_cost_climbing_stairs_dp_comp}\n</code></pre> min_cost_climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn minCostClimbingStairsDPComp(cost: []i32) i32 {\n    var n = cost.len - 1;\n    if (n == 1 or n == 2) {\n        return cost[n];\n    }\n    var a = cost[1];\n    var b = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (3..n + 1) |i| {\n        var tmp = b;\n        b = @min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/dp_problem_features/#1422","title":"14.2.2 \u00a0 \u65e0\u540e\u6548\u6027","text":"<p>\u65e0\u540e\u6548\u6027\u662f\u52a8\u6001\u89c4\u5212\u80fd\u591f\u6709\u6548\u89e3\u51b3\u95ee\u9898\u7684\u91cd\u8981\u7279\u6027\u4e4b\u4e00\uff0c\u5176\u5b9a\u4e49\u4e3a\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u786e\u5b9a\u7684\u72b6\u6001\uff0c\u5b83\u7684\u672a\u6765\u53d1\u5c55\u53ea\u4e0e\u5f53\u524d\u72b6\u6001\u6709\u5173\uff0c\u800c\u4e0e\u8fc7\u53bb\u7ecf\u5386\u7684\u6240\u6709\u72b6\u6001\u65e0\u5173\u3002</p> <p>\u4ee5\u722c\u697c\u68af\u95ee\u9898\u4e3a\u4f8b\uff0c\u7ed9\u5b9a\u72b6\u6001 \\(i\\) \uff0c\u5b83\u4f1a\u53d1\u5c55\u51fa\u72b6\u6001 \\(i+1\\) \u548c\u72b6\u6001 \\(i+2\\) \uff0c\u5206\u522b\u5bf9\u5e94\u8df3 \\(1\\) \u6b65\u548c\u8df3 \\(2\\) \u6b65\u3002\u5728\u505a\u51fa\u8fd9\u4e24\u79cd\u9009\u62e9\u65f6\uff0c\u6211\u4eec\u65e0\u987b\u8003\u8651\u72b6\u6001 \\(i\\) \u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5b83\u4eec\u5bf9\u72b6\u6001 \\(i\\) \u7684\u672a\u6765\u6ca1\u6709\u5f71\u54cd\u3002</p> <p>\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u7ed9\u722c\u697c\u68af\u95ee\u9898\u6dfb\u52a0\u4e00\u4e2a\u7ea6\u675f\uff0c\u60c5\u51b5\u5c31\u4e0d\u4e00\u6837\u4e86\u3002</p> <p>\u5e26\u7ea6\u675f\u722c\u697c\u68af</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u4f46\u4e0d\u80fd\u8fde\u7eed\u4e24\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f</p> <p>\u5982\u56fe 14-8 \u6240\u793a\uff0c\u722c\u4e0a\u7b2c \\(3\\) \u9636\u4ec5\u5269 \\(2\\) \u79cd\u53ef\u884c\u65b9\u6848\uff0c\u5176\u4e2d\u8fde\u7eed\u4e09\u6b21\u8df3 \\(1\\) \u9636\u7684\u65b9\u6848\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\uff0c\u56e0\u6b64\u88ab\u820d\u5f03\u3002</p> <p></p> <p> \u56fe 14-8 \u00a0 \u5e26\u7ea6\u675f\u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf </p> <p>\u5728\u8be5\u95ee\u9898\u4e2d\uff0c\u5982\u679c\u4e0a\u4e00\u8f6e\u662f\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\uff0c\u90a3\u4e48\u4e0b\u4e00\u8f6e\u5c31\u5fc5\u987b\u8df3 \\(2\\) \u9636\u3002\u8fd9\u610f\u5473\u7740\uff0c\u4e0b\u4e00\u6b65\u9009\u62e9\u4e0d\u80fd\u7531\u5f53\u524d\u72b6\u6001\uff08\u5f53\u524d\u6240\u5728\u697c\u68af\u9636\u6570\uff09\u72ec\u7acb\u51b3\u5b9a\uff0c\u8fd8\u548c\u524d\u4e00\u4e2a\u72b6\u6001\uff08\u4e0a\u4e00\u8f6e\u6240\u5728\u697c\u68af\u9636\u6570\uff09\u6709\u5173\u3002</p> <p>\u4e0d\u96be\u53d1\u73b0\uff0c\u6b64\u95ee\u9898\u5df2\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b \\(dp[i] = dp[i-1] + dp[i-2]\\) \u4e5f\u5931\u6548\u4e86\uff0c\u56e0\u4e3a \\(dp[i-1]\\) \u4ee3\u8868\u672c\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u4f46\u5176\u4e2d\u5305\u542b\u4e86\u8bb8\u591a\u201c\u4e0a\u4e00\u8f6e\u662f\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\u201d\u65b9\u6848\uff0c\u800c\u4e3a\u4e86\u6ee1\u8db3\u7ea6\u675f\uff0c\u6211\u4eec\u5c31\u4e0d\u80fd\u5c06 \\(dp[i-1]\\) \u76f4\u63a5\u8ba1\u5165 \\(dp[i]\\) \u4e2d\u3002</p> <p>\u4e3a\u6b64\uff0c\u6211\u4eec\u9700\u8981\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff1a\u72b6\u6001 \\([i, j]\\) \u8868\u793a\u5904\u5728\u7b2c \\(i\\) \u9636\u5e76\u4e14\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(j\\) \u9636\uff0c\u5176\u4e2d \\(j \\in \\{1, 2\\}\\) \u3002\u6b64\u72b6\u6001\u5b9a\u4e49\u6709\u6548\u5730\u533a\u5206\u4e86\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u8fd8\u662f \\(2\\) \u9636\uff0c\u6211\u4eec\u53ef\u4ee5\u636e\u6b64\u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u4ece\u4f55\u800c\u6765\u7684\u3002</p> <ul> <li>\u5f53\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u65f6\uff0c\u4e0a\u4e0a\u4e00\u8f6e\u53ea\u80fd\u9009\u62e9\u8df3 \\(2\\) \u9636\uff0c\u5373 \\(dp[i, 1]\\) \u53ea\u80fd\u4ece \\(dp[i-1, 2]\\) \u8f6c\u79fb\u8fc7\u6765\u3002</li> <li>\u5f53\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(2\\) \u9636\u65f6\uff0c\u4e0a\u4e0a\u4e00\u8f6e\u53ef\u9009\u62e9\u8df3 \\(1\\) \u9636\u6216\u8df3 \\(2\\) \u9636\uff0c\u5373 \\(dp[i, 2]\\) \u53ef\u4ee5\u4ece \\(dp[i-2, 1]\\) \u6216 \\(dp[i-2, 2]\\) \u8f6c\u79fb\u8fc7\u6765\u3002</li> </ul> <p>\u5982\u56fe 14-9 \u6240\u793a\uff0c\u5728\u8be5\u5b9a\u4e49\u4e0b\uff0c\\(dp[i, j]\\) \u8868\u793a\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u65b9\u6848\u6570\u3002\u6b64\u65f6\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ \\begin{cases} dp[i, 1] = dp[i-1, 2] \\\\ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] \\end{cases} \\] <p></p> <p> \u56fe 14-9 \u00a0 \u8003\u8651\u7ea6\u675f\u4e0b\u7684\u9012\u63a8\u5173\u7cfb </p> <p>\u6700\u7ec8\uff0c\u8fd4\u56de \\(dp[n, 1] + dp[n, 2]\\) \u5373\u53ef\uff0c\u4e24\u8005\u4e4b\u548c\u4ee3\u8868\u722c\u5230\u7b2c \\(n\\) \u9636\u7684\u65b9\u6848\u603b\u6570\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_constraint_dp.py<pre><code>def climbing_stairs_constraint_dp(n: int) -&gt; int:\n    \"\"\"\u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    if n == 1 or n == 2:\n        return 1\n    # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp = [[0] * 3 for _ in range(n + 1)]\n    # \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1], dp[1][2] = 1, 0\n    dp[2][1], dp[2][2] = 0, 1\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in range(3, n + 1):\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    return dp[n][1] + dp[n][2]\n</code></pre> climbing_stairs_constraint_dp.cpp<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(3, 0));\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.java<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[][] dp = new int[n + 1][3];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.cs<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[,] dp = new int[n + 1, 3];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1, 1] = 1;\n    dp[1, 2] = 0;\n    dp[2, 1] = 0;\n    dp[2, 2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i, 1] = dp[i - 1, 2];\n        dp[i, 2] = dp[i - 2, 1] + dp[i - 2, 2];\n    }\n    return dp[n, 1] + dp[n, 2];\n}\n</code></pre> climbing_stairs_constraint_dp.go<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsConstraintDP(n int) int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp := make([][3]int, n+1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i := 3; i &lt;= n; i++ {\n        dp[i][1] = dp[i-1][2]\n        dp[i][2] = dp[i-2][1] + dp[i-2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.swift<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsConstraintDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = Array(repeating: Array(repeating: 0, count: 3), count: n + 1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in 3 ... n {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.js<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsConstraintDP(n) {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = Array.from(new Array(n + 1), () =&gt; new Array(3));\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.ts<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsConstraintDP(n: number): number {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(3));\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.dart<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n  if (n == 1 || n == 2) {\n    return 1;\n  }\n  // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(3, 0));\n  // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n  dp[1][1] = 1;\n  dp[1][2] = 0;\n  dp[2][1] = 0;\n  dp[2][2] = 1;\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i][1] = dp[i - 1][2];\n    dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n  }\n  return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.rs<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_constraint_dp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return 1;\n    };\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    let mut dp = vec![vec![-1; 3]; n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in 3..=n {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.c<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(3, sizeof(int));\n    }\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    int res = dp[n][1] + dp[n][2];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> climbing_stairs_constraint_dp.kt<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfun climbingStairsConstraintDP(n: Int): Int {\n    if (n == 1 || n == 2) {\n        return 1\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    val dp = Array(n + 1) { IntArray(3) }\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (i in 3..n) {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.rb<pre><code>[class]{}-[func]{climbing_stairs_constraint_dp}\n</code></pre> climbing_stairs_constraint_dp.zig<pre><code>// \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\nfn climbingStairsConstraintDP(comptime n: usize) i32 {\n    if (n == 1 or n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = [_][3]i32{ [_]i32{ -1, -1, -1 } } ** (n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (3..n + 1) |i| {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5728\u4e0a\u9762\u7684\u6848\u4f8b\u4e2d\uff0c\u7531\u4e8e\u4ec5\u9700\u591a\u8003\u8651\u524d\u9762\u4e00\u4e2a\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u901a\u8fc7\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff0c\u4f7f\u5f97\u95ee\u9898\u91cd\u65b0\u6ee1\u8db3\u65e0\u540e\u6548\u6027\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u95ee\u9898\u5177\u6709\u975e\u5e38\u4e25\u91cd\u7684\u201c\u6709\u540e\u6548\u6027\u201d\u3002</p> <p>\u722c\u697c\u68af\u4e0e\u969c\u788d\u751f\u6210</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\u3002\u89c4\u5b9a\u5f53\u722c\u5230\u7b2c \\(i\\) \u9636\u65f6\uff0c\u7cfb\u7edf\u81ea\u52a8\u4f1a\u5728\u7b2c \\(2i\\) \u9636\u4e0a\u653e\u4e0a\u969c\u788d\u7269\uff0c\u4e4b\u540e\u6240\u6709\u8f6e\u90fd\u4e0d\u5141\u8bb8\u8df3\u5230\u7b2c \\(2i\\) \u9636\u4e0a\u3002\u4f8b\u5982\uff0c\u524d\u4e24\u8f6e\u5206\u522b\u8df3\u5230\u4e86\u7b2c \\(2\\)\u3001\\(3\\) \u9636\u4e0a\uff0c\u5219\u4e4b\u540e\u5c31\u4e0d\u80fd\u8df3\u5230\u7b2c \\(4\\)\u3001\\(6\\) \u9636\u4e0a\u3002\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f</p> <p>\u5728\u8fd9\u4e2a\u95ee\u9898\u4e2d\uff0c\u4e0b\u6b21\u8df3\u8dc3\u4f9d\u8d56\u8fc7\u53bb\u6240\u6709\u7684\u72b6\u6001\uff0c\u56e0\u4e3a\u6bcf\u4e00\u6b21\u8df3\u8dc3\u90fd\u4f1a\u5728\u66f4\u9ad8\u7684\u9636\u68af\u4e0a\u8bbe\u7f6e\u969c\u788d\uff0c\u5e76\u5f71\u54cd\u672a\u6765\u7684\u8df3\u8dc3\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u52a8\u6001\u89c4\u5212\u5f80\u5f80\u96be\u4ee5\u89e3\u51b3\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u590d\u6742\u7684\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff08\u4f8b\u5982\u65c5\u884c\u5546\u95ee\u9898\uff09\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9009\u62e9\u4f7f\u7528\u5176\u4ed6\u65b9\u6cd5\uff0c\u4f8b\u5982\u542f\u53d1\u5f0f\u641c\u7d22\u3001\u9057\u4f20\u7b97\u6cd5\u3001\u5f3a\u5316\u5b66\u4e60\u7b49\uff0c\u4ece\u800c\u5728\u6709\u9650\u65f6\u95f4\u5185\u5f97\u5230\u53ef\u7528\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/","title":"14.3 \u00a0 \u52a8\u6001\u89c4\u5212\u89e3\u9898\u601d\u8def","text":"<p>\u4e0a\u4e24\u8282\u4ecb\u7ecd\u4e86\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u4e3b\u8981\u7279\u5f81\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u4e00\u8d77\u63a2\u7a76\u4e24\u4e2a\u66f4\u52a0\u5b9e\u7528\u7684\u95ee\u9898\u3002</p> <ol> <li>\u5982\u4f55\u5224\u65ad\u4e00\u4e2a\u95ee\u9898\u662f\u4e0d\u662f\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff1f</li> <li>\u6c42\u89e3\u52a8\u6001\u89c4\u5212\u95ee\u9898\u8be5\u4ece\u4f55\u5904\u5165\u624b\uff0c\u5b8c\u6574\u6b65\u9aa4\u662f\u4ec0\u4e48\uff1f</li> </ol>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1431","title":"14.3.1 \u00a0 \u95ee\u9898\u5224\u65ad","text":"<p>\u603b\u7684\u6765\u8bf4\uff0c\u5982\u679c\u4e00\u4e2a\u95ee\u9898\u5305\u542b\u91cd\u53e0\u5b50\u95ee\u9898\u3001\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5e76\u6ee1\u8db3\u65e0\u540e\u6548\u6027\uff0c\u90a3\u4e48\u5b83\u901a\u5e38\u9002\u5408\u7528\u52a8\u6001\u89c4\u5212\u6c42\u89e3\u3002\u7136\u800c\uff0c\u6211\u4eec\u5f88\u96be\u4ece\u95ee\u9898\u63cf\u8ff0\u4e2d\u76f4\u63a5\u63d0\u53d6\u51fa\u8fd9\u4e9b\u7279\u6027\u3002\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f1a\u653e\u5bbd\u6761\u4ef6\uff0c\u5148\u89c2\u5bdf\u95ee\u9898\u662f\u5426\u9002\u5408\u4f7f\u7528\u56de\u6eaf\uff08\u7a77\u4e3e\uff09\u89e3\u51b3\u3002</p> <p>\u9002\u5408\u7528\u56de\u6eaf\u89e3\u51b3\u7684\u95ee\u9898\u901a\u5e38\u6ee1\u8db3\u201c\u51b3\u7b56\u6811\u6a21\u578b\u201d\uff0c\u8fd9\u79cd\u95ee\u9898\u53ef\u4ee5\u4f7f\u7528\u6811\u5f62\u7ed3\u6784\u6765\u63cf\u8ff0\uff0c\u5176\u4e2d\u6bcf\u4e00\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u51b3\u7b56\uff0c\u6bcf\u4e00\u6761\u8def\u5f84\u4ee3\u8868\u4e00\u4e2a\u51b3\u7b56\u5e8f\u5217\u3002</p> <p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u5982\u679c\u95ee\u9898\u5305\u542b\u660e\u786e\u7684\u51b3\u7b56\u6982\u5ff5\uff0c\u5e76\u4e14\u89e3\u662f\u901a\u8fc7\u4e00\u7cfb\u5217\u51b3\u7b56\u4ea7\u751f\u7684\uff0c\u90a3\u4e48\u5b83\u5c31\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\uff0c\u901a\u5e38\u53ef\u4ee5\u4f7f\u7528\u56de\u6eaf\u6765\u89e3\u51b3\u3002</p> <p>\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u52a8\u6001\u89c4\u5212\u95ee\u9898\u8fd8\u6709\u4e00\u4e9b\u5224\u65ad\u7684\u201c\u52a0\u5206\u9879\u201d\u3002</p> <ul> <li>\u95ee\u9898\u5305\u542b\u6700\u5927\uff08\u5c0f\uff09\u6216\u6700\u591a\uff08\u5c11\uff09\u7b49\u6700\u4f18\u5316\u63cf\u8ff0\u3002</li> <li>\u95ee\u9898\u7684\u72b6\u6001\u80fd\u591f\u4f7f\u7528\u4e00\u4e2a\u5217\u8868\u3001\u591a\u7ef4\u77e9\u9635\u6216\u6811\u6765\u8868\u793a\uff0c\u5e76\u4e14\u4e00\u4e2a\u72b6\u6001\u4e0e\u5176\u5468\u56f4\u7684\u72b6\u6001\u5b58\u5728\u9012\u63a8\u5173\u7cfb\u3002</li> </ul> <p>\u76f8\u5e94\u5730\uff0c\u4e5f\u5b58\u5728\u4e00\u4e9b\u201c\u51cf\u5206\u9879\u201d\u3002</p> <ul> <li>\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u800c\u4e0d\u662f\u627e\u51fa\u6700\u4f18\u89e3\u3002</li> <li>\u95ee\u9898\u63cf\u8ff0\u4e2d\u6709\u660e\u663e\u7684\u6392\u5217\u7ec4\u5408\u7684\u7279\u5f81\uff0c\u9700\u8981\u8fd4\u56de\u5177\u4f53\u7684\u591a\u4e2a\u65b9\u6848\u3002</li> </ul> <p>\u5982\u679c\u4e00\u4e2a\u95ee\u9898\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\uff0c\u5e76\u5177\u6709\u8f83\u4e3a\u660e\u663e\u7684\u201c\u52a0\u5206\u9879\u201d\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5047\u8bbe\u5b83\u662f\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff0c\u5e76\u5728\u6c42\u89e3\u8fc7\u7a0b\u4e2d\u9a8c\u8bc1\u5b83\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1432","title":"14.3.2 \u00a0 \u95ee\u9898\u6c42\u89e3\u6b65\u9aa4","text":"<p>\u52a8\u6001\u89c4\u5212\u7684\u89e3\u9898\u6d41\u7a0b\u4f1a\u56e0\u95ee\u9898\u7684\u6027\u8d28\u548c\u96be\u5ea6\u800c\u6709\u6240\u4e0d\u540c\uff0c\u4f46\u901a\u5e38\u9075\u5faa\u4ee5\u4e0b\u6b65\u9aa4\uff1a\u63cf\u8ff0\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u5efa\u7acb \\(dp\\) \u8868\uff0c\u63a8\u5bfc\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u7b49\u3002</p> <p>\u4e3a\u4e86\u66f4\u5f62\u8c61\u5730\u5c55\u793a\u89e3\u9898\u6b65\u9aa4\uff0c\u6211\u4eec\u4f7f\u7528\u4e00\u4e2a\u7ecf\u5178\u95ee\u9898\u201c\u6700\u5c0f\u8def\u5f84\u548c\u201d\u6765\u4e3e\u4f8b\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a \\(n \\times m\\) \u7684\u4e8c\u7ef4\u7f51\u683c <code>grid</code> \uff0c\u7f51\u683c\u4e2d\u7684\u6bcf\u4e2a\u5355\u5143\u683c\u5305\u542b\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a\u8be5\u5355\u5143\u683c\u7684\u4ee3\u4ef7\u3002\u673a\u5668\u4eba\u4ee5\u5de6\u4e0a\u89d2\u5355\u5143\u683c\u4e3a\u8d77\u59cb\u70b9\uff0c\u6bcf\u6b21\u53ea\u80fd\u5411\u4e0b\u6216\u8005\u5411\u53f3\u79fb\u52a8\u4e00\u6b65\uff0c\u76f4\u81f3\u5230\u8fbe\u53f3\u4e0b\u89d2\u5355\u5143\u683c\u3002\u8bf7\u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230\u53f3\u4e0b\u89d2\u7684\u6700\u5c0f\u8def\u5f84\u548c\u3002</p> <p>\u56fe 14-10 \u5c55\u793a\u4e86\u4e00\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u7f51\u683c\u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e3a \\(13\\) \u3002</p> <p></p> <p> \u56fe 14-10 \u00a0 \u6700\u5c0f\u8def\u5f84\u548c\u793a\u4f8b\u6570\u636e </p> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u672c\u9898\u7684\u6bcf\u4e00\u8f6e\u7684\u51b3\u7b56\u5c31\u662f\u4ece\u5f53\u524d\u683c\u5b50\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u3002\u8bbe\u5f53\u524d\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u4e3a \\([i, j]\\) \uff0c\u5219\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u540e\uff0c\u7d22\u5f15\u53d8\u4e3a \\([i+1, j]\\) \u6216 \\([i, j+1]\\) \u3002\u56e0\u6b64\uff0c\u72b6\u6001\u5e94\u5305\u542b\u884c\u7d22\u5f15\u548c\u5217\u7d22\u5f15\u4e24\u4e2a\u53d8\u91cf\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002</p> <p>\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u4ece\u8d77\u59cb\u70b9 \\([0, 0]\\) \u8d70\u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\uff0c\u89e3\u8bb0\u4e3a \\(dp[i, j]\\) \u3002</p> <p>\u81f3\u6b64\uff0c\u6211\u4eec\u5c31\u5f97\u5230\u4e86\u56fe 14-11 \u6240\u793a\u7684\u4e8c\u7ef4 \\(dp\\) \u77e9\u9635\uff0c\u5176\u5c3a\u5bf8\u4e0e\u8f93\u5165\u7f51\u683c \\(grid\\) \u76f8\u540c\u3002</p> <p></p> <p> \u56fe 14-11 \u00a0 \u72b6\u6001\u5b9a\u4e49\u4e0e dp \u8868 </p> <p>Note</p> <p>\u52a8\u6001\u89c4\u5212\u548c\u56de\u6eaf\u8fc7\u7a0b\u53ef\u4ee5\u63cf\u8ff0\u4e3a\u4e00\u4e2a\u51b3\u7b56\u5e8f\u5217\uff0c\u800c\u72b6\u6001\u7531\u6240\u6709\u51b3\u7b56\u53d8\u91cf\u6784\u6210\u3002\u5b83\u5e94\u5f53\u5305\u542b\u63cf\u8ff0\u89e3\u9898\u8fdb\u5ea6\u7684\u6240\u6709\u53d8\u91cf\uff0c\u5176\u5305\u542b\u4e86\u8db3\u591f\u7684\u4fe1\u606f\uff0c\u80fd\u591f\u7528\u6765\u63a8\u5bfc\u51fa\u4e0b\u4e00\u4e2a\u72b6\u6001\u3002</p> <p>\u6bcf\u4e2a\u72b6\u6001\u90fd\u5bf9\u5e94\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u6211\u4eec\u4f1a\u5b9a\u4e49\u4e00\u4e2a \\(dp\\) \u8868\u6765\u5b58\u50a8\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u72b6\u6001\u7684\u6bcf\u4e2a\u72ec\u7acb\u53d8\u91cf\u90fd\u662f \\(dp\\) \u8868\u7684\u4e00\u4e2a\u7ef4\u5ea6\u3002\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\\(dp\\) \u8868\u662f\u72b6\u6001\u548c\u5b50\u95ee\u9898\u7684\u89e3\u4e4b\u95f4\u7684\u6620\u5c04\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u5bf9\u4e8e\u72b6\u6001 \\([i, j]\\) \uff0c\u5b83\u53ea\u80fd\u4ece\u4e0a\u8fb9\u683c\u5b50 \\([i-1, j]\\) \u548c\u5de6\u8fb9\u683c\u5b50 \\([i, j-1]\\) \u8f6c\u79fb\u800c\u6765\u3002\u56e0\u6b64\u6700\u4f18\u5b50\u7ed3\u6784\u4e3a\uff1a\u5230\u8fbe \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u7531 \\([i, j-1]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e0e \\([i-1, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e2d\u8f83\u5c0f\u7684\u90a3\u4e00\u4e2a\u51b3\u5b9a\u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u63a8\u51fa\u56fe 14-12 \u6240\u793a\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a</p> \\[ dp[i, j] = \\min(dp[i-1, j], dp[i, j-1]) + grid[i, j] \\] <p></p> <p> \u56fe 14-12 \u00a0 \u6700\u4f18\u5b50\u7ed3\u6784\u4e0e\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b </p> <p>Note</p> <p>\u6839\u636e\u5b9a\u4e49\u597d\u7684 \\(dp\\) \u8868\uff0c\u601d\u8003\u539f\u95ee\u9898\u548c\u5b50\u95ee\u9898\u7684\u5173\u7cfb\uff0c\u627e\u51fa\u901a\u8fc7\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6765\u6784\u9020\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u7684\u65b9\u6cd5\uff0c\u5373\u6700\u4f18\u5b50\u7ed3\u6784\u3002</p> <p>\u4e00\u65e6\u6211\u4eec\u627e\u5230\u4e86\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528\u5b83\u6765\u6784\u5efa\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002</p> <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5728\u672c\u9898\u4e2d\uff0c\u5904\u5728\u9996\u884c\u7684\u72b6\u6001\u53ea\u80fd\u4ece\u5176\u5de6\u8fb9\u7684\u72b6\u6001\u5f97\u6765\uff0c\u5904\u5728\u9996\u5217\u7684\u72b6\u6001\u53ea\u80fd\u4ece\u5176\u4e0a\u8fb9\u7684\u72b6\u6001\u5f97\u6765\uff0c\u56e0\u6b64\u9996\u884c \\(i = 0\\) \u548c\u9996\u5217 \\(j = 0\\) \u662f\u8fb9\u754c\u6761\u4ef6\u3002</p> <p>\u5982\u56fe 14-13 \u6240\u793a\uff0c\u7531\u4e8e\u6bcf\u4e2a\u683c\u5b50\u662f\u7531\u5176\u5de6\u65b9\u683c\u5b50\u548c\u4e0a\u65b9\u683c\u5b50\u8f6c\u79fb\u800c\u6765\uff0c\u56e0\u6b64\u6211\u4eec\u4f7f\u7528\u5faa\u73af\u6765\u904d\u5386\u77e9\u9635\uff0c\u5916\u5faa\u73af\u904d\u5386\u5404\u884c\uff0c\u5185\u5faa\u73af\u904d\u5386\u5404\u5217\u3002</p> <p></p> <p> \u56fe 14-13 \u00a0 \u8fb9\u754c\u6761\u4ef6\u4e0e\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f </p> <p>Note</p> <p>\u8fb9\u754c\u6761\u4ef6\u5728\u52a8\u6001\u89c4\u5212\u4e2d\u7528\u4e8e\u521d\u59cb\u5316 \\(dp\\) \u8868\uff0c\u5728\u641c\u7d22\u4e2d\u7528\u4e8e\u526a\u679d\u3002</p> <p>\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f\u7684\u6838\u5fc3\u662f\u8981\u4fdd\u8bc1\u5728\u8ba1\u7b97\u5f53\u524d\u95ee\u9898\u7684\u89e3\u65f6\uff0c\u6240\u6709\u5b83\u4f9d\u8d56\u7684\u66f4\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u90fd\u5df2\u7ecf\u88ab\u6b63\u786e\u5730\u8ba1\u7b97\u51fa\u6765\u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u4eec\u5df2\u7ecf\u53ef\u4ee5\u76f4\u63a5\u5199\u51fa\u52a8\u6001\u89c4\u5212\u4ee3\u7801\u3002\u7136\u800c\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u4ece\u9876\u81f3\u5e95\u7684\u601d\u60f3\uff0c\u56e0\u6b64\u6309\u7167\u201c\u66b4\u529b\u641c\u7d22 \\(\\rightarrow\\) \u8bb0\u5fc6\u5316\u641c\u7d22 \\(\\rightarrow\\) \u52a8\u6001\u89c4\u5212\u201d\u7684\u987a\u5e8f\u5b9e\u73b0\u66f4\u52a0\u7b26\u5408\u601d\u7ef4\u4e60\u60ef\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"<p>\u4ece\u72b6\u6001 \\([i, j]\\) \u5f00\u59cb\u641c\u7d22\uff0c\u4e0d\u65ad\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u72b6\u6001 \\([i-1, j]\\) \u548c \\([i, j-1]\\) \uff0c\u9012\u5f52\u51fd\u6570\u5305\u62ec\u4ee5\u4e0b\u8981\u7d20\u3002</p> <ul> <li>\u9012\u5f52\u53c2\u6570\uff1a\u72b6\u6001 \\([i, j]\\) \u3002</li> <li>\u8fd4\u56de\u503c\uff1a\u4ece \\([0, 0]\\) \u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c \\(dp[i, j]\\) \u3002</li> <li>\u7ec8\u6b62\u6761\u4ef6\uff1a\u5f53 \\(i = 0\\) \u4e14 \\(j = 0\\) \u65f6\uff0c\u8fd4\u56de\u4ee3\u4ef7 \\(grid[0, 0]\\) \u3002</li> <li>\u526a\u679d\uff1a\u5f53 \\(i &lt; 0\\) \u65f6\u6216 \\(j &lt; 0\\) \u65f6\u7d22\u5f15\u8d8a\u754c\uff0c\u6b64\u65f6\u8fd4\u56de\u4ee3\u4ef7 \\(+\\infty\\) \uff0c\u4ee3\u8868\u4e0d\u53ef\u884c\u3002</li> </ul> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dfs(grid: list[list[int]], i: int, j: int) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22\"\"\"\n    # \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    up = min_path_sum_dfs(grid, i - 1, j)\n    left = min_path_sum_dfs(grid, i, j - 1)\n    # \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return min(left, up) + grid[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(vector&lt;vector&lt;int&gt;&gt; &amp;grid, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(int[][] grid, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint MinPathSumDFS(int[][] grid, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = MinPathSumDFS(grid, i - 1, j);\n    int left = MinPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return Math.Min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc minPathSumDFS(grid [][]int, i, j int) int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    up := minPathSumDFS(grid, i-1, j)\n    left := minPathSumDFS(grid, i, j-1)\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return int(math.Min(float64(left), float64(up))) + grid[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc minPathSumDFS(grid: [[Int]], i: Int, j: Int) -&gt; Int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    let up = minPathSumDFS(grid: grid, i: i - 1, j: j)\n    let left = minPathSumDFS(grid: grid, i: i, j: j - 1)\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction minPathSumDFS(grid, i, j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction minPathSumDFS(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i === 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(List&lt;List&lt;int&gt;&gt; grid, int i, int j) {\n  // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n  if (i &lt; 0 || j &lt; 0) {\n    // \u5728 Dart \u4e2d\uff0cint \u7c7b\u578b\u662f\u56fa\u5b9a\u8303\u56f4\u7684\u6574\u6570\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u65e0\u7a77\u5927\u201d\u7684\u503c\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n  int up = minPathSumDFS(grid, i - 1, j);\n  int left = minPathSumDFS(grid, i, j - 1);\n  // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n  return min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfn min_path_sum_dfs(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    let up = min_path_sum_dfs(grid, i - 1, j);\n    let left = min_path_sum_dfs(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    std::cmp::min(left, up) + grid[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(int grid[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfun minPathSumDFS(\n    grid: Array&lt;Array&lt;Int&gt;&gt;,\n    i: Int,\n    j: Int\n): Int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    val up = minPathSumDFS(grid, i - 1, j)\n    val left = minPathSumDFS(grid, i, j - 1)\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return (min(left.toDouble(), up.toDouble()) + grid[i][j]).toInt()\n}\n</code></pre> min_path_sum.rb<pre><code>[class]{}-[func]{min_path_sum_dfs}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22\nfn minPathSumDFS(grid: anytype, i: i32, j: i32) i32 {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 and j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 or j &lt; 0) {\n        return std.math.maxInt(i32);\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    var up = minPathSumDFS(grid, i - 1, j);\n    var left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 14-14 \u7ed9\u51fa\u4e86\u4ee5 \\(dp[2, 1]\\) \u4e3a\u6839\u8282\u70b9\u7684\u9012\u5f52\u6811\uff0c\u5176\u4e2d\u5305\u542b\u4e00\u4e9b\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5176\u6570\u91cf\u4f1a\u968f\u7740\u7f51\u683c <code>grid</code> \u7684\u5c3a\u5bf8\u53d8\u5927\u800c\u6025\u5267\u589e\u591a\u3002</p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9020\u6210\u91cd\u53e0\u5b50\u95ee\u9898\u7684\u539f\u56e0\u4e3a\uff1a\u5b58\u5728\u591a\u6761\u8def\u5f84\u53ef\u4ee5\u4ece\u5de6\u4e0a\u89d2\u5230\u8fbe\u67d0\u4e00\u5355\u5143\u683c\u3002</p> <p></p> <p> \u56fe 14-14 \u00a0 \u66b4\u529b\u641c\u7d22\u9012\u5f52\u6811 </p> <p>\u6bcf\u4e2a\u72b6\u6001\u90fd\u6709\u5411\u4e0b\u548c\u5411\u53f3\u4e24\u79cd\u9009\u62e9\uff0c\u4ece\u5de6\u4e0a\u89d2\u8d70\u5230\u53f3\u4e0b\u89d2\u603b\u5171\u9700\u8981 \\(m + n - 2\\) \u6b65\uff0c\u6240\u4ee5\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^{m + n})\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u8fd9\u79cd\u8ba1\u7b97\u65b9\u5f0f\u672a\u8003\u8651\u4e34\u8fd1\u7f51\u683c\u8fb9\u754c\u7684\u60c5\u51b5\uff0c\u5f53\u5230\u8fbe\u7f51\u7edc\u8fb9\u754c\u65f6\u53ea\u5269\u4e0b\u4e00\u79cd\u9009\u62e9\uff0c\u56e0\u6b64\u5b9e\u9645\u7684\u8def\u5f84\u6570\u91cf\u4f1a\u5c11\u4e00\u4e9b\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"<p>\u6211\u4eec\u5f15\u5165\u4e00\u4e2a\u548c\u7f51\u683c <code>grid</code> \u76f8\u540c\u5c3a\u5bf8\u7684\u8bb0\u5fc6\u5217\u8868 <code>mem</code> \uff0c\u7528\u4e8e\u8bb0\u5f55\u5404\u4e2a\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5e76\u5c06\u91cd\u53e0\u5b50\u95ee\u9898\u8fdb\u884c\u526a\u679d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dfs_mem(\n    grid: list[list[int]], mem: list[list[int]], i: int, j: int\n) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n    # \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][j] != -1:\n        return mem[i][j]\n    # \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n    left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n    # \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(vector&lt;vector&lt;int&gt;&gt; &amp;grid, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint MinPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = MinPathSumDFSMem(grid, mem, i - 1, j);\n    int left = MinPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = Math.Min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc minPathSumDFSMem(grid, mem [][]int, i, j int) int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    up := minPathSumDFSMem(grid, mem, i-1, j)\n    left := minPathSumDFSMem(grid, mem, i, j-1)\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = int(math.Min(float64(left), float64(up))) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc minPathSumDFSMem(grid: [[Int]], mem: inout [[Int]], i: Int, j: Int) -&gt; Int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    let up = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i - 1, j: j)\n    let left = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i, j: j - 1)\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction minPathSumDFSMem(grid, mem, i, j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] !== -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction minPathSumDFSMem(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(List&lt;List&lt;int&gt;&gt; grid, List&lt;List&lt;int&gt;&gt; mem, int i, int j) {\n  // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n  if (i &lt; 0 || j &lt; 0) {\n    // \u5728 Dart \u4e2d\uff0cint \u7c7b\u578b\u662f\u56fa\u5b9a\u8303\u56f4\u7684\u6574\u6570\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u65e0\u7a77\u5927\u201d\u7684\u503c\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n  if (mem[i][j] != -1) {\n    return mem[i][j];\n  }\n  // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n  int up = minPathSumDFSMem(grid, mem, i - 1, j);\n  int left = minPathSumDFSMem(grid, mem, i, j - 1);\n  // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n  mem[i][j] = min(left, up) + grid[i][j];\n  return mem[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn min_path_sum_dfs_mem(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i as usize][j as usize] != -1 {\n        return mem[i as usize][j as usize];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    let up = min_path_sum_dfs_mem(grid, mem, i - 1, j);\n    let left = min_path_sum_dfs_mem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];\n    mem[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(int grid[MAX_SIZE][MAX_SIZE], int mem[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfun minPathSumDFSMem(\n    grid: Array&lt;Array&lt;Int&gt;&gt;,\n    mem: Array&lt;Array&lt;Int&gt;&gt;,\n    i: Int,\n    j: Int\n): Int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j]\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    val up = minPathSumDFSMem(grid, mem, i - 1, j)\n    val left = minPathSumDFSMem(grid, mem, i, j - 1)\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = (min(left.toDouble(), up.toDouble()) + grid[i][j]).toInt()\n    return mem[i][j]\n}\n</code></pre> min_path_sum.rb<pre><code>[class]{}-[func]{min_path_sum_dfs_mem}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn minPathSumDFSMem(grid: anytype, mem: anytype, i: i32, j: i32) i32 {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 and j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 or j &lt; 0) {\n        return std.math.maxInt(i32);\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] != -1) {\n        return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    var up = minPathSumDFSMem(grid, mem, i - 1, j);\n    var left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] = @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n    return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 14-15 \u6240\u793a\uff0c\u5728\u5f15\u5165\u8bb0\u5fc6\u5316\u540e\uff0c\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\u53ea\u9700\u8ba1\u7b97\u4e00\u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u72b6\u6001\u603b\u6570\uff0c\u5373\u7f51\u683c\u5c3a\u5bf8 \\(O(nm)\\) \u3002</p> <p></p> <p> \u56fe 14-15 \u00a0 \u8bb0\u5fc6\u5316\u641c\u7d22\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"<p>\u57fa\u4e8e\u8fed\u4ee3\u5b9e\u73b0\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dp(grid: list[list[int]]) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n, m = len(grid), len(grid[0])\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * m for _ in range(n)]\n    dp[0][0] = grid[0][0]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in range(1, m):\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for i in range(1, n):\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in range(1, n):\n        for j in range(1, m):\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n    return dp[n - 1][m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n, vector&lt;int&gt;(m));\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n][m];\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint MinPathSumDP(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n, m];\n    dp[0, 0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0, j] = dp[0, j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i, 0] = dp[i - 1, 0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i, j] = Math.Min(dp[i, j - 1], dp[i - 1, j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1, m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDP(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        dp[i] = make([]int, m)\n    }\n    dp[0][0] = grid[0][0]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j := 1; j &lt; m; j++ {\n        dp[0][j] = dp[0][j-1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for i := 1; i &lt; n; i++ {\n        dp[i][0] = dp[i-1][0] + grid[i][0]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i := 1; i &lt; n; i++ {\n        for j := 1; j &lt; m; j++ {\n            dp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(dp[i-1][j]))) + grid[i][j]\n        }\n    }\n    return dp[n-1][m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDP(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: m), count: n)\n    dp[0][0] = grid[0][0]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in 1 ..&lt; m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for i in 1 ..&lt; n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in 1 ..&lt; n {\n        for j in 1 ..&lt; m {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDP(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt; n; i++) {\n        for (let j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDP(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt; n; i++) {\n        for (let j: number = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n, (i) =&gt; List.filled(m, 0));\n  dp[0][0] = grid[0][0];\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n  for (int j = 1; j &lt; m; j++) {\n    dp[0][j] = dp[0][j - 1] + grid[0][j];\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n  for (int i = 1; i &lt; n; i++) {\n    dp[i][0] = dp[i - 1][0] + grid[i][0];\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n  for (int i = 1; i &lt; n; i++) {\n    for (int j = 1; j &lt; m; j++) {\n      dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n    }\n  }\n  return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfn min_path_sum_dp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; m]; n];\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in 1..m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for i in 1..n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in 1..n {\n        for j in 1..m {\n            dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc(n * sizeof(int *));\n    for (int i = 0; i &lt; n; i++) {\n        dp[i] = calloc(m, sizeof(int));\n    }\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = myMin(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    int res = dp[n - 1][m - 1];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt; n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfun minPathSumDP(grid: Array&lt;Array&lt;Int&gt;&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n) { IntArray(m) }\n    dp[0][0] = grid[0][0]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (j in 1..&lt;m) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (i in 1..&lt;n) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (i in 1..&lt;n) {\n        for (j in 1..&lt;m) {\n            dp[i][j] =\n                (min(dp[i][j - 1].toDouble(), dp[i - 1][j].toDouble()) + grid[i][j]).toInt()\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>[class]{}-[func]{min_path_sum_dp}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212\nfn minPathSumDP(comptime grid: anytype) i32 {\n    comptime var n = grid.len;\n    comptime var m = grid[0].len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][m]i32{[_]i32{0} ** m} ** n;\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (1..m) |j| {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (1..n) |i| {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (1..n) |i| {\n        for (1..m) |j| {\n            dp[i][j] = @min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 14-16 \u5c55\u793a\u4e86\u6700\u5c0f\u8def\u5f84\u548c\u7684\u72b6\u6001\u8f6c\u79fb\u8fc7\u7a0b\uff0c\u5176\u904d\u5386\u4e86\u6574\u4e2a\u7f51\u683c\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nm)\\) \u3002</p> <p>\u6570\u7ec4 <code>dp</code> \u5927\u5c0f\u4e3a \\(n \\times m\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nm)\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-16 \u00a0 \u6700\u5c0f\u8def\u5f84\u548c\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#4","title":"4. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e\u6bcf\u4e2a\u683c\u5b50\u53ea\u4e0e\u5176\u5de6\u8fb9\u548c\u4e0a\u8fb9\u7684\u683c\u5b50\u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u53ea\u7528\u4e00\u4e2a\u5355\u884c\u6570\u7ec4\u6765\u5b9e\u73b0 \\(dp\\) \u8868\u3002</p> <p>\u8bf7\u6ce8\u610f\uff0c\u56e0\u4e3a\u6570\u7ec4 <code>dp</code> \u53ea\u80fd\u8868\u793a\u4e00\u884c\u7684\u72b6\u6001\uff0c\u6240\u4ee5\u6211\u4eec\u65e0\u6cd5\u63d0\u524d\u521d\u59cb\u5316\u9996\u5217\u72b6\u6001\uff0c\u800c\u662f\u5728\u904d\u5386\u6bcf\u884c\u65f6\u66f4\u65b0\u5b83\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig min_path_sum.py<pre><code>def min_path_sum_dp_comp(grid: list[list[int]]) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n, m = len(grid), len(grid[0])\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * m\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for j in range(1, m):\n        dp[j] = dp[j - 1] + grid[0][j]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in range(1, n):\n        # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in range(1, m):\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n    return dp[m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(m);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[m];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint MinPathSumDPComp(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[m];\n    dp[0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.Min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDPComp(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, m)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for j := 1; j &lt; m; j++ {\n        dp[j] = dp[j-1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i := 1; i &lt; n; i++ {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j := 1; j &lt; m; j++ {\n            dp[j] = int(math.Min(float64(dp[j-1]), float64(dp[j]))) + grid[i][j]\n        }\n    }\n    return dp[m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDPComp(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: m)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for j in 1 ..&lt; m {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in 1 ..&lt; n {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in 1 ..&lt; m {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDPComp(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = new Array(m);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (let i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDPComp(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = new Array(m);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (let i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(m, 0);\n  dp[0] = grid[0][0];\n  for (int j = 1; j &lt; m; j++) {\n    dp[j] = dp[j - 1] + grid[0][j];\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n  for (int i = 1; i &lt; n; i++) {\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    dp[0] = dp[0] + grid[i][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n    for (int j = 1; j &lt; m; j++) {\n      dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n    }\n  }\n  return dp[m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn min_path_sum_dp_comp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; m];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for j in 1..m {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in 1..n {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in 1..m {\n            dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    dp[m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(m, sizeof(int));\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = myMin(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    int res = dp[m - 1];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun minPathSumDPComp(grid: Array&lt;Array&lt;Int&gt;&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(m)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for (j in 1..&lt;m) {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (i in 1..&lt;n) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (j in 1..&lt;m) {\n            dp[j] = (min(dp[j - 1].toDouble(), dp[j].toDouble()) + grid[i][j]).toInt()\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>[class]{}-[func]{min_path_sum_dp_comp}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn minPathSumDPComp(comptime grid: anytype) i32 {\n    comptime var n = grid.len;\n    comptime var m = grid[0].len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** m;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (1..m) |j| {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (1..n) |i| {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        for (1..m) |j| {\n            dp[j] = @min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/","title":"14.6 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898","text":"<p>\u7f16\u8f91\u8ddd\u79bb\uff0c\u4e5f\u79f0 Levenshtein \u8ddd\u79bb\uff0c\u6307\u4e24\u4e2a\u5b57\u7b26\u4e32\u4e4b\u95f4\u4e92\u76f8\u8f6c\u6362\u7684\u6700\u5c11\u4fee\u6539\u6b21\u6570\uff0c\u901a\u5e38\u7528\u4e8e\u5728\u4fe1\u606f\u68c0\u7d22\u548c\u81ea\u7136\u8bed\u8a00\u5904\u7406\u4e2d\u5ea6\u91cf\u4e24\u4e2a\u5e8f\u5217\u7684\u76f8\u4f3c\u5ea6\u3002</p> <p>Question</p> <p>\u8f93\u5165\u4e24\u4e2a\u5b57\u7b26\u4e32 \\(s\\) \u548c \\(t\\) \uff0c\u8fd4\u56de\u5c06 \\(s\\) \u8f6c\u6362\u4e3a \\(t\\) \u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002</p> <p>\u4f60\u53ef\u4ee5\u5728\u4e00\u4e2a\u5b57\u7b26\u4e32\u4e2d\u8fdb\u884c\u4e09\u79cd\u7f16\u8f91\u64cd\u4f5c\uff1a\u63d2\u5165\u4e00\u4e2a\u5b57\u7b26\u3001\u5220\u9664\u4e00\u4e2a\u5b57\u7b26\u3001\u5c06\u5b57\u7b26\u66ff\u6362\u4e3a\u4efb\u610f\u4e00\u4e2a\u5b57\u7b26\u3002</p> <p>\u5982\u56fe 14-27 \u6240\u793a\uff0c\u5c06 <code>kitten</code> \u8f6c\u6362\u4e3a <code>sitting</code> \u9700\u8981\u7f16\u8f91 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u6362\u64cd\u4f5c\u4e0e 1 \u6b21\u6dfb\u52a0\u64cd\u4f5c\uff1b\u5c06 <code>hello</code> \u8f6c\u6362\u4e3a <code>algo</code> \u9700\u8981 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u6362\u64cd\u4f5c\u548c 1 \u6b21\u5220\u9664\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 14-27 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u53ef\u4ee5\u5f88\u81ea\u7136\u5730\u7528\u51b3\u7b56\u6811\u6a21\u578b\u6765\u89e3\u91ca\u3002\u5b57\u7b26\u4e32\u5bf9\u5e94\u6811\u8282\u70b9\uff0c\u4e00\u8f6e\u51b3\u7b56\uff08\u4e00\u6b21\u7f16\u8f91\u64cd\u4f5c\uff09\u5bf9\u5e94\u6811\u7684\u4e00\u6761\u8fb9\u3002</p> <p>\u5982\u56fe 14-28 \u6240\u793a\uff0c\u5728\u4e0d\u9650\u5236\u64cd\u4f5c\u7684\u60c5\u51b5\u4e0b\uff0c\u6bcf\u4e2a\u8282\u70b9\u90fd\u53ef\u4ee5\u6d3e\u751f\u51fa\u8bb8\u591a\u6761\u8fb9\uff0c\u6bcf\u6761\u8fb9\u5bf9\u5e94\u4e00\u79cd\u64cd\u4f5c\uff0c\u8fd9\u610f\u5473\u7740\u4ece <code>hello</code> \u8f6c\u6362\u5230 <code>algo</code> \u6709\u8bb8\u591a\u79cd\u53ef\u80fd\u7684\u8def\u5f84\u3002</p> <p>\u4ece\u51b3\u7b56\u6811\u7684\u89d2\u5ea6\u770b\uff0c\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u8282\u70b9 <code>hello</code> \u548c\u8282\u70b9 <code>algo</code> \u4e4b\u95f4\u7684\u6700\u77ed\u8def\u5f84\u3002</p> <p></p> <p> \u56fe 14-28 \u00a0 \u57fa\u4e8e\u51b3\u7b56\u6811\u6a21\u578b\u8868\u793a\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898 </p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u6bcf\u4e00\u8f6e\u7684\u51b3\u7b56\u662f\u5bf9\u5b57\u7b26\u4e32 \\(s\\) \u8fdb\u884c\u4e00\u6b21\u7f16\u8f91\u64cd\u4f5c\u3002</p> <p>\u6211\u4eec\u5e0c\u671b\u5728\u7f16\u8f91\u64cd\u4f5c\u7684\u8fc7\u7a0b\u4e2d\uff0c\u95ee\u9898\u7684\u89c4\u6a21\u9010\u6e10\u7f29\u5c0f\uff0c\u8fd9\u6837\u624d\u80fd\u6784\u5efa\u5b50\u95ee\u9898\u3002\u8bbe\u5b57\u7b26\u4e32 \\(s\\) \u548c \\(t\\) \u7684\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) \u548c \\(m\\) \uff0c\u6211\u4eec\u5148\u8003\u8651\u4e24\u5b57\u7b26\u4e32\u5c3e\u90e8\u7684\u5b57\u7b26 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u3002</p> <ul> <li>\u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u76f8\u540c\uff0c\u6211\u4eec\u53ef\u4ee5\u8df3\u8fc7\u5b83\u4eec\uff0c\u76f4\u63a5\u8003\u8651 \\(s[n-2]\\) \u548c \\(t[m-2]\\) \u3002</li> <li>\u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u4e0d\u540c\uff0c\u6211\u4eec\u9700\u8981\u5bf9 \\(s\\) \u8fdb\u884c\u4e00\u6b21\u7f16\u8f91\uff08\u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\uff09\uff0c\u4f7f\u5f97\u4e24\u5b57\u7b26\u4e32\u5c3e\u90e8\u7684\u5b57\u7b26\u76f8\u540c\uff0c\u4ece\u800c\u53ef\u4ee5\u8df3\u8fc7\u5b83\u4eec\uff0c\u8003\u8651\u89c4\u6a21\u66f4\u5c0f\u7684\u95ee\u9898\u3002</li> </ul> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u5728\u5b57\u7b26\u4e32 \\(s\\) \u4e2d\u8fdb\u884c\u7684\u6bcf\u4e00\u8f6e\u51b3\u7b56\uff08\u7f16\u8f91\u64cd\u4f5c\uff09\uff0c\u90fd\u4f1a\u4f7f\u5f97 \\(s\\) \u548c \\(t\\) \u4e2d\u5269\u4f59\u7684\u5f85\u5339\u914d\u5b57\u7b26\u53d1\u751f\u53d8\u5316\u3002\u56e0\u6b64\uff0c\u72b6\u6001\u4e3a\u5f53\u524d\u5728 \\(s\\) \u548c \\(t\\) \u4e2d\u8003\u8651\u7684\u7b2c \\(i\\) \u548c\u7b2c \\(j\\) \u4e2a\u5b57\u7b26\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002</p> <p>\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\uff1a\u5c06 \\(s\\) \u7684\u524d \\(i\\) \u4e2a\u5b57\u7b26\u66f4\u6539\u4e3a \\(t\\) \u7684\u524d \\(j\\) \u4e2a\u5b57\u7b26\u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002</p> <p>\u81f3\u6b64\uff0c\u5f97\u5230\u4e00\u4e2a\u5c3a\u5bf8\u4e3a \\((i+1) \\times (j+1)\\) \u7684\u4e8c\u7ef4 \\(dp\\) \u8868\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u8003\u8651\u5b50\u95ee\u9898 \\(dp[i, j]\\) \uff0c\u5176\u5bf9\u5e94\u7684\u4e24\u4e2a\u5b57\u7b26\u4e32\u7684\u5c3e\u90e8\u5b57\u7b26\u4e3a \\(s[i-1]\\) \u548c \\(t[j-1]\\) \uff0c\u53ef\u6839\u636e\u4e0d\u540c\u7f16\u8f91\u64cd\u4f5c\u5206\u4e3a\u56fe 14-29 \u6240\u793a\u7684\u4e09\u79cd\u60c5\u51b5\u3002</p> <ol> <li>\u5728 \\(s[i-1]\\) \u4e4b\u540e\u6dfb\u52a0 \\(t[j-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i, j-1]\\) \u3002</li> <li>\u5220\u9664 \\(s[i-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i-1, j]\\) \u3002</li> <li>\u5c06 \\(s[i-1]\\) \u66ff\u6362\u4e3a \\(t[j-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i-1, j-1]\\) \u3002</li> </ol> <p></p> <p> \u56fe 14-29 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u72b6\u6001\u8f6c\u79fb </p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u5f97\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\\(dp[i, j]\\) \u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(dp[i, j-1]\\)\u3001\\(dp[i-1, j]\\)\u3001\\(dp[i-1, j-1]\\) \u4e09\u8005\u4e2d\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\uff0c\u518d\u52a0\u4e0a\u672c\u6b21\u7684\u7f16\u8f91\u6b65\u6570 \\(1\\) \u3002\u5bf9\u5e94\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ dp[i, j] = \\min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1 \\] <p>\u8bf7\u6ce8\u610f\uff0c\u5f53 \\(s[i-1]\\) \u548c \\(t[j-1]\\) \u76f8\u540c\u65f6\uff0c\u65e0\u987b\u7f16\u8f91\u5f53\u524d\u5b57\u7b26\uff0c\u8fd9\u79cd\u60c5\u51b5\u4e0b\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ dp[i, j] = dp[i-1, j-1] \\] <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5f53\u4e24\u5b57\u7b26\u4e32\u90fd\u4e3a\u7a7a\u65f6\uff0c\u7f16\u8f91\u6b65\u6570\u4e3a \\(0\\) \uff0c\u5373 \\(dp[0, 0] = 0\\) \u3002\u5f53 \\(s\\) \u4e3a\u7a7a\u4f46 \\(t\\) \u4e0d\u4e3a\u7a7a\u65f6\uff0c\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(t\\) \u7684\u957f\u5ea6\uff0c\u5373\u9996\u884c \\(dp[0, j] = j\\) \u3002\u5f53 \\(s\\) \u4e0d\u4e3a\u7a7a\u4f46 \\(t\\) \u4e3a\u7a7a\u65f6\uff0c\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(s\\) \u7684\u957f\u5ea6\uff0c\u5373\u9996\u5217 \\(dp[i, 0] = i\\) \u3002</p> <p>\u89c2\u5bdf\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u89e3 \\(dp[i, j]\\) \u4f9d\u8d56\u5de6\u65b9\u3001\u4e0a\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u89e3\uff0c\u56e0\u6b64\u901a\u8fc7\u4e24\u5c42\u5faa\u73af\u6b63\u5e8f\u904d\u5386\u6574\u4e2a \\(dp\\) \u8868\u5373\u53ef\u3002</p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig edit_distance.py<pre><code>def edit_distance_dp(s: str, t: str) -&gt; int:\n    \"\"\"\u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n, m = len(s), len(t)\n    dp = [[0] * (m + 1) for _ in range(n + 1)]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i in range(1, n + 1):\n        dp[i][0] = i\n    for j in range(1, m + 1):\n        dp[0][j] = j\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in range(1, n + 1):\n        for j in range(1, m + 1):\n            if s[i - 1] == t[j - 1]:\n                # \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1]\n            else:\n                # \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1\n    return dp[n][m]\n</code></pre> edit_distance.cpp<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(m + 1, 0));\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.java<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[][] dp = new int[n + 1][m + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = Math.min(Math.min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.cs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint EditDistanceDP(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[,] dp = new int[n + 1, m + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i, 0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0, j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i, j] = dp[i - 1, j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i, j] = Math.Min(Math.Min(dp[i, j - 1], dp[i - 1, j]), dp[i - 1, j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n, m];\n}\n</code></pre> edit_distance.go<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDP(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, m+1)\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i := 1; i &lt;= n; i++ {\n        dp[i][0] = i\n    }\n    for j := 1; j &lt;= m; j++ {\n        dp[0][j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= m; j++ {\n            if s[i-1] == t[j-1] {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i-1][j-1]\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = MinInt(MinInt(dp[i][j-1], dp[i-1][j]), dp[i-1][j-1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.swift<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDP(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: Array(repeating: 0, count: m + 1), count: n + 1)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i in 1 ... n {\n        dp[i][0] = i\n    }\n    for j in 1 ... m {\n        dp[0][j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in 1 ... n {\n        for j in 1 ... m {\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.js<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDP(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(m + 1).fill(0));\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.ts<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDP(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: m + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.dart<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (_) =&gt; List.filled(m + 1, 0));\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n  for (int i = 1; i &lt;= n; i++) {\n    dp[i][0] = i;\n  }\n  for (int j = 1; j &lt;= m; j++) {\n    dp[0][j] = j;\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n  for (int i = 1; i &lt;= n; i++) {\n    for (int j = 1; j &lt;= m; j++) {\n      if (s[i - 1] == t[j - 1]) {\n        // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n        dp[i][j] = dp[i - 1][j - 1];\n      } else {\n        // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n        dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n      }\n    }\n  }\n  return dp[n][m];\n}\n</code></pre> edit_distance.rs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfn edit_distance_dp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![vec![0; m + 1]; n + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i in 1..=n {\n        dp[i][0] = i as i32;\n    }\n    for j in 1..m {\n        dp[0][j] = j as i32;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in 1..=n {\n        for j in 1..=m {\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] =\n                    std::cmp::min(std::cmp::min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    dp[n][m]\n}\n</code></pre> edit_distance.c<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(char *s, char *t, int n, int m) {\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(m + 1, sizeof(int));\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = myMin(myMin(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    int res = dp[n][m];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfun editDistanceDP(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = Array(n + 1) { IntArray(m + 1) }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (i in 1..n) {\n        dp[i][0] = i\n    }\n    for (j in 1..m) {\n        dp[0][j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (i in 1..n) {\n        for (j in 1..m) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] =\n                    (min(\n                        min(dp[i][j - 1].toDouble(), dp[i - 1][j].toDouble()),\n                        dp[i - 1][j - 1].toDouble()\n                    ) + 1).toInt()\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.rb<pre><code>[class]{}-[func]{edit_distance_dp}\n</code></pre> edit_distance.zig<pre><code>// \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212\nfn editDistanceDP(comptime s: []const u8, comptime t: []const u8) i32 {\n    comptime var n = s.len;\n    comptime var m = t.len;\n    var dp = [_][m + 1]i32{[_]i32{0} ** (m + 1)} ** (n + 1);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (1..n + 1) |i| {\n        dp[i][0] = @intCast(i);\n    }\n    for (1..m + 1) |j| {\n        dp[0][j] = @intCast(j);\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (1..n + 1) |i| {\n        for (1..m + 1) |j| {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = @min(@min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 14-30 \u6240\u793a\uff0c\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u7684\u72b6\u6001\u8f6c\u79fb\u8fc7\u7a0b\u4e0e\u80cc\u5305\u95ee\u9898\u975e\u5e38\u7c7b\u4f3c\uff0c\u90fd\u53ef\u4ee5\u770b\u4f5c\u586b\u5199\u4e00\u4e2a\u4e8c\u7ef4\u7f51\u683c\u7684\u8fc7\u7a0b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-30 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#3","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e \\(dp[i,j]\\) \u662f\u7531\u4e0a\u65b9 \\(dp[i-1, j]\\)\u3001\u5de6\u65b9 \\(dp[i, j-1]\\)\u3001\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \u8f6c\u79fb\u800c\u6765\u7684\uff0c\u800c\u6b63\u5e8f\u904d\u5386\u4f1a\u4e22\u5931\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \uff0c\u5012\u5e8f\u904d\u5386\u65e0\u6cd5\u63d0\u524d\u6784\u5efa \\(dp[i, j-1]\\) \uff0c\u56e0\u6b64\u4e24\u79cd\u904d\u5386\u987a\u5e8f\u90fd\u4e0d\u53ef\u53d6\u3002</p> <p>\u4e3a\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u53d8\u91cf <code>leftup</code> \u6765\u6682\u5b58\u5de6\u4e0a\u65b9\u7684\u89e3 \\(dp[i-1, j-1]\\) \uff0c\u4ece\u800c\u53ea\u9700\u8003\u8651\u5de6\u65b9\u548c\u4e0a\u65b9\u7684\u89e3\u3002\u6b64\u65f6\u7684\u60c5\u51b5\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u76f8\u540c\uff0c\u53ef\u4f7f\u7528\u6b63\u5e8f\u904d\u5386\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig edit_distance.py<pre><code>def edit_distance_dp_comp(s: str, t: str) -&gt; int:\n    \"\"\"\u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n, m = len(s), len(t)\n    dp = [0] * (m + 1)\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in range(1, m + 1):\n        dp[j] = j\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in range(1, n + 1):\n        # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        leftup = dp[0]  # \u6682\u5b58 dp[i-1, j-1]\n        dp[0] += 1\n        # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in range(1, m + 1):\n            temp = dp[j]\n            if s[i - 1] == t[j - 1]:\n                # \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup\n            else:\n                # \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = min(dp[j - 1], dp[j], leftup) + 1\n            leftup = temp  # \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n    return dp[m]\n</code></pre> edit_distance.cpp<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;int&gt; dp(m + 1, 0);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.java<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[] dp = new int[m + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = Math.min(Math.min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.cs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint EditDistanceDPComp(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[] dp = new int[m + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = Math.Min(Math.Min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.go<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDPComp(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([]int, m+1)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j := 1; j &lt;= m; j++ {\n        dp[j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i := 1; i &lt;= n; i++ {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        leftUp := dp[0] // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j := 1; j &lt;= m; j++ {\n            temp := dp[j]\n            if s[i-1] == t[j-1] {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftUp\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = MinInt(MinInt(dp[j-1], dp[j]), leftUp) + 1\n            }\n            leftUp = temp // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.swift<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDPComp(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: 0, count: m + 1)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in 1 ... m {\n        dp[j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in 1 ... n {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        var leftup = dp[0] // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in 1 ... m {\n            let temp = dp[j]\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n            }\n            leftup = temp // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.js<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDPComp(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (let i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        let leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.ts<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDPComp(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (let i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        let leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.dart<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;int&gt; dp = List.filled(m + 1, 0);\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n  for (int j = 1; j &lt;= m; j++) {\n    dp[j] = j;\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n  for (int i = 1; i &lt;= n; i++) {\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n    dp[0] = i;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n    for (int j = 1; j &lt;= m; j++) {\n      int temp = dp[j];\n      if (s[i - 1] == t[j - 1]) {\n        // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n        dp[j] = leftup;\n      } else {\n        // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n        dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n      }\n      leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n    }\n  }\n  return dp[m];\n}\n</code></pre> edit_distance.rs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn edit_distance_dp_comp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![0; m + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in 1..m {\n        dp[j] = j as i32;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in 1..=n {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        let mut leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i as i32;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in 1..=m {\n            let temp = dp[j];\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = std::cmp::min(std::cmp::min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    dp[m]\n}\n</code></pre> edit_distance.c<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(char *s, char *t, int n, int m) {\n    int *dp = calloc(m + 1, sizeof(int));\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = myMin(myMin(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    int res = dp[m];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun editDistanceDPComp(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = IntArray(m + 1)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (j in 1..m) {\n        dp[j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (i in 1..n) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        var leftup = dp[0] // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (j in 1..m) {\n            val temp = dp[j]\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = (min(min(dp[j - 1].toDouble(), dp[j].toDouble()), leftup.toDouble()) + 1).toInt()\n            }\n            leftup = temp // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.rb<pre><code>[class]{}-[func]{edit_distance_dp_comp}\n</code></pre> edit_distance.zig<pre><code>// \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn editDistanceDPComp(comptime s: []const u8, comptime t: []const u8) i32 {\n    comptime var n = s.len;\n    comptime var m = t.len;\n    var dp = [_]i32{0} ** (m + 1);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (1..m + 1) |j| {\n        dp[j] = @intCast(j);\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (1..n + 1) |i| {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        var leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = @intCast(i);\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (1..m + 1) |j| {\n            var temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = @min(@min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/","title":"14.1 \u00a0 \u521d\u63a2\u52a8\u6001\u89c4\u5212","text":"<p>\u52a8\u6001\u89c4\u5212\uff08dynamic programming\uff09\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u7b97\u6cd5\u8303\u5f0f\uff0c\u5b83\u5c06\u4e00\u4e2a\u95ee\u9898\u5206\u89e3\u4e3a\u4e00\u7cfb\u5217\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u5e76\u901a\u8fc7\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\u6765\u907f\u514d\u91cd\u590d\u8ba1\u7b97\uff0c\u4ece\u800c\u5927\u5e45\u63d0\u5347\u65f6\u95f4\u6548\u7387\u3002</p> <p>\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u4ece\u4e00\u4e2a\u7ecf\u5178\u4f8b\u9898\u5165\u624b\uff0c\u5148\u7ed9\u51fa\u5b83\u7684\u66b4\u529b\u56de\u6eaf\u89e3\u6cd5\uff0c\u89c2\u5bdf\u5176\u4e2d\u5305\u542b\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u518d\u9010\u6b65\u5bfc\u51fa\u66f4\u9ad8\u6548\u7684\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u3002</p> <p>\u722c\u697c\u68af</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f</p> <p>\u5982\u56fe 14-1 \u6240\u793a\uff0c\u5bf9\u4e8e\u4e00\u4e2a \\(3\\) \u9636\u697c\u68af\uff0c\u5171\u6709 \\(3\\) \u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\u3002</p> <p></p> <p> \u56fe 14-1 \u00a0 \u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf </p> <p>\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u65b9\u6848\u6570\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u56de\u6eaf\u6765\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u6027\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5c06\u722c\u697c\u68af\u60f3\u8c61\u4e3a\u4e00\u4e2a\u591a\u8f6e\u9009\u62e9\u7684\u8fc7\u7a0b\uff1a\u4ece\u5730\u9762\u51fa\u53d1\uff0c\u6bcf\u8f6e\u9009\u62e9\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u6bcf\u5f53\u5230\u8fbe\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u65b9\u6848\u6570\u91cf\u52a0 \\(1\\) \uff0c\u5f53\u8d8a\u8fc7\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u5176\u526a\u679d\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_backtrack.py<pre><code>def backtrack(choices: list[int], state: int, n: int, res: list[int]) -&gt; int:\n    \"\"\"\u56de\u6eaf\"\"\"\n    # \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if state == n:\n        res[0] += 1\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices:\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if state + choice &gt; n:\n            continue\n        # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res)\n        # \u56de\u9000\n\ndef climbing_stairs_backtrack(n: int) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\uff1a\u56de\u6eaf\"\"\"\n    choices = [1, 2]  # \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    state = 0  # \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    res = [0]  # \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res)\n    return res[0]\n</code></pre> climbing_stairs_backtrack.cpp<pre><code>/* \u56de\u6eaf */\nvoid backtrack(vector&lt;int&gt; &amp;choices, int state, int n, vector&lt;int&gt; &amp;res) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n)\n        res[0]++;\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (auto &amp;choice : choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n)\n            continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n    vector&lt;int&gt; choices = {1, 2}; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    int state = 0;                // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    vector&lt;int&gt; res = {0};        // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.java<pre><code>/* \u56de\u6eaf */\nvoid backtrack(List&lt;Integer&gt; choices, int state, int n, List&lt;Integer&gt; res) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n)\n        res.set(0, res.get(0) + 1);\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (Integer choice : choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n)\n            continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n    List&lt;Integer&gt; choices = Arrays.asList(1, 2); // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n    res.add(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.cs<pre><code>/* \u56de\u6eaf */\nvoid Backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n)\n        res[0]++;\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    foreach (int choice in choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n)\n            continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        Backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint ClimbingStairsBacktrack(int n) {\n    List&lt;int&gt; choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    List&lt;int&gt; res = [0]; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    Backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.go<pre><code>/* \u56de\u6eaf */\nfunc backtrack(choices []int, state, n int, res []int) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if state == n {\n        res[0] = res[0] + 1\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for _, choice := range choices {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if state+choice &gt; n {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state+choice, n, res)\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n int) int {\n    // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    choices := []int{1, 2}\n    // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    state := 0\n    res := make([]int, 1)\n    // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    res[0] = 0\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.swift<pre><code>/* \u56de\u6eaf */\nfunc backtrack(choices: [Int], state: Int, n: Int, res: inout [Int]) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if state == n {\n        res[0] += 1\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if state + choice &gt; n {\n            continue\n        }\n        backtrack(choices: choices, state: state + choice, n: n, res: &amp;res)\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n: Int) -&gt; Int {\n    let choices = [1, 2] // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    let state = 0 // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    var res: [Int] = []\n    res.append(0) // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices: choices, state: state, n: n, res: &amp;res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.js<pre><code>/* \u56de\u6eaf */\nfunction backtrack(choices, state, n, res) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state === n) res.set(0, res.get(0) + 1);\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n) continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n) {\n    const choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    const state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    const res = new Map();\n    res.set(0, 0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.ts<pre><code>/* \u56de\u6eaf */\nfunction backtrack(\n    choices: number[],\n    state: number,\n    n: number,\n    res: Map&lt;0, any&gt;\n): void {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state === n) res.set(0, res.get(0) + 1);\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n) continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n: number): number {\n    const choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    const state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    const res = new Map();\n    res.set(0, 0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.dart<pre><code>/* \u56de\u6eaf */\nvoid backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n  // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n  if (state == n) {\n    res[0]++;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  for (int choice in choices) {\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n    if (state + choice &gt; n) continue;\n    // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n    backtrack(choices, state + choice, n, res);\n    // \u56de\u9000\n  }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n  List&lt;int&gt; choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n  int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n  List&lt;int&gt; res = [];\n  res.add(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n  backtrack(choices, state, n, res);\n  return res[0];\n}\n</code></pre> climbing_stairs_backtrack.rs<pre><code>/* \u56de\u6eaf */\nfn backtrack(choices: &amp;[i32], state: i32, n: i32, res: &amp;mut [i32]) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if state == n {\n        res[0] = res[0] + 1;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for &amp;choice in choices {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if state + choice &gt; n {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfn climbing_stairs_backtrack(n: usize) -&gt; i32 {\n    let choices = vec![1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    let state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    let mut res = Vec::new();\n    res.push(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(&amp;choices, state, n as i32, &amp;mut res);\n    res[0]\n}\n</code></pre> climbing_stairs_backtrack.c<pre><code>/* \u56de\u6eaf */\nvoid backtrack(int *choices, int state, int n, int *res, int len) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n)\n        res[0]++;\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; len; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n)\n            continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res, len);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n    int choices[2] = {1, 2}; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    int state = 0;           // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    int *res = (int *)malloc(sizeof(int));\n    *res = 0; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    int len = sizeof(choices) / sizeof(int);\n    backtrack(choices, state, n, res, len);\n    int result = *res;\n    free(res);\n    return result;\n}\n</code></pre> climbing_stairs_backtrack.kt<pre><code>/* \u56de\u6eaf */\nfun backtrack(\n    choices: List&lt;Int&gt;,\n    state: Int,\n    n: Int,\n    res: MutableList&lt;Int&gt;\n) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n) res[0] = res[0] + 1\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (choice in choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n) continue\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res)\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfun climbingStairsBacktrack(n: Int): Int {\n    val choices = mutableListOf(1, 2) // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    val state = 0 // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    val res = ArrayList&lt;Int&gt;()\n    res.add(0) // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.rb<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{climbing_stairs_backtrack}\n</code></pre> climbing_stairs_backtrack.zig<pre><code>// \u56de\u6eaf\nfn backtrack(choices: []i32, state: i32, n: i32, res: std.ArrayList(i32)) void {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n) {\n        res.items[0] = res.items[0] + 1;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (choices) |choice| {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n// \u722c\u697c\u68af\uff1a\u56de\u6eaf\nfn climbingStairsBacktrack(n: usize) !i32 {\n    var choices = [_]i32{ 1, 2 }; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    var state: i32 = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    var res = std.ArrayList(i32).init(std.heap.page_allocator);\n    defer res.deinit();\n    try res.append(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(&amp;choices, state, @intCast(n), res);\n    return res.items[0];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1411","title":"14.1.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"<p>\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u5e76\u4e0d\u663e\u5f0f\u5730\u5bf9\u95ee\u9898\u8fdb\u884c\u62c6\u89e3\uff0c\u800c\u662f\u5c06\u6c42\u89e3\u95ee\u9898\u770b\u4f5c\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\uff0c\u901a\u8fc7\u8bd5\u63a2\u548c\u526a\u679d\uff0c\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u5c1d\u8bd5\u4ece\u95ee\u9898\u5206\u89e3\u7684\u89d2\u5ea6\u5206\u6790\u8fd9\u9053\u9898\u3002\u8bbe\u722c\u5230\u7b2c \\(i\\) \u9636\u5171\u6709 \\(dp[i]\\) \u79cd\u65b9\u6848\uff0c\u90a3\u4e48 \\(dp[i]\\) \u5c31\u662f\u539f\u95ee\u9898\uff0c\u5176\u5b50\u95ee\u9898\u5305\u62ec\uff1a</p> \\[ dp[i-1], dp[i-2], \\dots, dp[2], dp[1] \\] <p>\u7531\u4e8e\u6bcf\u8f6e\u53ea\u80fd\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u56e0\u6b64\u5f53\u6211\u4eec\u7ad9\u5728\u7b2c \\(i\\) \u9636\u697c\u68af\u4e0a\u65f6\uff0c\u4e0a\u4e00\u8f6e\u53ea\u53ef\u80fd\u7ad9\u5728\u7b2c \\(i - 1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u4e0a\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ea\u80fd\u4ece\u7b2c \\(i -1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u8fc8\u5411\u7b2c \\(i\\) \u9636\u3002</p> <p>\u7531\u6b64\u4fbf\u53ef\u5f97\u51fa\u4e00\u4e2a\u91cd\u8981\u63a8\u8bba\uff1a\u722c\u5230\u7b2c \\(i - 1\\) \u9636\u7684\u65b9\u6848\u6570\u52a0\u4e0a\u722c\u5230\u7b2c \\(i - 2\\) \u9636\u7684\u65b9\u6848\u6570\u5c31\u7b49\u4e8e\u722c\u5230\u7b2c \\(i\\) \u9636\u7684\u65b9\u6848\u6570\u3002\u516c\u5f0f\u5982\u4e0b\uff1a</p> \\[ dp[i] = dp[i-1] + dp[i-2] \\] <p>\u8fd9\u610f\u5473\u7740\u5728\u722c\u697c\u68af\u95ee\u9898\u4e2d\uff0c\u5404\u4e2a\u5b50\u95ee\u9898\u4e4b\u95f4\u5b58\u5728\u9012\u63a8\u5173\u7cfb\uff0c\u539f\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u7531\u5b50\u95ee\u9898\u7684\u89e3\u6784\u5efa\u5f97\u6765\u3002\u56fe 14-2 \u5c55\u793a\u4e86\u8be5\u9012\u63a8\u5173\u7cfb\u3002</p> <p></p> <p> \u56fe 14-2 \u00a0 \u65b9\u6848\u6570\u91cf\u9012\u63a8\u5173\u7cfb </p> <p>\u6211\u4eec\u53ef\u4ee5\u6839\u636e\u9012\u63a8\u516c\u5f0f\u5f97\u5230\u66b4\u529b\u641c\u7d22\u89e3\u6cd5\u3002\u4ee5 \\(dp[n]\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u9012\u5f52\u5730\u5c06\u4e00\u4e2a\u8f83\u5927\u95ee\u9898\u62c6\u89e3\u4e3a\u4e24\u4e2a\u8f83\u5c0f\u95ee\u9898\u7684\u548c\uff0c\u76f4\u81f3\u5230\u8fbe\u6700\u5c0f\u5b50\u95ee\u9898 \\(dp[1]\\) \u548c \\(dp[2]\\) \u65f6\u8fd4\u56de\u3002\u5176\u4e2d\uff0c\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u5373 \\(dp[1] = 1\\)\u3001\\(dp[2] = 2\\) \uff0c\u8868\u793a\u722c\u5230\u7b2c \\(1\\)\u3001\\(2\\) \u9636\u5206\u522b\u6709 \\(1\\)\u3001\\(2\\) \u79cd\u65b9\u6848\u3002</p> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u5b83\u548c\u6807\u51c6\u56de\u6eaf\u4ee3\u7801\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u4f46\u66f4\u52a0\u7b80\u6d01\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dfs.py<pre><code>def dfs(i: int) -&gt; int:\n    \"\"\"\u641c\u7d22\"\"\"\n    # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 or i == 2:\n        return i\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1) + dfs(i - 2)\n    return count\n\ndef climbing_stairs_dfs(n: int) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\uff1a\u641c\u7d22\"\"\"\n    return dfs(n)\n</code></pre> climbing_stairs_dfs.cpp<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.java<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.cs<pre><code>/* \u641c\u7d22 */\nint DFS(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1) + DFS(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint ClimbingStairsDFS(int n) {\n    return DFS(n);\n}\n</code></pre> climbing_stairs_dfs.go<pre><code>/* \u641c\u7d22 */\nfunc dfs(i int) int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfs(i-1) + dfs(i-2)\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunc climbingStairsDFS(n int) int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.swift<pre><code>/* \u641c\u7d22 */\nfunc dfs(i: Int) -&gt; Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1) + dfs(i: i - 2)\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunc climbingStairsDFS(n: Int) -&gt; Int {\n    dfs(i: n)\n}\n</code></pre> climbing_stairs_dfs.js<pre><code>/* \u641c\u7d22 */\nfunction dfs(i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunction climbingStairsDFS(n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.ts<pre><code>/* \u641c\u7d22 */\nfunction dfs(i: number): number {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunction climbingStairsDFS(n: number): number {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.dart<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n  // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n  if (i == 1 || i == 2) return i;\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1) + dfs(i - 2);\n  return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n  return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.rs<pre><code>/* \u641c\u7d22 */\nfn dfs(i: usize) -&gt; i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1) + dfs(i - 2);\n    count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfn climbing_stairs_dfs(n: usize) -&gt; i32 {\n    dfs(n)\n}\n</code></pre> climbing_stairs_dfs.c<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.kt<pre><code>/* \u641c\u7d22 */\nfun dfs(i: Int): Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2) return i\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1) + dfs(i - 2)\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfun climbingStairsDFS(n: Int): Int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.rb<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{climbing_stairs_dfs}\n</code></pre> climbing_stairs_dfs.zig<pre><code>// \u641c\u7d22\nfn dfs(i: usize) i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 or i == 2) {\n        return @intCast(i);\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    var count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n// \u722c\u697c\u68af\uff1a\u641c\u7d22\nfn climbingStairsDFS(comptime n: usize) i32 {\n    return dfs(n);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 14-3 \u5c55\u793a\u4e86\u66b4\u529b\u641c\u7d22\u5f62\u6210\u7684\u9012\u5f52\u6811\u3002\u5bf9\u4e8e\u95ee\u9898 \\(dp[n]\\) \uff0c\u5176\u9012\u5f52\u6811\u7684\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002\u6307\u6570\u9636\u5c5e\u4e8e\u7206\u70b8\u5f0f\u589e\u957f\uff0c\u5982\u679c\u6211\u4eec\u8f93\u5165\u4e00\u4e2a\u6bd4\u8f83\u5927\u7684 \\(n\\) \uff0c\u5219\u4f1a\u9677\u5165\u6f2b\u957f\u7684\u7b49\u5f85\u4e4b\u4e2d\u3002</p> <p></p> <p> \u56fe 14-3 \u00a0 \u722c\u697c\u68af\u5bf9\u5e94\u9012\u5f52\u6811 </p> <p>\u89c2\u5bdf\u56fe 14-3 \uff0c\u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u201c\u91cd\u53e0\u5b50\u95ee\u9898\u201d\u5bfc\u81f4\u7684\u3002\u4f8b\u5982 \\(dp[9]\\) \u88ab\u5206\u89e3\u4e3a \\(dp[8]\\) \u548c \\(dp[7]\\) \uff0c\\(dp[8]\\) \u88ab\u5206\u89e3\u4e3a \\(dp[7]\\) \u548c \\(dp[6]\\) \uff0c\u4e24\u8005\u90fd\u5305\u542b\u5b50\u95ee\u9898 \\(dp[7]\\) \u3002</p> <p>\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5b50\u95ee\u9898\u4e2d\u5305\u542b\u66f4\u5c0f\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5b50\u5b50\u5b59\u5b59\u65e0\u7a77\u5c3d\u4e5f\u3002\u7edd\u5927\u90e8\u5206\u8ba1\u7b97\u8d44\u6e90\u90fd\u6d6a\u8d39\u5728\u8fd9\u4e9b\u91cd\u53e0\u7684\u5b50\u95ee\u9898\u4e0a\u3002</p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1412","title":"14.1.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"<p>\u4e3a\u4e86\u63d0\u5347\u7b97\u6cd5\u6548\u7387\uff0c\u6211\u4eec\u5e0c\u671b\u6240\u6709\u7684\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u6570\u7ec4 <code>mem</code> \u6765\u8bb0\u5f55\u6bcf\u4e2a\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5e76\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u5c06\u91cd\u53e0\u5b50\u95ee\u9898\u526a\u679d\u3002</p> <ol> <li>\u5f53\u9996\u6b21\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u5c06\u5176\u8bb0\u5f55\u81f3 <code>mem[i]</code> \uff0c\u4ee5\u4fbf\u4e4b\u540e\u4f7f\u7528\u3002</li> <li>\u5f53\u518d\u6b21\u9700\u8981\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u4fbf\u53ef\u76f4\u63a5\u4ece <code>mem[i]</code> \u4e2d\u83b7\u53d6\u7ed3\u679c\uff0c\u4ece\u800c\u907f\u514d\u91cd\u590d\u8ba1\u7b97\u8be5\u5b50\u95ee\u9898\u3002</li> </ol> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dfs_mem.py<pre><code>def dfs(i: int, mem: list[int]) -&gt; int:\n    \"\"\"\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n    # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 or i == 2:\n        return i\n    # \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1:\n        return mem[i]\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    # \u8bb0\u5f55 dp[i]\n    mem[i] = count\n    return count\n\ndef climbing_stairs_dfs_mem(n: int) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n    # mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    mem = [-1] * (n + 1)\n    return dfs(n, mem)\n</code></pre> climbing_stairs_dfs_mem.cpp<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, vector&lt;int&gt; &amp;mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    vector&lt;int&gt; mem(n + 1, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.java<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int[] mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    int[] mem = new int[n + 1];\n    Arrays.fill(mem, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.cs<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint DFS(int i, int[] mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1, mem) + DFS(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint ClimbingStairsDFSMem(int n) {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    int[] mem = new int[n + 1];\n    Array.Fill(mem, -1);\n    return DFS(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.go<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc dfsMem(i int, mem []int) int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfsMem(i-1, mem) + dfsMem(i-2, mem)\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc climbingStairsDFSMem(n int) int {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    mem := make([]int, n+1)\n    for i := range mem {\n        mem[i] = -1\n    }\n    return dfsMem(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.swift<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc dfs(i: Int, mem: inout [Int]) -&gt; Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1, mem: &amp;mem) + dfs(i: i - 2, mem: &amp;mem)\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc climbingStairsDFSMem(n: Int) -&gt; Int {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    var mem = Array(repeating: -1, count: n + 1)\n    return dfs(i: n, mem: &amp;mem)\n}\n</code></pre> climbing_stairs_dfs_mem.js<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction dfs(i, mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction climbingStairsDFSMem(n) {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.ts<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction dfs(i: number, mem: number[]): number {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction climbingStairsDFSMem(n: number): number {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.dart<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, List&lt;int&gt; mem) {\n  // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n  if (i == 1 || i == 2) return i;\n  // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n  if (mem[i] != -1) return mem[i];\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n  // \u8bb0\u5f55 dp[i]\n  mem[i] = count;\n  return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n  // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n  List&lt;int&gt; mem = List.filled(n + 1, -1);\n  return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.rs<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn dfs(i: usize, mem: &amp;mut [i32]) -&gt; i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1 {\n        return mem[i];\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn climbing_stairs_dfs_mem(n: usize) -&gt; i32 {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    let mut mem = vec![-1; n + 1];\n    dfs(n, &amp;mut mem)\n}\n</code></pre> climbing_stairs_dfs_mem.c<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int *mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    int *mem = (int *)malloc((n + 1) * sizeof(int));\n    for (int i = 0; i &lt;= n; i++) {\n        mem[i] = -1;\n    }\n    int result = dfs(n, mem);\n    free(mem);\n    return result;\n}\n</code></pre> climbing_stairs_dfs_mem.kt<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfun dfs(i: Int, mem: IntArray): Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2) return i\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) return mem[i]\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfun climbingStairsDFSMem(n: Int): Int {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    val mem = IntArray(n + 1)\n    Arrays.fill(mem, -1)\n    return dfs(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.rb<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{climbing_stairs_dfs_mem}\n</code></pre> climbing_stairs_dfs_mem.zig<pre><code>// \u8bb0\u5fc6\u5316\u641c\u7d22\nfn dfs(i: usize, mem: []i32) i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 or i == 2) {\n        return @intCast(i);\n    }\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) {\n        return mem[i];\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    var count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n// \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn climbingStairsDFSMem(comptime n: usize) i32 {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    var mem = [_]i32{ -1 } ** (n + 1);\n    return dfs(n, &amp;mem);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u89c2\u5bdf\u56fe 14-4 \uff0c\u7ecf\u8fc7\u8bb0\u5fc6\u5316\u5904\u7406\u540e\uff0c\u6240\u6709\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u9700\u8ba1\u7b97\u4e00\u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u8fd9\u662f\u4e00\u4e2a\u5de8\u5927\u7684\u98de\u8dc3\u3002</p> <p></p> <p> \u56fe 14-4 \u00a0 \u8bb0\u5fc6\u5316\u641c\u7d22\u5bf9\u5e94\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1413","title":"14.1.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"<p>\u8bb0\u5fc6\u5316\u641c\u7d22\u662f\u4e00\u79cd\u201c\u4ece\u9876\u81f3\u5e95\u201d\u7684\u65b9\u6cd5\uff1a\u6211\u4eec\u4ece\u539f\u95ee\u9898\uff08\u6839\u8282\u70b9\uff09\u5f00\u59cb\uff0c\u9012\u5f52\u5730\u5c06\u8f83\u5927\u5b50\u95ee\u9898\u5206\u89e3\u4e3a\u8f83\u5c0f\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u89e3\u5df2\u77e5\u7684\u6700\u5c0f\u5b50\u95ee\u9898\uff08\u53f6\u8282\u70b9\uff09\u3002\u4e4b\u540e\uff0c\u901a\u8fc7\u56de\u6eaf\u9010\u5c42\u6536\u96c6\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u6784\u5efa\u51fa\u539f\u95ee\u9898\u7684\u89e3\u3002</p> <p>\u4e0e\u4e4b\u76f8\u53cd\uff0c\u52a8\u6001\u89c4\u5212\u662f\u4e00\u79cd\u201c\u4ece\u5e95\u81f3\u9876\u201d\u7684\u65b9\u6cd5\uff1a\u4ece\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u5f00\u59cb\uff0c\u8fed\u4ee3\u5730\u6784\u5efa\u66f4\u5927\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u76f4\u81f3\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002</p> <p>\u7531\u4e8e\u52a8\u6001\u89c4\u5212\u4e0d\u5305\u542b\u56de\u6eaf\u8fc7\u7a0b\uff0c\u56e0\u6b64\u53ea\u9700\u4f7f\u7528\u5faa\u73af\u8fed\u4ee3\u5b9e\u73b0\uff0c\u65e0\u987b\u4f7f\u7528\u9012\u5f52\u3002\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u521d\u59cb\u5316\u4e00\u4e2a\u6570\u7ec4 <code>dp</code> \u6765\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5b83\u8d77\u5230\u4e86\u4e0e\u8bb0\u5fc6\u5316\u641c\u7d22\u4e2d\u6570\u7ec4 <code>mem</code> \u76f8\u540c\u7684\u8bb0\u5f55\u4f5c\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dp.py<pre><code>def climbing_stairs_dp(n: int) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    if n == 1 or n == 2:\n        return n\n    # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp = [0] * (n + 1)\n    # \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1], dp[2] = 1, 2\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in range(3, n + 1):\n        dp[i] = dp[i - 1] + dp[i - 2]\n    return dp[n]\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    vector&lt;int&gt; dp(n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDP(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp := make([]int, n+1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1\n    dp[2] = 2\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = dp[i-1] + dp[i-2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = Array(repeating: 0, count: n + 1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1\n    dp[2] = 2\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in 3 ... n {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDP(n) {\n    if (n === 1 || n === 2) return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = new Array(n + 1).fill(-1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDP(n: number): number {\n    if (n === 1 || n === 2) return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = new Array(n + 1).fill(-1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n  if (n == 1 || n == 2) return n;\n  // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n  dp[1] = 1;\n  dp[2] = 2;\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = dp[i - 1] + dp[i - 2];\n  }\n  return dp[n];\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_dp(n: usize) -&gt; i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    let mut dp = vec![-1; n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in 3..=n {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    dp[n]\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int *dp = (int *)malloc((n + 1) * sizeof(int));\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    int result = dp[n];\n    free(dp);\n    return result;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfun climbingStairsDP(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    val dp = IntArray(n + 1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1\n    dp[2] = 2\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (i in 3..n) {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>[class]{}-[func]{climbing_stairs_dp}\n</code></pre> climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\nfn climbingStairsDP(comptime n: usize) i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (n == 1 or n == 2) {\n        return @intCast(n);\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = [_]i32{-1} ** (n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (3..n + 1) |i| {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 14-5 \u6a21\u62df\u4e86\u4ee5\u4e0a\u4ee3\u7801\u7684\u6267\u884c\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 14-5 \u00a0 \u722c\u697c\u68af\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u4e0e\u56de\u6eaf\u7b97\u6cd5\u4e00\u6837\uff0c\u52a8\u6001\u89c4\u5212\u4e5f\u4f7f\u7528\u201c\u72b6\u6001\u201d\u6982\u5ff5\u6765\u8868\u793a\u95ee\u9898\u6c42\u89e3\u7684\u7279\u5b9a\u9636\u6bb5\uff0c\u6bcf\u4e2a\u72b6\u6001\u90fd\u5bf9\u5e94\u4e00\u4e2a\u5b50\u95ee\u9898\u4ee5\u53ca\u76f8\u5e94\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002\u4f8b\u5982\uff0c\u722c\u697c\u68af\u95ee\u9898\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u5f53\u524d\u6240\u5728\u697c\u68af\u9636\u6570 \\(i\\) \u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5185\u5bb9\uff0c\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u52a8\u6001\u89c4\u5212\u7684\u5e38\u7528\u672f\u8bed\u3002</p> <ul> <li>\u5c06\u6570\u7ec4 <code>dp</code> \u79f0\u4e3a\\(dp\\)\uff08\u8868\uff09\uff0c\\(dp[i]\\) \u8868\u793a\u72b6\u6001 \\(i\\) \u5bf9\u5e94\u5b50\u95ee\u9898\u7684\u89e3\u3002</li> <li>\u5c06\u6700\u5c0f\u5b50\u95ee\u9898\u5bf9\u5e94\u7684\u72b6\u6001\uff08\u7b2c \\(1\\) \u9636\u548c\u7b2c \\(2\\) \u9636\u697c\u68af\uff09\u79f0\u4e3a\u521d\u59cb\u72b6\u6001\u3002</li> <li>\u5c06\u9012\u63a8\u516c\u5f0f \\(dp[i] = dp[i-1] + dp[i-2]\\) \u79f0\u4e3a\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1414","title":"14.1.4 \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7ec6\u5fc3\u7684\u8bfb\u8005\u53ef\u80fd\u53d1\u73b0\u4e86\uff0c\u7531\u4e8e \\(dp[i]\\) \u53ea\u4e0e \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u65e0\u987b\u4f7f\u7528\u4e00\u4e2a\u6570\u7ec4 <code>dp</code> \u6765\u5b58\u50a8\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u800c\u53ea\u9700\u4e24\u4e2a\u53d8\u91cf\u6eda\u52a8\u524d\u8fdb\u5373\u53ef\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig climbing_stairs_dp.py<pre><code>def climbing_stairs_dp_comp(n: int) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    if n == 1 or n == 2:\n        return n\n    a, b = 1, 2\n    for _ in range(3, n + 1):\n        a, b = b, a + b\n    return b\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDPComp(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    a, b := 1, 2\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i := 3; i &lt;= n; i++ {\n        a, b = b, a+b\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDPComp(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    var a = 1\n    var b = 2\n    for _ in 3 ... n {\n        (a, b) = (b, a + b)\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDPComp(n) {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDPComp(n: number): number {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n  if (n == 1 || n == 2) return n;\n  int a = 1, b = 2;\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = a + b;\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_dp_comp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    let (mut a, mut b) = (1, 2);\n    for _ in 3..=n {\n        let tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    b\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun climbingStairsDPComp(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    var a = 1\n    var b = 2\n    for (i in 3..n) {\n        val tmp = b\n        b += a\n        a = tmp\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>[class]{}-[func]{climbing_stairs_dp_comp}\n</code></pre> climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn climbingStairsDPComp(comptime n: usize) i32 {\n    if (n == 1 or n == 2) {\n        return @intCast(n);\n    }\n    var a: i32 = 1;\n    var b: i32 = 2;\n    for (3..n + 1) |_| {\n        var tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u7531\u4e8e\u7701\u53bb\u4e86\u6570\u7ec4 <code>dp</code> \u5360\u7528\u7684\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002</p> <p>\u5728\u52a8\u6001\u89c4\u5212\u95ee\u9898\u4e2d\uff0c\u5f53\u524d\u72b6\u6001\u5f80\u5f80\u4ec5\u4e0e\u524d\u9762\u6709\u9650\u4e2a\u72b6\u6001\u6709\u5173\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u53ea\u4fdd\u7559\u5fc5\u8981\u7684\u72b6\u6001\uff0c\u901a\u8fc7\u201c\u964d\u7ef4\u201d\u6765\u8282\u7701\u5185\u5b58\u7a7a\u95f4\u3002\u8fd9\u79cd\u7a7a\u95f4\u4f18\u5316\u6280\u5de7\u88ab\u79f0\u4e3a\u201c\u6eda\u52a8\u53d8\u91cf\u201d\u6216\u201c\u6eda\u52a8\u6570\u7ec4\u201d\u3002</p>"},{"location":"chapter_dynamic_programming/knapsack_problem/","title":"14.4 \u00a0 0-1 \u80cc\u5305\u95ee\u9898","text":"<p>\u80cc\u5305\u95ee\u9898\u662f\u4e00\u4e2a\u975e\u5e38\u597d\u7684\u52a8\u6001\u89c4\u5212\u5165\u95e8\u9898\u76ee\uff0c\u662f\u52a8\u6001\u89c4\u5212\u4e2d\u6700\u5e38\u89c1\u7684\u95ee\u9898\u5f62\u5f0f\u3002\u5176\u5177\u6709\u5f88\u591a\u53d8\u79cd\uff0c\u4f8b\u5982 0-1 \u80cc\u5305\u95ee\u9898\u3001\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u3001\u591a\u91cd\u80cc\u5305\u95ee\u9898\u7b49\u3002</p> <p>\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5148\u6765\u6c42\u89e3\u6700\u5e38\u89c1\u7684 0-1 \u80cc\u5305\u95ee\u9898\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ea\u80fd\u9009\u62e9\u4e00\u6b21\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002</p> <p>\u89c2\u5bdf\u56fe 14-17 \uff0c\u7531\u4e8e\u7269\u54c1\u7f16\u53f7 \\(i\\) \u4ece \\(1\\) \u5f00\u59cb\u8ba1\u6570\uff0c\u6570\u7ec4\u7d22\u5f15\u4ece \\(0\\) \u5f00\u59cb\u8ba1\u6570\uff0c\u56e0\u6b64\u7269\u54c1 \\(i\\) \u5bf9\u5e94\u91cd\u91cf \\(wgt[i-1]\\) \u548c\u4ef7\u503c \\(val[i-1]\\) \u3002</p> <p></p> <p> \u56fe 14-17 \u00a0 0-1 \u80cc\u5305\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u6211\u4eec\u53ef\u4ee5\u5c06 0-1 \u80cc\u5305\u95ee\u9898\u770b\u4f5c\u4e00\u4e2a\u7531 \\(n\\) \u8f6e\u51b3\u7b56\u7ec4\u6210\u7684\u8fc7\u7a0b\uff0c\u5bf9\u4e8e\u6bcf\u4e2a\u7269\u4f53\u90fd\u6709\u4e0d\u653e\u5165\u548c\u653e\u5165\u4e24\u79cd\u51b3\u7b56\uff0c\u56e0\u6b64\u8be5\u95ee\u9898\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\u3002</p> <p>\u8be5\u95ee\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u201c\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u201d\uff0c\u56e0\u6b64\u8f83\u5927\u6982\u7387\u662f\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u95ee\u9898\u3002</p> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u5bf9\u4e8e\u6bcf\u4e2a\u7269\u54c1\u6765\u8bf4\uff0c\u4e0d\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u4e0d\u53d8\uff1b\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u51cf\u5c0f\u3002\u7531\u6b64\u53ef\u5f97\u72b6\u6001\u5b9a\u4e49\uff1a\u5f53\u524d\u7269\u54c1\u7f16\u53f7 \\(i\\) \u548c\u5269\u4f59\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u8bb0\u4e3a \\([i, c]\\) \u3002</p> <p>\u72b6\u6001 \\([i, c]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u524d \\(i\\) \u4e2a\u7269\u54c1\u5728\u5269\u4f59\u5bb9\u91cf\u4e3a \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u4ef7\u503c\uff0c\u8bb0\u4e3a \\(dp[i, c]\\) \u3002</p> <p>\u5f85\u6c42\u89e3\u7684\u662f \\(dp[n, cap]\\) \uff0c\u56e0\u6b64\u9700\u8981\u4e00\u4e2a\u5c3a\u5bf8\u4e3a \\((n+1) \\times (cap+1)\\) \u7684\u4e8c\u7ef4 \\(dp\\) \u8868\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u5f53\u6211\u4eec\u505a\u51fa\u7269\u54c1 \\(i\\) \u7684\u51b3\u7b56\u540e\uff0c\u5269\u4f59\u7684\u662f\u524d \\(i-1\\) \u4e2a\u7269\u54c1\u7684\u51b3\u7b56\uff0c\u53ef\u5206\u4e3a\u4ee5\u4e0b\u4e24\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u4e0d\u53d8\uff0c\u72b6\u6001\u53d8\u5316\u4e3a \\([i-1, c]\\) \u3002</li> <li>\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u51cf\u5c11 \\(wgt[i-1]\\) \uff0c\u4ef7\u503c\u589e\u52a0 \\(val[i-1]\\) \uff0c\u72b6\u6001\u53d8\u5316\u4e3a \\([i-1, c-wgt[i-1]]\\) \u3002</li> </ul> <p>\u4e0a\u8ff0\u5206\u6790\u5411\u6211\u4eec\u63ed\u793a\u4e86\u672c\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u6700\u5927\u4ef7\u503c \\(dp[i, c]\\) \u7b49\u4e8e\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \u548c\u653e\u5165\u7269\u54c1 \\(i\\) \u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\u3002\u7531\u6b64\u53ef\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1]) \\] <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u82e5\u5f53\u524d\u7269\u54c1\u91cd\u91cf \\(wgt[i - 1]\\) \u8d85\u51fa\u5269\u4f59\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\u3002</p> <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5f53\u65e0\u7269\u54c1\u6216\u65e0\u5269\u4f59\u80cc\u5305\u5bb9\u91cf\u65f6\u6700\u5927\u4ef7\u503c\u4e3a \\(0\\) \uff0c\u5373\u9996\u5217 \\(dp[i, 0]\\) \u548c\u9996\u884c \\(dp[0, c]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002</p> <p>\u5f53\u524d\u72b6\u6001 \\([i, c]\\) \u4ece\u4e0a\u65b9\u7684\u72b6\u6001 \\([i-1, c]\\) \u548c\u5de6\u4e0a\u65b9\u7684\u72b6\u6001 \\([i-1, c-wgt[i-1]]\\) \u8f6c\u79fb\u800c\u6765\uff0c\u56e0\u6b64\u901a\u8fc7\u4e24\u5c42\u5faa\u73af\u6b63\u5e8f\u904d\u5386\u6574\u4e2a \\(dp\\) \u8868\u5373\u53ef\u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u4eec\u63a5\u4e0b\u6765\u6309\u987a\u5e8f\u5b9e\u73b0\u66b4\u529b\u641c\u7d22\u3001\u8bb0\u5fc6\u5316\u641c\u7d22\u3001\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u3002</p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"<p>\u641c\u7d22\u4ee3\u7801\u5305\u542b\u4ee5\u4e0b\u8981\u7d20\u3002</p> <ul> <li>\u9012\u5f52\u53c2\u6570\uff1a\u72b6\u6001 \\([i, c]\\) \u3002</li> <li>\u8fd4\u56de\u503c\uff1a\u5b50\u95ee\u9898\u7684\u89e3 \\(dp[i, c]\\) \u3002</li> <li>\u7ec8\u6b62\u6761\u4ef6\uff1a\u5f53\u7269\u54c1\u7f16\u53f7\u8d8a\u754c \\(i = 0\\) \u6216\u80cc\u5305\u5269\u4f59\u5bb9\u91cf\u4e3a \\(0\\) \u65f6\uff0c\u7ec8\u6b62\u9012\u5f52\u5e76\u8fd4\u56de\u4ef7\u503c \\(0\\) \u3002</li> <li>\u526a\u679d\uff1a\u82e5\u5f53\u524d\u7269\u54c1\u91cd\u91cf\u8d85\u51fa\u80cc\u5305\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dfs(wgt: list[int], val: list[int], i: int, c: int) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22\"\"\"\n    # \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 or c == 0:\n        return 0\n    # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs(wgt, val, i - 1, c)\n    # \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    no = knapsack_dfs(wgt, val, i - 1, c)\n    yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return max(no, yes)\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return max(no, yes);\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(int[] wgt, int[] val, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint KnapsackDFS(int[] weight, int[] val, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFS(weight, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = KnapsackDFS(weight, val, i - 1, c);\n    int yes = KnapsackDFS(weight, val, i - 1, c - weight[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return Math.Max(no, yes);\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc knapsackDFS(wgt, val []int, i, c int) int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i-1] &gt; c {\n        return knapsackDFS(wgt, val, i-1, c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    no := knapsackDFS(wgt, val, i-1, c)\n    yes := knapsackDFS(wgt, val, i-1, c-wgt[i-1]) + val[i-1]\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return int(math.Max(float64(no), float64(yes)))\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc knapsackDFS(wgt: [Int], val: [Int], i: Int, c: Int) -&gt; Int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c {\n        return knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    let no = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    let yes = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return max(no, yes)\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction knapsackDFS(wgt, val, i, c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction knapsackDFS(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    i: number,\n    c: number\n): number {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(List&lt;int&gt; wgt, List&lt;int&gt; val, int i, int c) {\n  // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFS(wgt, val, i - 1, c);\n  }\n  // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n  int no = knapsackDFS(wgt, val, i - 1, c);\n  int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n  return max(no, yes);\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfn knapsack_dfs(wgt: &amp;[i32], val: &amp;[i32], i: usize, c: usize) -&gt; i32 {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    let no = knapsack_dfs(wgt, val, i - 1, c);\n    let yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    std::cmp::max(no, yes)\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(int wgt[], int val[], int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return myMax(no, yes);\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfun knapsackDFS(\n    wgt: IntArray,\n    value: IntArray,\n    i: Int,\n    c: Int\n): Int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, value, i - 1, c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    val no = knapsackDFS(wgt, value, i - 1, c)\n    val yes = knapsackDFS(wgt, value, i - 1, c - wgt[i - 1]) + value[i - 1]\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return max(no.toDouble(), yes.toDouble()).toInt()\n}\n</code></pre> knapsack.rb<pre><code>[class]{}-[func]{knapsack_dfs}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22\nfn knapsackDFS(wgt: []i32, val: []i32, i: usize, c: usize) i32 {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 or c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    var no = knapsackDFS(wgt, val, i - 1, c);\n    var yes = knapsackDFS(wgt, val, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return @max(no, yes);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 14-18 \u6240\u793a\uff0c\u7531\u4e8e\u6bcf\u4e2a\u7269\u54c1\u90fd\u4f1a\u4ea7\u751f\u4e0d\u9009\u548c\u9009\u4e24\u6761\u641c\u7d22\u5206\u652f\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002</p> <p>\u89c2\u5bdf\u9012\u5f52\u6811\uff0c\u5bb9\u6613\u53d1\u73b0\u5176\u4e2d\u5b58\u5728\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u4f8b\u5982 \\(dp[1, 10]\\) \u7b49\u3002\u800c\u5f53\u7269\u54c1\u8f83\u591a\u3001\u80cc\u5305\u5bb9\u91cf\u8f83\u5927\uff0c\u5c24\u5176\u662f\u76f8\u540c\u91cd\u91cf\u7684\u7269\u54c1\u8f83\u591a\u65f6\uff0c\u91cd\u53e0\u5b50\u95ee\u9898\u7684\u6570\u91cf\u5c06\u4f1a\u5927\u5e45\u589e\u591a\u3002</p> <p></p> <p> \u56fe 14-18 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u66b4\u529b\u641c\u7d22\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"<p>\u4e3a\u4e86\u4fdd\u8bc1\u91cd\u53e0\u5b50\u95ee\u9898\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\uff0c\u6211\u4eec\u501f\u52a9\u8bb0\u5fc6\u5217\u8868 <code>mem</code> \u6765\u8bb0\u5f55\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5176\u4e2d <code>mem[i][c]</code> \u5bf9\u5e94 \\(dp[i, c]\\) \u3002</p> <p>\u5f15\u5165\u8bb0\u5fc6\u5316\u4e4b\u540e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u5b50\u95ee\u9898\u6570\u91cf\uff0c\u4e5f\u5c31\u662f \\(O(n \\times cap)\\) \u3002\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dfs_mem(\n    wgt: list[int], val: list[int], mem: list[list[int]], i: int, c: int\n) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n    # \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 or c == 0:\n        return 0\n    # \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1:\n        return mem[i][c]\n    # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    # \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(int[] wgt, int[] val, int[][] mem, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint KnapsackDFSMem(int[] weight, int[] val, int[][] mem, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFSMem(weight, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = KnapsackDFSMem(weight, val, mem, i - 1, c);\n    int yes = KnapsackDFSMem(weight, val, mem, i - 1, c - weight[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = Math.Max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc knapsackDFSMem(wgt, val []int, mem [][]int, i, c int) int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i-1] &gt; c {\n        return knapsackDFSMem(wgt, val, mem, i-1, c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    no := knapsackDFSMem(wgt, val, mem, i-1, c)\n    yes := knapsackDFSMem(wgt, val, mem, i-1, c-wgt[i-1]) + val[i-1]\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = int(math.Max(float64(no), float64(yes)))\n    return mem[i][c]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc knapsackDFSMem(wgt: [Int], val: [Int], mem: inout [[Int]], i: Int, c: Int) -&gt; Int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c {\n        return knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    let no = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    let yes = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction knapsackDFSMem(wgt, val, mem, i, c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction knapsackDFSMem(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    c: number\n): number {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(\n  List&lt;int&gt; wgt,\n  List&lt;int&gt; val,\n  List&lt;List&lt;int&gt;&gt; mem,\n  int i,\n  int c,\n) {\n  // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n  if (mem[i][c] != -1) {\n    return mem[i][c];\n  }\n  // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFSMem(wgt, val, mem, i - 1, c);\n  }\n  // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n  int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n  int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n  mem[i][c] = max(no, yes);\n  return mem[i][c];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn knapsack_dfs_mem(wgt: &amp;[i32], val: &amp;[i32], mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: usize, c: usize) -&gt; i32 {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1 {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    let no = knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    let yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = std::cmp::max(no, yes);\n    mem[i][c]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(int wgt[], int val[], int memCols, int **mem, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = myMax(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfun knapsackDFSMem(\n    wgt: IntArray,\n    value: IntArray,\n    mem: Array&lt;IntArray&gt;,\n    i: Int,\n    c: Int\n): Int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c]\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, value, mem, i - 1, c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    val no = knapsackDFSMem(wgt, value, mem, i - 1, c)\n    val yes = knapsackDFSMem(wgt, value, mem, i - 1, c - wgt[i - 1]) + value[i - 1]\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = max(no.toDouble(), yes.toDouble()).toInt()\n    return mem[i][c]\n}\n</code></pre> knapsack.rb<pre><code>[class]{}-[func]{knapsack_dfs_mem}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn knapsackDFSMem(wgt: []i32, val: []i32, mem: anytype, i: usize, c: usize) i32 {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 or c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    var no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    var yes = knapsackDFSMem(wgt, val, mem, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = @max(no, yes);\n    return mem[i][c];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 14-19 \u5c55\u793a\u4e86\u5728\u8bb0\u5fc6\u5316\u641c\u7d22\u4e2d\u88ab\u526a\u6389\u7684\u641c\u7d22\u5206\u652f\u3002</p> <p></p> <p> \u56fe 14-19 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u8bb0\u5fc6\u5316\u641c\u7d22\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"<p>\u52a8\u6001\u89c4\u5212\u5b9e\u8d28\u4e0a\u5c31\u662f\u5728\u72b6\u6001\u8f6c\u79fb\u4e2d\u586b\u5145 \\(dp\\) \u8868\u7684\u8fc7\u7a0b\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint KnapsackDP(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (weight[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i, c] = Math.Max(dp[i - 1, c - weight[i - 1]] + val[i - 1], dp[i - 1, c]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i-1][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array(n + 1)\n        .fill(0)\n        .map(() =&gt; Array(cap + 1).fill(0));\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n        dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfn knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = std::cmp::max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1] as usize] + val[i - 1],\n                );\n            }\n        }\n    }\n    dp[n][cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = myMax(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfun knapsackDP(\n    wgt: IntArray,\n    value: IntArray,\n    cap: Int\n): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c].toDouble(), (dp[i - 1][c - wgt[i - 1]] + value[i - 1]).toDouble())\n                    .toInt()\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.rb<pre><code>[class]{}-[func]{knapsack_dp}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\nfn knapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    comptime var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..cap + 1) |c| {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = @max(dp[i - 1][c], dp[i - 1][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 14-20 \u6240\u793a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u7531\u6570\u7ec4 <code>dp</code> \u5927\u5c0f\u51b3\u5b9a\uff0c\u5373 \\(O(n \\times cap)\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-20 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#4","title":"4. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e\u6bcf\u4e2a\u72b6\u6001\u90fd\u53ea\u4e0e\u5176\u4e0a\u4e00\u884c\u7684\u72b6\u6001\u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e24\u4e2a\u6570\u7ec4\u6eda\u52a8\u524d\u8fdb\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \u3002</p> <p>\u8fdb\u4e00\u6b65\u601d\u8003\uff0c\u6211\u4eec\u80fd\u5426\u4ec5\u7528\u4e00\u4e2a\u6570\u7ec4\u5b9e\u73b0\u7a7a\u95f4\u4f18\u5316\u5462\uff1f\u89c2\u5bdf\u53ef\u77e5\uff0c\u6bcf\u4e2a\u72b6\u6001\u90fd\u662f\u7531\u6b63\u4e0a\u65b9\u6216\u5de6\u4e0a\u65b9\u7684\u683c\u5b50\u8f6c\u79fb\u8fc7\u6765\u7684\u3002\u5047\u8bbe\u53ea\u6709\u4e00\u4e2a\u6570\u7ec4\uff0c\u5f53\u5f00\u59cb\u904d\u5386\u7b2c \\(i\\) \u884c\u65f6\uff0c\u8be5\u6570\u7ec4\u5b58\u50a8\u7684\u4ecd\u7136\u662f\u7b2c \\(i-1\\) \u884c\u7684\u72b6\u6001\u3002</p> <ul> <li>\u5982\u679c\u91c7\u53d6\u6b63\u5e8f\u904d\u5386\uff0c\u90a3\u4e48\u904d\u5386\u5230 \\(dp[i, j]\\) \u65f6\uff0c\u5de6\u4e0a\u65b9 \\(dp[i-1, 1]\\) ~ \\(dp[i-1, j-1]\\) \u503c\u53ef\u80fd\u5df2\u7ecf\u88ab\u8986\u76d6\uff0c\u6b64\u65f6\u5c31\u65e0\u6cd5\u5f97\u5230\u6b63\u786e\u7684\u72b6\u6001\u8f6c\u79fb\u7ed3\u679c\u3002</li> <li>\u5982\u679c\u91c7\u53d6\u5012\u5e8f\u904d\u5386\uff0c\u5219\u4e0d\u4f1a\u53d1\u751f\u8986\u76d6\u95ee\u9898\uff0c\u72b6\u6001\u8f6c\u79fb\u53ef\u4ee5\u6b63\u786e\u8fdb\u884c\u3002</li> </ul> <p>\u56fe 14-21 \u5c55\u793a\u4e86\u5728\u5355\u4e2a\u6570\u7ec4\u4e0b\u4ece\u7b2c \\(i = 1\\) \u884c\u8f6c\u6362\u81f3\u7b2c \\(i = 2\\) \u884c\u7684\u8fc7\u7a0b\u3002\u8bf7\u601d\u8003\u6b63\u5e8f\u904d\u5386\u548c\u5012\u5e8f\u904d\u5386\u7684\u533a\u522b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-21 \u00a0 0-1 \u80cc\u5305\u7684\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u4ec5\u9700\u5c06\u6570\u7ec4 <code>dp</code> \u7684\u7b2c\u4e00\u7ef4 \\(i\\) \u76f4\u63a5\u5220\u9664\uff0c\u5e76\u4e14\u628a\u5185\u5faa\u73af\u66f4\u6539\u4e3a\u5012\u5e8f\u904d\u5386\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig knapsack.py<pre><code>def knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * (cap + 1)\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        # \u5012\u5e8f\u904d\u5386\n        for c in range(cap, 0, -1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint KnapsackDPComp(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (int c = cap; c &gt; 0; c--) {\n            if (weight[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.Max(dp[c], dp[c - weight[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, cap+1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        // \u5012\u5e8f\u904d\u5386\n        for c := cap; c &gt;= 1; c-- {\n            if wgt[i-1] &lt;= c {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: cap + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1 ... n {\n        // \u5012\u5e8f\u904d\u5386\n        for c in (1 ... cap).reversed() {\n            if wgt[i - 1] &lt;= c {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array(cap + 1).fill(0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array(cap + 1).fill(0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    // \u5012\u5e8f\u904d\u5386\n    for (int c = cap; c &gt;= 1; c--) {\n      if (wgt[i - 1] &lt;= c) {\n        // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        // \u5012\u5e8f\u904d\u5386\n        for c in (1..=cap).rev() {\n            if wgt[i - 1] &lt;= c as i32 {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(cap + 1, sizeof(int));\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun knapsackDPComp(\n    wgt: IntArray,\n    value: IntArray,\n    cap: Int\n): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(cap + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for (i in 1..n) {\n        // \u5012\u5e8f\u904d\u5386\n        for (c in cap downTo 1) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] =\n                    max(dp[c].toDouble(), (dp[c - wgt[i - 1]] + value[i - 1]).toDouble()).toInt()\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.rb<pre><code>[class]{}-[func]{knapsack_dp_comp}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn knapsackDPComp(wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (cap + 1);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        // \u5012\u5e8f\u904d\u5386\n        var c = cap;\n        while (c &gt; 0) : (c -= 1) {\n            if (wgt[i - 1] &lt; c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/summary/","title":"14.7 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u52a8\u6001\u89c4\u5212\u5bf9\u95ee\u9898\u8fdb\u884c\u5206\u89e3\uff0c\u5e76\u901a\u8fc7\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\u6765\u89c4\u907f\u91cd\u590d\u8ba1\u7b97\uff0c\u63d0\u9ad8\u8ba1\u7b97\u6548\u7387\u3002</li> <li>\u4e0d\u8003\u8651\u65f6\u95f4\u7684\u524d\u63d0\u4e0b\uff0c\u6240\u6709\u52a8\u6001\u89c4\u5212\u95ee\u9898\u90fd\u53ef\u4ee5\u7528\u56de\u6eaf\uff08\u66b4\u529b\u641c\u7d22\uff09\u8fdb\u884c\u6c42\u89e3\uff0c\u4f46\u9012\u5f52\u6811\u4e2d\u5b58\u5728\u5927\u91cf\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u6548\u7387\u6781\u4f4e\u3002\u901a\u8fc7\u5f15\u5165\u8bb0\u5fc6\u5316\u5217\u8868\uff0c\u53ef\u4ee5\u5b58\u50a8\u6240\u6709\u8ba1\u7b97\u8fc7\u7684\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u4ece\u800c\u4fdd\u8bc1\u91cd\u53e0\u5b50\u95ee\u9898\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\u3002</li> <li>\u8bb0\u5fc6\u5316\u641c\u7d22\u662f\u4e00\u79cd\u4ece\u9876\u81f3\u5e95\u7684\u9012\u5f52\u5f0f\u89e3\u6cd5\uff0c\u800c\u4e0e\u4e4b\u5bf9\u5e94\u7684\u52a8\u6001\u89c4\u5212\u662f\u4e00\u79cd\u4ece\u5e95\u81f3\u9876\u7684\u9012\u63a8\u5f0f\u89e3\u6cd5\uff0c\u5176\u5982\u540c\u201c\u586b\u5199\u8868\u683c\u201d\u4e00\u6837\u3002\u7531\u4e8e\u5f53\u524d\u72b6\u6001\u4ec5\u4f9d\u8d56\u67d0\u4e9b\u5c40\u90e8\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u6d88\u9664 \\(dp\\) \u8868\u7684\u4e00\u4e2a\u7ef4\u5ea6\uff0c\u4ece\u800c\u964d\u4f4e\u7a7a\u95f4\u590d\u6742\u5ea6\u3002</li> <li>\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u901a\u7528\u7684\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u3001\u56de\u6eaf\u4e2d\u5177\u6709\u4e0d\u540c\u7684\u6027\u8d28\u3002</li> <li>\u52a8\u6001\u89c4\u5212\u95ee\u9898\u6709\u4e09\u5927\u7279\u6027\uff1a\u91cd\u53e0\u5b50\u95ee\u9898\u3001\u6700\u4f18\u5b50\u7ed3\u6784\u3001\u65e0\u540e\u6548\u6027\u3002</li> <li>\u5982\u679c\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u53ef\u4ee5\u4ece\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u5efa\u5f97\u6765\uff0c\u5219\u5b83\u5c31\u5177\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u3002</li> <li>\u65e0\u540e\u6548\u6027\u6307\u5bf9\u4e8e\u4e00\u4e2a\u72b6\u6001\uff0c\u5176\u672a\u6765\u53d1\u5c55\u53ea\u4e0e\u8be5\u72b6\u6001\u6709\u5173\uff0c\u800c\u4e0e\u8fc7\u53bb\u7ecf\u5386\u7684\u6240\u6709\u72b6\u6001\u65e0\u5173\u3002\u8bb8\u591a\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u4e0d\u5177\u6709\u65e0\u540e\u6548\u6027\uff0c\u65e0\u6cd5\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u5feb\u901f\u6c42\u89e3\u3002</li> </ul> <p>\u80cc\u5305\u95ee\u9898</p> <ul> <li>\u80cc\u5305\u95ee\u9898\u662f\u6700\u5178\u578b\u7684\u52a8\u6001\u89c4\u5212\u95ee\u9898\u4e4b\u4e00\uff0c\u5177\u6709 0-1 \u80cc\u5305\u3001\u5b8c\u5168\u80cc\u5305\u3001\u591a\u91cd\u80cc\u5305\u7b49\u53d8\u79cd\u3002</li> <li>0-1 \u80cc\u5305\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u524d \\(i\\) \u4e2a\u7269\u54c1\u5728\u5269\u4f59\u5bb9\u91cf\u4e3a \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u4ef7\u503c\u3002\u6839\u636e\u4e0d\u653e\u5165\u80cc\u5305\u548c\u653e\u5165\u80cc\u5305\u4e24\u79cd\u51b3\u7b56\uff0c\u53ef\u5f97\u5230\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5e76\u6784\u5efa\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002\u5728\u7a7a\u95f4\u4f18\u5316\u4e2d\uff0c\u7531\u4e8e\u6bcf\u4e2a\u72b6\u6001\u4f9d\u8d56\u6b63\u4e0a\u65b9\u548c\u5de6\u4e0a\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u9700\u8981\u5012\u5e8f\u904d\u5386\u5217\u8868\uff0c\u907f\u514d\u5de6\u4e0a\u65b9\u72b6\u6001\u88ab\u8986\u76d6\u3002</li> <li>\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u6bcf\u79cd\u7269\u54c1\u7684\u9009\u53d6\u6570\u91cf\u65e0\u9650\u5236\uff0c\u56e0\u6b64\u9009\u62e9\u653e\u5165\u7269\u54c1\u7684\u72b6\u6001\u8f6c\u79fb\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u4e0d\u540c\u3002\u7531\u4e8e\u72b6\u6001\u4f9d\u8d56\u6b63\u4e0a\u65b9\u548c\u6b63\u5de6\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u5728\u7a7a\u95f4\u4f18\u5316\u4e2d\u5e94\u5f53\u6b63\u5e8f\u904d\u5386\u3002</li> <li>\u96f6\u94b1\u5151\u6362\u95ee\u9898\u662f\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u4e00\u4e2a\u53d8\u79cd\u3002\u5b83\u4ece\u6c42\u201c\u6700\u5927\u201d\u4ef7\u503c\u53d8\u4e3a\u6c42\u201c\u6700\u5c0f\u201d\u786c\u5e01\u6570\u91cf\uff0c\u56e0\u6b64\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\max()\\) \u5e94\u6539\u4e3a \\(\\min()\\) \u3002\u4ece\u8ffd\u6c42\u201c\u4e0d\u8d85\u8fc7\u201d\u80cc\u5305\u5bb9\u91cf\u5230\u8ffd\u6c42\u201c\u6070\u597d\u201d\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u4f7f\u7528 \\(amt + 1\\) \u6765\u8868\u793a\u201c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u201d\u7684\u65e0\u6548\u89e3\u3002</li> <li>\u96f6\u94b1\u5151\u6362\u95ee\u9898 II \u4ece\u6c42\u201c\u6700\u5c11\u786c\u5e01\u6570\u91cf\u201d\u6539\u4e3a\u6c42\u201c\u786c\u5e01\u7ec4\u5408\u6570\u91cf\u201d\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u76f8\u5e94\u5730\u4ece \\(\\min()\\) \u6539\u4e3a\u6c42\u548c\u8fd0\u7b97\u7b26\u3002</li> </ul> <p>\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898</p> <ul> <li>\u7f16\u8f91\u8ddd\u79bb\uff08Levenshtein \u8ddd\u79bb\uff09\u7528\u4e8e\u8861\u91cf\u4e24\u4e2a\u5b57\u7b26\u4e32\u4e4b\u95f4\u7684\u76f8\u4f3c\u5ea6\uff0c\u5176\u5b9a\u4e49\u4e3a\u4ece\u4e00\u4e2a\u5b57\u7b26\u4e32\u5230\u53e6\u4e00\u4e2a\u5b57\u7b26\u4e32\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\uff0c\u7f16\u8f91\u64cd\u4f5c\u5305\u62ec\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66ff\u6362\u3002</li> <li>\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u5c06 \\(s\\) \u7684\u524d \\(i\\) \u4e2a\u5b57\u7b26\u66f4\u6539\u4e3a \\(t\\) \u7684\u524d \\(j\\) \u4e2a\u5b57\u7b26\u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002\u5f53 \\(s[i] \\ne t[j]\\) \u65f6\uff0c\u5177\u6709\u4e09\u79cd\u51b3\u7b56\uff1a\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66ff\u6362\uff0c\u5b83\u4eec\u90fd\u6709\u76f8\u5e94\u7684\u5269\u4f59\u5b50\u95ee\u9898\u3002\u636e\u6b64\u4fbf\u53ef\u4ee5\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\u4e0e\u6784\u5efa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002\u800c\u5f53 \\(s[i] = t[j]\\) \u65f6\uff0c\u65e0\u987b\u7f16\u8f91\u5f53\u524d\u5b57\u7b26\u3002</li> <li>\u5728\u7f16\u8f91\u8ddd\u79bb\u4e2d\uff0c\u72b6\u6001\u4f9d\u8d56\u5176\u6b63\u4e0a\u65b9\u3001\u6b63\u5de6\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u7a7a\u95f4\u4f18\u5316\u540e\u6b63\u5e8f\u6216\u5012\u5e8f\u904d\u5386\u90fd\u65e0\u6cd5\u6b63\u786e\u5730\u8fdb\u884c\u72b6\u6001\u8f6c\u79fb\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u5229\u7528\u4e00\u4e2a\u53d8\u91cf\u6682\u5b58\u5de6\u4e0a\u65b9\u72b6\u6001\uff0c\u4ece\u800c\u8f6c\u5316\u5230\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7b49\u4ef7\u7684\u60c5\u51b5\uff0c\u53ef\u4ee5\u5728\u7a7a\u95f4\u4f18\u5316\u540e\u8fdb\u884c\u6b63\u5e8f\u904d\u5386\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/","title":"14.5 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898","text":"<p>\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5148\u6c42\u89e3\u53e6\u4e00\u4e2a\u5e38\u89c1\u7684\u80cc\u5305\u95ee\u9898\uff1a\u5b8c\u5168\u80cc\u5305\uff0c\u518d\u4e86\u89e3\u5b83\u7684\u4e00\u79cd\u7279\u4f8b\uff1a\u96f6\u94b1\u5151\u6362\u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1451","title":"14.5.1 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002\u793a\u4f8b\u5982\u56fe 14-22 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 14-22 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u548c 0-1 \u80cc\u5305\u95ee\u9898\u975e\u5e38\u76f8\u4f3c\uff0c\u533a\u522b\u4ec5\u5728\u4e8e\u4e0d\u9650\u5236\u7269\u54c1\u7684\u9009\u62e9\u6b21\u6570\u3002</p> <ul> <li>\u5728 0-1 \u80cc\u5305\u95ee\u9898\u4e2d\uff0c\u6bcf\u79cd\u7269\u54c1\u53ea\u6709\u4e00\u4e2a\uff0c\u56e0\u6b64\u5c06\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u540e\uff0c\u53ea\u80fd\u4ece\u524d \\(i-1\\) \u4e2a\u7269\u54c1\u4e2d\u9009\u62e9\u3002</li> <li>\u5728\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u4e2d\uff0c\u6bcf\u79cd\u7269\u54c1\u7684\u6570\u91cf\u662f\u65e0\u9650\u7684\uff0c\u56e0\u6b64\u5c06\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u540e\uff0c\u4ecd\u53ef\u4ee5\u4ece\u524d \\(i\\) \u4e2a\u7269\u54c1\u4e2d\u9009\u62e9\u3002</li> </ul> <p>\u5728\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u89c4\u5b9a\u4e0b\uff0c\u72b6\u6001 \\([i, c]\\) \u7684\u53d8\u5316\u5206\u4e3a\u4e24\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u76f8\u540c\uff0c\u8f6c\u79fb\u81f3 \\([i-1, c]\\) \u3002</li> <li>\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u4e0d\u540c\uff0c\u8f6c\u79fb\u81f3 \\([i, c-wgt[i-1]]\\) \u3002</li> </ul> <p>\u4ece\u800c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u53d8\u4e3a\uff1a</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1]) \\]"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5bf9\u6bd4\u4e24\u9053\u9898\u76ee\u7684\u4ee3\u7801\uff0c\u72b6\u6001\u8f6c\u79fb\u4e2d\u6709\u4e00\u5904\u4ece \\(i-1\\) \u53d8\u4e3a \\(i\\) \uff0c\u5176\u4f59\u5b8c\u5168\u4e00\u81f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint UnboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i, c] = Math.Max(dp[i - 1, c], dp[i, c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n        dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfn unbounded_knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i][c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = myMax(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfun unboundedKnapsackDP(\n    wgt: IntArray,\n    value: IntArray,\n    cap: Int\n): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c].toDouble(), (dp[i][c - wgt[i - 1]] + value[i - 1]).toDouble())\n                    .toInt()\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>[class]{}-[func]{unbounded_knapsack_dp}\n</code></pre> unbounded_knapsack.zig<pre><code>// \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\nfn unboundedKnapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    comptime var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..cap + 1) |c| {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = @max(dp[i - 1][c], dp[i][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e\u5f53\u524d\u72b6\u6001\u662f\u4ece\u5de6\u8fb9\u548c\u4e0a\u8fb9\u7684\u72b6\u6001\u8f6c\u79fb\u800c\u6765\u7684\uff0c\u56e0\u6b64\u7a7a\u95f4\u4f18\u5316\u540e\u5e94\u8be5\u5bf9 \\(dp\\) \u8868\u4e2d\u7684\u6bcf\u4e00\u884c\u8fdb\u884c\u6b63\u5e8f\u904d\u5386\u3002</p> <p>\u8fd9\u4e2a\u904d\u5386\u987a\u5e8f\u4e0e 0-1 \u80cc\u5305\u6b63\u597d\u76f8\u53cd\u3002\u8bf7\u501f\u52a9\u56fe 14-23 \u6765\u7406\u89e3\u4e24\u8005\u7684\u533a\u522b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-23 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u5728\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u4ee3\u7801\u5b9e\u73b0\u6bd4\u8f83\u7b80\u5355\uff0c\u4ec5\u9700\u5c06\u6570\u7ec4 <code>dp</code> \u7684\u7b2c\u4e00\u7ef4\u5220\u9664\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * (cap + 1)\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        # \u6b63\u5e8f\u904d\u5386\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint UnboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.Max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, cap+1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: cap + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n        dp[c] = dp[c];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn unbounded_knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(cap + 1, sizeof(int));\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun unboundedKnapsackDPComp(\n    wgt: IntArray,\n    value: IntArray,\n    cap: Int\n): Int {\n    val n = wgt.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(cap + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] =\n                    max(dp[c].toDouble(), (dp[c - wgt[i - 1]] + value[i - 1]).toDouble()).toInt()\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>[class]{}-[func]{unbounded_knapsack_dp_comp}\n</code></pre> unbounded_knapsack.zig<pre><code>// \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn unboundedKnapsackDPComp(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    comptime var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (cap + 1);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..cap + 1) |c| {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1452","title":"14.5.2 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898","text":"<p>\u80cc\u5305\u95ee\u9898\u662f\u4e00\u5927\u7c7b\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u4ee3\u8868\uff0c\u5176\u62e5\u6709\u5f88\u591a\u53d8\u79cd\uff0c\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u80fd\u591f\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u3002\u5982\u679c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u56fe 14-24 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 14-24 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u96f6\u94b1\u5151\u6362\u53ef\u4ee5\u770b\u4f5c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u4e00\u79cd\u7279\u6b8a\u60c5\u51b5\uff0c\u4e24\u8005\u5177\u6709\u4ee5\u4e0b\u8054\u7cfb\u4e0e\u4e0d\u540c\u70b9\u3002</p> <ul> <li>\u4e24\u9053\u9898\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\uff0c\u201c\u7269\u54c1\u201d\u5bf9\u5e94\u201c\u786c\u5e01\u201d\u3001\u201c\u7269\u54c1\u91cd\u91cf\u201d\u5bf9\u5e94\u201c\u786c\u5e01\u9762\u503c\u201d\u3001\u201c\u80cc\u5305\u5bb9\u91cf\u201d\u5bf9\u5e94\u201c\u76ee\u6807\u91d1\u989d\u201d\u3002</li> <li>\u4f18\u5316\u76ee\u6807\u76f8\u53cd\uff0c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u662f\u8981\u6700\u5927\u5316\u7269\u54c1\u4ef7\u503c\uff0c\u96f6\u94b1\u5151\u6362\u95ee\u9898\u662f\u8981\u6700\u5c0f\u5316\u786c\u5e01\u6570\u91cf\u3002</li> <li>\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u662f\u6c42\u201c\u4e0d\u8d85\u8fc7\u201d\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u89e3\uff0c\u96f6\u94b1\u5151\u6362\u662f\u6c42\u201c\u6070\u597d\u201d\u51d1\u5230\u76ee\u6807\u91d1\u989d\u7684\u89e3\u3002</li> </ul> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u72b6\u6001 \\([i, a]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u524d \\(i\\) \u79cd\u786c\u5e01\u80fd\u591f\u51d1\u51fa\u91d1\u989d \\(a\\) \u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\uff0c\u8bb0\u4e3a \\(dp[i, a]\\) \u3002</p> <p>\u4e8c\u7ef4 \\(dp\\) \u8868\u7684\u5c3a\u5bf8\u4e3a \\((n+1) \\times (amt+1)\\) \u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u672c\u9898\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u5b58\u5728\u4ee5\u4e0b\u4e24\u70b9\u5dee\u5f02\u3002</p> <ul> <li>\u672c\u9898\u8981\u6c42\u6700\u5c0f\u503c\uff0c\u56e0\u6b64\u9700\u5c06\u8fd0\u7b97\u7b26 \\(\\max()\\) \u66f4\u6539\u4e3a \\(\\min()\\) \u3002</li> <li>\u4f18\u5316\u4e3b\u4f53\u662f\u786c\u5e01\u6570\u91cf\u800c\u975e\u5546\u54c1\u4ef7\u503c\uff0c\u56e0\u6b64\u5728\u9009\u4e2d\u786c\u5e01\u65f6\u6267\u884c \\(+1\\) \u5373\u53ef\u3002</li> </ul> \\[ dp[i, a] = \\min(dp[i-1, a], dp[i, a - coins[i-1]] + 1) \\] <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5f53\u76ee\u6807\u91d1\u989d\u4e3a \\(0\\) \u65f6\uff0c\u51d1\u51fa\u5b83\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u4e3a \\(0\\) \uff0c\u5373\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002</p> <p>\u5f53\u65e0\u786c\u5e01\u65f6\uff0c\u65e0\u6cd5\u51d1\u51fa\u4efb\u610f \\(&gt; 0\\) \u7684\u76ee\u6807\u91d1\u989d\uff0c\u5373\u662f\u65e0\u6548\u89e3\u3002\u4e3a\u4f7f\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\min()\\) \u51fd\u6570\u80fd\u591f\u8bc6\u522b\u5e76\u8fc7\u6ee4\u65e0\u6548\u89e3\uff0c\u6211\u4eec\u8003\u8651\u4f7f\u7528 \\(+ \\infty\\) \u6765\u8868\u793a\u5b83\u4eec\uff0c\u5373\u4ee4\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u4e8e \\(+ \\infty\\) \u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u5e76\u672a\u63d0\u4f9b \\(+ \\infty\\) \u53d8\u91cf\uff0c\u53ea\u80fd\u4f7f\u7528\u6574\u578b <code>int</code> \u7684\u6700\u5927\u503c\u6765\u4ee3\u66ff\u3002\u800c\u8fd9\u53c8\u4f1a\u5bfc\u81f4\u5927\u6570\u8d8a\u754c\uff1a\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(+ 1\\) \u64cd\u4f5c\u53ef\u80fd\u53d1\u751f\u6ea2\u51fa\u3002</p> <p>\u4e3a\u6b64\uff0c\u6211\u4eec\u91c7\u7528\u6570\u5b57 \\(amt + 1\\) \u6765\u8868\u793a\u65e0\u6548\u89e3\uff0c\u56e0\u4e3a\u51d1\u51fa \\(amt\\) \u7684\u786c\u5e01\u6570\u91cf\u6700\u591a\u4e3a \\(amt\\) \u3002\u6700\u540e\u8fd4\u56de\u524d\uff0c\u5224\u65ad \\(dp[n, amt]\\) \u662f\u5426\u7b49\u4e8e \\(amt + 1\\) \uff0c\u82e5\u662f\u5219\u8fd4\u56de \\(-1\\) \uff0c\u4ee3\u8868\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change.py<pre><code>def coin_change_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a in range(1, amt + 1):\n        dp[0][a] = MAX\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n    return dp[n][amt] if dp[n][amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][amt + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint CoinChangeDP(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, amt + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0, a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i, a] = Math.Min(dp[i - 1, a], dp[i, a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n, amt] != MAX ? dp[n, amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDP(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a := 1; a &lt;= amt; a++ {\n        dp[0][a] = max\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = int(math.Min(float64(dp[i-1][a]), float64(dp[i][a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a in 1 ... amt {\n        dp[0][a] = MAX\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDP(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n  for (int a = 1; a &lt;= amt; a++) {\n    dp[0][a] = MAX;\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n        dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfn coin_change_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a in 1..=amt {\n        dp[0][a] = max;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = std::cmp::min(dp[i - 1][a], dp[i][a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = myMin(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[n][amt] != MAX ? dp[n][amt] : -1;\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfun coinChangeDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (a in 1..amt) {\n        dp[0][a] = MAX\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a].toDouble(), (dp[i][a - coins[i - 1]] + 1).toDouble())\n                    .toInt()\n            }\n        }\n    }\n    return if (dp[n][amt] != MAX) dp[n][amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>[class]{}-[func]{coin_change_dp}\n</code></pre> coin_change.zig<pre><code>// \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212\nfn coinChangeDP(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    comptime var max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (1..amt + 1) |a| {\n        dp[0][a] = max;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = @min(dp[i - 1][a], dp[i][a - @as(usize, @intCast(coins[i - 1]))] + 1);\n            }\n        }\n    }\n    if (dp[n][amt] != max) {\n        return @intCast(dp[n][amt]);\n    } else {\n        return -1;\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u56fe 14-25 \u5c55\u793a\u4e86\u96f6\u94b1\u5151\u6362\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b\uff0c\u548c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u975e\u5e38\u76f8\u4f3c\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-25 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_1","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u96f6\u94b1\u5151\u6362\u7684\u7a7a\u95f4\u4f18\u5316\u7684\u5904\u7406\u65b9\u5f0f\u548c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u4e00\u81f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change.py<pre><code>def coin_change_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [MAX] * (amt + 1)\n    dp[0] = 0\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        # \u6b63\u5e8f\u904d\u5386\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n    return dp[amt] if dp[amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(amt + 1, MAX);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    Arrays.fill(dp, MAX);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint CoinChangeDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    Array.Fill(dp, MAX);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = Math.Min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDPComp(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, amt+1)\n    for i := 1; i &lt;= amt; i++ {\n        dp[i] = max\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        // \u5012\u5e8f\u904d\u5386\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = int(math.Min(float64(dp[a]), float64(dp[a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: MAX, count: amt + 1)\n    dp[0] = 0\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDPComp(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(amt + 1, MAX);\n  dp[0] = 0;\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n        dp[a] = dp[a];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n        dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn coin_change_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; amt + 1];\n    dp.fill(max);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = std::cmp::min(dp[a], dp[a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(amt + 1, sizeof(int));\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = myMin(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[amt] != MAX ? dp[amt] : -1;\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun coinChangeDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(amt + 1)\n    Arrays.fill(dp, MAX)\n    dp[0] = 0\n    // \u72b6\u6001\u8f6c\u79fb\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = min(dp[a].toDouble(), (dp[a - coins[i - 1]] + 1).toDouble()).toInt()\n            }\n        }\n    }\n    return if (dp[amt] != MAX) dp[amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>[class]{}-[func]{coin_change_dp_comp}\n</code></pre> coin_change.zig<pre><code>// \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn coinChangeDPComp(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    comptime var max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (amt + 1);\n    @memset(&amp;dp, max);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = @min(dp[a], dp[a - @as(usize, @intCast(coins[i - 1]))] + 1);\n            }\n        }\n    }\n    if (dp[amt] != max) {\n        return @intCast(dp[amt]);\n    } else {\n        return -1;\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1453-ii","title":"14.5.3 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898 II","text":"<p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u786c\u5e01\u7ec4\u5408\u6570\u91cf\u3002\u793a\u4f8b\u5982\u56fe 14-26 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 14-26 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898 II \u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_2","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u76f8\u6bd4\u4e8e\u4e0a\u4e00\u9898\uff0c\u672c\u9898\u76ee\u6807\u662f\u6c42\u7ec4\u5408\u6570\u91cf\uff0c\u56e0\u6b64\u5b50\u95ee\u9898\u53d8\u4e3a\uff1a\u524d \\(i\\) \u79cd\u786c\u5e01\u80fd\u591f\u51d1\u51fa\u91d1\u989d \\(a\\) \u7684\u7ec4\u5408\u6570\u91cf\u3002\u800c \\(dp\\) \u8868\u4ecd\u7136\u662f\u5c3a\u5bf8\u4e3a \\((n+1) \\times (amt + 1)\\) \u7684\u4e8c\u7ef4\u77e9\u9635\u3002</p> <p>\u5f53\u524d\u72b6\u6001\u7684\u7ec4\u5408\u6570\u91cf\u7b49\u4e8e\u4e0d\u9009\u5f53\u524d\u786c\u5e01\u4e0e\u9009\u5f53\u524d\u786c\u5e01\u8fd9\u4e24\u79cd\u51b3\u7b56\u7684\u7ec4\u5408\u6570\u91cf\u4e4b\u548c\u3002\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]] \\] <p>\u5f53\u76ee\u6807\u91d1\u989d\u4e3a \\(0\\) \u65f6\uff0c\u65e0\u987b\u9009\u62e9\u4efb\u4f55\u786c\u5e01\u5373\u53ef\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u5e94\u5c06\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u521d\u59cb\u5316\u4e3a \\(1\\) \u3002\u5f53\u65e0\u786c\u5e01\u65f6\uff0c\u65e0\u6cd5\u51d1\u51fa\u4efb\u4f55 \\(&gt;0\\) \u7684\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change_ii.py<pre><code>def coin_change_ii_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(coins)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # \u521d\u59cb\u5316\u9996\u5217\n    for i in range(n + 1):\n        dp[i][0] = 1\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n    return dp[n][amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(int[] coins, int amt) {\n    int n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][amt + 1];\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint CoinChangeIIDP(int[] coins, int amt) {\n    int n = coins.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, amt + 1];\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i, 0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i, a] = dp[i - 1, a] + dp[i, a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n, amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDP(coins []int, amt int) int {\n    n := len(coins)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // \u521d\u59cb\u5316\u9996\u5217\n    for i := 0; i &lt;= n; i++ {\n        dp[i][0] = 1\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i-1][a] + dp[i][a-coins[i-1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // \u521d\u59cb\u5316\u9996\u5217\n    for i in 0 ... n {\n        dp[i][0] = 1\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDP(coins, amt) {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // \u521d\u59cb\u5316\u9996\u5217\n  for (int i = 0; i &lt;= n; i++) {\n    dp[i][0] = 1;\n  }\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[n][amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfn coin_change_ii_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // \u521d\u59cb\u5316\u9996\u5217\n    for i in 0..=n {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[n][amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[n][amt];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfun coinChangeIIDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (i in 0..n) {\n        dp[i][0] = 1\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>[class]{}-[func]{coin_change_ii_dp}\n</code></pre> coin_change_ii.zig<pre><code>// \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212\nfn coinChangeIIDP(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (0..n + 1) |i| {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - @as(usize, @intCast(coins[i - 1]))];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_2","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7a7a\u95f4\u4f18\u5316\u5904\u7406\u65b9\u5f0f\u76f8\u540c\uff0c\u5220\u9664\u786c\u5e01\u7ef4\u5ea6\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change_ii.py<pre><code>def coin_change_ii_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(coins)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * (amt + 1)\n    dp[0] = 1\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        # \u6b63\u5e8f\u904d\u5386\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n    return dp[amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(amt + 1, 0);\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint CoinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDPComp(coins []int, amt int) int {\n    n := len(coins)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, amt+1)\n    dp[0] = 1\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        // \u5012\u5e8f\u904d\u5386\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a-coins[i-1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: amt + 1)\n    dp[0] = 1\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDPComp(coins, amt) {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(amt + 1, 0);\n  dp[0] = 1;\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n        dp[a] = dp[a];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n        dp[a] = dp[a] + dp[a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn coin_change_ii_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; amt + 1];\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(amt + 1, sizeof(int));\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[amt];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfun coinChangeIIDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // \u521d\u59cb\u5316 dp \u8868\n    val dp = IntArray(amt + 1)\n    dp[0] = 1\n    // \u72b6\u6001\u8f6c\u79fb\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>[class]{}-[func]{coin_change_ii_dp_comp}\n</code></pre> coin_change_ii.zig<pre><code>// \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn coinChangeIIDPComp(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (amt + 1);\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = dp[a] + dp[a - @as(usize, @intCast(coins[i - 1]))];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_graph/","title":"\u7b2c 9 \u7ae0 \u00a0 \u56fe","text":"<p>Abstract</p> <p>\u5728\u751f\u547d\u65c5\u9014\u4e2d\uff0c\u6211\u4eec\u5c31\u50cf\u662f\u4e00\u4e2a\u4e2a\u8282\u70b9\uff0c\u88ab\u65e0\u6570\u770b\u4e0d\u89c1\u7684\u8fb9\u76f8\u8fde\u3002</p> <p>\u6bcf\u4e00\u6b21\u7684\u76f8\u8bc6\u4e0e\u76f8\u79bb\uff0c\u90fd\u5728\u8fd9\u5f20\u5de8\u5927\u7684\u7f51\u7edc\u56fe\u4e2d\u7559\u4e0b\u72ec\u7279\u7684\u5370\u8bb0\u3002</p>"},{"location":"chapter_graph/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>9.1 \u00a0 \u56fe</li> <li>9.2 \u00a0 \u56fe\u57fa\u7840\u64cd\u4f5c</li> <li>9.3 \u00a0 \u56fe\u7684\u904d\u5386</li> <li>9.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_graph/graph/","title":"9.1 \u00a0 \u56fe","text":"<p>\u56fe\uff08graph\uff09\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u7531\u9876\u70b9\uff08vertex\uff09\u548c\u8fb9\uff08edge\uff09\u7ec4\u6210\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u56fe \\(G\\) \u62bd\u8c61\u5730\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9 \\(V\\) \u548c\u4e00\u7ec4\u8fb9 \\(E\\) \u7684\u96c6\u5408\u3002\u4ee5\u4e0b\u793a\u4f8b\u5c55\u793a\u4e86\u4e00\u4e2a\u5305\u542b 5 \u4e2a\u9876\u70b9\u548c 7 \u6761\u8fb9\u7684\u56fe\u3002</p> \\[ \\begin{aligned} V &amp; = \\{ 1, 2, 3, 4, 5 \\} \\newline E &amp; = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G &amp; = \\{ V, E \\} \\newline \\end{aligned} \\] <p>\u5982\u679c\u5c06\u9876\u70b9\u770b\u4f5c\u8282\u70b9\uff0c\u5c06\u8fb9\u770b\u4f5c\u8fde\u63a5\u5404\u4e2a\u8282\u70b9\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u56fe\u770b\u4f5c\u4e00\u79cd\u4ece\u94fe\u8868\u62d3\u5c55\u800c\u6765\u7684\u6570\u636e\u7ed3\u6784\u3002\u5982\u56fe 9-1 \u6240\u793a\uff0c\u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u56e0\u800c\u66f4\u4e3a\u590d\u6742\u3002</p> <p></p> <p> \u56fe 9-1 \u00a0 \u94fe\u8868\u3001\u6811\u3001\u56fe\u4e4b\u95f4\u7684\u5173\u7cfb </p>"},{"location":"chapter_graph/graph/#911","title":"9.1.1 \u00a0 \u56fe\u7684\u5e38\u89c1\u7c7b\u578b\u4e0e\u672f\u8bed","text":"<p>\u6839\u636e\u8fb9\u662f\u5426\u5177\u6709\u65b9\u5411\uff0c\u53ef\u5206\u4e3a\u65e0\u5411\u56fe\uff08undirected graph\uff09\u548c\u6709\u5411\u56fe\uff08directed graph\uff09\uff0c\u5982\u56fe 9-2 \u6240\u793a\u3002</p> <ul> <li>\u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u8fb9\u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u7684\u201c\u53cc\u5411\u201d\u8fde\u63a5\u5173\u7cfb\uff0c\u4f8b\u5982\u5fae\u4fe1\u6216 QQ \u4e2d\u7684\u201c\u597d\u53cb\u5173\u7cfb\u201d\u3002</li> <li>\u5728\u6709\u5411\u56fe\u4e2d\uff0c\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u5373 \\(A \\rightarrow B\\) \u548c \\(A \\leftarrow B\\) \u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u4f8b\u5982\u5fae\u535a\u6216\u6296\u97f3\u4e0a\u7684\u201c\u5173\u6ce8\u201d\u4e0e\u201c\u88ab\u5173\u6ce8\u201d\u5173\u7cfb\u3002</li> </ul> <p></p> <p> \u56fe 9-2 \u00a0 \u6709\u5411\u56fe\u4e0e\u65e0\u5411\u56fe </p> <p>\u6839\u636e\u6240\u6709\u9876\u70b9\u662f\u5426\u8fde\u901a\uff0c\u53ef\u5206\u4e3a\u8fde\u901a\u56fe\uff08connected graph\uff09\u548c\u975e\u8fde\u901a\u56fe\uff08disconnected graph\uff09\uff0c\u5982\u56fe 9-3 \u6240\u793a\u3002</p> <ul> <li>\u5bf9\u4e8e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u53ef\u4ee5\u5230\u8fbe\u5176\u4f59\u4efb\u610f\u9876\u70b9\u3002</li> <li>\u5bf9\u4e8e\u975e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\u3002</li> </ul> <p></p> <p> \u56fe 9-3 \u00a0 \u8fde\u901a\u56fe\u4e0e\u975e\u8fde\u901a\u56fe </p> <p>\u6211\u4eec\u8fd8\u53ef\u4ee5\u4e3a\u8fb9\u6dfb\u52a0\u201c\u6743\u91cd\u201d\u53d8\u91cf\uff0c\u4ece\u800c\u5f97\u5230\u5982\u56fe 9-4 \u6240\u793a\u7684\u6709\u6743\u56fe\uff08weighted graph\uff09\u3002\u4f8b\u5982\u5728\u300a\u738b\u8005\u8363\u8000\u300b\u7b49\u624b\u6e38\u4e2d\uff0c\u7cfb\u7edf\u4f1a\u6839\u636e\u5171\u540c\u6e38\u620f\u65f6\u95f4\u6765\u8ba1\u7b97\u73a9\u5bb6\u4e4b\u95f4\u7684\u201c\u4eb2\u5bc6\u5ea6\u201d\uff0c\u8fd9\u79cd\u4eb2\u5bc6\u5ea6\u7f51\u7edc\u5c31\u53ef\u4ee5\u7528\u6709\u6743\u56fe\u6765\u8868\u793a\u3002</p> <p></p> <p> \u56fe 9-4 \u00a0 \u6709\u6743\u56fe\u4e0e\u65e0\u6743\u56fe </p> <p>\u56fe\u6570\u636e\u7ed3\u6784\u5305\u542b\u4ee5\u4e0b\u5e38\u7528\u672f\u8bed\u3002</p> <ul> <li>\u90bb\u63a5\uff08adjacency\uff09\uff1a\u5f53\u4e24\u9876\u70b9\u4e4b\u95f4\u5b58\u5728\u8fb9\u76f8\u8fde\u65f6\uff0c\u79f0\u8fd9\u4e24\u9876\u70b9\u201c\u90bb\u63a5\u201d\u3002\u5728\u56fe 9-4 \u4e2d\uff0c\u9876\u70b9 1 \u7684\u90bb\u63a5\u9876\u70b9\u4e3a\u9876\u70b9 2\u30013\u30015\u3002</li> <li>\u8def\u5f84\uff08path\uff09\uff1a\u4ece\u9876\u70b9 A \u5230\u9876\u70b9 B \u7ecf\u8fc7\u7684\u8fb9\u6784\u6210\u7684\u5e8f\u5217\u88ab\u79f0\u4e3a\u4ece A \u5230 B \u7684\u201c\u8def\u5f84\u201d\u3002\u5728\u56fe 9-4 \u4e2d\uff0c\u8fb9\u5e8f\u5217 1-5-2-4 \u662f\u9876\u70b9 1 \u5230\u9876\u70b9 4 \u7684\u4e00\u6761\u8def\u5f84\u3002</li> <li>\u5ea6\uff08degree\uff09\uff1a\u4e00\u4e2a\u9876\u70b9\u62e5\u6709\u7684\u8fb9\u6570\u3002\u5bf9\u4e8e\u6709\u5411\u56fe\uff0c\u5165\u5ea6\uff08in-degree\uff09\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u6307\u5411\u8be5\u9876\u70b9\uff0c\u51fa\u5ea6\uff08out-degree\uff09\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u4ece\u8be5\u9876\u70b9\u6307\u51fa\u3002</li> </ul>"},{"location":"chapter_graph/graph/#912","title":"9.1.2 \u00a0 \u56fe\u7684\u8868\u793a","text":"<p>\u56fe\u7684\u5e38\u7528\u8868\u793a\u65b9\u5f0f\u5305\u62ec\u201c\u90bb\u63a5\u77e9\u9635\u201d\u548c\u201c\u90bb\u63a5\u8868\u201d\u3002\u4ee5\u4e0b\u4f7f\u7528\u65e0\u5411\u56fe\u8fdb\u884c\u4e3e\u4f8b\u3002</p>"},{"location":"chapter_graph/graph/#1","title":"1. \u00a0 \u90bb\u63a5\u77e9\u9635","text":"<p>\u8bbe\u56fe\u7684\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u90bb\u63a5\u77e9\u9635\uff08adjacency matrix\uff09\u4f7f\u7528\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u662f\u5426\u5b58\u5728\u8fb9\u3002</p> <p>\u5982\u56fe 9-5 \u6240\u793a\uff0c\u8bbe\u90bb\u63a5\u77e9\u9635\u4e3a \\(M\\)\u3001\u9876\u70b9\u5217\u8868\u4e3a \\(V\\) \uff0c\u90a3\u4e48\u77e9\u9635\u5143\u7d20 \\(M[i, j] = 1\\) \u8868\u793a\u9876\u70b9 \\(V[i]\\) \u5230\u9876\u70b9 \\(V[j]\\) \u4e4b\u95f4\u5b58\u5728\u8fb9\uff0c\u53cd\u4e4b \\(M[i, j] = 0\\) \u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u65e0\u8fb9\u3002</p> <p></p> <p> \u56fe 9-5 \u00a0 \u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a </p> <p>\u90bb\u63a5\u77e9\u9635\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u9876\u70b9\u4e0d\u80fd\u4e0e\u81ea\u8eab\u76f8\u8fde\uff0c\u56e0\u6b64\u90bb\u63a5\u77e9\u9635\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u6ca1\u6709\u610f\u4e49\u3002</li> <li>\u5bf9\u4e8e\u65e0\u5411\u56fe\uff0c\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u7b49\u4ef7\uff0c\u6b64\u65f6\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\u3002</li> <li>\u5c06\u90bb\u63a5\u77e9\u9635\u7684\u5143\u7d20\u4ece \\(1\\) \u548c \\(0\\) \u66ff\u6362\u4e3a\u6743\u91cd\uff0c\u5219\u53ef\u8868\u793a\u6709\u6743\u56fe\u3002</li> </ul> <p>\u4f7f\u7528\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u8bbf\u95ee\u77e9\u9635\u5143\u7d20\u4ee5\u83b7\u53d6\u8fb9\uff0c\u56e0\u6b64\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002\u7136\u800c\uff0c\u77e9\u9635\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u5185\u5b58\u5360\u7528\u8f83\u591a\u3002</p>"},{"location":"chapter_graph/graph/#2","title":"2. \u00a0 \u90bb\u63a5\u8868","text":"<p>\u90bb\u63a5\u8868\uff08adjacency list\uff09\u4f7f\u7528 \\(n\\) \u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u94fe\u8868\u8282\u70b9\u8868\u793a\u9876\u70b9\u3002\u7b2c \\(i\\) \u4e2a\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff08\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u9876\u70b9\uff09\u3002\u56fe 9-6 \u5c55\u793a\u4e86\u4e00\u4e2a\u4f7f\u7528\u90bb\u63a5\u8868\u5b58\u50a8\u7684\u56fe\u7684\u793a\u4f8b\u3002</p> <p></p> <p> \u56fe 9-6 \u00a0 \u56fe\u7684\u90bb\u63a5\u8868\u8868\u793a </p> <p>\u90bb\u63a5\u8868\u4ec5\u5b58\u50a8\u5b9e\u9645\u5b58\u5728\u7684\u8fb9\uff0c\u800c\u8fb9\u7684\u603b\u6570\u901a\u5e38\u8fdc\u5c0f\u4e8e \\(n^2\\) \uff0c\u56e0\u6b64\u5b83\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002\u7136\u800c\uff0c\u5728\u90bb\u63a5\u8868\u4e2d\u9700\u8981\u901a\u8fc7\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u5176\u65f6\u95f4\u6548\u7387\u4e0d\u5982\u90bb\u63a5\u77e9\u9635\u3002</p> <p>\u89c2\u5bdf\u56fe 9-6 \uff0c\u90bb\u63a5\u8868\u7ed3\u6784\u4e0e\u54c8\u5e0c\u8868\u4e2d\u7684\u201c\u94fe\u5f0f\u5730\u5740\u201d\u975e\u5e38\u76f8\u4f3c\uff0c\u56e0\u6b64\u6211\u4eec\u4e5f\u53ef\u4ee5\u91c7\u7528\u7c7b\u4f3c\u7684\u65b9\u6cd5\u6765\u4f18\u5316\u6548\u7387\u3002\u6bd4\u5982\u5f53\u94fe\u8868\u8f83\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u5316\u4e3a AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u6548\u7387\u4ece \\(O(n)\\) \u4f18\u5316\u81f3 \\(O(\\log n)\\) \uff1b\u8fd8\u53ef\u4ee5\u628a\u94fe\u8868\u8f6c\u6362\u4e3a\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u964d\u81f3 \\(O(1)\\) \u3002</p>"},{"location":"chapter_graph/graph/#913","title":"9.1.3 \u00a0 \u56fe\u7684\u5e38\u89c1\u5e94\u7528","text":"<p>\u5982\u8868 9-1 \u6240\u793a\uff0c\u8bb8\u591a\u73b0\u5b9e\u7cfb\u7edf\u53ef\u4ee5\u7528\u56fe\u6765\u5efa\u6a21\uff0c\u76f8\u5e94\u7684\u95ee\u9898\u4e5f\u53ef\u4ee5\u7ea6\u5316\u4e3a\u56fe\u8ba1\u7b97\u95ee\u9898\u3002</p> <p> \u8868 9-1 \u00a0 \u73b0\u5b9e\u751f\u6d3b\u4e2d\u5e38\u89c1\u7684\u56fe </p> \u9876\u70b9 \u8fb9 \u56fe\u8ba1\u7b97\u95ee\u9898 \u793e\u4ea4\u7f51\u7edc \u7528\u6237 \u597d\u53cb\u5173\u7cfb \u6f5c\u5728\u597d\u53cb\u63a8\u8350 \u5730\u94c1\u7ebf\u8def \u7ad9\u70b9 \u7ad9\u70b9\u95f4\u7684\u8fde\u901a\u6027 \u6700\u77ed\u8def\u7ebf\u63a8\u8350 \u592a\u9633\u7cfb \u661f\u4f53 \u661f\u4f53\u95f4\u7684\u4e07\u6709\u5f15\u529b\u4f5c\u7528 \u884c\u661f\u8f68\u9053\u8ba1\u7b97"},{"location":"chapter_graph/graph_operations/","title":"9.2 \u00a0 \u56fe\u7684\u57fa\u7840\u64cd\u4f5c","text":"<p>\u56fe\u7684\u57fa\u7840\u64cd\u4f5c\u53ef\u5206\u4e3a\u5bf9\u201c\u8fb9\u201d\u7684\u64cd\u4f5c\u548c\u5bf9\u201c\u9876\u70b9\u201d\u7684\u64cd\u4f5c\u3002\u5728\u201c\u90bb\u63a5\u77e9\u9635\u201d\u548c\u201c\u90bb\u63a5\u8868\u201d\u4e24\u79cd\u8868\u793a\u65b9\u6cd5\u4e0b\uff0c\u5b9e\u73b0\u65b9\u5f0f\u6709\u6240\u4e0d\u540c\u3002</p>"},{"location":"chapter_graph/graph_operations/#921","title":"9.2.1 \u00a0 \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u7684\u5b9e\u73b0","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \u7684\u65e0\u5411\u56fe\uff0c\u5219\u5404\u79cd\u64cd\u4f5c\u7684\u5b9e\u73b0\u65b9\u5f0f\u5982\u56fe 9-7 \u6240\u793a\u3002</p> <ul> <li>\u6dfb\u52a0\u6216\u5220\u9664\u8fb9\uff1a\u76f4\u63a5\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u4fee\u6539\u6307\u5b9a\u7684\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u800c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u9700\u8981\u540c\u65f6\u66f4\u65b0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002</li> <li>\u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u7684\u5c3e\u90e8\u6dfb\u52a0\u4e00\u884c\u4e00\u5217\uff0c\u5e76\u5168\u90e8\u586b \\(0\\) \u5373\u53ef\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> <li>\u5220\u9664\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u4e00\u884c\u4e00\u5217\u3002\u5f53\u5220\u9664\u9996\u884c\u9996\u5217\u65f6\u8fbe\u5230\u6700\u5dee\u60c5\u51b5\uff0c\u9700\u8981\u5c06 \\((n-1)^2\\) \u4e2a\u5143\u7d20\u201c\u5411\u5de6\u4e0a\u79fb\u52a8\u201d\uff0c\u4ece\u800c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> <li>\u521d\u59cb\u5316\uff1a\u4f20\u5165 \\(n\\) \u4e2a\u9876\u70b9\uff0c\u521d\u59cb\u5316\u957f\u5ea6\u4e3a \\(n\\) \u7684\u9876\u70b9\u5217\u8868 <code>vertices</code> \uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b\u521d\u59cb\u5316 \\(n \\times n\\) \u5927\u5c0f\u7684\u90bb\u63a5\u77e9\u9635 <code>adjMat</code> \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> </ul> \u521d\u59cb\u5316\u90bb\u63a5\u77e9\u9635\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9 <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-7 \u00a0 \u90bb\u63a5\u77e9\u9635\u7684\u521d\u59cb\u5316\u3001\u589e\u5220\u8fb9\u3001\u589e\u5220\u9876\u70b9 </p> <p>\u4ee5\u4e0b\u662f\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u7684\u5b9e\u73b0\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_adjacency_matrix.py<pre><code>class GraphAdjMat:\n    \"\"\"\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n\n    def __init__(self, vertices: list[int], edges: list[list[int]]):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        # \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n        self.vertices: list[int] = []\n        # \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n        self.adj_mat: list[list[int]] = []\n        # \u6dfb\u52a0\u9876\u70b9\n        for val in vertices:\n            self.add_vertex(val)\n        # \u6dfb\u52a0\u8fb9\n        # \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for e in edges:\n            self.add_edge(e[0], e[1])\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\n        return len(self.vertices)\n\n    def add_vertex(self, val: int):\n        \"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\n        n = self.size()\n        # \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        self.vertices.append(val)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        new_row = [0] * n\n        self.adj_mat.append(new_row)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for row in self.adj_mat:\n            row.append(0)\n\n    def remove_vertex(self, index: int):\n        \"\"\"\u5220\u9664\u9876\u70b9\"\"\"\n        if index &gt;= self.size():\n            raise IndexError()\n        # \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        self.vertices.pop(index)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        self.adj_mat.pop(index)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for row in self.adj_mat:\n            row.pop(index)\n\n    def add_edge(self, i: int, j: int):\n        \"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n        # \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        # \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        # \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        self.adj_mat[i][j] = 1\n        self.adj_mat[j][i] = 1\n\n    def remove_edge(self, i: int, j: int):\n        \"\"\"\u5220\u9664\u8fb9\"\"\"\n        # \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        # \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        self.adj_mat[i][j] = 0\n        self.adj_mat[j][i] = 0\n\n    def print(self):\n        \"\"\"\u6253\u5370\u90bb\u63a5\u77e9\u9635\"\"\"\n        print(\"\u9876\u70b9\u5217\u8868 =\", self.vertices)\n        print(\"\u90bb\u63a5\u77e9\u9635 =\")\n        print_matrix(self.adj_mat)\n</code></pre> graph_adjacency_matrix.cpp<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    vector&lt;int&gt; vertices;       // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    vector&lt;vector&lt;int&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    GraphAdjMat(const vector&lt;int&gt; &amp;vertices, const vector&lt;vector&lt;int&gt;&gt; &amp;edges) {\n        // \u6dfb\u52a0\u9876\u70b9\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (const vector&lt;int&gt; &amp;edge : edges) {\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int size() const {\n        return vertices.size();\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    void addVertex(int val) {\n        int n = size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.push_back(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        adjMat.emplace_back(vector&lt;int&gt;(n, 0));\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.push_back(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    void removeVertex(int index) {\n        if (index &gt;= size()) {\n            throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.erase(vertices.begin() + index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.erase(adjMat.begin() + index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.erase(row.begin() + index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    void addEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    void removeEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n        }\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    void print() {\n        cout &lt;&lt; \"\u9876\u70b9\u5217\u8868 = \";\n        printVector(vertices);\n        cout &lt;&lt; \"\u90bb\u63a5\u77e9\u9635 =\" &lt;&lt; endl;\n        printVectorMatrix(adjMat);\n    }\n};\n</code></pre> graph_adjacency_matrix.java<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    List&lt;Integer&gt; vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    List&lt;List&lt;Integer&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = new ArrayList&lt;&gt;();\n        this.adjMat = new ArrayList&lt;&gt;();\n        // \u6dfb\u52a0\u9876\u70b9\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (int[] e : edges) {\n            addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    public int size() {\n        return vertices.size();\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void addVertex(int val) {\n        int n = size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.add(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        List&lt;Integer&gt; newRow = new ArrayList&lt;&gt;(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.add(0);\n        }\n        adjMat.add(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.add(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void removeVertex(int index) {\n        if (index &gt;= size())\n            throw new IndexOutOfBoundsException();\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.remove(index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void addEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat.get(i).set(j, 1);\n        adjMat.get(j).set(i, 1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void removeEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        adjMat.get(i).set(j, 0);\n        adjMat.get(j).set(i, 0);\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    public void print() {\n        System.out.print(\"\u9876\u70b9\u5217\u8868 = \");\n        System.out.println(vertices);\n        System.out.println(\"\u90bb\u63a5\u77e9\u9635 =\");\n        PrintUtil.printMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.cs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    List&lt;int&gt; vertices;     // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    List&lt;List&lt;int&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u6dfb\u52a0\u9876\u70b9\n        foreach (int val in vertices) {\n            AddVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        foreach (int[] e in edges) {\n            AddEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int Size() {\n        return vertices.Count;\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void AddVertex(int val) {\n        int n = Size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.Add(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        List&lt;int&gt; newRow = new(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.Add(0);\n        }\n        adjMat.Add(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.Add(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void RemoveVertex(int index) {\n        if (index &gt;= Size())\n            throw new IndexOutOfRangeException();\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.RemoveAt(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.RemoveAt(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.RemoveAt(index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void AddEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void RemoveEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    public void Print() {\n        Console.Write(\"\u9876\u70b9\u5217\u8868 = \");\n        PrintUtil.PrintList(vertices);\n        Console.WriteLine(\"\u90bb\u63a5\u77e9\u9635 =\");\n        PrintUtil.PrintMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.go<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjMat struct {\n    // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    vertices []int\n    // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    adjMat [][]int\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n    // \u6dfb\u52a0\u9876\u70b9\n    n := len(vertices)\n    adjMat := make([][]int, n)\n    for i := range adjMat {\n        adjMat[i] = make([]int, n)\n    }\n    // \u521d\u59cb\u5316\u56fe\n    g := &amp;graphAdjMat{\n        vertices: vertices,\n        adjMat:   adjMat,\n    }\n    // \u6dfb\u52a0\u8fb9\n    // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    for i := range edges {\n        g.addEdge(edges[i][0], edges[i][1])\n    }\n    return g\n}\n\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjMat) size() int {\n    return len(g.vertices)\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjMat) addVertex(val int) {\n    n := g.size()\n    // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n    g.vertices = append(g.vertices, val)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n    newRow := make([]int, n)\n    g.adjMat = append(g.adjMat, newRow)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i], 0)\n    }\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjMat) removeVertex(index int) {\n    if index &gt;= g.size() {\n        return\n    }\n    // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n    g.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n    g.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n    }\n}\n\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) addEdge(i, j int) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n    g.adjMat[i][j] = 1\n    g.adjMat[j][i] = 1\n}\n\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    g.adjMat[i][j] = 0\n    g.adjMat[j][i] = 0\n}\n\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc (g *graphAdjMat) print() {\n    fmt.Printf(\"\\t\u9876\u70b9\u5217\u8868 = %v\\n\", g.vertices)\n    fmt.Printf(\"\\t\u90bb\u63a5\u77e9\u9635 = \\n\")\n    for i := range g.adjMat {\n        fmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n    }\n}\n</code></pre> graph_adjacency_matrix.swift<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    private var vertices: [Int] // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    private var adjMat: [[Int]] // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init(vertices: [Int], edges: [[Int]]) {\n        self.vertices = []\n        adjMat = []\n        // \u6dfb\u52a0\u9876\u70b9\n        for val in vertices {\n            addVertex(val: val)\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for e in edges {\n            addEdge(i: e[0], j: e[1])\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    func size() -&gt; Int {\n        vertices.count\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    func addVertex(val: Int) {\n        let n = size()\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.append(val)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        let newRow = Array(repeating: 0, count: n)\n        adjMat.append(newRow)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for i in adjMat.indices {\n            adjMat[i].append(0)\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    func removeVertex(index: Int) {\n        if index &gt;= size() {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.remove(at: index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.remove(at: index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for i in adjMat.indices {\n            adjMat[i].remove(at: index)\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    func addEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    func removeEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    func print() {\n        Swift.print(\"\u9876\u70b9\u5217\u8868 = \", terminator: \"\")\n        Swift.print(vertices)\n        Swift.print(\"\u90bb\u63a5\u77e9\u9635 =\")\n        PrintUtil.printMatrix(matrix: adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.js<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    constructor(vertices, edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u6dfb\u52a0\u9876\u70b9\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size() {\n        return this.vertices.length;\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(val) {\n        const n = this.size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        this.vertices.push(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        const newRow = [];\n        for (let j = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(index) {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        this.vertices.splice(index, 1);\n\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        this.adjMat.splice(index, 1);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    addEdge(i, j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    removeEdge(i, j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    print() {\n        console.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\n        console.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.ts<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    vertices: number[]; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    adjMat: number[][]; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    constructor(vertices: number[], edges: number[][]) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u6dfb\u52a0\u9876\u70b9\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size(): number {\n        return this.vertices.length;\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(val: number): void {\n        const n: number = this.size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        this.vertices.push(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        const newRow: number[] = [];\n        for (let j: number = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(index: number): void {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        this.vertices.splice(index, 1);\n\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        this.adjMat.splice(index, 1);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    addEdge(i: number, j: number): void {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    removeEdge(i: number, j: number): void {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    print(): void {\n        console.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\n        console.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.dart<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n  List&lt;int&gt; vertices = []; // \u9876\u70b9\u5143\u7d20\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n  List&lt;List&lt;int&gt;&gt; adjMat = []; //\u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  GraphAdjMat(List&lt;int&gt; vertices, List&lt;List&lt;int&gt;&gt; edges) {\n    this.vertices = [];\n    this.adjMat = [];\n    // \u6dfb\u52a0\u9876\u70b9\n    for (int val in vertices) {\n      addVertex(val);\n    }\n    // \u6dfb\u52a0\u8fb9\n    // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    for (List&lt;int&gt; e in edges) {\n      addEdge(e[0], e[1]);\n    }\n  }\n\n  /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n  int size() {\n    return vertices.length;\n  }\n\n  /* \u6dfb\u52a0\u9876\u70b9 */\n  void addVertex(int val) {\n    int n = size();\n    // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n    vertices.add(val);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n    List&lt;int&gt; newRow = List.filled(n, 0, growable: true);\n    adjMat.add(newRow);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n    for (List&lt;int&gt; row in adjMat) {\n      row.add(0);\n    }\n  }\n\n  /* \u5220\u9664\u9876\u70b9 */\n  void removeVertex(int index) {\n    if (index &gt;= size()) {\n      throw IndexError;\n    }\n    // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n    vertices.removeAt(index);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n    adjMat.removeAt(index);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n    for (List&lt;int&gt; row in adjMat) {\n      row.removeAt(index);\n    }\n  }\n\n  /* \u6dfb\u52a0\u8fb9 */\n  // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n  void addEdge(int i, int j) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n    adjMat[i][j] = 1;\n    adjMat[j][i] = 1;\n  }\n\n  /* \u5220\u9664\u8fb9 */\n  // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n  void removeEdge(int i, int j) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    adjMat[i][j] = 0;\n    adjMat[j][i] = 0;\n  }\n\n  /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n  void printAdjMat() {\n    print(\"\u9876\u70b9\u5217\u8868 = $vertices\");\n    print(\"\u90bb\u63a5\u77e9\u9635 = \");\n    printMatrix(adjMat);\n  }\n}\n</code></pre> graph_adjacency_matrix.rs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjMat {\n    // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    pub vertices: Vec&lt;i32&gt;,\n    // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    pub adj_mat: Vec&lt;Vec&lt;i32&gt;&gt;,\n}\n\nimpl GraphAdjMat {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(vertices: Vec&lt;i32&gt;, edges: Vec&lt;[usize; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjMat {\n            vertices: vec![],\n            adj_mat: vec![],\n        };\n        // \u6dfb\u52a0\u9876\u70b9\n        for val in vertices {\n            graph.add_vertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for edge in edges {\n            graph.add_edge(edge[0], edge[1])\n        }\n\n        graph\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    pub fn size(&amp;self) -&gt; usize {\n        self.vertices.len()\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    pub fn add_vertex(&amp;mut self, val: i32) {\n        let n = self.size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        self.vertices.push(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        self.adj_mat.push(vec![0; n]);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for row in &amp;mut self.adj_mat {\n            row.push(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    pub fn remove_vertex(&amp;mut self, index: usize) {\n        if index &gt;= self.size() {\n            panic!(\"index error\")\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        self.vertices.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        self.adj_mat.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for row in &amp;mut self.adj_mat {\n            row.remove(index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    pub fn add_edge(&amp;mut self, i: usize, j: usize) {\n        // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        self.adj_mat[i][j] = 1;\n        self.adj_mat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    pub fn remove_edge(&amp;mut self, i: usize, j: usize) {\n        // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        self.adj_mat[i][j] = 0;\n        self.adj_mat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    pub fn print(&amp;self) {\n        println!(\"\u9876\u70b9\u5217\u8868 = {:?}\", self.vertices);\n        println!(\"\u90bb\u63a5\u77e9\u9635 =\");\n        println!(\"[\");\n        for row in &amp;self.adj_mat {\n            println!(\"  {:?},\", row);\n        }\n        println!(\"]\")\n    }\n}\n</code></pre> graph_adjacency_matrix.c<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7ed3\u6784\u4f53 */\ntypedef struct {\n    int vertices[MAX_SIZE];\n    int adjMat[MAX_SIZE][MAX_SIZE];\n    int size;\n} GraphAdjMat;\n\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjMat *newGraphAdjMat() {\n    GraphAdjMat *graph = (GraphAdjMat *)malloc(sizeof(GraphAdjMat));\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        for (int j = 0; j &lt; MAX_SIZE; j++) {\n            graph-&gt;adjMat[i][j] = 0;\n        }\n    }\n    return graph;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjMat(GraphAdjMat *graph) {\n    free(graph);\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjMat *graph, int val) {\n    if (graph-&gt;size == MAX_SIZE) {\n        fprintf(stderr, \"\u56fe\u7684\u9876\u70b9\u6570\u91cf\u5df2\u8fbe\u6700\u5927\u503c\\n\");\n        return;\n    }\n    // \u6dfb\u52a0\u7b2c n \u4e2a\u9876\u70b9\uff0c\u5e76\u5c06\u7b2c n \u884c\u548c\u5217\u7f6e\u96f6\n    int n = graph-&gt;size;\n    graph-&gt;vertices[n] = val;\n    for (int i = 0; i &lt;= n; i++) {\n        graph-&gt;adjMat[n][i] = graph-&gt;adjMat[i][n] = 0;\n    }\n    graph-&gt;size++;\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjMat *graph, int index) {\n    if (index &lt; 0 || index &gt;= graph-&gt;size) {\n        fprintf(stderr, \"\u9876\u70b9\u7d22\u5f15\u8d8a\u754c\\n\");\n        return;\n    }\n    // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        graph-&gt;vertices[i] = graph-&gt;vertices[i + 1];\n    }\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        for (int j = 0; j &lt; graph-&gt;size; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i + 1][j];\n        }\n    }\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        for (int j = index; j &lt; graph-&gt;size - 1; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i][j + 1];\n        }\n    }\n    graph-&gt;size--;\n}\n\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 1;\n    graph-&gt;adjMat[j][i] = 1;\n}\n\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 0;\n    graph-&gt;adjMat[j][i] = 0;\n}\n\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid printGraphAdjMat(GraphAdjMat *graph) {\n    printf(\"\u9876\u70b9\u5217\u8868 = \");\n    printArray(graph-&gt;vertices, graph-&gt;size);\n    printf(\"\u90bb\u63a5\u77e9\u9635 =\\n\");\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        printArray(graph-&gt;adjMat[i], graph-&gt;size);\n    }\n}\n</code></pre> graph_adjacency_matrix.kt<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat(vertices: IntArray, edges: Array&lt;IntArray&gt;) {\n    val vertices: MutableList&lt;Int&gt; = ArrayList() // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    val adjMat: MutableList&lt;MutableList&lt;Int&gt;&gt; = ArrayList() // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    init {\n        // \u6dfb\u52a0\u9876\u70b9\n        for (vertex in vertices) {\n            addVertex(vertex)\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (edge in edges) {\n            addEdge(edge[0], edge[1])\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    fun size(): Int {\n        return vertices.size\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    fun addVertex(value: Int) {\n        val n = size()\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.add(value)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        val newRow: MutableList&lt;Int&gt; = mutableListOf()\n        for (j in 0..&lt;n) {\n            newRow.add(0)\n        }\n        adjMat.add(newRow)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (row in adjMat) {\n            row.add(0)\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    fun removeVertex(index: Int) {\n        if (index &gt;= size()) throw IndexOutOfBoundsException()\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.removeAt(index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.removeAt(index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (row in adjMat) {\n            row.removeAt(index)\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    fun addEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) throw java.lang.IndexOutOfBoundsException()\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    fun removeEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) throw java.lang.IndexOutOfBoundsException()\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    fun print() {\n        print(\"\u9876\u70b9\u5217\u8868 = \")\n        println(vertices);\n        println(\"\u90bb\u63a5\u77e9\u9635 =\");\n        printMatrix(adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.rb<pre><code>[class]{GraphAdjMat}-[func]{}\n</code></pre> graph_adjacency_matrix.zig<pre><code>[class]{GraphAdjMat}-[func]{}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_graph/graph_operations/#922","title":"9.2.2 \u00a0 \u57fa\u4e8e\u90bb\u63a5\u8868\u7684\u5b9e\u73b0","text":"<p>\u8bbe\u65e0\u5411\u56fe\u7684\u9876\u70b9\u603b\u6570\u4e3a \\(n\\)\u3001\u8fb9\u603b\u6570\u4e3a \\(m\\) \uff0c\u5219\u53ef\u6839\u636e\u56fe 9-8 \u6240\u793a\u7684\u65b9\u6cd5\u5b9e\u73b0\u5404\u79cd\u64cd\u4f5c\u3002</p> <ul> <li>\u6dfb\u52a0\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u7684\u672b\u5c3e\u6dfb\u52a0\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u4e3a\u662f\u65e0\u5411\u56fe\uff0c\u6240\u4ee5\u9700\u8981\u540c\u65f6\u6dfb\u52a0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002</li> <li>\u5220\u9664\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u4e2d\u67e5\u627e\u5e76\u5220\u9664\u6307\u5b9a\u8fb9\uff0c\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\u3002\u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u9700\u8981\u540c\u65f6\u5220\u9664\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002</li> <li>\u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u94fe\u8868\uff0c\u5e76\u5c06\u65b0\u589e\u9876\u70b9\u4f5c\u4e3a\u94fe\u8868\u5934\u8282\u70b9\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002</li> <li>\u5220\u9664\u9876\u70b9\uff1a\u9700\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5220\u9664\u5305\u542b\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002</li> <li>\u521d\u59cb\u5316\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u521b\u5efa \\(n\\) \u4e2a\u9876\u70b9\u548c \\(2m\\) \u6761\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002</li> </ul> \u521d\u59cb\u5316\u90bb\u63a5\u8868\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9 <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-8 \u00a0 \u90bb\u63a5\u8868\u7684\u521d\u59cb\u5316\u3001\u589e\u5220\u8fb9\u3001\u589e\u5220\u9876\u70b9 </p> <p>\u4ee5\u4e0b\u662f\u90bb\u63a5\u8868\u7684\u4ee3\u7801\u5b9e\u73b0\u3002\u5bf9\u6bd4\u56fe 9-8 \uff0c\u5b9e\u9645\u4ee3\u7801\u6709\u4ee5\u4e0b\u4e0d\u540c\u3002</p> <ul> <li>\u4e3a\u4e86\u65b9\u4fbf\u6dfb\u52a0\u4e0e\u5220\u9664\u9876\u70b9\uff0c\u4ee5\u53ca\u7b80\u5316\u4ee3\u7801\uff0c\u6211\u4eec\u4f7f\u7528\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u6765\u4ee3\u66ff\u94fe\u8868\u3002</li> <li>\u4f7f\u7528\u54c8\u5e0c\u8868\u6765\u5b58\u50a8\u90bb\u63a5\u8868\uff0c<code>key</code> \u4e3a\u9876\u70b9\u5b9e\u4f8b\uff0c<code>value</code> \u4e3a\u8be5\u9876\u70b9\u7684\u90bb\u63a5\u9876\u70b9\u5217\u8868\uff08\u94fe\u8868\uff09\u3002</li> </ul> <p>\u53e6\u5916\uff0c\u6211\u4eec\u5728\u90bb\u63a5\u8868\u4e2d\u4f7f\u7528 <code>Vertex</code> \u7c7b\u6765\u8868\u793a\u9876\u70b9\uff0c\u8fd9\u6837\u505a\u7684\u539f\u56e0\u662f\uff1a\u5982\u679c\u4e0e\u90bb\u63a5\u77e9\u9635\u4e00\u6837\uff0c\u7528\u5217\u8868\u7d22\u5f15\u6765\u533a\u5206\u4e0d\u540c\u9876\u70b9\uff0c\u90a3\u4e48\u5047\u8bbe\u8981\u5220\u9664\u7d22\u5f15\u4e3a \\(i\\) \u7684\u9876\u70b9\uff0c\u5219\u9700\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5c06\u6240\u6709\u5927\u4e8e \\(i\\) \u7684\u7d22\u5f15\u5168\u90e8\u51cf \\(1\\) \uff0c\u6548\u7387\u5f88\u4f4e\u3002\u800c\u5982\u679c\u6bcf\u4e2a\u9876\u70b9\u90fd\u662f\u552f\u4e00\u7684 <code>Vertex</code> \u5b9e\u4f8b\uff0c\u5220\u9664\u67d0\u4e00\u9876\u70b9\u4e4b\u540e\u5c31\u65e0\u987b\u6539\u52a8\u5176\u4ed6\u9876\u70b9\u4e86\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_adjacency_list.py<pre><code>class GraphAdjList:\n    \"\"\"\u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n\n    def __init__(self, edges: list[list[Vertex]]):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        # \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        self.adj_list = dict[Vertex, list[Vertex]]()\n        # \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for edge in edges:\n            self.add_vertex(edge[0])\n            self.add_vertex(edge[1])\n            self.add_edge(edge[0], edge[1])\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\n        return len(self.adj_list)\n\n    def add_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # \u6dfb\u52a0\u8fb9 vet1 - vet2\n        self.adj_list[vet1].append(vet2)\n        self.adj_list[vet2].append(vet1)\n\n    def remove_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"\u5220\u9664\u8fb9\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # \u5220\u9664\u8fb9 vet1 - vet2\n        self.adj_list[vet1].remove(vet2)\n        self.adj_list[vet2].remove(vet1)\n\n    def add_vertex(self, vet: Vertex):\n        \"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\n        if vet in self.adj_list:\n            return\n        # \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        self.adj_list[vet] = []\n\n    def remove_vertex(self, vet: Vertex):\n        \"\"\"\u5220\u9664\u9876\u70b9\"\"\"\n        if vet not in self.adj_list:\n            raise ValueError()\n        # \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        self.adj_list.pop(vet)\n        # \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for vertex in self.adj_list:\n            if vet in self.adj_list[vertex]:\n                self.adj_list[vertex].remove(vet)\n\n    def print(self):\n        \"\"\"\u6253\u5370\u90bb\u63a5\u8868\"\"\"\n        print(\"\u90bb\u63a5\u8868 =\")\n        for vertex in self.adj_list:\n            tmp = [v.val for v in self.adj_list[vertex]]\n            print(f\"{vertex.val}: {tmp},\")\n</code></pre> graph_adjacency_list.cpp<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n  public:\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    unordered_map&lt;Vertex *, vector&lt;Vertex *&gt;&gt; adjList;\n\n    /* \u5728 vector \u4e2d\u5220\u9664\u6307\u5b9a\u8282\u70b9 */\n    void remove(vector&lt;Vertex *&gt; &amp;vec, Vertex *vet) {\n        for (int i = 0; i &lt; vec.size(); i++) {\n            if (vec[i] == vet) {\n                vec.erase(vec.begin() + i);\n                break;\n            }\n        }\n    }\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    GraphAdjList(const vector&lt;vector&lt;Vertex *&gt;&gt; &amp;edges) {\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (const vector&lt;Vertex *&gt; &amp;edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int size() {\n        return adjList.size();\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    void addEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1].push_back(vet2);\n        adjList[vet2].push_back(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    void removeEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        remove(adjList[vet1], vet2);\n        remove(adjList[vet2], vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    void addVertex(Vertex *vet) {\n        if (adjList.count(vet))\n            return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList[vet] = vector&lt;Vertex *&gt;();\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    void removeVertex(Vertex *vet) {\n        if (!adjList.count(vet))\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.erase(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (auto &amp;adj : adjList) {\n            remove(adj.second, vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    void print() {\n        cout &lt;&lt; \"\u90bb\u63a5\u8868 =\" &lt;&lt; endl;\n        for (auto &amp;adj : adjList) {\n            const auto &amp;key = adj.first;\n            const auto &amp;vec = adj.second;\n            cout &lt;&lt; key-&gt;val &lt;&lt; \": \";\n            printVector(vetsToVals(vec));\n        }\n    }\n};\n</code></pre> graph_adjacency_list.java<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public GraphAdjList(Vertex[][] edges) {\n        this.adjList = new HashMap&lt;&gt;();\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (Vertex[] edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    public int size() {\n        return adjList.size();\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    public void addEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList.get(vet1).add(vet2);\n        adjList.get(vet2).add(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    public void removeEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList.get(vet1).remove(vet2);\n        adjList.get(vet2).remove(vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void addVertex(Vertex vet) {\n        if (adjList.containsKey(vet))\n            return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList.put(vet, new ArrayList&lt;&gt;());\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void removeVertex(Vertex vet) {\n        if (!adjList.containsKey(vet))\n            throw new IllegalArgumentException();\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.remove(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (List&lt;Vertex&gt; list : adjList.values()) {\n            list.remove(vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    public void print() {\n        System.out.println(\"\u90bb\u63a5\u8868 =\");\n        for (Map.Entry&lt;Vertex, List&lt;Vertex&gt;&gt; pair : adjList.entrySet()) {\n            List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n            for (Vertex vertex : pair.getValue())\n                tmp.add(vertex.val);\n            System.out.println(pair.getKey().val + \": \" + tmp + \",\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.cs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    public Dictionary&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* \u6784\u9020\u51fd\u6570 */\n    public GraphAdjList(Vertex[][] edges) {\n        adjList = [];\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        foreach (Vertex[] edge in edges) {\n            AddVertex(edge[0]);\n            AddVertex(edge[1]);\n            AddEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int Size() {\n        return adjList.Count;\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    public void AddEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1].Add(vet2);\n        adjList[vet2].Add(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    public void RemoveEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList[vet1].Remove(vet2);\n        adjList[vet2].Remove(vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void AddVertex(Vertex vet) {\n        if (adjList.ContainsKey(vet))\n            return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList.Add(vet, []);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void RemoveVertex(Vertex vet) {\n        if (!adjList.ContainsKey(vet))\n            throw new InvalidOperationException();\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.Remove(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        foreach (List&lt;Vertex&gt; list in adjList.Values) {\n            list.Remove(vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    public void Print() {\n        Console.WriteLine(\"\u90bb\u63a5\u8868 =\");\n        foreach (KeyValuePair&lt;Vertex, List&lt;Vertex&gt;&gt; pair in adjList) {\n            List&lt;int&gt; tmp = [];\n            foreach (Vertex vertex in pair.Value)\n                tmp.Add(vertex.val);\n            Console.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.go<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjList struct {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    adjList map[Vertex][]Vertex\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\n    g := &amp;graphAdjList{\n        adjList: make(map[Vertex][]Vertex),\n    }\n    // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n    for _, edge := range edges {\n        g.addVertex(edge[0])\n        g.addVertex(edge[1])\n        g.addEdge(edge[0], edge[1])\n    }\n    return g\n}\n\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjList) size() int {\n    return len(g.adjList)\n}\n\n/* \u6dfb\u52a0\u8fb9 */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // \u6dfb\u52a0\u8fb9 vet1 - vet2, \u6dfb\u52a0\u533f\u540d struct{},\n    g.adjList[vet1] = append(g.adjList[vet1], vet2)\n    g.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n\n/* \u5220\u9664\u8fb9 */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // \u5220\u9664\u8fb9 vet1 - vet2\n    g.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2)\n    g.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1)\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if ok {\n        return\n    }\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n    g.adjList[vet] = make([]Vertex, 0)\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if !ok {\n        panic(\"error\")\n    }\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n    delete(g.adjList, vet)\n    // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n    for v, list := range g.adjList {\n        g.adjList[v] = DeleteSliceElms(list, vet)\n    }\n}\n\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nfunc (g *graphAdjList) print() {\n    var builder strings.Builder\n    fmt.Printf(\"\u90bb\u63a5\u8868 = \\n\")\n    for k, v := range g.adjList {\n        builder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\n        for _, vet := range v {\n            builder.WriteString(strconv.Itoa(vet.Val) + \" \")\n        }\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> graph_adjacency_list.swift<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    public private(set) var adjList: [Vertex: [Vertex]]\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public init(edges: [[Vertex]]) {\n        adjList = [:]\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for edge in edges {\n            addVertex(vet: edge[0])\n            addVertex(vet: edge[1])\n            addEdge(vet1: edge[0], vet2: edge[1])\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    public func size() -&gt; Int {\n        adjList.count\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    public func addEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"\u53c2\u6570\u9519\u8bef\")\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1]?.append(vet2)\n        adjList[vet2]?.append(vet1)\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    public func removeEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"\u53c2\u6570\u9519\u8bef\")\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList[vet1]?.removeAll { $0 == vet2 }\n        adjList[vet2]?.removeAll { $0 == vet1 }\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public func addVertex(vet: Vertex) {\n        if adjList[vet] != nil {\n            return\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList[vet] = []\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public func removeVertex(vet: Vertex) {\n        if adjList[vet] == nil {\n            fatalError(\"\u53c2\u6570\u9519\u8bef\")\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.removeValue(forKey: vet)\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for key in adjList.keys {\n            adjList[key]?.removeAll { $0 == vet }\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    public func print() {\n        Swift.print(\"\u90bb\u63a5\u8868 =\")\n        for (vertex, list) in adjList {\n            let list = list.map { $0.val }\n            Swift.print(\"\\(vertex.val): \\(list),\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.js<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    adjList;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(edges) {\n        this.adjList = new Map();\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size() {\n        return this.adjList.size;\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    addEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    removeEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(vet) {\n        if (this.adjList.has(vet)) return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        this.adjList.set(vet, []);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(vet) {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        this.adjList.delete(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (const set of this.adjList.values()) {\n            const index = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    print() {\n        console.log('\u90bb\u63a5\u8868 =');\n        for (const [key, value] of this.adjList) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.ts<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    adjList: Map&lt;Vertex, Vertex[]&gt;;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(edges: Vertex[][]) {\n        this.adjList = new Map();\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size(): number {\n        return this.adjList.size;\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    addEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    removeEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(vet: Vertex): void {\n        if (this.adjList.has(vet)) return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        this.adjList.set(vet, []);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(vet: Vertex): void {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        this.adjList.delete(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (const set of this.adjList.values()) {\n            const index: number = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    print(): void {\n        console.log('\u90bb\u63a5\u8868 =');\n        for (const [key, value] of this.adjList.entries()) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.dart<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n  // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n  Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList = {};\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  GraphAdjList(List&lt;List&lt;Vertex&gt;&gt; edges) {\n    for (List&lt;Vertex&gt; edge in edges) {\n      addVertex(edge[0]);\n      addVertex(edge[1]);\n      addEdge(edge[0], edge[1]);\n    }\n  }\n\n  /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n  int size() {\n    return adjList.length;\n  }\n\n  /* \u6dfb\u52a0\u8fb9 */\n  void addEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // \u6dfb\u52a0\u8fb9 vet1 - vet2\n    adjList[vet1]!.add(vet2);\n    adjList[vet2]!.add(vet1);\n  }\n\n  /* \u5220\u9664\u8fb9 */\n  void removeEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // \u5220\u9664\u8fb9 vet1 - vet2\n    adjList[vet1]!.remove(vet2);\n    adjList[vet2]!.remove(vet1);\n  }\n\n  /* \u6dfb\u52a0\u9876\u70b9 */\n  void addVertex(Vertex vet) {\n    if (adjList.containsKey(vet)) return;\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n    adjList[vet] = [];\n  }\n\n  /* \u5220\u9664\u9876\u70b9 */\n  void removeVertex(Vertex vet) {\n    if (!adjList.containsKey(vet)) {\n      throw ArgumentError;\n    }\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n    adjList.remove(vet);\n    // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n    adjList.forEach((key, value) {\n      value.remove(vet);\n    });\n  }\n\n  /* \u6253\u5370\u90bb\u63a5\u8868 */\n  void printAdjList() {\n    print(\"\u90bb\u63a5\u8868 =\");\n    adjList.forEach((key, value) {\n      List&lt;int&gt; tmp = [];\n      for (Vertex vertex in value) {\n        tmp.add(vertex.val);\n      }\n      print(\"${key.val}: $tmp,\");\n    });\n  }\n}\n</code></pre> graph_adjacency_list.rs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    pub adj_list: HashMap&lt;Vertex, Vec&lt;Vertex&gt;&gt;,\n}\n\nimpl GraphAdjList {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(edges: Vec&lt;[Vertex; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjList {\n            adj_list: HashMap::new(),\n        };\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for edge in edges {\n            graph.add_vertex(edge[0]);\n            graph.add_vertex(edge[1]);\n            graph.add_edge(edge[0], edge[1]);\n        }\n\n        graph\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    #[allow(unused)]\n    pub fn size(&amp;self) -&gt; usize {\n        self.adj_list.len()\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    pub fn add_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n        {\n            panic!(\"value error\");\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        self.adj_list.get_mut(&amp;vet1).unwrap().push(vet2);\n        self.adj_list.get_mut(&amp;vet2).unwrap().push(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    #[allow(unused)]\n    pub fn remove_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n        {\n            panic!(\"value error\");\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        self.adj_list\n            .get_mut(&amp;vet1)\n            .unwrap()\n            .retain(|&amp;vet| vet != vet2);\n        self.adj_list\n            .get_mut(&amp;vet2)\n            .unwrap()\n            .retain(|&amp;vet| vet != vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    pub fn add_vertex(&amp;mut self, vet: Vertex) {\n        if self.adj_list.contains_key(&amp;vet) {\n            return;\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        self.adj_list.insert(vet, vec![]);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    #[allow(unused)]\n    pub fn remove_vertex(&amp;mut self, vet: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet) {\n            panic!(\"value error\");\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        self.adj_list.remove(&amp;vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for list in self.adj_list.values_mut() {\n            list.retain(|&amp;v| v != vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    pub fn print(&amp;self) {\n        println!(\"\u90bb\u63a5\u8868 =\");\n        for (vertex, list) in &amp;self.adj_list {\n            let list = list.iter().map(|vertex| vertex.val).collect::&lt;Vec&lt;i32&gt;&gt;();\n            println!(\"{}: {:?},\", vertex.val, list);\n        }\n    }\n}\n</code></pre> graph_adjacency_list.c<pre><code>/* \u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct AdjListNode {\n    Vertex *vertex;           // \u9876\u70b9\n    struct AdjListNode *next; // \u540e\u7ee7\u8282\u70b9\n} AdjListNode;\n\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* \u6dfb\u52a0\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid addEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *node = (AdjListNode *)malloc(sizeof(AdjListNode));\n    node-&gt;vertex = vet;\n    // \u5934\u63d2\u6cd5\n    node-&gt;next = head-&gt;next;\n    head-&gt;next = node;\n}\n\n/* \u5220\u9664\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid removeEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *pre = head;\n    AdjListNode *cur = head-&gt;next;\n    // \u5728\u94fe\u8868\u4e2d\u641c\u7d22 vet \u5bf9\u5e94\u8282\u70b9\n    while (cur != NULL &amp;&amp; cur-&gt;vertex != vet) {\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n    if (cur == NULL)\n        return;\n    // \u5c06 vet \u5bf9\u5e94\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\n    pre-&gt;next = cur-&gt;next;\n    // \u91ca\u653e\u5185\u5b58\n    free(cur);\n}\n\n/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntypedef struct {\n    AdjListNode *heads[MAX_SIZE]; // \u8282\u70b9\u6570\u7ec4\n    int size;                     // \u8282\u70b9\u6570\u91cf\n} GraphAdjList;\n\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjList *newGraphAdjList() {\n    GraphAdjList *graph = (GraphAdjList *)malloc(sizeof(GraphAdjList));\n    if (!graph) {\n        return NULL;\n    }\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        graph-&gt;heads[i] = NULL;\n    }\n    return graph;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjList(GraphAdjList *graph) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        AdjListNode *cur = graph-&gt;heads[i];\n        while (cur != NULL) {\n            AdjListNode *next = cur-&gt;next;\n            if (cur != graph-&gt;heads[i]) {\n                free(cur);\n            }\n            cur = next;\n        }\n        free(graph-&gt;heads[i]-&gt;vertex);\n        free(graph-&gt;heads[i]);\n    }\n    free(graph);\n}\n\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL &amp;&amp; head1 != head2);\n    // \u6dfb\u52a0\u8fb9 vet1 - vet2\n    addEdgeHelper(head1, vet2);\n    addEdgeHelper(head2, vet1);\n}\n\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL);\n    // \u5220\u9664\u8fb9 vet1 - vet2\n    removeEdgeHelper(head1, head2-&gt;vertex);\n    removeEdgeHelper(head2, head1-&gt;vertex);\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjList *graph, Vertex *vet) {\n    assert(graph != NULL &amp;&amp; graph-&gt;size &lt; MAX_SIZE);\n    AdjListNode *head = (AdjListNode *)malloc(sizeof(AdjListNode));\n    head-&gt;vertex = vet;\n    head-&gt;next = NULL;\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n    graph-&gt;heads[graph-&gt;size++] = head;\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjList *graph, Vertex *vet) {\n    AdjListNode *node = findNode(graph, vet);\n    assert(node != NULL);\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n    AdjListNode *cur = node, *pre = NULL;\n    while (cur) {\n        pre = cur;\n        cur = cur-&gt;next;\n        free(pre);\n    }\n    // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        cur = graph-&gt;heads[i];\n        pre = NULL;\n        while (cur) {\n            pre = cur;\n            cur = cur-&gt;next;\n            if (cur &amp;&amp; cur-&gt;vertex == vet) {\n                pre-&gt;next = cur-&gt;next;\n                free(cur);\n                break;\n            }\n        }\n    }\n    // \u5c06\u8be5\u9876\u70b9\u4e4b\u540e\u7684\u9876\u70b9\u5411\u524d\u79fb\u52a8\uff0c\u4ee5\u586b\u8865\u7a7a\u7f3a\n    int i;\n    for (i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i] == node)\n            break;\n    }\n    for (int j = i; j &lt; graph-&gt;size - 1; j++) {\n        graph-&gt;heads[j] = graph-&gt;heads[j + 1];\n    }\n    graph-&gt;size--;\n    free(vet);\n}\n</code></pre> graph_adjacency_list.kt<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList(edges: Array&lt;Array&lt;Vertex?&gt;&gt;) {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    val adjList: MutableMap&lt;Vertex, MutableList&lt;Vertex&gt;&gt; = HashMap()\n\n    /* \u6784\u9020\u51fd\u6570 */\n    init {\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (edge in edges) {\n            addVertex(edge[0]!!);\n            addVertex(edge[1]!!);\n            addEdge(edge[0]!!, edge[1]!!);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    fun size(): Int {\n        return adjList.size\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    fun addEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1]?.add(vet2)\n        adjList[vet2]?.add(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    fun removeEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList[vet1]?.remove(vet2);\n        adjList[vet2]?.remove(vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    fun addVertex(vet: Vertex) {\n        if (adjList.containsKey(vet))\n            return\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList[vet] = mutableListOf()\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    fun removeVertex(vet: Vertex) {\n        if (!adjList.containsKey(vet))\n            throw IllegalArgumentException()\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.remove(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (list in adjList.values) {\n            list.remove(vet)\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    fun print() {\n        println(\"\u90bb\u63a5\u8868 =\")\n        for (pair in adjList.entries) {\n            val tmp = ArrayList&lt;Int&gt;()\n            for (vertex in pair.value) {\n                tmp.add(vertex.value)\n            }\n            println(\"${pair.key.value}: $tmp,\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.rb<pre><code>[class]{GraphAdjList}-[func]{}\n</code></pre> graph_adjacency_list.zig<pre><code>[class]{GraphAdjList}-[func]{}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_graph/graph_operations/#923","title":"9.2.3 \u00a0 \u6548\u7387\u5bf9\u6bd4","text":"<p>\u8bbe\u56fe\u4e2d\u5171\u6709 \\(n\\) \u4e2a\u9876\u70b9\u548c \\(m\\) \u6761\u8fb9\uff0c\u8868 9-2 \u5bf9\u6bd4\u4e86\u90bb\u63a5\u77e9\u9635\u548c\u90bb\u63a5\u8868\u7684\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u3002</p> <p> \u8868 9-2 \u00a0 \u90bb\u63a5\u77e9\u9635\u4e0e\u90bb\u63a5\u8868\u5bf9\u6bd4 </p> \u90bb\u63a5\u77e9\u9635 \u90bb\u63a5\u8868\uff08\u94fe\u8868\uff09 \u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09 \u5224\u65ad\u662f\u5426\u90bb\u63a5 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u8fb9 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u8fb9 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u9876\u70b9 \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u9876\u70b9 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) \u5185\u5b58\u7a7a\u95f4\u5360\u7528 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\) <p>\u89c2\u5bdf\u8868 9-2 \uff0c\u4f3c\u4e4e\u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09\u7684\u65f6\u95f4\u6548\u7387\u4e0e\u7a7a\u95f4\u6548\u7387\u6700\u4f18\u3002\u4f46\u5b9e\u9645\u4e0a\uff0c\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u64cd\u4f5c\u8fb9\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u53ea\u9700\u4e00\u6b21\u6570\u7ec4\u8bbf\u95ee\u6216\u8d4b\u503c\u64cd\u4f5c\u5373\u53ef\u3002\u7efc\u5408\u6765\u770b\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u4e86\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u7684\u539f\u5219\uff0c\u800c\u90bb\u63a5\u8868\u4f53\u73b0\u4e86\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u7684\u539f\u5219\u3002</p>"},{"location":"chapter_graph/graph_traversal/","title":"9.3 \u00a0 \u56fe\u7684\u904d\u5386","text":"<p>\u6811\u4ee3\u8868\u7684\u662f\u201c\u4e00\u5bf9\u591a\u201d\u7684\u5173\u7cfb\uff0c\u800c\u56fe\u5219\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u53ef\u4ee5\u8868\u793a\u4efb\u610f\u7684\u201c\u591a\u5bf9\u591a\u201d\u5173\u7cfb\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u6811\u770b\u4f5c\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\u3002\u663e\u7136\uff0c\u6811\u7684\u904d\u5386\u64cd\u4f5c\u4e5f\u662f\u56fe\u7684\u904d\u5386\u64cd\u4f5c\u7684\u4e00\u79cd\u7279\u4f8b\u3002</p> <p>\u56fe\u548c\u6811\u90fd\u9700\u8981\u5e94\u7528\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u904d\u5386\u64cd\u4f5c\u3002\u56fe\u7684\u904d\u5386\u65b9\u5f0f\u4e5f\u53ef\u5206\u4e3a\u4e24\u79cd\uff1a\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u548c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002</p>"},{"location":"chapter_graph/graph_traversal/#931","title":"9.3.1 \u00a0 \u5e7f\u5ea6\u4f18\u5148\u904d\u5386","text":"<p>\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u7684\u904d\u5386\u65b9\u5f0f\uff0c\u4ece\u67d0\u4e2a\u8282\u70b9\u51fa\u53d1\uff0c\u59cb\u7ec8\u4f18\u5148\u8bbf\u95ee\u8ddd\u79bb\u6700\u8fd1\u7684\u9876\u70b9\uff0c\u5e76\u4e00\u5c42\u5c42\u5411\u5916\u6269\u5f20\u3002\u5982\u56fe 9-9 \u6240\u793a\uff0c\u4ece\u5de6\u4e0a\u89d2\u9876\u70b9\u51fa\u53d1\uff0c\u9996\u5148\u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u7136\u540e\u904d\u5386\u4e0b\u4e00\u4e2a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u8bbf\u95ee\u5b8c\u6bd5\u3002</p> <p></p> <p> \u56fe 9-9 \u00a0 \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 </p>"},{"location":"chapter_graph/graph_traversal/#1","title":"1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"<p>BFS \u901a\u5e38\u501f\u52a9\u961f\u5217\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002\u961f\u5217\u5177\u6709\u201c\u5148\u5165\u5148\u51fa\u201d\u7684\u6027\u8d28\uff0c\u8fd9\u4e0e BFS \u7684\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u601d\u60f3\u5f02\u66f2\u540c\u5de5\u3002</p> <ol> <li>\u5c06\u904d\u5386\u8d77\u59cb\u9876\u70b9 <code>startVet</code> \u52a0\u5165\u961f\u5217\uff0c\u5e76\u5f00\u542f\u5faa\u73af\u3002</li> <li>\u5728\u5faa\u73af\u7684\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\uff0c\u5f39\u51fa\u961f\u9996\u9876\u70b9\u5e76\u8bb0\u5f55\u8bbf\u95ee\uff0c\u7136\u540e\u5c06\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u52a0\u5165\u5230\u961f\u5217\u5c3e\u90e8\u3002</li> <li>\u5faa\u73af\u6b65\u9aa4 <code>2.</code> \uff0c\u76f4\u5230\u6240\u6709\u9876\u70b9\u88ab\u8bbf\u95ee\u5b8c\u6bd5\u540e\u7ed3\u675f\u3002</li> </ol> <p>\u4e3a\u4e86\u9632\u6b62\u91cd\u590d\u904d\u5386\u9876\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>visited</code> \u6765\u8bb0\u5f55\u54ea\u4e9b\u8282\u70b9\u5df2\u88ab\u8bbf\u95ee\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_bfs.py<pre><code>def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\n    # \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    # \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res = []\n    # \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited = set[Vertex]([start_vet])\n    # \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    que = deque[Vertex]([start_vet])\n    # \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while len(que) &gt; 0:\n        vet = que.popleft()  # \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        # \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for adj_vet in graph.adj_list[vet]:\n            if adj_vet in visited:\n                continue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            que.append(adj_vet)  # \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adj_vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    # \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n</code></pre> graph_bfs.cpp<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector&lt;Vertex *&gt; graphBFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    vector&lt;Vertex *&gt; res;\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    unordered_set&lt;Vertex *&gt; visited = {startVet};\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    queue&lt;Vertex *&gt; que;\n    que.push(startVet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!que.empty()) {\n        Vertex *vet = que.front();\n        que.pop();          // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push_back(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (auto adjVet : graph.adjList[vet]) {\n            if (visited.count(adjVet))\n                continue;            // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            que.push(adjVet);        // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.emplace(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.java<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    visited.add(startVet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    Queue&lt;Vertex&gt; que = new LinkedList&lt;&gt;();\n    que.offer(startVet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!que.isEmpty()) {\n        Vertex vet = que.poll(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.add(vet);            // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (Vertex adjVet : graph.adjList.get(vet)) {\n            if (visited.contains(adjVet))\n                continue;        // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            que.offer(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.cs<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; GraphBFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    HashSet&lt;Vertex&gt; visited = [startVet];\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    Queue&lt;Vertex&gt; que = new();\n    que.Enqueue(startVet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (que.Count &gt; 0) {\n        Vertex vet = que.Dequeue(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.Add(vet);               // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        foreach (Vertex adjVet in graph.adjList[vet]) {\n            if (visited.Contains(adjVet)) {\n                continue;          // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.Enqueue(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.Add(adjVet);   // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.go<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res := make([]Vertex, 0)\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited := make(map[Vertex]struct{})\n    visited[startVet] = struct{}{}\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS, \u4f7f\u7528\u5207\u7247\u6a21\u62df\u961f\u5217\n    queue := make([]Vertex, 0)\n    queue = append(queue, startVet)\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    for len(queue) &gt; 0 {\n        // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        vet := queue[0]\n        queue = queue[1:]\n        // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        res = append(res, vet)\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for _, adjVet := range g.adjList[vet] {\n            _, isExist := visited[adjVet]\n            // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            if !isExist {\n                queue = append(queue, adjVet)\n                visited[adjVet] = struct{}{}\n            }\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.swift<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    var res: [Vertex] = []\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    var visited: Set&lt;Vertex&gt; = [startVet]\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    var que: [Vertex] = [startVet]\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while !que.isEmpty {\n        let vet = que.removeFirst() // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for adjVet in graph.adjList[vet] ?? [] {\n            if visited.contains(adjVet) {\n                continue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.append(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.insert(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.js<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph, startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited = new Set();\n    visited.add(startVet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    const que = [startVet];\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (que.length) {\n        const vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.ts<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res: Vertex[] = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited: Set&lt;Vertex&gt; = new Set();\n    visited.add(startVet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    const que = [startVet];\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (que.length) {\n        const vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\n            visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.dart<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n  // \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n  // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n  List&lt;Vertex&gt; res = [];\n  // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n  Set&lt;Vertex&gt; visited = {};\n  visited.add(startVet);\n  // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n  Queue&lt;Vertex&gt; que = Queue();\n  que.add(startVet);\n  // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n  while (que.isNotEmpty) {\n    Vertex vet = que.removeFirst(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n    res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (Vertex adjVet in graph.adjList[vet]!) {\n      if (visited.contains(adjVet)) {\n        continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n      }\n      que.add(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n      visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    }\n  }\n  // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n  return res;\n}\n</code></pre> graph_bfs.rs<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    let mut res = vec![];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    let mut visited = HashSet::new();\n    visited.insert(start_vet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    let mut que = VecDeque::new();\n    que.push_back(start_vet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while !que.is_empty() {\n        let vet = que.pop_front().unwrap(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n            for &amp;adj_vet in adj_vets {\n                if visited.contains(&amp;adj_vet) {\n                    continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n                }\n                que.push_back(adj_vet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n                visited.insert(adj_vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n            }\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res\n}\n</code></pre> graph_bfs.c<pre><code>/* \u8282\u70b9\u961f\u5217\u7ed3\u6784\u4f53 */\ntypedef struct {\n    Vertex *vertices[MAX_SIZE];\n    int front, rear, size;\n} Queue;\n\n/* \u6784\u9020\u51fd\u6570 */\nQueue *newQueue() {\n    Queue *q = (Queue *)malloc(sizeof(Queue));\n    q-&gt;front = q-&gt;rear = q-&gt;size = 0;\n    return q;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nint isEmpty(Queue *q) {\n    return q-&gt;size == 0;\n}\n\n/* \u5165\u961f\u64cd\u4f5c */\nvoid enqueue(Queue *q, Vertex *vet) {\n    q-&gt;vertices[q-&gt;rear] = vet;\n    q-&gt;rear = (q-&gt;rear + 1) % MAX_SIZE;\n    q-&gt;size++;\n}\n\n/* \u51fa\u961f\u64cd\u4f5c */\nVertex *dequeue(Queue *q) {\n    Vertex *vet = q-&gt;vertices[q-&gt;front];\n    q-&gt;front = (q-&gt;front + 1) % MAX_SIZE;\n    q-&gt;size--;\n    return vet;\n}\n\n/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **visited, int size, Vertex *vet) {\n    // \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\n    for (int i = 0; i &lt; size; i++) {\n        if (visited[i] == vet)\n            return 1;\n    }\n    return 0;\n}\n\n/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphBFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize, Vertex **visited, int *visitedSize) {\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    Queue *queue = newQueue();\n    enqueue(queue, startVet);\n    visited[(*visitedSize)++] = startVet;\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!isEmpty(queue)) {\n        Vertex *vet = dequeue(queue); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res[(*resSize)++] = vet;      // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        AdjListNode *node = findNode(graph, vet);\n        while (node != NULL) {\n            // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            if (!isVisited(visited, *visitedSize, node-&gt;vertex)) {\n                enqueue(queue, node-&gt;vertex);             // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n                visited[(*visitedSize)++] = node-&gt;vertex; // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n            }\n            node = node-&gt;next;\n        }\n    }\n    // \u91ca\u653e\u5185\u5b58\n    free(queue);\n}\n</code></pre> graph_bfs.kt<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfun graphBFS(graph: GraphAdjList, startVet: Vertex): List&lt;Vertex&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    val res: MutableList&lt;Vertex&gt; = ArrayList()\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    val visited: MutableSet&lt;Vertex&gt; = HashSet()\n    visited.add(startVet)\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    val que: Queue&lt;Vertex&gt; = LinkedList()\n    que.offer(startVet)\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!que.isEmpty()) {\n        val vet = que.poll() // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.add(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (adjVet in graph.adjList[vet]!!) {\n            if (visited.contains(adjVet)) continue  // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n\n            que.offer(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.rb<pre><code>[class]{}-[func]{graph_bfs}\n</code></pre> graph_bfs.zig<pre><code>[class]{}-[func]{graphBFS}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u4ee3\u7801\u76f8\u5bf9\u62bd\u8c61\uff0c\u5efa\u8bae\u5bf9\u7167\u56fe 9-10 \u6765\u52a0\u6df1\u7406\u89e3\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-10 \u00a0 \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u6b65\u9aa4 </p> <p>\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f</p> <p>\u4e0d\u552f\u4e00\u3002\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u53ea\u8981\u6c42\u6309\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u987a\u5e8f\u904d\u5386\uff0c\u800c\u591a\u4e2a\u76f8\u540c\u8ddd\u79bb\u7684\u9876\u70b9\u7684\u904d\u5386\u987a\u5e8f\u5141\u8bb8\u88ab\u4efb\u610f\u6253\u4e71\u3002\u4ee5\u56fe 9-10 \u4e3a\u4f8b\uff0c\u9876\u70b9 \\(1\\)\u3001\\(3\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\uff0c\u9876\u70b9 \\(2\\)\u3001\\(4\\)\u3001\\(6\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u4e5f\u53ef\u4ee5\u4efb\u610f\u4ea4\u6362\u3002</p>"},{"location":"chapter_graph/graph_traversal/#2","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u9876\u70b9\u90fd\u4f1a\u5165\u961f\u5e76\u51fa\u961f\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u5728\u904d\u5386\u90bb\u63a5\u9876\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002</p> <p>\u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5217\u8868 <code>res</code> \uff0c\u54c8\u5e0c\u8868 <code>visited</code> \uff0c\u961f\u5217 <code>que</code> \u4e2d\u7684\u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002</p>"},{"location":"chapter_graph/graph_traversal/#932","title":"9.3.2 \u00a0 \u6df1\u5ea6\u4f18\u5148\u904d\u5386","text":"<p>\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u518d\u56de\u5934\u7684\u904d\u5386\u65b9\u5f0f\u3002\u5982\u56fe 9-11 \u6240\u793a\uff0c\u4ece\u5de6\u4e0a\u89d2\u9876\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u5f53\u524d\u9876\u70b9\u7684\u67d0\u4e2a\u90bb\u63a5\u9876\u70b9\uff0c\u76f4\u5230\u8d70\u5230\u5c3d\u5934\u65f6\u8fd4\u56de\uff0c\u518d\u7ee7\u7eed\u8d70\u5230\u5c3d\u5934\u5e76\u8fd4\u56de\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u904d\u5386\u5b8c\u6210\u3002</p> <p></p> <p> \u56fe 9-11 \u00a0 \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386 </p>"},{"location":"chapter_graph/graph_traversal/#1_1","title":"1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"<p>\u8fd9\u79cd\u201c\u8d70\u5230\u5c3d\u5934\u518d\u8fd4\u56de\u201d\u7684\u7b97\u6cd5\u8303\u5f0f\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002\u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u5728\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u4e2d\uff0c\u6211\u4eec\u4e5f\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>visited</code> \u6765\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\uff0c\u4ee5\u907f\u514d\u91cd\u590d\u8bbf\u95ee\u9876\u70b9\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig graph_dfs.py<pre><code>def dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n    \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570\"\"\"\n    res.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    # \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for adjVet in graph.adj_list[vet]:\n        if adjVet in visited:\n            continue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        # \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet)\n\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\n    # \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    # \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res = []\n    # \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited = set[Vertex]()\n    dfs(graph, visited, res, start_vet)\n    return res\n</code></pre> graph_dfs.cpp<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList &amp;graph, unordered_set&lt;Vertex *&gt; &amp;visited, vector&lt;Vertex *&gt; &amp;res, Vertex *vet) {\n    res.push_back(vet);   // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.emplace(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (Vertex *adjVet : graph.adjList[vet]) {\n        if (visited.count(adjVet))\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector&lt;Vertex *&gt; graphDFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    vector&lt;Vertex *&gt; res;\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    unordered_set&lt;Vertex *&gt; visited;\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.java<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList graph, Set&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (Vertex adjVet : graph.adjList.get(vet)) {\n        if (visited.contains(adjVet))\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.cs<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid DFS(GraphAdjList graph, HashSet&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.Add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.Add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    foreach (Vertex adjVet in graph.adjList[vet]) {\n        if (visited.Contains(adjVet)) {\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9                             \n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        DFS(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; GraphDFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    HashSet&lt;Vertex&gt; visited = [];\n    DFS(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.go<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n    // append \u64cd\u4f5c\u4f1a\u8fd4\u56de\u65b0\u7684\u7684\u5f15\u7528\uff0c\u5fc5\u987b\u8ba9\u539f\u5f15\u7528\u91cd\u65b0\u8d4b\u503c\u4e3a\u65b0slice\u7684\u5f15\u7528\n    *res = append(*res, vet)\n    visited[vet] = struct{}{}\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for _, adjVet := range g.adjList[vet] {\n        _, isExist := visited[adjVet]\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        if !isExist {\n            dfs(g, visited, res, adjVet)\n        }\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res := make([]Vertex, 0)\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited := make(map[Vertex]struct{})\n    dfs(g, visited, &amp;res, startVet)\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_dfs.swift<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfunc dfs(graph: GraphAdjList, visited: inout Set&lt;Vertex&gt;, res: inout [Vertex], vet: Vertex) {\n    res.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.insert(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for adjVet in graph.adjList[vet] ?? [] {\n        if visited.contains(adjVet) {\n            continue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: adjVet)\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    var res: [Vertex] = []\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    var visited: Set&lt;Vertex&gt; = []\n    dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: startVet)\n    return res\n}\n</code></pre> graph_dfs.js<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction dfs(graph, visited, res, vet) {\n    res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph, startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.ts<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfunction dfs(\n    graph: GraphAdjList,\n    visited: Set&lt;Vertex&gt;,\n    res: Vertex[],\n    vet: Vertex\n): void {\n    res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res: Vertex[] = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited: Set&lt;Vertex&gt; = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.dart<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(\n  GraphAdjList graph,\n  Set&lt;Vertex&gt; visited,\n  List&lt;Vertex&gt; res,\n  Vertex vet,\n) {\n  res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n  visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n  // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n  for (Vertex adjVet in graph.adjList[vet]!) {\n    if (visited.contains(adjVet)) {\n      continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n    }\n    // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n    dfs(graph, visited, res, adjVet);\n  }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n  // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n  List&lt;Vertex&gt; res = [];\n  // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n  Set&lt;Vertex&gt; visited = {};\n  dfs(graph, visited, res, startVet);\n  return res;\n}\n</code></pre> graph_dfs.rs<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfn dfs(graph: &amp;GraphAdjList, visited: &amp;mut HashSet&lt;Vertex&gt;, res: &amp;mut Vec&lt;Vertex&gt;, vet: Vertex) {\n    res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.insert(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n                         // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n        for &amp;adj_vet in adj_vets {\n            if visited.contains(&amp;adj_vet) {\n                continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n            dfs(graph, visited, res, adj_vet);\n        }\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    let mut res = vec![];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    let mut visited = HashSet::new();\n    dfs(&amp;graph, &amp;mut visited, &amp;mut res, start_vet);\n\n    res\n}\n</code></pre> graph_dfs.c<pre><code>/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **res, int size, Vertex *vet) {\n    // \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\n    for (int i = 0; i &lt; size; i++) {\n        if (res[i] == vet) {\n            return 1;\n        }\n    }\n    return 0;\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList *graph, Vertex **res, int *resSize, Vertex *vet) {\n    // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    res[(*resSize)++] = vet;\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    AdjListNode *node = findNode(graph, vet);\n    while (node != NULL) {\n        // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        if (!isVisited(res, *resSize, node-&gt;vertex)) {\n            // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n            dfs(graph, res, resSize, node-&gt;vertex);\n        }\n        node = node-&gt;next;\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphDFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize) {\n    dfs(graph, res, resSize, startVet);\n}\n</code></pre> graph_dfs.kt<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u8f85\u52a9\u51fd\u6570 */\nfun dfs(\n    graph: GraphAdjList,\n    visited: MutableSet&lt;Vertex?&gt;,\n    res: MutableList&lt;Vertex?&gt;,\n    vet: Vertex?\n) {\n    res.add(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (adjVet in graph.adjList[vet]!!) {\n        if (visited.contains(adjVet)) continue  // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet)\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfun graphDFS(\n    graph: GraphAdjList,\n    startVet: Vertex?\n): List&lt;Vertex?&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    val res: MutableList&lt;Vertex?&gt; = ArrayList()\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    val visited: MutableSet&lt;Vertex?&gt; = HashSet()\n    dfs(graph, visited, res, startVet)\n    return res\n}\n</code></pre> graph_dfs.rb<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{graph_dfs}\n</code></pre> graph_dfs.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{graphDFS}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u56fe 9-12 \u6240\u793a\u3002</p> <ul> <li>\u76f4\u865a\u7ebf\u4ee3\u8868\u5411\u4e0b\u9012\u63a8\uff0c\u8868\u793a\u5f00\u542f\u4e86\u4e00\u4e2a\u65b0\u7684\u9012\u5f52\u65b9\u6cd5\u6765\u8bbf\u95ee\u65b0\u9876\u70b9\u3002</li> <li>\u66f2\u865a\u7ebf\u4ee3\u8868\u5411\u4e0a\u56de\u6eaf\uff0c\u8868\u793a\u6b64\u9012\u5f52\u65b9\u6cd5\u5df2\u7ecf\u8fd4\u56de\uff0c\u56de\u6eaf\u5230\u4e86\u5f00\u542f\u6b64\u65b9\u6cd5\u7684\u4f4d\u7f6e\u3002</li> </ul> <p>\u4e3a\u4e86\u52a0\u6df1\u7406\u89e3\uff0c\u5efa\u8bae\u5c06\u56fe 9-12 \u4e0e\u4ee3\u7801\u7ed3\u5408\u8d77\u6765\uff0c\u5728\u8111\u4e2d\u6a21\u62df\uff08\u6216\u8005\u7528\u7b14\u753b\u4e0b\u6765\uff09\u6574\u4e2a DFS \u8fc7\u7a0b\uff0c\u5305\u62ec\u6bcf\u4e2a\u9012\u5f52\u65b9\u6cd5\u4f55\u65f6\u5f00\u542f\u3001\u4f55\u65f6\u8fd4\u56de\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-12 \u00a0 \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u6b65\u9aa4 </p> <p>\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f</p> <p>\u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5e8f\u5217\u7684\u987a\u5e8f\u4e5f\u4e0d\u662f\u552f\u4e00\u7684\u3002\u7ed9\u5b9a\u67d0\u9876\u70b9\uff0c\u5148\u5f80\u54ea\u4e2a\u65b9\u5411\u63a2\u7d22\u90fd\u53ef\u4ee5\uff0c\u5373\u90bb\u63a5\u9876\u70b9\u7684\u987a\u5e8f\u53ef\u4ee5\u4efb\u610f\u6253\u4e71\uff0c\u90fd\u662f\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002</p> <p>\u4ee5\u6811\u7684\u904d\u5386\u4e3a\u4f8b\uff0c\u201c\u6839 \\(\\rightarrow\\) \u5de6 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u53f3 \\(\\rightarrow\\) \u6839\u201d\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\uff0c\u5b83\u4eec\u5c55\u793a\u4e86\u4e09\u79cd\u904d\u5386\u4f18\u5148\u7ea7\uff0c\u7136\u800c\u8fd9\u4e09\u8005\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002</p>"},{"location":"chapter_graph/graph_traversal/#2_1","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u9876\u70b9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(1\\) \u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002</p> <p>\u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5217\u8868 <code>res</code> \uff0c\u54c8\u5e0c\u8868 <code>visited</code> \u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u6700\u5927\u4e3a \\(|V|\\) \uff0c\u56e0\u6b64\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002</p>"},{"location":"chapter_graph/summary/","title":"9.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_graph/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u56fe\u7531\u9876\u70b9\u548c\u8fb9\u7ec4\u6210\uff0c\u53ef\u4ee5\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9\u548c\u4e00\u7ec4\u8fb9\u6784\u6210\u7684\u96c6\u5408\u3002</li> <li>\u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u56e0\u800c\u66f4\u4e3a\u590d\u6742\u3002</li> <li>\u6709\u5411\u56fe\u7684\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u8fde\u901a\u56fe\u4e2d\u7684\u4efb\u610f\u9876\u70b9\u5747\u53ef\u8fbe\uff0c\u6709\u6743\u56fe\u7684\u6bcf\u6761\u8fb9\u90fd\u5305\u542b\u6743\u91cd\u53d8\u91cf\u3002</li> <li>\u90bb\u63a5\u77e9\u9635\u5229\u7528\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u6709\u8fb9\u6216\u65e0\u8fb9\u3002\u90bb\u63a5\u77e9\u9635\u5728\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u4e0a\u6548\u7387\u5f88\u9ad8\uff0c\u4f46\u7a7a\u95f4\u5360\u7528\u8f83\u591a\u3002</li> <li>\u90bb\u63a5\u8868\u4f7f\u7528\u591a\u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u7b2c \\(i\\) \u4e2a\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u3002\u90bb\u63a5\u8868\u76f8\u5bf9\u4e8e\u90bb\u63a5\u77e9\u9635\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\uff0c\u4f46\u7531\u4e8e\u9700\u8981\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u65f6\u95f4\u6548\u7387\u8f83\u4f4e\u3002</li> <li>\u5f53\u90bb\u63a5\u8868\u4e2d\u7684\u94fe\u8868\u8fc7\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u5176\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6216\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002</li> <li>\u4ece\u7b97\u6cd5\u601d\u60f3\u7684\u89d2\u5ea6\u5206\u6790\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u4e86\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff0c\u90bb\u63a5\u8868\u4f53\u73b0\u4e86\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002</li> <li>\u56fe\u53ef\u7528\u4e8e\u5efa\u6a21\u5404\u7c7b\u73b0\u5b9e\u7cfb\u7edf\uff0c\u5982\u793e\u4ea4\u7f51\u7edc\u3001\u5730\u94c1\u7ebf\u8def\u7b49\u3002</li> <li>\u6811\u662f\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\uff0c\u6811\u7684\u904d\u5386\u4e5f\u662f\u56fe\u7684\u904d\u5386\u7684\u4e00\u79cd\u7279\u4f8b\u3002</li> <li>\u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u3001\u5c42\u5c42\u6269\u5f20\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u901a\u5e38\u501f\u52a9\u961f\u5217\u5b9e\u73b0\u3002</li> <li>\u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u65f6\u518d\u56de\u6eaf\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002</li> </ul>"},{"location":"chapter_graph/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u8def\u5f84\u7684\u5b9a\u4e49\u662f\u9876\u70b9\u5e8f\u5217\u8fd8\u662f\u8fb9\u5e8f\u5217\uff1f</p> <p>\u7ef4\u57fa\u767e\u79d1\u4e0a\u4e0d\u540c\u8bed\u8a00\u7248\u672c\u7684\u5b9a\u4e49\u4e0d\u4e00\u81f4\uff1a\u82f1\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u8fb9\u5e8f\u5217\u201d\uff0c\u800c\u4e2d\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u201d\u3002\u4ee5\u4e0b\u662f\u82f1\u6587\u7248\u539f\u6587\uff1aIn graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.</p> <p>\u5728\u672c\u6587\u4e2d\uff0c\u8def\u5f84\u88ab\u89c6\u4e3a\u4e00\u4e2a\u8fb9\u5e8f\u5217\uff0c\u800c\u4e0d\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u3002\u8fd9\u662f\u56e0\u4e3a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u53ef\u80fd\u5b58\u5728\u591a\u6761\u8fb9\u8fde\u63a5\uff0c\u6b64\u65f6\u6bcf\u6761\u8fb9\u90fd\u5bf9\u5e94\u4e00\u6761\u8def\u5f84\u3002</p> <p>Q\uff1a\u975e\u8fde\u901a\u56fe\u4e2d\u662f\u5426\u4f1a\u6709\u65e0\u6cd5\u904d\u5386\u5230\u7684\u70b9\uff1f</p> <p>\u5728\u975e\u8fde\u901a\u56fe\u4e2d\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\u3002\u904d\u5386\u975e\u8fde\u901a\u56fe\u9700\u8981\u8bbe\u7f6e\u591a\u4e2a\u8d77\u70b9\uff0c\u4ee5\u904d\u5386\u5230\u56fe\u7684\u6240\u6709\u8fde\u901a\u5206\u91cf\u3002</p> <p>Q\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\uff0c\u201c\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u6240\u6709\u9876\u70b9\u201d\u7684\u9876\u70b9\u987a\u5e8f\u662f\u5426\u6709\u8981\u6c42\uff1f</p> <p>\u53ef\u4ee5\u662f\u4efb\u610f\u987a\u5e8f\u3002\u4f46\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u53ef\u80fd\u9700\u8981\u6309\u7167\u6307\u5b9a\u89c4\u5219\u6765\u6392\u5e8f\uff0c\u6bd4\u5982\u6309\u7167\u9876\u70b9\u6dfb\u52a0\u7684\u6b21\u5e8f\uff0c\u6216\u8005\u6309\u7167\u9876\u70b9\u503c\u5927\u5c0f\u7684\u987a\u5e8f\u7b49\uff0c\u8fd9\u6837\u6709\u52a9\u4e8e\u5feb\u901f\u67e5\u627e\u201c\u5e26\u6709\u67d0\u79cd\u6781\u503c\u201d\u7684\u9876\u70b9\u3002</p>"},{"location":"chapter_greedy/","title":"\u7b2c 15 \u7ae0 \u00a0 \u8d2a\u5fc3","text":"<p>Abstract</p> <p>\u5411\u65e5\u8475\u671d\u7740\u592a\u9633\u8f6c\u52a8\uff0c\u65f6\u523b\u8ffd\u6c42\u81ea\u8eab\u6210\u957f\u7684\u6700\u5927\u53ef\u80fd\u3002</p> <p>\u8d2a\u5fc3\u7b56\u7565\u5728\u4e00\u8f6e\u8f6e\u7684\u7b80\u5355\u9009\u62e9\u4e2d\uff0c\u9010\u6b65\u5bfc\u5411\u6700\u4f73\u7b54\u6848\u3002</p>"},{"location":"chapter_greedy/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>15.1 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5</li> <li>15.2 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898</li> <li>15.3 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898</li> <li>15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898</li> <li>15.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_greedy/fractional_knapsack_problem/","title":"15.2 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ea\u80fd\u9009\u62e9\u4e00\u6b21\uff0c\u4f46\u53ef\u4ee5\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u4ef7\u503c\u6839\u636e\u9009\u62e9\u7684\u91cd\u91cf\u6bd4\u4f8b\u8ba1\u7b97\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002\u793a\u4f8b\u5982\u56fe 15-3 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 15-3 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u5206\u6570\u80cc\u5305\u95ee\u9898\u548c 0-1 \u80cc\u5305\u95ee\u9898\u6574\u4f53\u4e0a\u975e\u5e38\u76f8\u4f3c\uff0c\u72b6\u6001\u5305\u542b\u5f53\u524d\u7269\u54c1 \\(i\\) \u548c\u5bb9\u91cf \\(c\\) \uff0c\u76ee\u6807\u662f\u6c42\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u6700\u5927\u4ef7\u503c\u3002</p> <p>\u4e0d\u540c\u70b9\u5728\u4e8e\uff0c\u672c\u9898\u5141\u8bb8\u53ea\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\u3002\u5982\u56fe 15-4 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5bf9\u7269\u54c1\u4efb\u610f\u5730\u8fdb\u884c\u5207\u5206\uff0c\u5e76\u6309\u7167\u91cd\u91cf\u6bd4\u4f8b\u6765\u8ba1\u7b97\u76f8\u5e94\u4ef7\u503c\u3002</p> <ol> <li>\u5bf9\u4e8e\u7269\u54c1 \\(i\\) \uff0c\u5b83\u5728\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u4ef7\u503c\u4e3a \\(val[i-1] / wgt[i-1]\\) \uff0c\u7b80\u79f0\u5355\u4f4d\u4ef7\u503c\u3002</li> <li>\u5047\u8bbe\u653e\u5165\u4e00\u90e8\u5206\u7269\u54c1 \\(i\\) \uff0c\u91cd\u91cf\u4e3a \\(w\\) \uff0c\u5219\u80cc\u5305\u589e\u52a0\u7684\u4ef7\u503c\u4e3a \\(w \\times val[i-1] / wgt[i-1]\\) \u3002</li> </ol> <p></p> <p> \u56fe 15-4 \u00a0 \u7269\u54c1\u5728\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u4ef7\u503c </p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"<p>\u6700\u5927\u5316\u80cc\u5305\u5185\u7269\u54c1\u603b\u4ef7\u503c\uff0c\u672c\u8d28\u4e0a\u662f\u6700\u5927\u5316\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u7269\u54c1\u4ef7\u503c\u3002\u7531\u6b64\u4fbf\u53ef\u63a8\u7406\u51fa\u56fe 15-5 \u6240\u793a\u7684\u8d2a\u5fc3\u7b56\u7565\u3002</p> <ol> <li>\u5c06\u7269\u54c1\u6309\u7167\u5355\u4f4d\u4ef7\u503c\u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\u3002</li> <li>\u904d\u5386\u6240\u6709\u7269\u54c1\uff0c\u6bcf\u8f6e\u8d2a\u5fc3\u5730\u9009\u62e9\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\u7684\u7269\u54c1\u3002</li> <li>\u82e5\u5269\u4f59\u80cc\u5305\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u4f7f\u7528\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u586b\u6ee1\u80cc\u5305\u3002</li> </ol> <p></p> <p> \u56fe 15-5 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u8d2a\u5fc3\u7b56\u7565 </p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u6211\u4eec\u5efa\u7acb\u4e86\u4e00\u4e2a\u7269\u54c1\u7c7b <code>Item</code> \uff0c\u4ee5\u4fbf\u5c06\u7269\u54c1\u6309\u7167\u5355\u4f4d\u4ef7\u503c\u8fdb\u884c\u6392\u5e8f\u3002\u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u5f53\u80cc\u5305\u5df2\u6ee1\u65f6\u8df3\u51fa\u5e76\u8fd4\u56de\u89e3\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig fractional_knapsack.py<pre><code>class Item:\n    \"\"\"\u7269\u54c1\"\"\"\n\n    def __init__(self, w: int, v: int):\n        self.w = w  # \u7269\u54c1\u91cd\u91cf\n        self.v = v  # \u7269\u54c1\u4ef7\u503c\n\ndef fractional_knapsack(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"\u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3\"\"\"\n    # \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    items = [Item(w, v) for w, v in zip(wgt, val)]\n    # \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    items.sort(key=lambda item: item.v / item.w, reverse=True)\n    # \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    res = 0\n    for item in items:\n        if item.w &lt;= cap:\n            # \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v\n            cap -= item.w\n        else:\n            # \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (item.v / item.w) * cap\n            # \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break\n    return res\n</code></pre> fractional_knapsack.cpp<pre><code>/* \u7269\u54c1 */\nclass Item {\n  public:\n    int w; // \u7269\u54c1\u91cd\u91cf\n    int v; // \u7269\u54c1\u4ef7\u503c\n\n    Item(int w, int v) : w(w), v(v) {\n    }\n};\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    vector&lt;Item&gt; items;\n    for (int i = 0; i &lt; wgt.size(); i++) {\n        items.push_back(Item(wgt[i], val[i]));\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    sort(items.begin(), items.end(), [](Item &amp;a, Item &amp;b) { return (double)a.v / a.w &gt; (double)b.v / b.w; });\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    double res = 0;\n    for (auto &amp;item : items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (double)item.v / item.w * cap;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.java<pre><code>/* \u7269\u54c1 */\nclass Item {\n    int w; // \u7269\u54c1\u91cd\u91cf\n    int v; // \u7269\u54c1\u4ef7\u503c\n\n    public Item(int w, int v) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    Item[] items = new Item[wgt.length];\n    for (int i = 0; i &lt; wgt.length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    Arrays.sort(items, Comparator.comparingDouble(item -&gt; -((double) item.v / item.w)));\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    double res = 0;\n    for (Item item : items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (double) item.v / item.w * cap;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.cs<pre><code>/* \u7269\u54c1 */\nclass Item(int w, int v) {\n    public int w = w; // \u7269\u54c1\u91cd\u91cf\n    public int v = v; // \u7269\u54c1\u4ef7\u503c\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble FractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    Item[] items = new Item[wgt.Length];\n    for (int i = 0; i &lt; wgt.Length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    Array.Sort(items, (x, y) =&gt; (y.v / y.w).CompareTo(x.v / x.w));\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    double res = 0;\n    foreach (Item item in items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (double)item.v / item.w * cap;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.go<pre><code>/* \u7269\u54c1 */\ntype Item struct {\n    w int // \u7269\u54c1\u91cd\u91cf\n    v int // \u7269\u54c1\u4ef7\u503c\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunc fractionalKnapsack(wgt []int, val []int, cap int) float64 {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    items := make([]Item, len(wgt))\n    for i := 0; i &lt; len(wgt); i++ {\n        items[i] = Item{wgt[i], val[i]}\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    sort.Slice(items, func(i, j int) bool {\n        return float64(items[i].v)/float64(items[i].w) &gt; float64(items[j].v)/float64(items[j].w)\n    })\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    res := 0.0\n    for _, item := range items {\n        if item.w &lt;= cap {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += float64(item.v)\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += float64(item.v) / float64(item.w) * float64(cap)\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.swift<pre><code>/* \u7269\u54c1 */\nclass Item {\n    var w: Int // \u7269\u54c1\u91cd\u91cf\n    var v: Int // \u7269\u54c1\u4ef7\u503c\n\n    init(w: Int, v: Int) {\n        self.w = w\n        self.v = v\n    }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunc fractionalKnapsack(wgt: [Int], val: [Int], cap: Int) -&gt; Double {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    var items = zip(wgt, val).map { Item(w: $0, v: $1) }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    items.sort { -(Double($0.v) / Double($0.w)) &lt; -(Double($1.v) / Double($1.w)) }\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    var res = 0.0\n    var cap = cap\n    for item in items {\n        if item.w &lt;= cap {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += Double(item.v)\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += Double(item.v) / Double(item.w) * Double(cap)\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.js<pre><code>/* \u7269\u54c1 */\nclass Item {\n    constructor(w, v) {\n        this.w = w; // \u7269\u54c1\u91cd\u91cf\n        this.v = v; // \u7269\u54c1\u4ef7\u503c\n    }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunction fractionalKnapsack(wgt, val, cap) {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    const items = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (item.v / item.w) * cap;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.ts<pre><code>/* \u7269\u54c1 */\nclass Item {\n    w: number; // \u7269\u54c1\u91cd\u91cf\n    v: number; // \u7269\u54c1\u4ef7\u503c\n\n    constructor(w: number, v: number) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunction fractionalKnapsack(wgt: number[], val: number[], cap: number): number {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    const items: Item[] = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (item.v / item.w) * cap;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.dart<pre><code>/* \u7269\u54c1 */\nclass Item {\n  int w; // \u7269\u54c1\u91cd\u91cf\n  int v; // \u7269\u54c1\u4ef7\u503c\n\n  Item(this.w, this.v);\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n  List&lt;Item&gt; items = List.generate(wgt.length, (i) =&gt; Item(wgt[i], val[i]));\n  // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n  items.sort((a, b) =&gt; (b.v / b.w).compareTo(a.v / a.w));\n  // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n  double res = 0;\n  for (Item item in items) {\n    if (item.w &lt;= cap) {\n      // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n      res += item.v;\n      cap -= item.w;\n    } else {\n      // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n      res += item.v / item.w * cap;\n      // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n      break;\n    }\n  }\n  return res;\n}\n</code></pre> fractional_knapsack.rs<pre><code>/* \u7269\u54c1 */\nstruct Item {\n    w: i32, // \u7269\u54c1\u91cd\u91cf\n    v: i32, // \u7269\u54c1\u4ef7\u503c\n}\n\nimpl Item {\n    fn new(w: i32, v: i32) -&gt; Self {\n        Self { w, v }\n    }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfn fractional_knapsack(wgt: &amp;[i32], val: &amp;[i32], mut cap: i32) -&gt; f64 {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    let mut items = wgt\n        .iter()\n        .zip(val.iter())\n        .map(|(&amp;w, &amp;v)| Item::new(w, v))\n        .collect::&lt;Vec&lt;Item&gt;&gt;();\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    items.sort_by(|a, b| {\n        (b.v as f64 / b.w as f64)\n            .partial_cmp(&amp;(a.v as f64 / a.w as f64))\n            .unwrap()\n    });\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    let mut res = 0.0;\n    for item in &amp;items {\n        if item.w &lt;= cap {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v as f64;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += item.v as f64 / item.w as f64 * cap as f64;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    res\n}\n</code></pre> fractional_knapsack.c<pre><code>/* \u7269\u54c1 */\ntypedef struct {\n    int w; // \u7269\u54c1\u91cd\u91cf\n    int v; // \u7269\u54c1\u4ef7\u503c\n} Item;\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfloat fractionalKnapsack(int wgt[], int val[], int itemCount, int cap) {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    Item *items = malloc(sizeof(Item) * itemCount);\n    for (int i = 0; i &lt; itemCount; i++) {\n        items[i] = (Item){.w = wgt[i], .v = val[i]};\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    qsort(items, (size_t)itemCount, sizeof(Item), sortByValueDensity);\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    float res = 0.0;\n    for (int i = 0; i &lt; itemCount; i++) {\n        if (items[i].w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += items[i].v;\n            cap -= items[i].w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (float)cap / items[i].w * items[i].v;\n            cap = 0;\n            break;\n        }\n    }\n    free(items);\n    return res;\n}\n</code></pre> fractional_knapsack.kt<pre><code>/* \u7269\u54c1 */\nclass Item(\n    val w: Int, // \u7269\u54c1\n    val v: Int // \u7269\u54c1\u4ef7\u503c\n)\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfun fractionalKnapsack(\n    wgt: IntArray,\n    value: IntArray,\n    c: Int\n): Double {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    var cap = c\n    val items = arrayOfNulls&lt;Item&gt;(wgt.size)\n    for (i in wgt.indices) {\n        items[i] = Item(wgt[i], value[i])\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    Arrays.sort(items, Comparator.comparingDouble { item: Item -&gt; -(item.v.toDouble() / item.w) })\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    var res = 0.0\n    for (item in items) {\n        if (item!!.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v.toDouble()\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += item.v.toDouble() / item.w * cap\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break\n        }\n    }\n    return res\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfun fractionalKnapsack(\n    wgt: IntArray,\n    value: IntArray,\n    c: Int\n): Double {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    var cap = c\n    val items = arrayOfNulls&lt;Item&gt;(wgt.size)\n    for (i in wgt.indices) {\n        items[i] = Item(wgt[i], value[i])\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    Arrays.sort(items, Comparator.comparingDouble { item: Item -&gt; -(item.v.toDouble() / item.w) })\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    var res = 0.0\n    for (item in items) {\n        if (item!!.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v.toDouble()\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += item.v.toDouble() / item.w * cap\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.rb<pre><code>[class]{Item}-[func]{}\n\n[class]{}-[func]{fractional_knapsack}\n</code></pre> fractional_knapsack.zig<pre><code>[class]{Item}-[func]{}\n\n[class]{}-[func]{fractionalKnapsack}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u9664\u6392\u5e8f\u4e4b\u5916\uff0c\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u9700\u8981\u904d\u5386\u6574\u4e2a\u7269\u54c1\u5217\u8868\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u7269\u54c1\u6570\u91cf\u3002</p> <p>\u7531\u4e8e\u521d\u59cb\u5316\u4e86\u4e00\u4e2a <code>Item</code> \u5bf9\u8c61\u5217\u8868\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"<p>\u91c7\u7528\u53cd\u8bc1\u6cd5\u3002\u5047\u8bbe\u7269\u54c1 \\(x\\) \u662f\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\u7684\u7269\u54c1\uff0c\u4f7f\u7528\u67d0\u7b97\u6cd5\u6c42\u5f97\u6700\u5927\u4ef7\u503c\u4e3a <code>res</code> \uff0c\u4f46\u8be5\u89e3\u4e2d\u4e0d\u5305\u542b\u7269\u54c1 \\(x\\) \u3002</p> <p>\u73b0\u5728\u4ece\u80cc\u5305\u4e2d\u62ff\u51fa\u5355\u4f4d\u91cd\u91cf\u7684\u4efb\u610f\u7269\u54c1\uff0c\u5e76\u66ff\u6362\u4e3a\u5355\u4f4d\u91cd\u91cf\u7684\u7269\u54c1 \\(x\\) \u3002\u7531\u4e8e\u7269\u54c1 \\(x\\) \u7684\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\uff0c\u56e0\u6b64\u66ff\u6362\u540e\u7684\u603b\u4ef7\u503c\u4e00\u5b9a\u5927\u4e8e <code>res</code> \u3002\u8fd9\u4e0e <code>res</code> \u662f\u6700\u4f18\u89e3\u77db\u76fe\uff0c\u8bf4\u660e\u6700\u4f18\u89e3\u4e2d\u5fc5\u987b\u5305\u542b\u7269\u54c1 \\(x\\) \u3002</p> <p>\u5bf9\u4e8e\u8be5\u89e3\u4e2d\u7684\u5176\u4ed6\u7269\u54c1\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u6784\u5efa\u51fa\u4e0a\u8ff0\u77db\u76fe\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u5355\u4f4d\u4ef7\u503c\u66f4\u5927\u7684\u7269\u54c1\u603b\u662f\u66f4\u4f18\u9009\u62e9\uff0c\u8fd9\u8bf4\u660e\u8d2a\u5fc3\u7b56\u7565\u662f\u6709\u6548\u7684\u3002</p> <p>\u5982\u56fe 15-6 \u6240\u793a\uff0c\u5982\u679c\u5c06\u7269\u54c1\u91cd\u91cf\u548c\u7269\u54c1\u5355\u4f4d\u4ef7\u503c\u5206\u522b\u770b\u4f5c\u4e00\u5f20\u4e8c\u7ef4\u56fe\u8868\u7684\u6a2a\u8f74\u548c\u7eb5\u8f74\uff0c\u5219\u5206\u6570\u80cc\u5305\u95ee\u9898\u53ef\u8f6c\u5316\u4e3a\u201c\u6c42\u5728\u6709\u9650\u6a2a\u8f74\u533a\u95f4\u4e0b\u56f4\u6210\u7684\u6700\u5927\u9762\u79ef\u201d\u3002\u8fd9\u4e2a\u7c7b\u6bd4\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u4ece\u51e0\u4f55\u89d2\u5ea6\u7406\u89e3\u8d2a\u5fc3\u7b56\u7565\u7684\u6709\u6548\u6027\u3002</p> <p></p> <p> \u56fe 15-6 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u51e0\u4f55\u8868\u793a </p>"},{"location":"chapter_greedy/greedy_algorithm/","title":"15.1 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5","text":"<p>\u8d2a\u5fc3\u7b97\u6cd5\uff08greedy algorithm\uff09\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u89e3\u51b3\u4f18\u5316\u95ee\u9898\u7684\u7b97\u6cd5\uff0c\u5176\u57fa\u672c\u601d\u60f3\u662f\u5728\u95ee\u9898\u7684\u6bcf\u4e2a\u51b3\u7b56\u9636\u6bb5\uff0c\u90fd\u9009\u62e9\u5f53\u524d\u770b\u8d77\u6765\u6700\u4f18\u7684\u9009\u62e9\uff0c\u5373\u8d2a\u5fc3\u5730\u505a\u51fa\u5c40\u90e8\u6700\u4f18\u7684\u51b3\u7b56\uff0c\u4ee5\u671f\u83b7\u5f97\u5168\u5c40\u6700\u4f18\u89e3\u3002\u8d2a\u5fc3\u7b97\u6cd5\u7b80\u6d01\u4e14\u9ad8\u6548\uff0c\u5728\u8bb8\u591a\u5b9e\u9645\u95ee\u9898\u4e2d\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\u3002</p> <p>\u8d2a\u5fc3\u7b97\u6cd5\u548c\u52a8\u6001\u89c4\u5212\u90fd\u5e38\u7528\u4e8e\u89e3\u51b3\u4f18\u5316\u95ee\u9898\u3002\u5b83\u4eec\u4e4b\u95f4\u5b58\u5728\u4e00\u4e9b\u76f8\u4f3c\u4e4b\u5904\uff0c\u6bd4\u5982\u90fd\u4f9d\u8d56\u6700\u4f18\u5b50\u7ed3\u6784\u6027\u8d28\uff0c\u4f46\u5de5\u4f5c\u539f\u7406\u4e0d\u540c\u3002</p> <ul> <li>\u52a8\u6001\u89c4\u5212\u4f1a\u6839\u636e\u4e4b\u524d\u9636\u6bb5\u7684\u6240\u6709\u51b3\u7b56\u6765\u8003\u8651\u5f53\u524d\u51b3\u7b56\uff0c\u5e76\u4f7f\u7528\u8fc7\u53bb\u5b50\u95ee\u9898\u7684\u89e3\u6765\u6784\u5efa\u5f53\u524d\u5b50\u95ee\u9898\u7684\u89e3\u3002</li> <li>\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4f1a\u8003\u8651\u8fc7\u53bb\u7684\u51b3\u7b56\uff0c\u800c\u662f\u4e00\u8def\u5411\u524d\u5730\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u4e0d\u65ad\u7f29\u5c0f\u95ee\u9898\u8303\u56f4\uff0c\u76f4\u81f3\u95ee\u9898\u88ab\u89e3\u51b3\u3002</li> </ul> <p>\u6211\u4eec\u5148\u901a\u8fc7\u4f8b\u9898\u201c\u96f6\u94b1\u5151\u6362\u201d\u4e86\u89e3\u8d2a\u5fc3\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002\u8fd9\u9053\u9898\u5df2\u7ecf\u5728\u201c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u201d\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u76f8\u4fe1\u4f60\u5bf9\u5b83\u5e76\u4e0d\u964c\u751f\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u80fd\u591f\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u3002\u5982\u679c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002</p> <p>\u672c\u9898\u91c7\u53d6\u7684\u8d2a\u5fc3\u7b56\u7565\u5982\u56fe 15-1 \u6240\u793a\u3002\u7ed9\u5b9a\u76ee\u6807\u91d1\u989d\uff0c\u6211\u4eec\u8d2a\u5fc3\u5730\u9009\u62e9\u4e0d\u5927\u4e8e\u4e14\u6700\u63a5\u8fd1\u5b83\u7684\u786c\u5e01\uff0c\u4e0d\u65ad\u5faa\u73af\u8be5\u6b65\u9aa4\uff0c\u76f4\u81f3\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u4e3a\u6b62\u3002</p> <p></p> <p> \u56fe 15-1 \u00a0 \u96f6\u94b1\u5151\u6362\u7684\u8d2a\u5fc3\u7b56\u7565 </p> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig coin_change_greedy.py<pre><code>def coin_change_greedy(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3\"\"\"\n    # \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    i = len(coins) - 1\n    count = 0\n    # \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while amt &gt; 0:\n        # \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while i &gt; 0 and coins[i] &gt; amt:\n            i -= 1\n        # \u9009\u62e9 coins[i]\n        amt -= coins[i]\n        count += 1\n    # \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return count if amt == 0 else -1\n</code></pre> coin_change_greedy.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(vector&lt;int&gt; &amp;coins, int amt) {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    int i = coins.size() - 1;\n    int count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.java<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(int[] coins, int amt) {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    int i = coins.length - 1;\n    int count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.cs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint CoinChangeGreedy(int[] coins, int amt) {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    int i = coins.Length - 1;\n    int count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.go<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunc coinChangeGreedy(coins []int, amt int) int {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    i := len(coins) - 1\n    count := 0\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    for amt &gt; 0 {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        for i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i--\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i]\n        count++\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    if amt != 0 {\n        return -1\n    }\n    return count\n}\n</code></pre> coin_change_greedy.swift<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunc coinChangeGreedy(coins: [Int], amt: Int) -&gt; Int {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    var i = coins.count - 1\n    var count = 0\n    var amt = amt\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while amt &gt; 0 {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i]\n        count += 1\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt == 0 ? count : -1\n}\n</code></pre> coin_change_greedy.js<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunction coinChangeGreedy(coins, amt) {\n    // \u5047\u8bbe coins \u6570\u7ec4\u6709\u5e8f\n    let i = coins.length - 1;\n    let count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.ts<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunction coinChangeGreedy(coins: number[], amt: number): number {\n    // \u5047\u8bbe coins \u6570\u7ec4\u6709\u5e8f\n    let i = coins.length - 1;\n    let count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.dart<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(List&lt;int&gt; coins, int amt) {\n  // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n  int i = coins.length - 1;\n  int count = 0;\n  // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n  while (amt &gt; 0) {\n    // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n    while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n      i--;\n    }\n    // \u9009\u62e9 coins[i]\n    amt -= coins[i];\n    count++;\n  }\n  // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n  return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.rs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfn coin_change_greedy(coins: &amp;[i32], mut amt: i32) -&gt; i32 {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    let mut i = coins.len() - 1;\n    let mut count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while amt &gt; 0 {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count += 1;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    if amt == 0 {\n        count\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change_greedy.c<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(int *coins, int size, int amt) {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    int i = size - 1;\n    int count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.kt<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfun coinChangeGreedy(coins: IntArray, amt: Int): Int {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    var am = amt\n    var i = coins.size - 1\n    var count = 0\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (am &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; am) {\n            i--\n        }\n        // \u9009\u62e9 coins[i]\n        am -= coins[i]\n        count++\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return if (am == 0) count else -1\n}\n</code></pre> coin_change_greedy.rb<pre><code>[class]{}-[func]{coin_change_greedy}\n</code></pre> coin_change_greedy.zig<pre><code>[class]{}-[func]{coinChangeGreedy}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u4f60\u53ef\u80fd\u4f1a\u4e0d\u7531\u5730\u53d1\u51fa\u611f\u53f9\uff1aSo clean \uff01\u8d2a\u5fc3\u7b97\u6cd5\u4ec5\u7528\u7ea6\u5341\u884c\u4ee3\u7801\u5c31\u89e3\u51b3\u4e86\u96f6\u94b1\u5151\u6362\u95ee\u9898\u3002</p>"},{"location":"chapter_greedy/greedy_algorithm/#1511","title":"15.1.1 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5\u7684\u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4ec5\u64cd\u4f5c\u76f4\u63a5\u3001\u5b9e\u73b0\u7b80\u5355\uff0c\u800c\u4e14\u901a\u5e38\u6548\u7387\u4e5f\u5f88\u9ad8\u3002\u5728\u4ee5\u4e0a\u4ee3\u7801\u4e2d\uff0c\u8bb0\u786c\u5e01\u6700\u5c0f\u9762\u503c\u4e3a \\(\\min(coins)\\) \uff0c\u5219\u8d2a\u5fc3\u9009\u62e9\u6700\u591a\u5faa\u73af \\(amt / \\min(coins)\\) \u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(amt / \\min(coins))\\) \u3002\u8fd9\u6bd4\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\times amt)\\) \u5c0f\u4e86\u4e00\u4e2a\u6570\u91cf\u7ea7\u3002</p> <p>\u7136\u800c\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u786c\u5e01\u9762\u503c\u7ec4\u5408\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u5e76\u4e0d\u80fd\u627e\u5230\u6700\u4f18\u89e3\u3002\u56fe 15-2 \u7ed9\u51fa\u4e86\u4e24\u4e2a\u793a\u4f8b\u3002</p> <ul> <li>\u6b63\u4f8b \\(coins = [1, 5, 10, 20, 50, 100]\\)\uff1a\u5728\u8be5\u786c\u5e01\u7ec4\u5408\u4e0b\uff0c\u7ed9\u5b9a\u4efb\u610f \\(amt\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u90fd\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u53cd\u4f8b \\(coins = [1, 20, 50]\\)\uff1a\u5047\u8bbe \\(amt = 60\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 10\\) \u7684\u5151\u6362\u7ec4\u5408\uff0c\u5171\u8ba1 \\(11\\) \u679a\u786c\u5e01\uff0c\u4f46\u52a8\u6001\u89c4\u5212\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3 \\(20 + 20 + 20\\) \uff0c\u4ec5\u9700 \\(3\\) \u679a\u786c\u5e01\u3002</li> <li>\u53cd\u4f8b \\(coins = [1, 49, 50]\\)\uff1a\u5047\u8bbe \\(amt = 98\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 48\\) \u7684\u5151\u6362\u7ec4\u5408\uff0c\u5171\u8ba1 \\(49\\) \u679a\u786c\u5e01\uff0c\u4f46\u52a8\u6001\u89c4\u5212\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3 \\(49 + 49\\) \uff0c\u4ec5\u9700 \\(2\\) \u679a\u786c\u5e01\u3002</li> </ul> <p></p> <p> \u56fe 15-2 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5\u65e0\u6cd5\u627e\u51fa\u6700\u4f18\u89e3\u7684\u793a\u4f8b </p> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5bf9\u4e8e\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u65e0\u6cd5\u4fdd\u8bc1\u627e\u5230\u5168\u5c40\u6700\u4f18\u89e3\uff0c\u5e76\u4e14\u6709\u53ef\u80fd\u627e\u5230\u975e\u5e38\u5dee\u7684\u89e3\u3002\u5b83\u66f4\u9002\u5408\u7528\u52a8\u6001\u89c4\u5212\u89e3\u51b3\u3002</p> <p>\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u9002\u7528\u60c5\u51b5\u5206\u4ee5\u4e0b\u4e24\u79cd\u3002</p> <ol> <li>\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1a\u8d2a\u5fc3\u7b97\u6cd5\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u5f80\u5f80\u662f\u6700\u4f18\u9009\u62e9\uff0c\u56e0\u4e3a\u5b83\u5f80\u5f80\u6bd4\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u66f4\u9ad8\u6548\u3002</li> <li>\u53ef\u4ee5\u627e\u5230\u8fd1\u4f3c\u6700\u4f18\u89e3\uff1a\u8d2a\u5fc3\u7b97\u6cd5\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u4e5f\u662f\u53ef\u7528\u7684\u3002\u5bf9\u4e8e\u5f88\u591a\u590d\u6742\u95ee\u9898\u6765\u8bf4\uff0c\u5bfb\u627e\u5168\u5c40\u6700\u4f18\u89e3\u975e\u5e38\u56f0\u96be\uff0c\u80fd\u4ee5\u8f83\u9ad8\u6548\u7387\u627e\u5230\u6b21\u4f18\u89e3\u4e5f\u662f\u975e\u5e38\u4e0d\u9519\u7684\u3002</li> </ol>"},{"location":"chapter_greedy/greedy_algorithm/#1512","title":"15.1.2 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5\u7279\u6027","text":"<p>\u90a3\u4e48\u95ee\u9898\u6765\u4e86\uff0c\u4ec0\u4e48\u6837\u7684\u95ee\u9898\u9002\u5408\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u5462\uff1f\u6216\u8005\u8bf4\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1f</p> <p>\u76f8\u8f83\u4e8e\u52a8\u6001\u89c4\u5212\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u4f7f\u7528\u6761\u4ef6\u66f4\u52a0\u82db\u523b\uff0c\u5176\u4e3b\u8981\u5173\u6ce8\u95ee\u9898\u7684\u4e24\u4e2a\u6027\u8d28\u3002</p> <ul> <li>\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\uff1a\u53ea\u6709\u5f53\u5c40\u90e8\u6700\u4f18\u9009\u62e9\u59cb\u7ec8\u53ef\u4ee5\u5bfc\u81f4\u5168\u5c40\u6700\u4f18\u89e3\u65f6\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u624d\u80fd\u4fdd\u8bc1\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u5305\u542b\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u3002</li> </ul> <p>\u6700\u4f18\u5b50\u7ed3\u6784\u5df2\u7ecf\u5728\u201c\u52a8\u6001\u89c4\u5212\u201d\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u8fd9\u91cc\u4e0d\u518d\u8d58\u8ff0\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u4e00\u4e9b\u95ee\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784\u5e76\u4e0d\u660e\u663e\uff0c\u4f46\u4ecd\u7136\u53ef\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u89e3\u51b3\u3002</p> <p>\u6211\u4eec\u4e3b\u8981\u63a2\u7a76\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u7684\u5224\u65ad\u65b9\u6cd5\u3002\u867d\u7136\u5b83\u7684\u63cf\u8ff0\u770b\u4e0a\u53bb\u6bd4\u8f83\u7b80\u5355\uff0c\u4f46\u5b9e\u9645\u4e0a\u5bf9\u4e8e\u8bb8\u591a\u95ee\u9898\uff0c\u8bc1\u660e\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u5e76\u975e\u6613\u4e8b\u3002</p> <p>\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff0c\u6211\u4eec\u867d\u7136\u80fd\u591f\u5bb9\u6613\u5730\u4e3e\u51fa\u53cd\u4f8b\uff0c\u5bf9\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u8fdb\u884c\u8bc1\u4f2a\uff0c\u4f46\u8bc1\u5b9e\u7684\u96be\u5ea6\u8f83\u5927\u3002\u5982\u679c\u95ee\uff1a\u6ee1\u8db3\u4ec0\u4e48\u6761\u4ef6\u7684\u786c\u5e01\u7ec4\u5408\u53ef\u4ee5\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\uff1f\u6211\u4eec\u5f80\u5f80\u53ea\u80fd\u51ed\u501f\u76f4\u89c9\u6216\u4e3e\u4f8b\u5b50\u6765\u7ed9\u51fa\u4e00\u4e2a\u6a21\u68f1\u4e24\u53ef\u7684\u7b54\u6848\uff0c\u800c\u96be\u4ee5\u7ed9\u51fa\u4e25\u8c28\u7684\u6570\u5b66\u8bc1\u660e\u3002</p> <p>Quote</p> <p>\u6709\u4e00\u7bc7\u8bba\u6587\u7ed9\u51fa\u4e86\u4e00\u4e2a \\(O(n^3)\\) \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u7b97\u6cd5\uff0c\u7528\u4e8e\u5224\u65ad\u4e00\u4e2a\u786c\u5e01\u7ec4\u5408\u80fd\u5426\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u627e\u51fa\u4efb\u610f\u91d1\u989d\u7684\u6700\u4f18\u89e3\u3002</p> <p>Pearson, D. A polynomial-time algorithm for the change-making problem[J]. Operations Research Letters, 2005, 33(3): 231-234.</p>"},{"location":"chapter_greedy/greedy_algorithm/#1513","title":"15.1.3 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5\u89e3\u9898\u6b65\u9aa4","text":"<p>\u8d2a\u5fc3\u95ee\u9898\u7684\u89e3\u51b3\u6d41\u7a0b\u5927\u4f53\u53ef\u5206\u4e3a\u4ee5\u4e0b\u4e09\u6b65\u3002</p> <ol> <li>\u95ee\u9898\u5206\u6790\uff1a\u68b3\u7406\u4e0e\u7406\u89e3\u95ee\u9898\u7279\u6027\uff0c\u5305\u62ec\u72b6\u6001\u5b9a\u4e49\u3001\u4f18\u5316\u76ee\u6807\u548c\u7ea6\u675f\u6761\u4ef6\u7b49\u3002\u8fd9\u4e00\u6b65\u5728\u56de\u6eaf\u548c\u52a8\u6001\u89c4\u5212\u4e2d\u90fd\u6709\u6d89\u53ca\u3002</li> <li>\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\uff1a\u786e\u5b9a\u5982\u4f55\u5728\u6bcf\u4e00\u6b65\u4e2d\u505a\u51fa\u8d2a\u5fc3\u9009\u62e9\u3002\u8fd9\u4e2a\u7b56\u7565\u80fd\u591f\u5728\u6bcf\u4e00\u6b65\u51cf\u5c0f\u95ee\u9898\u7684\u89c4\u6a21\uff0c\u5e76\u6700\u7ec8\u89e3\u51b3\u6574\u4e2a\u95ee\u9898\u3002</li> <li>\u6b63\u786e\u6027\u8bc1\u660e\uff1a\u901a\u5e38\u9700\u8981\u8bc1\u660e\u95ee\u9898\u5177\u6709\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u3002\u8fd9\u4e2a\u6b65\u9aa4\u53ef\u80fd\u9700\u8981\u7528\u5230\u6570\u5b66\u8bc1\u660e\uff0c\u4f8b\u5982\u5f52\u7eb3\u6cd5\u6216\u53cd\u8bc1\u6cd5\u7b49\u3002</li> </ol> <p>\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u662f\u6c42\u89e3\u95ee\u9898\u7684\u6838\u5fc3\u6b65\u9aa4\uff0c\u4f46\u5b9e\u65bd\u8d77\u6765\u53ef\u80fd\u5e76\u4e0d\u5bb9\u6613\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u4e0d\u540c\u95ee\u9898\u7684\u8d2a\u5fc3\u7b56\u7565\u7684\u5dee\u5f02\u8f83\u5927\u3002\u5bf9\u4e8e\u8bb8\u591a\u95ee\u9898\u6765\u8bf4\uff0c\u8d2a\u5fc3\u7b56\u7565\u6bd4\u8f83\u6d45\u663e\uff0c\u6211\u4eec\u901a\u8fc7\u4e00\u4e9b\u5927\u6982\u7684\u601d\u8003\u4e0e\u5c1d\u8bd5\u5c31\u80fd\u5f97\u51fa\u3002\u800c\u5bf9\u4e8e\u4e00\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u8d2a\u5fc3\u7b56\u7565\u53ef\u80fd\u975e\u5e38\u9690\u853d\uff0c\u8fd9\u79cd\u60c5\u51b5\u5c31\u975e\u5e38\u8003\u9a8c\u4e2a\u4eba\u7684\u89e3\u9898\u7ecf\u9a8c\u4e0e\u7b97\u6cd5\u80fd\u529b\u4e86\u3002</li> <li>\u67d0\u4e9b\u8d2a\u5fc3\u7b56\u7565\u5177\u6709\u8f83\u5f3a\u7684\u8ff7\u60d1\u6027\u3002\u5f53\u6211\u4eec\u6ee1\u6000\u4fe1\u5fc3\u8bbe\u8ba1\u597d\u8d2a\u5fc3\u7b56\u7565\uff0c\u5199\u51fa\u89e3\u9898\u4ee3\u7801\u5e76\u63d0\u4ea4\u8fd0\u884c\uff0c\u5f88\u53ef\u80fd\u53d1\u73b0\u90e8\u5206\u6d4b\u8bd5\u6837\u4f8b\u65e0\u6cd5\u901a\u8fc7\u3002\u8fd9\u662f\u56e0\u4e3a\u8bbe\u8ba1\u7684\u8d2a\u5fc3\u7b56\u7565\u53ea\u662f\u201c\u90e8\u5206\u6b63\u786e\u201d\u7684\uff0c\u4e0a\u6587\u4ecb\u7ecd\u7684\u96f6\u94b1\u5151\u6362\u5c31\u662f\u4e00\u4e2a\u5178\u578b\u6848\u4f8b\u3002</li> </ul> <p>\u4e3a\u4e86\u4fdd\u8bc1\u6b63\u786e\u6027\uff0c\u6211\u4eec\u5e94\u8be5\u5bf9\u8d2a\u5fc3\u7b56\u7565\u8fdb\u884c\u4e25\u8c28\u7684\u6570\u5b66\u8bc1\u660e\uff0c\u901a\u5e38\u9700\u8981\u7528\u5230\u53cd\u8bc1\u6cd5\u6216\u6570\u5b66\u5f52\u7eb3\u6cd5\u3002</p> <p>\u7136\u800c\uff0c\u6b63\u786e\u6027\u8bc1\u660e\u4e5f\u5f88\u53ef\u80fd\u4e0d\u662f\u4e00\u4ef6\u6613\u4e8b\u3002\u5982\u82e5\u6ca1\u6709\u5934\u7eea\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9009\u62e9\u9762\u5411\u6d4b\u8bd5\u7528\u4f8b\u8fdb\u884c\u4ee3\u7801\u8c03\u8bd5\uff0c\u4e00\u6b65\u6b65\u4fee\u6539\u4e0e\u9a8c\u8bc1\u8d2a\u5fc3\u7b56\u7565\u3002</p>"},{"location":"chapter_greedy/greedy_algorithm/#1514","title":"15.1.4 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5\u5178\u578b\u4f8b\u9898","text":"<p>\u8d2a\u5fc3\u7b97\u6cd5\u5e38\u5e38\u5e94\u7528\u5728\u6ee1\u8db3\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u4f18\u5316\u95ee\u9898\u4e2d\uff0c\u4ee5\u4e0b\u5217\u4e3e\u4e86\u4e00\u4e9b\u5178\u578b\u7684\u8d2a\u5fc3\u7b97\u6cd5\u95ee\u9898\u3002</p> <ul> <li>\u786c\u5e01\u627e\u96f6\u95ee\u9898\uff1a\u5728\u67d0\u4e9b\u786c\u5e01\u7ec4\u5408\u4e0b\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u603b\u662f\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u533a\u95f4\u8c03\u5ea6\u95ee\u9898\uff1a\u5047\u8bbe\u4f60\u6709\u4e00\u4e9b\u4efb\u52a1\uff0c\u6bcf\u4e2a\u4efb\u52a1\u5728\u4e00\u6bb5\u65f6\u95f4\u5185\u8fdb\u884c\uff0c\u4f60\u7684\u76ee\u6807\u662f\u5b8c\u6210\u5c3d\u53ef\u80fd\u591a\u7684\u4efb\u52a1\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9009\u62e9\u7ed3\u675f\u65f6\u95f4\u6700\u65e9\u7684\u4efb\u52a1\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u5c31\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u5206\u6570\u80cc\u5305\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u7269\u54c1\u548c\u4e00\u4e2a\u8f7d\u91cd\u91cf\uff0c\u4f60\u7684\u76ee\u6807\u662f\u9009\u62e9\u4e00\u7ec4\u7269\u54c1\uff0c\u4f7f\u5f97\u603b\u91cd\u91cf\u4e0d\u8d85\u8fc7\u8f7d\u91cd\u91cf\uff0c\u4e14\u603b\u4ef7\u503c\u6700\u5927\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9009\u62e9\u6027\u4ef7\u6bd4\u6700\u9ad8\uff08\u4ef7\u503c / \u91cd\u91cf\uff09\u7684\u7269\u54c1\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u5728\u4e00\u4e9b\u60c5\u51b5\u4e0b\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u80a1\u7968\u4e70\u5356\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u80a1\u7968\u7684\u5386\u53f2\u4ef7\u683c\uff0c\u4f60\u53ef\u4ee5\u8fdb\u884c\u591a\u6b21\u4e70\u5356\uff0c\u4f46\u5982\u679c\u4f60\u5df2\u7ecf\u6301\u6709\u80a1\u7968\uff0c\u90a3\u4e48\u5728\u5356\u51fa\u4e4b\u524d\u4e0d\u80fd\u518d\u4e70\uff0c\u76ee\u6807\u662f\u83b7\u53d6\u6700\u5927\u5229\u6da6\u3002</li> <li>\u970d\u592b\u66fc\u7f16\u7801\uff1a\u970d\u592b\u66fc\u7f16\u7801\u662f\u4e00\u79cd\u7528\u4e8e\u65e0\u635f\u6570\u636e\u538b\u7f29\u7684\u8d2a\u5fc3\u7b97\u6cd5\u3002\u901a\u8fc7\u6784\u5efa\u970d\u592b\u66fc\u6811\uff0c\u6bcf\u6b21\u9009\u62e9\u51fa\u73b0\u9891\u7387\u6700\u4f4e\u7684\u4e24\u4e2a\u8282\u70b9\u5408\u5e76\uff0c\u6700\u540e\u5f97\u5230\u7684\u970d\u592b\u66fc\u6811\u7684\u5e26\u6743\u8def\u5f84\u957f\u5ea6\uff08\u7f16\u7801\u957f\u5ea6\uff09\u6700\u5c0f\u3002</li> <li>Dijkstra \u7b97\u6cd5\uff1a\u5b83\u662f\u4e00\u79cd\u89e3\u51b3\u7ed9\u5b9a\u6e90\u9876\u70b9\u5230\u5176\u4f59\u5404\u9876\u70b9\u7684\u6700\u77ed\u8def\u5f84\u95ee\u9898\u7684\u8d2a\u5fc3\u7b97\u6cd5\u3002</li> </ul>"},{"location":"chapter_greedy/max_capacity_problem/","title":"15.3 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898","text":"<p>Question</p> <p>\u8f93\u5165\u4e00\u4e2a\u6570\u7ec4 \\(ht\\) \uff0c\u5176\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u4ee3\u8868\u4e00\u4e2a\u5782\u76f4\u9694\u677f\u7684\u9ad8\u5ea6\u3002\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u9694\u677f\uff0c\u4ee5\u53ca\u5b83\u4eec\u4e4b\u95f4\u7684\u7a7a\u95f4\u53ef\u4ee5\u7ec4\u6210\u4e00\u4e2a\u5bb9\u5668\u3002</p> <p>\u5bb9\u5668\u7684\u5bb9\u91cf\u7b49\u4e8e\u9ad8\u5ea6\u548c\u5bbd\u5ea6\u7684\u4e58\u79ef\uff08\u9762\u79ef\uff09\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u8f83\u77ed\u7684\u9694\u677f\u51b3\u5b9a\uff0c\u5bbd\u5ea6\u662f\u4e24\u4e2a\u9694\u677f\u7684\u6570\u7ec4\u7d22\u5f15\u4e4b\u5dee\u3002</p> <p>\u8bf7\u5728\u6570\u7ec4\u4e2d\u9009\u62e9\u4e24\u4e2a\u9694\u677f\uff0c\u4f7f\u5f97\u7ec4\u6210\u7684\u5bb9\u5668\u7684\u5bb9\u91cf\u6700\u5927\uff0c\u8fd4\u56de\u6700\u5927\u5bb9\u91cf\u3002\u793a\u4f8b\u5982\u56fe 15-7 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 15-7 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u5bb9\u5668\u7531\u4efb\u610f\u4e24\u4e2a\u9694\u677f\u56f4\u6210\uff0c\u56e0\u6b64\u672c\u9898\u7684\u72b6\u6001\u4e3a\u4e24\u4e2a\u9694\u677f\u7684\u7d22\u5f15\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002</p> <p>\u6839\u636e\u9898\u610f\uff0c\u5bb9\u91cf\u7b49\u4e8e\u9ad8\u5ea6\u4e58\u4ee5\u5bbd\u5ea6\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u77ed\u677f\u51b3\u5b9a\uff0c\u5bbd\u5ea6\u662f\u4e24\u9694\u677f\u7684\u6570\u7ec4\u7d22\u5f15\u4e4b\u5dee\u3002\u8bbe\u5bb9\u91cf\u4e3a \\(cap[i, j]\\) \uff0c\u5219\u53ef\u5f97\u8ba1\u7b97\u516c\u5f0f\uff1a</p> \\[ cap[i, j] = \\min(ht[i], ht[j]) \\times (j - i) \\] <p>\u8bbe\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u4e24\u4e2a\u9694\u677f\u7684\u7ec4\u5408\u6570\u91cf\uff08\u72b6\u6001\u603b\u6570\uff09\u4e3a \\(C_n^2 = \\frac{n(n - 1)}{2}\\) \u4e2a\u3002\u6700\u76f4\u63a5\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u7a77\u4e3e\u6240\u6709\u72b6\u6001\uff0c\u4ece\u800c\u6c42\u5f97\u6700\u5927\u5bb9\u91cf\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_greedy/max_capacity_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"<p>\u8fd9\u9053\u9898\u8fd8\u6709\u66f4\u9ad8\u6548\u7387\u7684\u89e3\u6cd5\u3002\u5982\u56fe 15-8 \u6240\u793a\uff0c\u73b0\u9009\u53d6\u4e00\u4e2a\u72b6\u6001 \\([i, j]\\) \uff0c\u5176\u6ee1\u8db3\u7d22\u5f15 \\(i &lt; j\\) \u4e14\u9ad8\u5ea6 \\(ht[i] &lt; ht[j]\\) \uff0c\u5373 \\(i\\) \u4e3a\u77ed\u677f\u3001\\(j\\) \u4e3a\u957f\u677f\u3002</p> <p></p> <p> \u56fe 15-8 \u00a0 \u521d\u59cb\u72b6\u6001 </p> <p>\u5982\u56fe 15-9 \u6240\u793a\uff0c\u82e5\u6b64\u65f6\u5c06\u957f\u677f \\(j\\) \u5411\u77ed\u677f \\(i\\) \u9760\u8fd1\uff0c\u5219\u5bb9\u91cf\u4e00\u5b9a\u53d8\u5c0f\u3002</p> <p>\u8fd9\u662f\u56e0\u4e3a\u5728\u79fb\u52a8\u957f\u677f \\(j\\) \u540e\uff0c\u5bbd\u5ea6 \\(j-i\\) \u80af\u5b9a\u53d8\u5c0f\uff1b\u800c\u9ad8\u5ea6\u7531\u77ed\u677f\u51b3\u5b9a\uff0c\u56e0\u6b64\u9ad8\u5ea6\u53ea\u53ef\u80fd\u4e0d\u53d8\uff08 \\(i\\) \u4ecd\u4e3a\u77ed\u677f\uff09\u6216\u53d8\u5c0f\uff08\u79fb\u52a8\u540e\u7684 \\(j\\) \u6210\u4e3a\u77ed\u677f\uff09\u3002</p> <p></p> <p> \u56fe 15-9 \u00a0 \u5411\u5185\u79fb\u52a8\u957f\u677f\u540e\u7684\u72b6\u6001 </p> <p>\u53cd\u5411\u601d\u8003\uff0c\u6211\u4eec\u53ea\u6709\u5411\u5185\u6536\u7f29\u77ed\u677f \\(i\\) \uff0c\u624d\u6709\u53ef\u80fd\u4f7f\u5bb9\u91cf\u53d8\u5927\u3002\u56e0\u4e3a\u867d\u7136\u5bbd\u5ea6\u4e00\u5b9a\u53d8\u5c0f\uff0c\u4f46\u9ad8\u5ea6\u53ef\u80fd\u4f1a\u53d8\u5927\uff08\u79fb\u52a8\u540e\u7684\u77ed\u677f \\(i\\) \u53ef\u80fd\u4f1a\u53d8\u957f\uff09\u3002\u4f8b\u5982\u5728\u56fe 15-10 \u4e2d\uff0c\u79fb\u52a8\u77ed\u677f\u540e\u9762\u79ef\u53d8\u5927\u3002</p> <p></p> <p> \u56fe 15-10 \u00a0 \u5411\u5185\u79fb\u52a8\u77ed\u677f\u540e\u7684\u72b6\u6001 </p> <p>\u7531\u6b64\u4fbf\u53ef\u63a8\u51fa\u672c\u9898\u7684\u8d2a\u5fc3\u7b56\u7565\uff1a\u521d\u59cb\u5316\u4e24\u6307\u9488\uff0c\u4f7f\u5176\u5206\u5217\u5bb9\u5668\u4e24\u7aef\uff0c\u6bcf\u8f6e\u5411\u5185\u6536\u7f29\u77ed\u677f\u5bf9\u5e94\u7684\u6307\u9488\uff0c\u76f4\u81f3\u4e24\u6307\u9488\u76f8\u9047\u3002</p> <p>\u56fe 15-11 \u5c55\u793a\u4e86\u8d2a\u5fc3\u7b56\u7565\u7684\u6267\u884c\u8fc7\u7a0b\u3002</p> <ol> <li>\u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5206\u5217\u6570\u7ec4\u4e24\u7aef\u3002</li> <li>\u8ba1\u7b97\u5f53\u524d\u72b6\u6001\u7684\u5bb9\u91cf \\(cap[i, j]\\) \uff0c\u5e76\u66f4\u65b0\u6700\u5927\u5bb9\u91cf\u3002</li> <li>\u6bd4\u8f83\u677f \\(i\\) \u548c \u677f \\(j\\) \u7684\u9ad8\u5ea6\uff0c\u5e76\u5c06\u77ed\u677f\u5411\u5185\u79fb\u52a8\u4e00\u683c\u3002</li> <li>\u5faa\u73af\u6267\u884c\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\uff0c\u76f4\u81f3 \\(i\\) \u548c \\(j\\) \u76f8\u9047\u65f6\u7ed3\u675f\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 15-11 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898\u7684\u8d2a\u5fc3\u8fc7\u7a0b </p>"},{"location":"chapter_greedy/max_capacity_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u4ee3\u7801\u5faa\u73af\u6700\u591a \\(n\\) \u8f6e\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> <p>\u53d8\u91cf \\(i\\)\u3001\\(j\\)\u3001\\(res\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig max_capacity.py<pre><code>def max_capacity(ht: list[int]) -&gt; int:\n    \"\"\"\u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3\"\"\"\n    # \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    i, j = 0, len(ht) - 1\n    # \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    res = 0\n    # \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while i &lt; j:\n        # \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        # \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if ht[i] &lt; ht[j]:\n            i += 1\n        else:\n            j -= 1\n    return res\n</code></pre> max_capacity.cpp<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(vector&lt;int&gt; &amp;ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    int i = 0, j = ht.size() - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    int res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int cap = min(ht[i], ht[j]) * (j - i);\n        res = max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.java<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(int[] ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    int i = 0, j = ht.length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    int res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.cs<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint MaxCapacity(int[] ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    int i = 0, j = ht.Length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    int res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int cap = Math.Min(ht[i], ht[j]) * (j - i);\n        res = Math.Max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.go<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunc maxCapacity(ht []int) int {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    i, j := 0, len(ht)-1\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    res := 0\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    for i &lt; j {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        capacity := int(math.Min(float64(ht[i]), float64(ht[j]))) * (j - i)\n        res = int(math.Max(float64(res), float64(capacity)))\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if ht[i] &lt; ht[j] {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.swift<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunc maxCapacity(ht: [Int]) -&gt; Int {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    var i = ht.startIndex, j = ht.endIndex - 1\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    var res = 0\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while i &lt; j {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        let cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if ht[i] &lt; ht[j] {\n            i += 1\n        } else {\n            j -= 1\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.js<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunction maxCapacity(ht) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    let i = 0,\n        j = ht.length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    let res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        const cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.ts<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunction maxCapacity(ht: number[]): number {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    let i = 0,\n        j = ht.length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    let res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        const cap: number = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.dart<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(List&lt;int&gt; ht) {\n  // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n  int i = 0, j = ht.length - 1;\n  // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n  int res = 0;\n  // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n  while (i &lt; j) {\n    // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n    int cap = min(ht[i], ht[j]) * (j - i);\n    res = max(res, cap);\n    // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n    if (ht[i] &lt; ht[j]) {\n      i++;\n    } else {\n      j--;\n    }\n  }\n  return res;\n}\n</code></pre> max_capacity.rs<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfn max_capacity(ht: &amp;[i32]) -&gt; i32 {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    let mut i = 0;\n    let mut j = ht.len() - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    let mut res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while i &lt; j {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        let cap = std::cmp::min(ht[i], ht[j]) * (j - i) as i32;\n        res = std::cmp::max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if ht[i] &lt; ht[j] {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    res\n}\n</code></pre> max_capacity.c<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(int ht[], int htLength) {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    int i = 0;\n    int j = htLength - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    int res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int capacity = myMin(ht[i], ht[j]) * (j - i);\n        res = myMax(res, capacity);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.kt<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfun maxCapacity(ht: IntArray): Int {\n    // \u521d\u59cb\u5316 i, j\uff0c\u4f7f\u5176\u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    var i = 0\n    var j = ht.size - 1\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    var res = 0\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        val cap = (min(ht[i].toDouble(), ht[j].toDouble()) * (j - i)).toInt()\n        res = max(res.toDouble(), cap.toDouble()).toInt()\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.rb<pre><code>[class]{}-[func]{max_capacity}\n</code></pre> max_capacity.zig<pre><code>[class]{}-[func]{maxCapacity}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_greedy/max_capacity_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"<p>\u4e4b\u6240\u4ee5\u8d2a\u5fc3\u6bd4\u7a77\u4e3e\u66f4\u5feb\uff0c\u662f\u56e0\u4e3a\u6bcf\u8f6e\u7684\u8d2a\u5fc3\u9009\u62e9\u90fd\u4f1a\u201c\u8df3\u8fc7\u201d\u4e00\u4e9b\u72b6\u6001\u3002</p> <p>\u6bd4\u5982\u5728\u72b6\u6001 \\(cap[i, j]\\) \u4e0b\uff0c\\(i\\) \u4e3a\u77ed\u677f\u3001\\(j\\) \u4e3a\u957f\u677f\u3002\u82e5\u8d2a\u5fc3\u5730\u5c06\u77ed\u677f \\(i\\) \u5411\u5185\u79fb\u52a8\u4e00\u683c\uff0c\u4f1a\u5bfc\u81f4\u56fe 15-12 \u6240\u793a\u7684\u72b6\u6001\u88ab\u201c\u8df3\u8fc7\u201d\u3002\u8fd9\u610f\u5473\u7740\u4e4b\u540e\u65e0\u6cd5\u9a8c\u8bc1\u8fd9\u4e9b\u72b6\u6001\u7684\u5bb9\u91cf\u5927\u5c0f\u3002</p> \\[ cap[i, i+1], cap[i, i+2], \\dots, cap[i, j-2], cap[i, j-1] \\] <p></p> <p> \u56fe 15-12 \u00a0 \u79fb\u52a8\u77ed\u677f\u5bfc\u81f4\u88ab\u8df3\u8fc7\u7684\u72b6\u6001 </p> <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u8fd9\u4e9b\u88ab\u8df3\u8fc7\u7684\u72b6\u6001\u5b9e\u9645\u4e0a\u5c31\u662f\u5c06\u957f\u677f \\(j\\) \u5411\u5185\u79fb\u52a8\u7684\u6240\u6709\u72b6\u6001\u3002\u524d\u9762\u6211\u4eec\u5df2\u7ecf\u8bc1\u660e\u5185\u79fb\u957f\u677f\u4e00\u5b9a\u4f1a\u5bfc\u81f4\u5bb9\u91cf\u53d8\u5c0f\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u88ab\u8df3\u8fc7\u7684\u72b6\u6001\u90fd\u4e0d\u53ef\u80fd\u662f\u6700\u4f18\u89e3\uff0c\u8df3\u8fc7\u5b83\u4eec\u4e0d\u4f1a\u5bfc\u81f4\u9519\u8fc7\u6700\u4f18\u89e3\u3002</p> <p>\u4ee5\u4e0a\u5206\u6790\u8bf4\u660e\uff0c\u79fb\u52a8\u77ed\u677f\u7684\u64cd\u4f5c\u662f\u201c\u5b89\u5168\u201d\u7684\uff0c\u8d2a\u5fc3\u7b56\u7565\u662f\u6709\u6548\u7684\u3002</p>"},{"location":"chapter_greedy/max_product_cutting_problem/","title":"15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570 \\(n\\) \uff0c\u5c06\u5176\u5207\u5206\u4e3a\u81f3\u5c11\u4e24\u4e2a\u6b63\u6574\u6570\u7684\u548c\uff0c\u6c42\u5207\u5206\u540e\u6240\u6709\u6574\u6570\u7684\u4e58\u79ef\u6700\u5927\u662f\u591a\u5c11\uff0c\u5982\u56fe 15-13 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 15-13 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u7684\u95ee\u9898\u5b9a\u4e49 </p> <p>\u5047\u8bbe\u6211\u4eec\u5c06 \\(n\\) \u5207\u5206\u4e3a \\(m\\) \u4e2a\u6574\u6570\u56e0\u5b50\uff0c\u5176\u4e2d\u7b2c \\(i\\) \u4e2a\u56e0\u5b50\u8bb0\u4e3a \\(n_i\\) \uff0c\u5373</p> \\[ n = \\sum_{i=1}^{m}n_i \\] <p>\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u5f97\u6240\u6709\u6574\u6570\u56e0\u5b50\u7684\u6700\u5927\u4e58\u79ef\uff0c\u5373</p> \\[ \\max(\\prod_{i=1}^{m}n_i) \\] <p>\u6211\u4eec\u9700\u8981\u601d\u8003\u7684\u662f\uff1a\u5207\u5206\u6570\u91cf \\(m\\) \u5e94\u8be5\u591a\u5927\uff0c\u6bcf\u4e2a \\(n_i\\) \u5e94\u8be5\u662f\u591a\u5c11\uff1f</p>"},{"location":"chapter_greedy/max_product_cutting_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"<p>\u6839\u636e\u7ecf\u9a8c\uff0c\u4e24\u4e2a\u6574\u6570\u7684\u4e58\u79ef\u5f80\u5f80\u6bd4\u5b83\u4eec\u7684\u52a0\u548c\u66f4\u5927\u3002\u5047\u8bbe\u4ece \\(n\\) \u4e2d\u5206\u51fa\u4e00\u4e2a\u56e0\u5b50 \\(2\\) \uff0c\u5219\u5b83\u4eec\u7684\u4e58\u79ef\u4e3a \\(2(n-2)\\) \u3002\u6211\u4eec\u5c06\u8be5\u4e58\u79ef\u4e0e \\(n\\) \u4f5c\u6bd4\u8f83\uff1a</p> \\[ \\begin{aligned} 2(n-2) &amp; \\geq n \\newline 2n - n - 4 &amp; \\geq 0 \\newline n &amp; \\geq 4 \\end{aligned} \\] <p>\u5982\u56fe 15-14 \u6240\u793a\uff0c\u5f53 \\(n \\geq 4\\) \u65f6\uff0c\u5207\u5206\u51fa\u4e00\u4e2a \\(2\\) \u540e\u4e58\u79ef\u4f1a\u53d8\u5927\uff0c\u8fd9\u8bf4\u660e\u5927\u4e8e\u7b49\u4e8e \\(4\\) \u7684\u6574\u6570\u90fd\u5e94\u8be5\u88ab\u5207\u5206\u3002</p> <p>\u8d2a\u5fc3\u7b56\u7565\u4e00\uff1a\u5982\u679c\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b \\(\\geq 4\\) \u7684\u56e0\u5b50\uff0c\u90a3\u4e48\u5b83\u5c31\u5e94\u8be5\u88ab\u7ee7\u7eed\u5207\u5206\u3002\u6700\u7ec8\u7684\u5207\u5206\u65b9\u6848\u53ea\u5e94\u51fa\u73b0 \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u8fd9\u4e09\u79cd\u56e0\u5b50\u3002</p> <p></p> <p> \u56fe 15-14 \u00a0 \u5207\u5206\u5bfc\u81f4\u4e58\u79ef\u53d8\u5927 </p> <p>\u63a5\u4e0b\u6765\u601d\u8003\u54ea\u4e2a\u56e0\u5b50\u662f\u6700\u4f18\u7684\u3002\u5728 \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u8fd9\u4e09\u4e2a\u56e0\u5b50\u4e2d\uff0c\u663e\u7136 \\(1\\) \u662f\u6700\u5dee\u7684\uff0c\u56e0\u4e3a \\(1 \\times (n-1) &lt; n\\) \u6052\u6210\u7acb\uff0c\u5373\u5207\u5206\u51fa \\(1\\) \u53cd\u800c\u4f1a\u5bfc\u81f4\u4e58\u79ef\u51cf\u5c0f\u3002</p> <p>\u5982\u56fe 15-15 \u6240\u793a\uff0c\u5f53 \\(n = 6\\) \u65f6\uff0c\u6709 \\(3 \\times 3 &gt; 2 \\times 2 \\times 2\\) \u3002\u8fd9\u610f\u5473\u7740\u5207\u5206\u51fa \\(3\\) \u6bd4\u5207\u5206\u51fa \\(2\\) \u66f4\u4f18\u3002</p> <p>\u8d2a\u5fc3\u7b56\u7565\u4e8c\uff1a\u5728\u5207\u5206\u65b9\u6848\u4e2d\uff0c\u6700\u591a\u53ea\u5e94\u5b58\u5728\u4e24\u4e2a \\(2\\) \u3002\u56e0\u4e3a\u4e09\u4e2a \\(2\\) \u603b\u662f\u53ef\u4ee5\u66ff\u6362\u4e3a\u4e24\u4e2a \\(3\\) \uff0c\u4ece\u800c\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002</p> <p></p> <p> \u56fe 15-15 \u00a0 \u6700\u4f18\u5207\u5206\u56e0\u5b50 </p> <p>\u7efc\u4e0a\u6240\u8ff0\uff0c\u53ef\u63a8\u7406\u51fa\u4ee5\u4e0b\u8d2a\u5fc3\u7b56\u7565\u3002</p> <ol> <li>\u8f93\u5165\u6574\u6570 \\(n\\) \uff0c\u4ece\u5176\u4e0d\u65ad\u5730\u5207\u5206\u51fa\u56e0\u5b50 \\(3\\) \uff0c\u76f4\u81f3\u4f59\u6570\u4e3a \\(0\\)\u3001\\(1\\)\u3001\\(2\\) \u3002</li> <li>\u5f53\u4f59\u6570\u4e3a \\(0\\) \u65f6\uff0c\u4ee3\u8868 \\(n\\) \u662f \\(3\\) \u7684\u500d\u6570\uff0c\u56e0\u6b64\u4e0d\u505a\u4efb\u4f55\u5904\u7406\u3002</li> <li>\u5f53\u4f59\u6570\u4e3a \\(2\\) \u65f6\uff0c\u4e0d\u7ee7\u7eed\u5212\u5206\uff0c\u4fdd\u7559\u3002</li> <li>\u5f53\u4f59\u6570\u4e3a \\(1\\) \u65f6\uff0c\u7531\u4e8e \\(2 \\times 2 &gt; 1 \\times 3\\) \uff0c\u56e0\u6b64\u5e94\u5c06\u6700\u540e\u4e00\u4e2a \\(3\\) \u66ff\u6362\u4e3a \\(2\\) \u3002</li> </ol>"},{"location":"chapter_greedy/max_product_cutting_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5982\u56fe 15-16 \u6240\u793a\uff0c\u6211\u4eec\u65e0\u987b\u901a\u8fc7\u5faa\u73af\u6765\u5207\u5206\u6574\u6570\uff0c\u800c\u53ef\u4ee5\u5229\u7528\u5411\u4e0b\u6574\u9664\u8fd0\u7b97\u5f97\u5230 \\(3\\) \u7684\u4e2a\u6570 \\(a\\) \uff0c\u7528\u53d6\u6a21\u8fd0\u7b97\u5f97\u5230\u4f59\u6570 \\(b\\) \uff0c\u6b64\u65f6\u6709\uff1a</p> \\[ n = 3 a + b \\] <p>\u8bf7\u6ce8\u610f\uff0c\u5bf9\u4e8e \\(n \\leq 3\\) \u7684\u8fb9\u754c\u60c5\u51b5\uff0c\u5fc5\u987b\u62c6\u5206\u51fa\u4e00\u4e2a \\(1\\) \uff0c\u4e58\u79ef\u4e3a \\(1 \\times (n - 1)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig max_product_cutting.py<pre><code>def max_product_cutting(n: int) -&gt; int:\n    \"\"\"\u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3\"\"\"\n    # \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if n &lt;= 3:\n        return 1 * (n - 1)\n    # \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    a, b = n // 3, n % 3\n    if b == 1:\n        # \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return int(math.pow(3, a - 1)) * 2 * 2\n    if b == 2:\n        # \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return int(math.pow(3, a)) * 2\n    # \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return int(math.pow(3, a))\n</code></pre> max_product_cutting.cpp<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return (int)pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return (int)pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return (int)pow(3, a);\n}\n</code></pre> max_product_cutting.java<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return (int) Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return (int) Math.pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return (int) Math.pow(3, a);\n}\n</code></pre> max_product_cutting.cs<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint MaxProductCutting(int n) {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return (int)Math.Pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return (int)Math.Pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return (int)Math.Pow(3, a);\n}\n</code></pre> max_product_cutting.go<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunc maxProductCutting(n int) int {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    a := n / 3\n    b := n % 3\n    if b == 1 {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return int(math.Pow(3, float64(a-1))) * 2 * 2\n    }\n    if b == 2 {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return int(math.Pow(3, float64(a))) * 2\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return int(math.Pow(3, float64(a)))\n}\n</code></pre> max_product_cutting.swift<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunc maxProductCutting(n: Int) -&gt; Int {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    let a = n / 3\n    let b = n % 3\n    if b == 1 {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return pow(3, a - 1) * 2 * 2\n    }\n    if b == 2 {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return pow(3, a) * 2\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return pow(3, a)\n}\n</code></pre> max_product_cutting.js<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunction maxProductCutting(n) {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    let a = Math.floor(n / 3);\n    let b = n % 3;\n    if (b === 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return Math.pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.ts<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunction maxProductCutting(n: number): number {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    let a: number = Math.floor(n / 3);\n    let b: number = n % 3;\n    if (b === 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return Math.pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.dart<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n  // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n  if (n &lt;= 3) {\n    return 1 * (n - 1);\n  }\n  // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n  int a = n ~/ 3;\n  int b = n % 3;\n  if (b == 1) {\n    // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n    return (pow(3, a - 1) * 2 * 2).toInt();\n  }\n  if (b == 2) {\n    // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return (pow(3, a) * 2).toInt();\n  }\n  // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n  return pow(3, a).toInt();\n}\n</code></pre> max_product_cutting.rs<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfn max_product_cutting(n: i32) -&gt; i32 {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if n &lt;= 3 {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    let a = n / 3;\n    let b = n % 3;\n    if b == 1 {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        3_i32.pow(a as u32 - 1) * 2 * 2\n    } else if b == 2 {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        3_i32.pow(a as u32) * 2\n    } else {\n        // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        3_i32.pow(a as u32)\n    }\n}\n</code></pre> max_product_cutting.c<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return pow(3, a);\n}\n</code></pre> max_product_cutting.kt<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfun maxProductCutting(n: Int): Int {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1)\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    val a = n / 3\n    val b = n % 3\n    if (b == 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return 3.0.pow((a - 1).toDouble()).toInt() * 2 * 2\n    }\n    if (b == 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return 3.0.pow(a.toDouble()).toInt() * 2 * 2\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return 3.0.pow(a.toDouble()).toInt()\n}\n</code></pre> max_product_cutting.rb<pre><code>[class]{}-[func]{max_product_cutting}\n</code></pre> max_product_cutting.zig<pre><code>[class]{}-[func]{maxProductCutting}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p></p> <p> \u56fe 15-16 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u7684\u8ba1\u7b97\u65b9\u6cd5 </p> <p>\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u7f16\u7a0b\u8bed\u8a00\u7684\u5e42\u8fd0\u7b97\u7684\u5b9e\u73b0\u65b9\u6cd5\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u5e38\u7528\u7684\u5e42\u8ba1\u7b97\u51fd\u6570\u6709\u4e09\u79cd\u3002</p> <ul> <li>\u8fd0\u7b97\u7b26 <code>**</code> \u548c\u51fd\u6570 <code>pow()</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log\u2061 a)\\) \u3002</li> <li>\u51fd\u6570 <code>math.pow()</code> \u5185\u90e8\u8c03\u7528 C \u8bed\u8a00\u5e93\u7684 <code>pow()</code> \u51fd\u6570\uff0c\u5176\u6267\u884c\u6d6e\u70b9\u53d6\u5e42\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</li> </ul> <p>\u53d8\u91cf \\(a\\) \u548c \\(b\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</p>"},{"location":"chapter_greedy/max_product_cutting_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"<p>\u4f7f\u7528\u53cd\u8bc1\u6cd5\uff0c\u53ea\u5206\u6790 \\(n \\geq 3\\) \u7684\u60c5\u51b5\u3002</p> <ol> <li>\u6240\u6709\u56e0\u5b50 \\(\\leq 3\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728 \\(\\geq 4\\) \u7684\u56e0\u5b50 \\(x\\) \uff0c\u90a3\u4e48\u4e00\u5b9a\u53ef\u4ee5\u5c06\u5176\u7ee7\u7eed\u5212\u5206\u4e3a \\(2(x-2)\\) \uff0c\u4ece\u800c\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</li> <li>\u5207\u5206\u65b9\u6848\u4e0d\u5305\u542b \\(1\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728\u4e00\u4e2a\u56e0\u5b50 \\(1\\) \uff0c\u90a3\u4e48\u5b83\u4e00\u5b9a\u53ef\u4ee5\u5408\u5e76\u5165\u53e6\u5916\u4e00\u4e2a\u56e0\u5b50\u4e2d\uff0c\u4ee5\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</li> <li>\u5207\u5206\u65b9\u6848\u6700\u591a\u5305\u542b\u4e24\u4e2a \\(2\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b\u4e09\u4e2a \\(2\\) \uff0c\u90a3\u4e48\u4e00\u5b9a\u53ef\u4ee5\u66ff\u6362\u4e3a\u4e24\u4e2a \\(3\\) \uff0c\u4e58\u79ef\u66f4\u5927\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</li> </ol>"},{"location":"chapter_greedy/summary/","title":"15.5 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u8d2a\u5fc3\u7b97\u6cd5\u901a\u5e38\u7528\u4e8e\u89e3\u51b3\u6700\u4f18\u5316\u95ee\u9898\uff0c\u5176\u539f\u7406\u662f\u5728\u6bcf\u4e2a\u51b3\u7b56\u9636\u6bb5\u90fd\u505a\u51fa\u5c40\u90e8\u6700\u4f18\u7684\u51b3\u7b56\uff0c\u4ee5\u671f\u83b7\u5f97\u5168\u5c40\u6700\u4f18\u89e3\u3002</li> <li>\u8d2a\u5fc3\u7b97\u6cd5\u4f1a\u8fed\u4ee3\u5730\u505a\u51fa\u4e00\u4e2a\u53c8\u4e00\u4e2a\u7684\u8d2a\u5fc3\u9009\u62e9\uff0c\u6bcf\u8f6e\u90fd\u5c06\u95ee\u9898\u8f6c\u5316\u6210\u4e00\u4e2a\u89c4\u6a21\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u5230\u95ee\u9898\u88ab\u89e3\u51b3\u3002</li> <li>\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4ec5\u5b9e\u73b0\u7b80\u5355\uff0c\u8fd8\u5177\u6709\u5f88\u9ad8\u7684\u89e3\u9898\u6548\u7387\u3002\u76f8\u6bd4\u4e8e\u52a8\u6001\u89c4\u5212\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u66f4\u4f4e\u3002</li> <li>\u5728\u96f6\u94b1\u5151\u6362\u95ee\u9898\u4e2d\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u786c\u5e01\u7ec4\u5408\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1b\u5bf9\u4e8e\u53e6\u5916\u4e00\u4e9b\u786c\u5e01\u7ec4\u5408\u5219\u4e0d\u7136\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ef\u80fd\u627e\u5230\u5f88\u5dee\u7684\u89e3\u3002</li> <li>\u9002\u5408\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u7684\u95ee\u9898\u5177\u6709\u4e24\u5927\u6027\u8d28\uff1a\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u3002\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u4ee3\u8868\u8d2a\u5fc3\u7b56\u7565\u7684\u6709\u6548\u6027\u3002</li> <li>\u5bf9\u4e8e\u67d0\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u7684\u8bc1\u660e\u5e76\u4e0d\u7b80\u5355\u3002\u76f8\u5bf9\u6765\u8bf4\uff0c\u8bc1\u4f2a\u66f4\u52a0\u5bb9\u6613\uff0c\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\u3002</li> <li>\u6c42\u89e3\u8d2a\u5fc3\u95ee\u9898\u4e3b\u8981\u5206\u4e3a\u4e09\u6b65\uff1a\u95ee\u9898\u5206\u6790\u3001\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u3001\u6b63\u786e\u6027\u8bc1\u660e\u3002\u5176\u4e2d\uff0c\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u662f\u6838\u5fc3\u6b65\u9aa4\uff0c\u6b63\u786e\u6027\u8bc1\u660e\u5f80\u5f80\u662f\u96be\u70b9\u3002</li> <li>\u5206\u6570\u80cc\u5305\u95ee\u9898\u5728 0-1 \u80cc\u5305\u7684\u57fa\u7840\u4e0a\uff0c\u5141\u8bb8\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u56e0\u6b64\u53ef\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u3002\u8d2a\u5fc3\u7b56\u7565\u7684\u6b63\u786e\u6027\u53ef\u4ee5\u4f7f\u7528\u53cd\u8bc1\u6cd5\u6765\u8bc1\u660e\u3002</li> <li>\u6700\u5927\u5bb9\u91cf\u95ee\u9898\u53ef\u4f7f\u7528\u7a77\u4e3e\u6cd5\u6c42\u89e3\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u901a\u8fc7\u8bbe\u8ba1\u8d2a\u5fc3\u7b56\u7565\uff0c\u6bcf\u8f6e\u5411\u5185\u79fb\u52a8\u77ed\u677f\uff0c\u53ef\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>\u5728\u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u5148\u540e\u63a8\u7406\u51fa\u4e24\u4e2a\u8d2a\u5fc3\u7b56\u7565\uff1a\\(\\geq 4\\) \u7684\u6574\u6570\u90fd\u5e94\u8be5\u7ee7\u7eed\u5207\u5206\uff0c\u6700\u4f18\u5207\u5206\u56e0\u5b50\u4e3a \\(3\\) \u3002\u4ee3\u7801\u4e2d\u5305\u542b\u5e42\u8fd0\u7b97\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u5e42\u8fd0\u7b97\u5b9e\u73b0\u65b9\u6cd5\uff0c\u901a\u5e38\u4e3a \\(O(1)\\) \u6216 \\(O(\\log n)\\) \u3002</li> </ul>"},{"location":"chapter_hashing/","title":"\u7b2c 6 \u7ae0 \u00a0 \u54c8\u5e0c\u8868","text":"<p>Abstract</p> <p>\u5728\u8ba1\u7b97\u673a\u4e16\u754c\u4e2d\uff0c\u54c8\u5e0c\u8868\u5982\u540c\u4e00\u4f4d\u806a\u6167\u7684\u56fe\u4e66\u7ba1\u7406\u5458\u3002</p> <p>\u4ed6\u77e5\u9053\u5982\u4f55\u8ba1\u7b97\u7d22\u4e66\u53f7\uff0c\u4ece\u800c\u53ef\u4ee5\u5feb\u901f\u627e\u5230\u76ee\u6807\u56fe\u4e66\u3002</p>"},{"location":"chapter_hashing/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>6.1 \u00a0 \u54c8\u5e0c\u8868</li> <li>6.2 \u00a0 \u54c8\u5e0c\u51b2\u7a81</li> <li>6.3 \u00a0 \u54c8\u5e0c\u7b97\u6cd5</li> <li>6.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_hashing/hash_algorithm/","title":"6.3 \u00a0 \u54c8\u5e0c\u7b97\u6cd5","text":"<p>\u524d\u4e24\u8282\u4ecb\u7ecd\u4e86\u54c8\u5e0c\u8868\u7684\u5de5\u4f5c\u539f\u7406\u548c\u54c8\u5e0c\u51b2\u7a81\u7684\u5904\u7406\u65b9\u6cd5\u3002\u7136\u800c\u65e0\u8bba\u662f\u5f00\u653e\u5bfb\u5740\u8fd8\u662f\u94fe\u5f0f\u5730\u5740\uff0c\u5b83\u4eec\u53ea\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u53d1\u751f\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\uff0c\u800c\u65e0\u6cd5\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u3002</p> <p>\u5982\u679c\u54c8\u5e0c\u51b2\u7a81\u8fc7\u4e8e\u9891\u7e41\uff0c\u54c8\u5e0c\u8868\u7684\u6027\u80fd\u5219\u4f1a\u6025\u5267\u52a3\u5316\u3002\u5982\u56fe 6-8 \u6240\u793a\uff0c\u5bf9\u4e8e\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\uff0c\u7406\u60f3\u60c5\u51b5\u4e0b\u952e\u503c\u5bf9\u5747\u5300\u5206\u5e03\u5728\u5404\u4e2a\u6876\u4e2d\uff0c\u8fbe\u5230\u6700\u4f73\u67e5\u8be2\u6548\u7387\uff1b\u6700\u5dee\u60c5\u51b5\u4e0b\u6240\u6709\u952e\u503c\u5bf9\u90fd\u5b58\u50a8\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002</p> <p></p> <p> \u56fe 6-8 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u7684\u6700\u4f73\u60c5\u51b5\u4e0e\u6700\u5dee\u60c5\u51b5 </p> <p>\u952e\u503c\u5bf9\u7684\u5206\u5e03\u60c5\u51b5\u7531\u54c8\u5e0c\u51fd\u6570\u51b3\u5b9a\u3002\u56de\u5fc6\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u6b65\u9aa4\uff0c\u5148\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u518d\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\uff1a</p> <pre><code>index = hash(key) % capacity\n</code></pre> <p>\u89c2\u5bdf\u4ee5\u4e0a\u516c\u5f0f\uff0c\u5f53\u54c8\u5e0c\u8868\u5bb9\u91cf <code>capacity</code> \u56fa\u5b9a\u65f6\uff0c\u54c8\u5e0c\u7b97\u6cd5 <code>hash()</code> \u51b3\u5b9a\u4e86\u8f93\u51fa\u503c\uff0c\u8fdb\u800c\u51b3\u5b9a\u4e86\u952e\u503c\u5bf9\u5728\u54c8\u5e0c\u8868\u4e2d\u7684\u5206\u5e03\u60c5\u51b5\u3002</p> <p>\u8fd9\u610f\u5473\u7740\uff0c\u4e3a\u4e86\u964d\u4f4e\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u6982\u7387\uff0c\u6211\u4eec\u5e94\u5f53\u5c06\u6ce8\u610f\u529b\u96c6\u4e2d\u5728\u54c8\u5e0c\u7b97\u6cd5 <code>hash()</code> \u7684\u8bbe\u8ba1\u4e0a\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#631","title":"6.3.1 \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u76ee\u6807","text":"<p>\u4e3a\u4e86\u5b9e\u73b0\u201c\u65e2\u5feb\u53c8\u7a33\u201d\u7684\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u5177\u5907\u4ee5\u4e0b\u7279\u70b9\u3002</p> <ul> <li>\u786e\u5b9a\u6027\uff1a\u5bf9\u4e8e\u76f8\u540c\u7684\u8f93\u5165\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u59cb\u7ec8\u4ea7\u751f\u76f8\u540c\u7684\u8f93\u51fa\u3002\u8fd9\u6837\u624d\u80fd\u786e\u4fdd\u54c8\u5e0c\u8868\u662f\u53ef\u9760\u7684\u3002</li> <li>\u6548\u7387\u9ad8\uff1a\u8ba1\u7b97\u54c8\u5e0c\u503c\u7684\u8fc7\u7a0b\u5e94\u8be5\u8db3\u591f\u5feb\u3002\u8ba1\u7b97\u5f00\u9500\u8d8a\u5c0f\uff0c\u54c8\u5e0c\u8868\u7684\u5b9e\u7528\u6027\u8d8a\u9ad8\u3002</li> <li>\u5747\u5300\u5206\u5e03\uff1a\u54c8\u5e0c\u7b97\u6cd5\u5e94\u4f7f\u5f97\u952e\u503c\u5bf9\u5747\u5300\u5206\u5e03\u5728\u54c8\u5e0c\u8868\u4e2d\u3002\u5206\u5e03\u8d8a\u5747\u5300\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9664\u4e86\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u54c8\u5e0c\u8868\uff0c\u8fd8\u5e7f\u6cdb\u5e94\u7528\u4e8e\u5176\u4ed6\u9886\u57df\u4e2d\u3002</p> <ul> <li>\u5bc6\u7801\u5b58\u50a8\uff1a\u4e3a\u4e86\u4fdd\u62a4\u7528\u6237\u5bc6\u7801\u7684\u5b89\u5168\uff0c\u7cfb\u7edf\u901a\u5e38\u4e0d\u4f1a\u76f4\u63a5\u5b58\u50a8\u7528\u6237\u7684\u660e\u6587\u5bc6\u7801\uff0c\u800c\u662f\u5b58\u50a8\u5bc6\u7801\u7684\u54c8\u5e0c\u503c\u3002\u5f53\u7528\u6237\u8f93\u5165\u5bc6\u7801\u65f6\uff0c\u7cfb\u7edf\u4f1a\u5bf9\u8f93\u5165\u7684\u5bc6\u7801\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u7136\u540e\u4e0e\u5b58\u50a8\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u5bc6\u7801\u5c31\u88ab\u89c6\u4e3a\u6b63\u786e\u3002</li> <li>\u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5\uff1a\u6570\u636e\u53d1\u9001\u65b9\u53ef\u4ee5\u8ba1\u7b97\u6570\u636e\u7684\u54c8\u5e0c\u503c\u5e76\u5c06\u5176\u4e00\u540c\u53d1\u9001\uff1b\u63a5\u6536\u65b9\u53ef\u4ee5\u91cd\u65b0\u8ba1\u7b97\u63a5\u6536\u5230\u7684\u6570\u636e\u7684\u54c8\u5e0c\u503c\uff0c\u5e76\u4e0e\u63a5\u6536\u5230\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u6570\u636e\u5c31\u88ab\u89c6\u4e3a\u5b8c\u6574\u3002</li> </ul> <p>\u5bf9\u4e8e\u5bc6\u7801\u5b66\u7684\u76f8\u5173\u5e94\u7528\uff0c\u4e3a\u4e86\u9632\u6b62\u4ece\u54c8\u5e0c\u503c\u63a8\u5bfc\u51fa\u539f\u59cb\u5bc6\u7801\u7b49\u9006\u5411\u5de5\u7a0b\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9700\u8981\u5177\u5907\u66f4\u9ad8\u7b49\u7ea7\u7684\u5b89\u5168\u7279\u6027\u3002</p> <ul> <li>\u5355\u5411\u6027\uff1a\u65e0\u6cd5\u901a\u8fc7\u54c8\u5e0c\u503c\u53cd\u63a8\u51fa\u5173\u4e8e\u8f93\u5165\u6570\u636e\u7684\u4efb\u4f55\u4fe1\u606f\u3002</li> <li>\u6297\u78b0\u649e\u6027\uff1a\u5e94\u5f53\u6781\u96be\u627e\u5230\u4e24\u4e2a\u4e0d\u540c\u7684\u8f93\u5165\uff0c\u4f7f\u5f97\u5b83\u4eec\u7684\u54c8\u5e0c\u503c\u76f8\u540c\u3002</li> <li>\u96ea\u5d29\u6548\u5e94\uff1a\u8f93\u5165\u7684\u5fae\u5c0f\u53d8\u5316\u5e94\u5f53\u5bfc\u81f4\u8f93\u51fa\u7684\u663e\u8457\u4e14\u4e0d\u53ef\u9884\u6d4b\u7684\u53d8\u5316\u3002</li> </ul> <p>\u8bf7\u6ce8\u610f\uff0c\u201c\u5747\u5300\u5206\u5e03\u201d\u4e0e\u201c\u6297\u78b0\u649e\u6027\u201d\u662f\u4e24\u4e2a\u72ec\u7acb\u7684\u6982\u5ff5\uff0c\u6ee1\u8db3\u5747\u5300\u5206\u5e03\u4e0d\u4e00\u5b9a\u6ee1\u8db3\u6297\u78b0\u649e\u6027\u3002\u4f8b\u5982\uff0c\u5728\u968f\u673a\u8f93\u5165 <code>key</code> \u4e0b\uff0c\u54c8\u5e0c\u51fd\u6570 <code>key % 100</code> \u53ef\u4ee5\u4ea7\u751f\u5747\u5300\u5206\u5e03\u7684\u8f93\u51fa\u3002\u7136\u800c\u8be5\u54c8\u5e0c\u7b97\u6cd5\u8fc7\u4e8e\u7b80\u5355\uff0c\u6240\u6709\u540e\u4e24\u4f4d\u76f8\u7b49\u7684 <code>key</code> \u7684\u8f93\u51fa\u90fd\u76f8\u540c\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u4ece\u54c8\u5e0c\u503c\u53cd\u63a8\u51fa\u53ef\u7528\u7684 <code>key</code> \uff0c\u4ece\u800c\u7834\u89e3\u5bc6\u7801\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#632","title":"6.3.2 \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1","text":"<p>\u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u662f\u4e00\u4e2a\u9700\u8981\u8003\u8651\u8bb8\u591a\u56e0\u7d20\u7684\u590d\u6742\u95ee\u9898\u3002\u7136\u800c\u5bf9\u4e8e\u67d0\u4e9b\u8981\u6c42\u4e0d\u9ad8\u7684\u573a\u666f\uff0c\u6211\u4eec\u4e5f\u80fd\u8bbe\u8ba1\u4e00\u4e9b\u7b80\u5355\u7684\u54c8\u5e0c\u7b97\u6cd5\u3002</p> <ul> <li>\u52a0\u6cd5\u54c8\u5e0c\uff1a\u5bf9\u8f93\u5165\u7684\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u8fdb\u884c\u76f8\u52a0\uff0c\u5c06\u5f97\u5230\u7684\u603b\u548c\u4f5c\u4e3a\u54c8\u5e0c\u503c\u3002</li> <li>\u4e58\u6cd5\u54c8\u5e0c\uff1a\u5229\u7528\u4e58\u6cd5\u7684\u4e0d\u76f8\u5173\u6027\uff0c\u6bcf\u8f6e\u4e58\u4ee5\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5404\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u54c8\u5e0c\u503c\u4e2d\u3002</li> <li>\u5f02\u6216\u54c8\u5e0c\uff1a\u5c06\u8f93\u5165\u6570\u636e\u7684\u6bcf\u4e2a\u5143\u7d20\u901a\u8fc7\u5f02\u6216\u64cd\u4f5c\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\u3002</li> <li>\u65cb\u8f6c\u54c8\u5e0c\uff1a\u5c06\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\uff0c\u6bcf\u6b21\u7d2f\u79ef\u4e4b\u524d\u90fd\u4f1a\u5bf9\u54c8\u5e0c\u503c\u8fdb\u884c\u65cb\u8f6c\u64cd\u4f5c\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig simple_hash.py<pre><code>def add_hash(key: str) -&gt; int:\n    \"\"\"\u52a0\u6cd5\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash += ord(c)\n    return hash % modulus\n\ndef mul_hash(key: str) -&gt; int:\n    \"\"\"\u4e58\u6cd5\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = 31 * hash + ord(c)\n    return hash % modulus\n\ndef xor_hash(key: str) -&gt; int:\n    \"\"\"\u5f02\u6216\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash ^= ord(c)\n    return hash % modulus\n\ndef rot_hash(key: str) -&gt; int:\n    \"\"\"\u65cb\u8f6c\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ ord(c)\n    return hash % modulus\n</code></pre> simple_hash.cpp<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (31 * hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash ^= (int)c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.java<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (31 * hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\n    int hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash ^= (int) c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n</code></pre> simple_hash.cs<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint AddHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint MulHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (31 * hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint XorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash ^= c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint RotHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.go<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (31*hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key string) int {\n    hash := 0\n    modulus := 1000000007\n    for _, b := range []byte(key) {\n        fmt.Println(int(b))\n        hash ^= int(b)\n        hash = (31*hash + int(b)) % modulus\n    }\n    return hash &amp; modulus\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ int64(b)) % modulus\n    }\n    return int(hash)\n}\n</code></pre> simple_hash.swift<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (31 * hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash ^= Int(scalar.value)\n        }\n    }\n    return hash &amp; MODULUS\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n</code></pre> simple_hash.js<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.ts<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.dart<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (31 * hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash ^= key.codeUnitAt(i);\n  }\n  return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n</code></pre> simple_hash.rs<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfn add_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfn mul_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (31 * hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfn xor_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash ^= c as i64;\n    }\n\n    (hash &amp; MODULUS) as i32\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfn rot_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n</code></pre> simple_hash.c<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (31 * hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(char *key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash ^= (unsigned char)key[i];\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (unsigned char)key[i]) % MODULUS;\n    }\n\n    return (int)hash;\n}\n</code></pre> simple_hash.kt<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfun addHash(key: String): Int {\n    var hash = 0L\n    for (c in key.toCharArray()) {\n        hash = (hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfun mulHash(key: String): Int {\n    var hash = 0L\n    for (c in key.toCharArray()) {\n        hash = (31 * hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfun xorHash(key: String): Int {\n    var hash = 0\n    for (c in key.toCharArray()) {\n        hash = hash xor c.code\n    }\n    return hash and MODULUS\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfun rotHash(key: String): Int {\n    var hash = 0L\n    for (c in key.toCharArray()) {\n        hash = ((hash shl 4) xor (hash shr 28) xor c.code.toLong()) % MODULUS\n    }\n    return hash.toInt()\n}\n</code></pre> simple_hash.rb<pre><code>[class]{}-[func]{add_hash}\n\n[class]{}-[func]{mul_hash}\n\n[class]{}-[func]{xor_hash}\n\n[class]{}-[func]{rot_hash}\n</code></pre> simple_hash.zig<pre><code>[class]{}-[func]{addHash}\n\n[class]{}-[func]{mulHash}\n\n[class]{}-[func]{xorHash}\n\n[class]{}-[func]{rotHash}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u6bcf\u79cd\u54c8\u5e0c\u7b97\u6cd5\u7684\u6700\u540e\u4e00\u6b65\u90fd\u662f\u5bf9\u5927\u8d28\u6570 \\(1000000007\\) \u53d6\u6a21\uff0c\u4ee5\u786e\u4fdd\u54c8\u5e0c\u503c\u5728\u5408\u9002\u7684\u8303\u56f4\u5185\u3002\u503c\u5f97\u601d\u8003\u7684\u662f\uff0c\u4e3a\u4ec0\u4e48\u8981\u5f3a\u8c03\u5bf9\u8d28\u6570\u53d6\u6a21\uff0c\u6216\u8005\u8bf4\u5bf9\u5408\u6570\u53d6\u6a21\u7684\u5f0a\u7aef\u662f\u4ec0\u4e48\uff1f\u8fd9\u662f\u4e00\u4e2a\u6709\u8da3\u7684\u95ee\u9898\u3002</p> <p>\u5148\u629b\u51fa\u7ed3\u8bba\uff1a\u4f7f\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u53ef\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u7684\u5747\u5300\u5206\u5e03\u3002\u56e0\u4e3a\u8d28\u6570\u4e0d\u4e0e\u5176\u4ed6\u6570\u5b57\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u53ef\u4ee5\u51cf\u5c11\u56e0\u53d6\u6a21\u64cd\u4f5c\u800c\u4ea7\u751f\u7684\u5468\u671f\u6027\u6a21\u5f0f\uff0c\u4ece\u800c\u907f\u514d\u54c8\u5e0c\u51b2\u7a81\u3002</p> <p>\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u5408\u6570 \\(9\\) \u4f5c\u4e3a\u6a21\u6570\uff0c\u5b83\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\uff0c\u90a3\u4e48\u6240\u6709\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\u7684 <code>key</code> \u90fd\u4f1a\u88ab\u6620\u5c04\u5230 \\(0\\)\u3001\\(3\\)\u3001\\(6\\) \u8fd9\u4e09\u4e2a\u54c8\u5e0c\u503c\u3002</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 9 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\dots \\} \\end{aligned} \\] <p>\u5982\u679c\u8f93\u5165 <code>key</code> \u6070\u597d\u6ee1\u8db3\u8fd9\u79cd\u7b49\u5dee\u6570\u5217\u7684\u6570\u636e\u5206\u5e03\uff0c\u90a3\u4e48\u54c8\u5e0c\u503c\u5c31\u4f1a\u51fa\u73b0\u805a\u5806\uff0c\u4ece\u800c\u52a0\u91cd\u54c8\u5e0c\u51b2\u7a81\u3002\u73b0\u5728\uff0c\u5047\u8bbe\u5c06 <code>modulus</code> \u66ff\u6362\u4e3a\u8d28\u6570 \\(13\\) \uff0c\u7531\u4e8e <code>key</code> \u548c <code>modulus</code> \u4e4b\u95f4\u4e0d\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u56e0\u6b64\u8f93\u51fa\u7684\u54c8\u5e0c\u503c\u7684\u5747\u5300\u6027\u4f1a\u660e\u663e\u63d0\u5347\u3002</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 13 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\dots \\} \\end{aligned} \\] <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5982\u679c\u80fd\u591f\u4fdd\u8bc1 <code>key</code> \u662f\u968f\u673a\u5747\u5300\u5206\u5e03\u7684\uff0c\u90a3\u4e48\u9009\u62e9\u8d28\u6570\u6216\u8005\u5408\u6570\u4f5c\u4e3a\u6a21\u6570\u90fd\u53ef\u4ee5\uff0c\u5b83\u4eec\u90fd\u80fd\u8f93\u51fa\u5747\u5300\u5206\u5e03\u7684\u54c8\u5e0c\u503c\u3002\u800c\u5f53 <code>key</code> \u7684\u5206\u5e03\u5b58\u5728\u67d0\u79cd\u5468\u671f\u6027\u65f6\uff0c\u5bf9\u5408\u6570\u53d6\u6a21\u66f4\u5bb9\u6613\u51fa\u73b0\u805a\u96c6\u73b0\u8c61\u3002</p> <p>\u603b\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u901a\u5e38\u9009\u53d6\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u5e76\u4e14\u8fd9\u4e2a\u8d28\u6570\u6700\u597d\u8db3\u591f\u5927\uff0c\u4ee5\u5c3d\u53ef\u80fd\u6d88\u9664\u5468\u671f\u6027\u6a21\u5f0f\uff0c\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#633","title":"6.3.3 \u00a0 \u5e38\u89c1\u54c8\u5e0c\u7b97\u6cd5","text":"<p>\u4e0d\u96be\u53d1\u73b0\uff0c\u4ee5\u4e0a\u4ecb\u7ecd\u7684\u7b80\u5355\u54c8\u5e0c\u7b97\u6cd5\u90fd\u6bd4\u8f83\u201c\u8106\u5f31\u201d\uff0c\u8fdc\u8fdc\u6ca1\u6709\u8fbe\u5230\u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u76ee\u6807\u3002\u4f8b\u5982\uff0c\u7531\u4e8e\u52a0\u6cd5\u548c\u5f02\u6216\u6ee1\u8db3\u4ea4\u6362\u5f8b\uff0c\u56e0\u6b64\u52a0\u6cd5\u54c8\u5e0c\u548c\u5f02\u6216\u54c8\u5e0c\u65e0\u6cd5\u533a\u5206\u5185\u5bb9\u76f8\u540c\u4f46\u987a\u5e8f\u4e0d\u540c\u7684\u5b57\u7b26\u4e32\uff0c\u8fd9\u53ef\u80fd\u4f1a\u52a0\u5267\u54c8\u5e0c\u51b2\u7a81\uff0c\u5e76\u5f15\u8d77\u4e00\u4e9b\u5b89\u5168\u95ee\u9898\u3002</p> <p>\u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u7528\u4e00\u4e9b\u6807\u51c6\u54c8\u5e0c\u7b97\u6cd5\uff0c\u4f8b\u5982 MD5\u3001SHA-1\u3001SHA-2 \u548c SHA-3 \u7b49\u3002\u5b83\u4eec\u53ef\u4ee5\u5c06\u4efb\u610f\u957f\u5ea6\u7684\u8f93\u5165\u6570\u636e\u6620\u5c04\u5230\u6052\u5b9a\u957f\u5ea6\u7684\u54c8\u5e0c\u503c\u3002</p> <p>\u8fd1\u4e00\u4e2a\u4e16\u7eaa\u4ee5\u6765\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5904\u5728\u4e0d\u65ad\u5347\u7ea7\u4e0e\u4f18\u5316\u7684\u8fc7\u7a0b\u4e2d\u3002\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u52aa\u529b\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u6027\u80fd\uff0c\u53e6\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u548c\u9ed1\u5ba2\u5219\u81f4\u529b\u4e8e\u5bfb\u627e\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u95ee\u9898\u3002\u8868 6-2 \u5c55\u793a\u4e86\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5\u3002</p> <ul> <li>MD5 \u548c SHA-1 \u5df2\u591a\u6b21\u88ab\u6210\u529f\u653b\u51fb\uff0c\u56e0\u6b64\u5b83\u4eec\u88ab\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u5f03\u7528\u3002</li> <li>SHA-2 \u7cfb\u5217\u4e2d\u7684 SHA-256 \u662f\u6700\u5b89\u5168\u7684\u54c8\u5e0c\u7b97\u6cd5\u4e4b\u4e00\uff0c\u4ecd\u672a\u51fa\u73b0\u6210\u529f\u7684\u653b\u51fb\u6848\u4f8b\uff0c\u56e0\u6b64\u5e38\u7528\u5728\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u4e2d\u3002</li> <li>SHA-3 \u76f8\u8f83 SHA-2 \u7684\u5b9e\u73b0\u5f00\u9500\u66f4\u4f4e\u3001\u8ba1\u7b97\u6548\u7387\u66f4\u9ad8\uff0c\u4f46\u76ee\u524d\u4f7f\u7528\u8986\u76d6\u5ea6\u4e0d\u5982 SHA-2 \u7cfb\u5217\u3002</li> </ul> <p> \u8868 6-2 \u00a0 \u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5 </p> MD5 SHA-1 SHA-2 SHA-3 \u63a8\u51fa\u65f6\u95f4 1992 1995 2002 2008 \u8f93\u51fa\u957f\u5ea6 128 bit 160 bit 256/512 bit 224/256/384/512 bit \u54c8\u5e0c\u51b2\u7a81 \u8f83\u591a \u8f83\u591a \u5f88\u5c11 \u5f88\u5c11 \u5b89\u5168\u7b49\u7ea7 \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u9ad8 \u9ad8 \u5e94\u7528 \u5df2\u88ab\u5f03\u7528\uff0c\u4ecd\u7528\u4e8e\u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5 \u5df2\u88ab\u5f03\u7528 \u52a0\u5bc6\u8d27\u5e01\u4ea4\u6613\u9a8c\u8bc1\u3001\u6570\u5b57\u7b7e\u540d\u7b49 \u53ef\u7528\u4e8e\u66ff\u4ee3 SHA-2"},{"location":"chapter_hashing/hash_algorithm/#634","title":"6.3.4 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u54c8\u5e0c\u503c","text":"<p>\u6211\u4eec\u77e5\u9053\uff0c\u54c8\u5e0c\u8868\u7684 <code>key</code> \u53ef\u4ee5\u662f\u6574\u6570\u3001\u5c0f\u6570\u6216\u5b57\u7b26\u4e32\u7b49\u6570\u636e\u7c7b\u578b\u3002\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u8fd9\u4e9b\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u7684\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u8c03\u7528 <code>hash()</code> \u51fd\u6570\u6765\u8ba1\u7b97\u5404\u79cd\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\u3002</p> <ul> <li>\u6574\u6570\u548c\u5e03\u5c14\u91cf\u7684\u54c8\u5e0c\u503c\u5c31\u662f\u5176\u672c\u8eab\u3002</li> <li>\u6d6e\u70b9\u6570\u548c\u5b57\u7b26\u4e32\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u8f83\u4e3a\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u8bf7\u81ea\u884c\u5b66\u4e60\u3002</li> <li>\u5143\u7ec4\u7684\u54c8\u5e0c\u503c\u662f\u5bf9\u5176\u4e2d\u6bcf\u4e00\u4e2a\u5143\u7d20\u8fdb\u884c\u54c8\u5e0c\uff0c\u7136\u540e\u5c06\u8fd9\u4e9b\u54c8\u5e0c\u503c\u7ec4\u5408\u8d77\u6765\uff0c\u5f97\u5230\u5355\u4e00\u7684\u54c8\u5e0c\u503c\u3002</li> <li>\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u57fa\u4e8e\u5176\u5185\u5b58\u5730\u5740\u751f\u6210\u3002\u901a\u8fc7\u91cd\u5199\u5bf9\u8c61\u7684\u54c8\u5e0c\u65b9\u6cd5\uff0c\u53ef\u5b9e\u73b0\u57fa\u4e8e\u5185\u5bb9\u751f\u6210\u54c8\u5e0c\u503c\u3002</li> </ul> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u54c8\u5e0c\u503c\u8ba1\u7b97\u51fd\u6570\u7684\u5b9a\u4e49\u548c\u65b9\u6cd5\u4e0d\u540c\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig built_in_hash.py<pre><code>num = 3\nhash_num = hash(num)\n# \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nbol = True\nhash_bol = hash(bol)\n# \u5e03\u5c14\u91cf True \u7684\u54c8\u5e0c\u503c\u4e3a 1\n\ndec = 3.14159\nhash_dec = hash(dec)\n# \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 326484311674566659\n\nstr = \"Hello \u7b97\u6cd5\"\nhash_str = hash(str)\n# \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 4617003410720528961\n\ntup = (12836, \"\u5c0f\u54c8\")\nhash_tup = hash(tup)\n# \u5143\u7ec4 (12836, '\u5c0f\u54c8') \u7684\u54c8\u5e0c\u503c\u4e3a 1029005403108185979\n\nobj = ListNode(0)\nhash_obj = hash(obj)\n# \u8282\u70b9\u5bf9\u8c61 &lt;ListNode object at 0x1058fd810&gt; \u7684\u54c8\u5e0c\u503c\u4e3a 274267521\n</code></pre> built_in_hash.cpp<pre><code>int num = 3;\nsize_t hashNum = hash&lt;int&gt;()(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nbool bol = true;\nsize_t hashBol = hash&lt;bool&gt;()(bol);\n// \u5e03\u5c14\u91cf 1 \u7684\u54c8\u5e0c\u503c\u4e3a 1\n\ndouble dec = 3.14159;\nsize_t hashDec = hash&lt;double&gt;()(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 4614256650576692846\n\nstring str = \"Hello \u7b97\u6cd5\";\nsize_t hashStr = hash&lt;string&gt;()(str);\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 15466937326284535026\n\n// \u5728 C++ \u4e2d\uff0c\u5185\u7f6e std:hash() \u4ec5\u63d0\u4f9b\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\n// \u6570\u7ec4\u3001\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u9700\u8981\u81ea\u884c\u5b9e\u73b0\n</code></pre> built_in_hash.java<pre><code>int num = 3;\nint hashNum = Integer.hashCode(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\n\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729\n\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -727081396\n\nObject[] arr = { 12836, \"\u5c0f\u54c8\" };\nint hashTup = Arrays.hashCode(arr);\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 1151158\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode@7dc5e7b4 \u7684\u54c8\u5e0c\u503c\u4e3a 2110121908\n</code></pre> built_in_hash.cs<pre><code>int num = 3;\nint hashNum = num.GetHashCode();\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3;\n\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1;\n\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729;\n\nstring str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.GetHashCode();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -586107568;\n\nobject[] arr = [12836, \"\u5c0f\u54c8\"];\nint hashTup = arr.GetHashCode();\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 42931033;\n\nListNode obj = new(0);\nint hashObj = obj.GetHashCode();\n// \u8282\u70b9\u5bf9\u8c61 0 \u7684\u54c8\u5e0c\u503c\u4e3a 39053774;\n</code></pre> built_in_hash.go<pre><code>// Go \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.swift<pre><code>let num = 3\nlet hashNum = num.hashValue\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 9047044699613009734\n\nlet bol = true\nlet hashBol = bol.hashValue\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a -4431640247352757451\n\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -2465384235396674631\n\nlet str = \"Hello \u7b97\u6cd5\"\nlet hashStr = str.hashValue\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -7850626797806988787\n\nlet arr = [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")]\nlet hashTup = arr.hashValue\n// \u6570\u7ec4 [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")] \u7684\u54c8\u5e0c\u503c\u4e3a -2308633508154532996\n\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode \u7684\u54c8\u5e0c\u503c\u4e3a -2434780518035996159\n</code></pre> built_in_hash.js<pre><code>// JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.ts<pre><code>// TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.dart<pre><code>int num = 3;\nint hashNum = num.hashCode;\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 34803\n\nbool bol = true;\nint hashBol = bol.hashCode;\n// \u5e03\u5c14\u503c true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\n\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 2570631074981783\n\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode;\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 468167534\n\nList arr = [12836, \"\u5c0f\u54c8\"];\nint hashArr = arr.hashCode;\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 976512528\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// \u8282\u70b9\u5bf9\u8c61 Instance of 'ListNode' \u7684\u54c8\u5e0c\u503c\u4e3a 1033450432\n</code></pre> built_in_hash.rs<pre><code>use std::collections::hash_map::DefaultHasher;\nuse std::hash::{Hash, Hasher};\n\nlet num = 3;\nlet mut num_hasher = DefaultHasher::new();\nnum.hash(&amp;mut num_hasher);\nlet hash_num = num_hasher.finish();\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 568126464209439262\n\nlet bol = true;\nlet mut bol_hasher = DefaultHasher::new();\nbol.hash(&amp;mut bol_hasher);\nlet hash_bol = bol_hasher.finish();\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 4952851536318644461\n\nlet dec: f32 = 3.14159;\nlet mut dec_hasher = DefaultHasher::new();\ndec.to_bits().hash(&amp;mut dec_hasher);\nlet hash_dec = dec_hasher.finish();\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 2566941990314602357\n\nlet str = \"Hello \u7b97\u6cd5\";\nlet mut str_hasher = DefaultHasher::new();\nstr.hash(&amp;mut str_hasher);\nlet hash_str = str_hasher.finish();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 16092673739211250988\n\nlet arr = (&amp;12836, &amp;\"\u5c0f\u54c8\");\nlet mut tup_hasher = DefaultHasher::new();\narr.hash(&amp;mut tup_hasher);\nlet hash_tup = tup_hasher.finish();\n// \u5143\u7ec4 (12836, \"\u5c0f\u54c8\") \u7684\u54c8\u5e0c\u503c\u4e3a 1885128010422702749\n\nlet node = ListNode::new(42);\nlet mut hasher = DefaultHasher::new();\nnode.borrow().val.hash(&amp;mut hasher);\nlet hash = hasher.finish();\n// \u8282\u70b9\u5bf9\u8c61 RefCell { value: ListNode { val: 42, next: None } } \u7684\u54c8\u5e0c\u503c\u4e3a15387811073369036852\n</code></pre> built_in_hash.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.kt<pre><code>val num = 3\nval hashNum = num.hashCode()\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nval bol = true\nval hashBol = bol.hashCode()\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\n\nval dec = 3.14159\nval hashDec = dec.hashCode()\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729\n\nval str = \"Hello \u7b97\u6cd5\"\nval hashStr = str.hashCode()\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -727081396\n\nval arr = arrayOf&lt;Any&gt;(12836, \"\u5c0f\u54c8\")\nval hashTup = arr.hashCode()\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 189568618\n\nval obj = ListNode(0)\nval hashObj = obj.hashCode()\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode@1d81eb93 \u7684\u54c8\u5e0c\u503c\u4e3a 495053715\n</code></pre> built_in_hash.rb<pre><code>\n</code></pre> built_in_hash.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5728\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u624d\u53ef\u4f5c\u4e3a\u54c8\u5e0c\u8868\u7684 <code>key</code> \u3002\u5047\u5982\u6211\u4eec\u5c06\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4f5c\u4e3a <code>key</code> \uff0c\u5f53\u5217\u8868\u7684\u5185\u5bb9\u53d1\u751f\u53d8\u5316\u65f6\uff0c\u5b83\u7684\u54c8\u5e0c\u503c\u4e5f\u968f\u4e4b\u6539\u53d8\uff0c\u6211\u4eec\u5c31\u65e0\u6cd5\u5728\u54c8\u5e0c\u8868\u4e2d\u67e5\u8be2\u5230\u539f\u5148\u7684 <code>value</code> \u4e86\u3002</p> <p>\u867d\u7136\u81ea\u5b9a\u4e49\u5bf9\u8c61\uff08\u6bd4\u5982\u94fe\u8868\u8282\u70b9\uff09\u7684\u6210\u5458\u53d8\u91cf\u662f\u53ef\u53d8\u7684\uff0c\u4f46\u5b83\u662f\u53ef\u54c8\u5e0c\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u901a\u5e38\u662f\u57fa\u4e8e\u5185\u5b58\u5730\u5740\u751f\u6210\u7684\uff0c\u5373\u4f7f\u5bf9\u8c61\u7684\u5185\u5bb9\u53d1\u751f\u4e86\u53d8\u5316\uff0c\u4f46\u5b83\u7684\u5185\u5b58\u5730\u5740\u4e0d\u53d8\uff0c\u54c8\u5e0c\u503c\u4ecd\u7136\u662f\u4e0d\u53d8\u7684\u3002</p> <p>\u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u53d1\u73b0\u5728\u4e0d\u540c\u63a7\u5236\u53f0\u4e2d\u8fd0\u884c\u7a0b\u5e8f\u65f6\uff0c\u8f93\u51fa\u7684\u54c8\u5e0c\u503c\u662f\u4e0d\u540c\u7684\u3002\u8fd9\u662f\u56e0\u4e3a Python \u89e3\u91ca\u5668\u5728\u6bcf\u6b21\u542f\u52a8\u65f6\uff0c\u90fd\u4f1a\u4e3a\u5b57\u7b26\u4e32\u54c8\u5e0c\u51fd\u6570\u52a0\u5165\u4e00\u4e2a\u968f\u673a\u7684\u76d0\uff08salt\uff09\u503c\u3002\u8fd9\u79cd\u505a\u6cd5\u53ef\u4ee5\u6709\u6548\u9632\u6b62 HashDoS \u653b\u51fb\uff0c\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u3002</p>"},{"location":"chapter_hashing/hash_collision/","title":"6.2 \u00a0 \u54c8\u5e0c\u51b2\u7a81","text":"<p>\u4e0a\u4e00\u8282\u63d0\u5230\uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u5165\u7a7a\u95f4\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff0c\u56e0\u6b64\u7406\u8bba\u4e0a\u54c8\u5e0c\u51b2\u7a81\u662f\u4e0d\u53ef\u907f\u514d\u7684\u3002\u6bd4\u5982\uff0c\u8f93\u5165\u7a7a\u95f4\u4e3a\u5168\u4f53\u6574\u6570\uff0c\u8f93\u51fa\u7a7a\u95f4\u4e3a\u6570\u7ec4\u5bb9\u91cf\u5927\u5c0f\uff0c\u5219\u5fc5\u7136\u6709\u591a\u4e2a\u6574\u6570\u6620\u5c04\u81f3\u540c\u4e00\u6876\u7d22\u5f15\u3002</p> <p>\u54c8\u5e0c\u51b2\u7a81\u4f1a\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u9519\u8bef\uff0c\u4e25\u91cd\u5f71\u54cd\u54c8\u5e0c\u8868\u7684\u53ef\u7528\u6027\u3002\u4e3a\u4e86\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6bcf\u5f53\u9047\u5230\u54c8\u5e0c\u51b2\u7a81\u65f6\uff0c\u6211\u4eec\u5c31\u8fdb\u884c\u54c8\u5e0c\u8868\u6269\u5bb9\uff0c\u76f4\u81f3\u51b2\u7a81\u6d88\u5931\u4e3a\u6b62\u3002\u6b64\u65b9\u6cd5\u7b80\u5355\u7c97\u66b4\u4e14\u6709\u6548\uff0c\u4f46\u6548\u7387\u592a\u4f4e\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u8981\u8fdb\u884c\u5927\u91cf\u7684\u6570\u636e\u642c\u8fd0\u4e0e\u54c8\u5e0c\u503c\u8ba1\u7b97\u3002\u4e3a\u4e86\u63d0\u5347\u6548\u7387\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u7b56\u7565\u3002</p> <ol> <li>\u6539\u826f\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u4f7f\u5f97\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\u3002</li> <li>\u4ec5\u5728\u5fc5\u8981\u65f6\uff0c\u5373\u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u624d\u6267\u884c\u6269\u5bb9\u64cd\u4f5c\u3002</li> </ol> <p>\u54c8\u5e0c\u8868\u7684\u7ed3\u6784\u6539\u826f\u65b9\u6cd5\u4e3b\u8981\u5305\u62ec\u201c\u94fe\u5f0f\u5730\u5740\u201d\u548c\u201c\u5f00\u653e\u5bfb\u5740\u201d\u3002</p>"},{"location":"chapter_hashing/hash_collision/#621","title":"6.2.1 \u00a0 \u94fe\u5f0f\u5730\u5740","text":"<p>\u5728\u539f\u59cb\u54c8\u5e0c\u8868\u4e2d\uff0c\u6bcf\u4e2a\u6876\u4ec5\u80fd\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u94fe\u5f0f\u5730\u5740\uff08separate chaining\uff09\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u6362\u4e3a\u94fe\u8868\uff0c\u5c06\u952e\u503c\u5bf9\u4f5c\u4e3a\u94fe\u8868\u8282\u70b9\uff0c\u5c06\u6240\u6709\u53d1\u751f\u51b2\u7a81\u7684\u952e\u503c\u5bf9\u90fd\u5b58\u50a8\u5728\u540c\u4e00\u94fe\u8868\u4e2d\u3002\u56fe 6-5 \u5c55\u793a\u4e86\u4e00\u4e2a\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\u7684\u4f8b\u5b50\u3002</p> <p></p> <p> \u56fe 6-5 \u00a0 \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 </p> <p>\u57fa\u4e8e\u94fe\u5f0f\u5730\u5740\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\u7684\u64cd\u4f5c\u65b9\u6cd5\u53d1\u751f\u4e86\u4ee5\u4e0b\u53d8\u5316\u3002</p> <ul> <li>\u67e5\u8be2\u5143\u7d20\uff1a\u8f93\u5165 <code>key</code> \uff0c\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u6876\u7d22\u5f15\uff0c\u5373\u53ef\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u904d\u5386\u94fe\u8868\u5e76\u5bf9\u6bd4 <code>key</code> \u4ee5\u67e5\u627e\u76ee\u6807\u952e\u503c\u5bf9\u3002</li> <li>\u6dfb\u52a0\u5143\u7d20\uff1a\u9996\u5148\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u5c06\u8282\u70b9\uff08\u952e\u503c\u5bf9\uff09\u6dfb\u52a0\u5230\u94fe\u8868\u4e2d\u3002</li> <li>\u5220\u9664\u5143\u7d20\uff1a\u6839\u636e\u54c8\u5e0c\u51fd\u6570\u7684\u7ed3\u679c\u8bbf\u95ee\u94fe\u8868\u5934\u90e8\uff0c\u63a5\u7740\u904d\u5386\u94fe\u8868\u4ee5\u67e5\u627e\u76ee\u6807\u8282\u70b9\u5e76\u5c06\u5176\u5220\u9664\u3002</li> </ul> <p>\u94fe\u5f0f\u5730\u5740\u5b58\u5728\u4ee5\u4e0b\u5c40\u9650\u6027\u3002</p> <ul> <li>\u5360\u7528\u7a7a\u95f4\u589e\u5927\uff1a\u94fe\u8868\u5305\u542b\u8282\u70b9\u6307\u9488\uff0c\u5b83\u76f8\u6bd4\u6570\u7ec4\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u67e5\u8be2\u6548\u7387\u964d\u4f4e\uff1a\u56e0\u4e3a\u9700\u8981\u7ebf\u6027\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u5bf9\u5e94\u5143\u7d20\u3002</li> </ul> <p>\u4ee5\u4e0b\u4ee3\u7801\u7ed9\u51fa\u4e86\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\u7684\u7b80\u5355\u5b9e\u73b0\uff0c\u9700\u8981\u6ce8\u610f\u4e24\u70b9\u3002</p> <ul> <li>\u4f7f\u7528\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4ee3\u66ff\u94fe\u8868\uff0c\u4ece\u800c\u7b80\u5316\u4ee3\u7801\u3002\u5728\u8fd9\u79cd\u8bbe\u5b9a\u4e0b\uff0c\u54c8\u5e0c\u8868\uff08\u6570\u7ec4\uff09\u5305\u542b\u591a\u4e2a\u6876\uff0c\u6bcf\u4e2a\u6876\u90fd\u662f\u4e00\u4e2a\u5217\u8868\u3002</li> <li>\u4ee5\u4e0b\u5b9e\u73b0\u5305\u542b\u54c8\u5e0c\u8868\u6269\u5bb9\u65b9\u6cd5\u3002\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(\\frac{2}{3}\\) \u65f6\uff0c\u6211\u4eec\u5c06\u54c8\u5e0c\u8868\u6269\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map_chaining.py<pre><code>class HashMapChaining:\n    \"\"\"\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\n        self.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\n        self.load_thres = 2.0 / 3.0  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        self.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\n        self.buckets = [[] for _ in range(self.capacity)]  # \u6876\u6570\u7ec4\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\n        return self.size / self.capacity\n\n    def get(self, key: int) -&gt; str | None:\n        \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for pair in bucket:\n            if pair.key == key:\n                return pair.val\n        # \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n        # \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for pair in bucket:\n            if pair.key == key:\n                pair.val = val\n                return\n        # \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        pair = Pair(key, val)\n        bucket.append(pair)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for pair in bucket:\n            if pair.key == key:\n                bucket.remove(pair)\n                self.size -= 1\n                break\n\n    def extend(self):\n        \"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n        # \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        buckets = self.buckets\n        # \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio\n        self.buckets = [[] for _ in range(self.capacity)]\n        self.size = 0\n        # \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for bucket in buckets:\n            for pair in bucket:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n        for bucket in self.buckets:\n            res = []\n            for pair in bucket:\n                res.append(str(pair.key) + \" -&gt; \" + pair.val)\n            print(res)\n</code></pre> hash_map_chaining.cpp<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n  private:\n    int size;                       // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity;                   // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres;               // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio;                // \u6269\u5bb9\u500d\u6570\n    vector&lt;vector&lt;Pair *&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    HashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3.0), extendRatio(2) {\n        buckets.resize(capacity);\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~HashMapChaining() {\n        for (auto &amp;bucket : buckets) {\n            for (Pair *pair : bucket) {\n                // \u91ca\u653e\u5185\u5b58\n                delete pair;\n            }\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double loadFactor() {\n        return (double)size / (double)capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    string get(int key) {\n        int index = hashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                return pair-&gt;val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n        return \"\";\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                pair-&gt;val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        buckets[index].push_back(new Pair(key, val));\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        auto &amp;bucket = buckets[index];\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (int i = 0; i &lt; bucket.size(); i++) {\n            if (bucket[i]-&gt;key == key) {\n                Pair *tmp = bucket[i];\n                bucket.erase(bucket.begin() + i); // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n                delete tmp;                       // \u91ca\u653e\u5185\u5b58\n                size--;\n                return;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        vector&lt;vector&lt;Pair *&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets.clear();\n        buckets.resize(capacity);\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (auto &amp;bucket : bucketsTmp) {\n            for (Pair *pair : bucket) {\n                put(pair-&gt;key, pair-&gt;val);\n                // \u91ca\u653e\u5185\u5b58\n                delete pair;\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (auto &amp;bucket : buckets) {\n            cout &lt;&lt; \"[\";\n            for (Pair *pair : bucket) {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; \", \";\n            }\n            cout &lt;&lt; \"]\\n\";\n        }\n    }\n};\n</code></pre> hash_map_chaining.java<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio; // \u6269\u5bb9\u500d\u6570\n    List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    String get(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, String val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        Pair pair = new Pair(key, val);\n        bucket.add(pair);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (List&lt;Pair&gt; bucket : bucketsTmp) {\n            for (Pair pair : bucket) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (List&lt;Pair&gt; bucket : buckets) {\n            List&lt;String&gt; res = new ArrayList&lt;&gt;();\n            for (Pair pair : bucket) {\n                res.add(pair.key + \" -&gt; \" + pair.val);\n            }\n            System.out.println(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.cs<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio; // \u6269\u5bb9\u500d\u6570\n    List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        int index = HashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        buckets[index].Add(new Pair(key, val));\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        foreach (Pair pair in buckets[index].ToList()) {\n            if (pair.key == key) {\n                buckets[index].Remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void Extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        foreach (List&lt;Pair&gt; bucket in bucketsTmp) {\n            foreach (Pair pair in bucket) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void Print() {\n        foreach (List&lt;Pair&gt; bucket in buckets) {\n            List&lt;string&gt; res = [];\n            foreach (Pair pair in bucket) {\n                res.Add(pair.key + \" -&gt; \" + pair.val);\n            }\n            foreach (string kv in res) {\n                Console.WriteLine(kv);\n            }\n        }\n    }\n}\n</code></pre> hash_map_chaining.go<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntype hashMapChaining struct {\n    size        int      // \u952e\u503c\u5bf9\u6570\u91cf\n    capacity    int      // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    loadThres   float64  // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    extendRatio int      // \u6269\u5bb9\u500d\u6570\n    buckets     [][]pair // \u6876\u6570\u7ec4\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapChaining() *hashMapChaining {\n    buckets := make([][]pair, 4)\n    for i := 0; i &lt; 4; i++ {\n        buckets[i] = make([]pair, 0)\n    }\n    return &amp;hashMapChaining{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     buckets,\n    }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapChaining) hashFunc(key int) int {\n    return key % m.capacity\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapChaining) loadFactor() float64 {\n    return float64(m.size) / float64(m.capacity)\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapChaining) get(key int) string {\n    idx := m.hashFunc(key)\n    bucket := m.buckets[idx]\n    // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    for _, p := range bucket {\n        if p.key == key {\n            return p.val\n        }\n    }\n    // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n    return \"\"\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapChaining) put(key int, val string) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if m.loadFactor() &gt; m.loadThres {\n        m.extend()\n    }\n    idx := m.hashFunc(key)\n    // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n    for i := range m.buckets[idx] {\n        if m.buckets[idx][i].key == key {\n            m.buckets[idx][i].val = val\n            return\n        }\n    }\n    // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n    p := pair{\n        key: key,\n        val: val,\n    }\n    m.buckets[idx] = append(m.buckets[idx], p)\n    m.size += 1\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapChaining) remove(key int) {\n    idx := m.hashFunc(key)\n    // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n    for i, p := range m.buckets[idx] {\n        if p.key == key {\n            // \u5207\u7247\u5220\u9664\n            m.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\n            m.size -= 1\n            break\n        }\n    }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    tmpBuckets := make([][]pair, len(m.buckets))\n    for i := 0; i &lt; len(m.buckets); i++ {\n        tmpBuckets[i] = make([]pair, len(m.buckets[i]))\n        copy(tmpBuckets[i], m.buckets[i])\n    }\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    m.capacity *= m.extendRatio\n    m.buckets = make([][]pair, m.capacity)\n    for i := 0; i &lt; m.capacity; i++ {\n        m.buckets[i] = make([]pair, 0)\n    }\n    m.size = 0\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for _, bucket := range tmpBuckets {\n        for _, p := range bucket {\n            m.put(p.key, p.val)\n        }\n    }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) print() {\n    var builder strings.Builder\n\n    for _, bucket := range m.buckets {\n        builder.WriteString(\"[\")\n        for _, p := range bucket {\n            builder.WriteString(strconv.Itoa(p.key) + \" -&gt; \" + p.val + \" \")\n        }\n        builder.WriteString(\"]\")\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> hash_map_chaining.swift<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n    var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    var loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    var extendRatio: Int // \u6269\u5bb9\u500d\u6570\n    var buckets: [[Pair]] // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: [], count: capacity)\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for pair in bucket {\n            if pair.key == key {\n                return pair.val\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de nil\n        return nil\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val\n                return\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        let pair = Pair(key: key, val: val)\n        buckets[index].append(pair)\n        size += 1\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (pairIndex, pair) in bucket.enumerated() {\n            if pair.key == key {\n                buckets[index].remove(at: pairIndex)\n                size -= 1\n                break\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    func extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        buckets = Array(repeating: [], count: capacity)\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for bucket in bucketsTmp {\n            for pair in bucket {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    func print() {\n        for bucket in buckets {\n            let res = bucket.map { \"\\($0.key) -&gt; \\($0.val)\" }\n            Swift.print(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.js<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    #size; // \u952e\u503c\u5bf9\u6570\u91cf\n    #capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    #loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    #extendRatio; // \u6269\u5bb9\u500d\u6570\n    #buckets; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key) {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key, val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key) {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    #extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print() {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.ts<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    #size: number; // \u952e\u503c\u5bf9\u6570\u91cf\n    #capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    #loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    #extendRatio: number; // \u6269\u5bb9\u500d\u6570\n    #buckets: Pair[][]; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key: number): number {\n        return key % this.#capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    #loadFactor(): number {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key: number): string | null {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key: number, val: string): void {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key: number): void {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    #extend(): void {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print(): void {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.dart<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n  late int size; // \u952e\u503c\u5bf9\u6570\u91cf\n  late int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n  late double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n  late int extendRatio; // \u6269\u5bb9\u500d\u6570\n  late List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  HashMapChaining() {\n    size = 0;\n    capacity = 4;\n    loadThres = 2.0 / 3.0;\n    extendRatio = 2;\n    buckets = List.generate(capacity, (_) =&gt; []);\n  }\n\n  /* \u54c8\u5e0c\u51fd\u6570 */\n  int hashFunc(int key) {\n    return key % capacity;\n  }\n\n  /* \u8d1f\u8f7d\u56e0\u5b50 */\n  double loadFactor() {\n    return size / capacity;\n  }\n\n  /* \u67e5\u8be2\u64cd\u4f5c */\n  String? get(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        return pair.val;\n      }\n    }\n    // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n    return null;\n  }\n\n  /* \u6dfb\u52a0\u64cd\u4f5c */\n  void put(int key, String val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor() &gt; loadThres) {\n      extend();\n    }\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        pair.val = val;\n        return;\n      }\n    }\n    // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n    Pair pair = Pair(key, val);\n    bucket.add(pair);\n    size++;\n  }\n\n  /* \u5220\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        bucket.remove(pair);\n        size--;\n        break;\n      }\n    }\n  }\n\n  /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n  void extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    capacity *= extendRatio;\n    buckets = List.generate(capacity, (_) =&gt; []);\n    size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (List&lt;Pair&gt; bucket in bucketsTmp) {\n      for (Pair pair in bucket) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* \u6253\u5370\u54c8\u5e0c\u8868 */\n  void printHashMap() {\n    for (List&lt;Pair&gt; bucket in buckets) {\n      List&lt;String&gt; res = [];\n      for (Pair pair in bucket) {\n        res.add(\"${pair.key} -&gt; ${pair.val}\");\n      }\n      print(res);\n    }\n  }\n}\n</code></pre> hash_map_chaining.rs<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapChaining {\n    size: i32,\n    capacity: i32,\n    load_thres: f32,\n    extend_ratio: i32,\n    buckets: Vec&lt;Vec&lt;Pair&gt;&gt;,\n}\n\nimpl HashMapChaining {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![vec![]; 4],\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % self.capacity as usize\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fn load_factor(&amp;self) -&gt; f32 {\n        self.size as f32 / self.capacity as f32\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fn remove(&amp;mut self, key: i32) -&gt; Option&lt;String&gt; {\n        let index = self.hash_func(key);\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for i in 0..bucket.len() {\n            if bucket[i].key == key {\n                let pair = bucket.remove(i);\n                self.size -= 1;\n                return Some(pair.val);\n            }\n        }\n\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de None\n        None\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fn extend(&amp;mut self) {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let buckets_tmp = std::mem::replace(&amp;mut self.buckets, vec![]);\n\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![Vec::new(); self.capacity as usize];\n        self.size = 0;\n\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for bucket in buckets_tmp {\n            for pair in bucket {\n                self.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fn print(&amp;self) {\n        for bucket in &amp;self.buckets {\n            let mut res = Vec::new();\n            for pair in bucket {\n                res.push(format!(\"{} -&gt; {}\", pair.key, pair.val));\n            }\n            println!(\"{:?}\", res);\n        }\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n\n        let index = self.hash_func(key);\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val.clone();\n                return;\n            }\n        }\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        let pair = Pair {\n            key,\n            val: val.clone(),\n        };\n        bucket.push(pair);\n        self.size += 1;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        let index = self.hash_func(key);\n        let bucket = &amp;self.buckets[index];\n\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for pair in bucket {\n            if pair.key == key {\n                return Some(&amp;pair.val);\n            }\n        }\n\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de None\n        None\n    }\n}\n</code></pre> hash_map_chaining.c<pre><code>/* \u94fe\u8868\u8282\u70b9 */\ntypedef struct Node {\n    Pair *pair;\n    struct Node *next;\n} Node;\n\n/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\n    int size;         // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity;     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio;  // \u6269\u5bb9\u500d\u6570\n    Node **buckets;   // \u6876\u6570\u7ec4\n} HashMapChaining;\n\n/* \u6784\u9020\u51fd\u6570 */\nHashMapChaining *newHashMapChaining() {\n    HashMapChaining *hashMap = (HashMapChaining *)malloc(sizeof(HashMapChaining));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    return hashMap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapChaining(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        while (cur) {\n            Node *tmp = cur;\n            cur = cur-&gt;next;\n            free(tmp-&gt;pair);\n            free(tmp);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap);\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapChaining *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapChaining *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            return cur-&gt;pair-&gt;val;\n        }\n        cur = cur-&gt;next;\n    }\n    return \"\"; // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapChaining *hashMap, int key, const char *val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    int index = hashFunc(hashMap, key);\n    // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            strcpy(cur-&gt;pair-&gt;val, val); // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n            return;\n        }\n        cur = cur-&gt;next;\n    }\n    // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n    Pair *newPair = (Pair *)malloc(sizeof(Pair));\n    newPair-&gt;key = key;\n    strcpy(newPair-&gt;val, val);\n    Node *newNode = (Node *)malloc(sizeof(Node));\n    newNode-&gt;pair = newPair;\n    newNode-&gt;next = hashMap-&gt;buckets[index];\n    hashMap-&gt;buckets[index] = newNode;\n    hashMap-&gt;size++;\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapChaining *hashMap) {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    int oldCapacity = hashMap-&gt;capacity;\n    Node **oldBuckets = hashMap-&gt;buckets;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    hashMap-&gt;size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Node *cur = oldBuckets[i];\n        while (cur) {\n            put(hashMap, cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            Node *temp = cur;\n            cur = cur-&gt;next;\n            // \u91ca\u653e\u5185\u5b58\n            free(temp-&gt;pair);\n            free(temp);\n        }\n    }\n\n    free(oldBuckets);\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    Node *cur = hashMap-&gt;buckets[index];\n    Node *pre = NULL;\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n            if (pre) {\n                pre-&gt;next = cur-&gt;next;\n            } else {\n                hashMap-&gt;buckets[index] = cur-&gt;next;\n            }\n            // \u91ca\u653e\u5185\u5b58\n            free(cur-&gt;pair);\n            free(cur);\n            hashMap-&gt;size--;\n            return;\n        }\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        printf(\"[\");\n        while (cur) {\n            printf(\"%d -&gt; %s, \", cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            cur = cur-&gt;next;\n        }\n        printf(\"]\\n\");\n    }\n}\n</code></pre> hash_map_chaining.kt<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining() {\n    var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n    var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    val loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    val extendRatio: Int // \u6269\u5bb9\u500d\u6570\n    var buckets: MutableList&lt;MutableList&lt;Pair&gt;&gt; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = ArrayList(capacity)\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (pair in bucket) {\n            if (pair.key == key) return pair.value\n        }\n        // \u82e5\u672a\u627e\u5230 key \uff0c\u5219\u8fd4\u56de null\n        return null\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (pair in bucket) {\n            if (pair.key == key) {\n                pair.value = value\n                return\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        val pair = Pair(key, value)\n        bucket.add(pair)\n        size++\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (pair in bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair)\n                size--\n                break\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fun extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        val bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        // mutablelist \u65e0\u56fa\u5b9a\u5927\u5c0f\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (bucket in bucketsTmp) {\n            for (pair in bucket) {\n                put(pair.key, pair.value)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (bucket in buckets) {\n            val res = mutableListOf&lt;String&gt;()\n            for (pair in bucket) {\n                val k = pair.key\n                val v = pair.value\n                res.add(\"$k -&gt; $v\")\n            }\n            println(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.rb<pre><code>[class]{HashMapChaining}-[func]{}\n</code></pre> hash_map_chaining.zig<pre><code>[class]{HashMapChaining}-[func]{}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5f53\u94fe\u8868\u5f88\u957f\u65f6\uff0c\u67e5\u8be2\u6548\u7387 \\(O(n)\\) \u5f88\u5dee\u3002\u6b64\u65f6\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u201cAVL \u6811\u201d\u6216\u201c\u7ea2\u9ed1\u6811\u201d\uff0c\u4ece\u800c\u5c06\u67e5\u8be2\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002</p>"},{"location":"chapter_hashing/hash_collision/#622","title":"6.2.2 \u00a0 \u5f00\u653e\u5bfb\u5740","text":"<p>\u5f00\u653e\u5bfb\u5740\uff08open addressing\uff09\u4e0d\u5f15\u5165\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u800c\u662f\u901a\u8fc7\u201c\u591a\u6b21\u63a2\u6d4b\u201d\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\uff0c\u63a2\u6d4b\u65b9\u5f0f\u4e3b\u8981\u5305\u62ec\u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u548c\u591a\u6b21\u54c8\u5e0c\u7b49\u3002</p> <p>\u4e0b\u9762\u4ee5\u7ebf\u6027\u63a2\u6d4b\u4e3a\u4f8b\uff0c\u4ecb\u7ecd\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\u7684\u5de5\u4f5c\u673a\u5236\u3002</p>"},{"location":"chapter_hashing/hash_collision/#1","title":"1. \u00a0 \u7ebf\u6027\u63a2\u6d4b","text":"<p>\u7ebf\u6027\u63a2\u6d4b\u91c7\u7528\u56fa\u5b9a\u6b65\u957f\u7684\u7ebf\u6027\u641c\u7d22\u6765\u8fdb\u884c\u63a2\u6d4b\uff0c\u5176\u64cd\u4f5c\u65b9\u6cd5\u4e0e\u666e\u901a\u54c8\u5e0c\u8868\u6709\u6240\u4e0d\u540c\u3002</p> <ul> <li>\u63d2\u5165\u5143\u7d20\uff1a\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u82e5\u53d1\u73b0\u6876\u5185\u5df2\u6709\u5143\u7d20\uff0c\u5219\u4ece\u51b2\u7a81\u4f4d\u7f6e\u5411\u540e\u7ebf\u6027\u904d\u5386\uff08\u6b65\u957f\u901a\u5e38\u4e3a \\(1\\) \uff09\uff0c\u76f4\u81f3\u627e\u5230\u7a7a\u6876\uff0c\u5c06\u5143\u7d20\u63d2\u5165\u5176\u4e2d\u3002</li> <li>\u67e5\u627e\u5143\u7d20\uff1a\u82e5\u53d1\u73b0\u54c8\u5e0c\u51b2\u7a81\uff0c\u5219\u4f7f\u7528\u76f8\u540c\u6b65\u957f\u5411\u540e\u8fdb\u884c\u7ebf\u6027\u904d\u5386\uff0c\u76f4\u5230\u627e\u5230\u5bf9\u5e94\u5143\u7d20\uff0c\u8fd4\u56de <code>value</code> \u5373\u53ef\uff1b\u5982\u679c\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u76ee\u6807\u5143\u7d20\u4e0d\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8fd4\u56de <code>None</code> \u3002</li> </ul> <p>\u56fe 6-6 \u5c55\u793a\u4e86\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u7684\u952e\u503c\u5bf9\u5206\u5e03\u3002\u6839\u636e\u6b64\u54c8\u5e0c\u51fd\u6570\uff0c\u6700\u540e\u4e24\u4f4d\u76f8\u540c\u7684 <code>key</code> \u90fd\u4f1a\u88ab\u6620\u5c04\u5230\u76f8\u540c\u7684\u6876\u3002\u800c\u901a\u8fc7\u7ebf\u6027\u63a2\u6d4b\uff0c\u5b83\u4eec\u88ab\u4f9d\u6b21\u5b58\u50a8\u5728\u8be5\u6876\u4ee5\u53ca\u4e4b\u4e0b\u7684\u6876\u4e2d\u3002</p> <p></p> <p> \u56fe 6-6 \u00a0 \u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u7684\u952e\u503c\u5bf9\u5206\u5e03 </p> <p>\u7136\u800c\uff0c\u7ebf\u6027\u63a2\u6d4b\u5bb9\u6613\u4ea7\u751f\u201c\u805a\u96c6\u73b0\u8c61\u201d\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u6570\u7ec4\u4e2d\u8fde\u7eed\u88ab\u5360\u7528\u7684\u4f4d\u7f6e\u8d8a\u957f\uff0c\u8fd9\u4e9b\u8fde\u7eed\u4f4d\u7f6e\u53d1\u751f\u54c8\u5e0c\u51b2\u7a81\u7684\u53ef\u80fd\u6027\u8d8a\u5927\uff0c\u4ece\u800c\u8fdb\u4e00\u6b65\u4fc3\u4f7f\u8be5\u4f4d\u7f6e\u7684\u805a\u5806\u751f\u957f\uff0c\u5f62\u6210\u6076\u6027\u5faa\u73af\uff0c\u6700\u7ec8\u5bfc\u81f4\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u6548\u7387\u52a3\u5316\u3002</p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6211\u4eec\u4e0d\u80fd\u5728\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\u4e2d\u76f4\u63a5\u5220\u9664\u5143\u7d20\u3002\u8fd9\u662f\u56e0\u4e3a\u5220\u9664\u5143\u7d20\u4f1a\u5728\u6570\u7ec4\u5185\u4ea7\u751f\u4e00\u4e2a\u7a7a\u6876 <code>None</code> \uff0c\u800c\u5f53\u67e5\u8be2\u5143\u7d20\u65f6\uff0c\u7ebf\u6027\u63a2\u6d4b\u5230\u8be5\u7a7a\u6876\u5c31\u4f1a\u8fd4\u56de\uff0c\u56e0\u6b64\u5728\u8be5\u7a7a\u6876\u4e4b\u4e0b\u7684\u5143\u7d20\u90fd\u65e0\u6cd5\u518d\u88ab\u8bbf\u95ee\u5230\uff0c\u7a0b\u5e8f\u53ef\u80fd\u8bef\u5224\u8fd9\u4e9b\u5143\u7d20\u4e0d\u5b58\u5728\uff0c\u5982\u56fe 6-7 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 6-7 \u00a0 \u5728\u5f00\u653e\u5bfb\u5740\u4e2d\u5220\u9664\u5143\u7d20\u5bfc\u81f4\u7684\u67e5\u8be2\u95ee\u9898 </p> <p>\u4e3a\u4e86\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u61d2\u5220\u9664\uff08lazy deletion\uff09\u673a\u5236\uff1a\u5b83\u4e0d\u76f4\u63a5\u4ece\u54c8\u5e0c\u8868\u4e2d\u79fb\u9664\u5143\u7d20\uff0c\u800c\u662f\u5229\u7528\u4e00\u4e2a\u5e38\u91cf <code>TOMBSTONE</code> \u6765\u6807\u8bb0\u8fd9\u4e2a\u6876\u3002\u5728\u8be5\u673a\u5236\u4e0b\uff0c<code>None</code> \u548c <code>TOMBSTONE</code> \u90fd\u4ee3\u8868\u7a7a\u6876\uff0c\u90fd\u53ef\u4ee5\u653e\u7f6e\u952e\u503c\u5bf9\u3002\u4f46\u4e0d\u540c\u7684\u662f\uff0c\u7ebf\u6027\u63a2\u6d4b\u5230 <code>TOMBSTONE</code> \u65f6\u5e94\u8be5\u7ee7\u7eed\u904d\u5386\uff0c\u56e0\u4e3a\u5176\u4e4b\u4e0b\u53ef\u80fd\u8fd8\u5b58\u5728\u952e\u503c\u5bf9\u3002</p> <p>\u7136\u800c\uff0c\u61d2\u5220\u9664\u53ef\u80fd\u4f1a\u52a0\u901f\u54c8\u5e0c\u8868\u7684\u6027\u80fd\u9000\u5316\u3002\u8fd9\u662f\u56e0\u4e3a\u6bcf\u6b21\u5220\u9664\u64cd\u4f5c\u90fd\u4f1a\u4ea7\u751f\u4e00\u4e2a\u5220\u9664\u6807\u8bb0\uff0c\u968f\u7740 <code>TOMBSTONE</code> \u7684\u589e\u52a0\uff0c\u641c\u7d22\u65f6\u95f4\u4e5f\u4f1a\u589e\u52a0\uff0c\u56e0\u4e3a\u7ebf\u6027\u63a2\u6d4b\u53ef\u80fd\u9700\u8981\u8df3\u8fc7\u591a\u4e2a <code>TOMBSTONE</code> \u624d\u80fd\u627e\u5230\u76ee\u6807\u5143\u7d20\u3002</p> <p>\u4e3a\u6b64\uff0c\u8003\u8651\u5728\u7ebf\u6027\u63a2\u6d4b\u4e2d\u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a <code>TOMBSTONE</code> \u7684\u7d22\u5f15\uff0c\u5e76\u5c06\u641c\u7d22\u5230\u7684\u76ee\u6807\u5143\u7d20\u4e0e\u8be5 <code>TOMBSTONE</code> \u4ea4\u6362\u4f4d\u7f6e\u3002\u8fd9\u6837\u505a\u7684\u597d\u5904\u662f\u5f53\u6bcf\u6b21\u67e5\u8be2\u6216\u6dfb\u52a0\u5143\u7d20\u65f6\uff0c\u5143\u7d20\u4f1a\u88ab\u79fb\u52a8\u81f3\u8ddd\u79bb\u7406\u60f3\u4f4d\u7f6e\uff08\u63a2\u6d4b\u8d77\u59cb\u70b9\uff09\u66f4\u8fd1\u7684\u6876\uff0c\u4ece\u800c\u4f18\u5316\u67e5\u8be2\u6548\u7387\u3002</p> <p>\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u4e2a\u5305\u542b\u61d2\u5220\u9664\u7684\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u3002\u4e3a\u4e86\u66f4\u52a0\u5145\u5206\u5730\u4f7f\u7528\u54c8\u5e0c\u8868\u7684\u7a7a\u95f4\uff0c\u6211\u4eec\u5c06\u54c8\u5e0c\u8868\u770b\u4f5c\u4e00\u4e2a\u201c\u73af\u5f62\u6570\u7ec4\u201d\uff0c\u5f53\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map_open_addressing.py<pre><code>class HashMapOpenAddressing:\n    \"\"\"\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\n        self.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\n        self.load_thres = 2.0 / 3.0  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        self.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\n        self.buckets: list[Pair | None] = [None] * self.capacity  # \u6876\u6570\u7ec4\n        self.TOMBSTONE = Pair(-1, \"-1\")  # \u5220\u9664\u6807\u8bb0\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\n        return self.size / self.capacity\n\n    def find_bucket(self, key: int) -&gt; int:\n        \"\"\"\u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\"\"\"\n        index = self.hash_func(key)\n        first_tombstone = -1\n        # \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while self.buckets[index] is not None:\n            # \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if self.buckets[index].key == key:\n                # \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if first_tombstone != -1:\n                    self.buckets[first_tombstone] = self.buckets[index]\n                    self.buckets[index] = self.TOMBSTONE\n                    return first_tombstone  # \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                return index  # \u8fd4\u56de\u6876\u7d22\u5f15\n            # \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if first_tombstone == -1 and self.buckets[index] is self.TOMBSTONE:\n                first_tombstone = index\n            # \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % self.capacity\n        # \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return index if first_tombstone == -1 else first_tombstone\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n        # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            return self.buckets[index].val\n        # \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n        # \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index].val = val\n            return\n        # \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        self.buckets[index] = Pair(key, val)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n        # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index] = self.TOMBSTONE\n            self.size -= 1\n\n    def extend(self):\n        \"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n        # \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        buckets_tmp = self.buckets\n        # \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio\n        self.buckets = [None] * self.capacity\n        self.size = 0\n        # \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for pair in buckets_tmp:\n            if pair not in [None, self.TOMBSTONE]:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n        for pair in self.buckets:\n            if pair is None:\n                print(\"None\")\n            elif pair is self.TOMBSTONE:\n                print(\"TOMBSTONE\")\n            else:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> hash_map_open_addressing.cpp<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n  private:\n    int size;                             // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity = 4;                     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    const double loadThres = 2.0 / 3.0;     // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    const int extendRatio = 2;            // \u6269\u5bb9\u500d\u6570\n    vector&lt;Pair *&gt; buckets;               // \u6876\u6570\u7ec4\n    Pair *TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    HashMapOpenAddressing() : size(0), buckets(capacity, nullptr) {\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~HashMapOpenAddressing() {\n        for (Pair *pair : buckets) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                delete pair;\n            }\n        }\n        delete TOMBSTONE;\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double loadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != nullptr) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index]-&gt;key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    string get(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]-&gt;val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n        return \"\";\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]-&gt;val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            delete buckets[index];\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        vector&lt;Pair *&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = vector&lt;Pair *&gt;(capacity, nullptr);\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (Pair *pair : bucketsTmp) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                put(pair-&gt;key, pair-&gt;val);\n                delete pair;\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (Pair *pair : buckets) {\n            if (pair == nullptr) {\n                cout &lt;&lt; \"nullptr\" &lt;&lt; endl;\n            } else if (pair == TOMBSTONE) {\n                cout &lt;&lt; \"TOMBSTONE\" &lt;&lt; endl;\n            } else {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; endl;\n            }\n        }\n    }\n};\n</code></pre> hash_map_open_addressing.java<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    private int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    private int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    private final double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    private final int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n    private Pair[] buckets; // \u6876\u6570\u7ec4\n    private final Pair TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    private double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    private int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index].key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public String get(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void put(int key, String val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void remove(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    private void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        Pair[] bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (Pair pair : bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void print() {\n        for (Pair pair : buckets) {\n            if (pair == null) {\n                System.out.println(\"null\");\n            } else if (pair == TOMBSTONE) {\n                System.out.println(\"TOMBSTONE\");\n            } else {\n                System.out.println(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.cs<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n    Pair[] buckets; // \u6876\u6570\u7ec4\n    Pair TOMBSTONE = new(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    int FindBucket(int key) {\n        int index = HashFunc(key);\n        int firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index].key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public string? Get(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void Extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        Pair[] bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        foreach (Pair pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void Print() {\n        foreach (Pair pair in buckets) {\n            if (pair == null) {\n                Console.WriteLine(\"null\");\n            } else if (pair == TOMBSTONE) {\n                Console.WriteLine(\"TOMBSTONE\");\n            } else {\n                Console.WriteLine(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.go<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntype hashMapOpenAddressing struct {\n    size        int     // \u952e\u503c\u5bf9\u6570\u91cf\n    capacity    int     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    loadThres   float64 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    extendRatio int     // \u6269\u5bb9\u500d\u6570\n    buckets     []*pair // \u6876\u6570\u7ec4\n    TOMBSTONE   *pair   // \u5220\u9664\u6807\u8bb0\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\n    return &amp;hashMapOpenAddressing{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     make([]*pair, 4),\n        TOMBSTONE:   &amp;pair{-1, \"-1\"},\n    }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (h *hashMapOpenAddressing) hashFunc(key int) int {\n    return key % h.capacity // \u6839\u636e\u952e\u8ba1\u7b97\u54c8\u5e0c\u503c\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (h *hashMapOpenAddressing) loadFactor() float64 {\n    return float64(h.size) / float64(h.capacity) // \u8ba1\u7b97\u5f53\u524d\u8d1f\u8f7d\u56e0\u5b50\n}\n\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nfunc (h *hashMapOpenAddressing) findBucket(key int) int {\n    index := h.hashFunc(key) // \u83b7\u53d6\u521d\u59cb\u7d22\u5f15\n    firstTombstone := -1     // \u8bb0\u5f55\u9047\u5230\u7684\u7b2c\u4e00\u4e2aTOMBSTONE\u7684\u4f4d\u7f6e\n    for h.buckets[index] != nil {\n        if h.buckets[index].key == key {\n            if firstTombstone != -1 {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                h.buckets[firstTombstone] = h.buckets[index]\n                h.buckets[index] = h.TOMBSTONE\n                return firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n            }\n            return index // \u8fd4\u56de\u627e\u5230\u7684\u7d22\u5f15\n        }\n        if firstTombstone == -1 &amp;&amp; h.buckets[index] == h.TOMBSTONE {\n            firstTombstone = index // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\u7684\u4f4d\u7f6e\n        }\n        index = (index + 1) % h.capacity // \u7ebf\u6027\u63a2\u6d4b\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n    if firstTombstone != -1 {\n        return firstTombstone\n    }\n    return index\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) get(key int) string {\n    index := h.findBucket(key) // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        return h.buckets[index].val // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    }\n    return \"\" // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de \"\"\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) put(key int, val string) {\n    if h.loadFactor() &gt; h.loadThres {\n        h.extend() // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    }\n    index := h.findBucket(key) // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] == nil || h.buckets[index] == h.TOMBSTONE {\n        h.buckets[index] = &amp;pair{key, val} // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        h.size++\n    } else {\n        h.buckets[index].val = val // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val\n    }\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (h *hashMapOpenAddressing) remove(key int) {\n    index := h.findBucket(key) // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        h.buckets[index] = h.TOMBSTONE // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        h.size--\n    }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (h *hashMapOpenAddressing) extend() {\n    oldBuckets := h.buckets               // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    h.capacity *= h.extendRatio           // \u66f4\u65b0\u5bb9\u91cf\n    h.buckets = make([]*pair, h.capacity) // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    h.size = 0                            // \u91cd\u7f6e\u5927\u5c0f\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for _, pair := range oldBuckets {\n        if pair != nil &amp;&amp; pair != h.TOMBSTONE {\n            h.put(pair.key, pair.val)\n        }\n    }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (h *hashMapOpenAddressing) print() {\n    for _, pair := range h.buckets {\n        if pair == nil {\n            fmt.Println(\"nil\")\n        } else if pair == h.TOMBSTONE {\n            fmt.Println(\"TOMBSTONE\")\n        } else {\n            fmt.Printf(\"%d -&gt; %s\\n\", pair.key, pair.val)\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.swift<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n    var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    var loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    var extendRatio: Int // \u6269\u5bb9\u500d\u6570\n    var buckets: [Pair?] // \u6876\u6570\u7ec4\n    var TOMBSTONE: Pair // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: nil, count: capacity)\n        TOMBSTONE = Pair(key: -1, val: \"-1\")\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    func findBucket(key: Int) -&gt; Int {\n        var index = hashFunc(key: key)\n        var firstTombstone = -1\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while buckets[index] != nil {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if buckets[index]!.key == key {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if firstTombstone != -1 {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE {\n                firstTombstone = index\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            return buckets[index]!.val\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return nil\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index]!.val = val\n            return\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = Pair(key: key, val: val)\n        size += 1\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index] = TOMBSTONE\n            size -= 1\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    func extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        buckets = Array(repeating: nil, count: capacity)\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for pair in bucketsTmp {\n            if let pair, pair != TOMBSTONE {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    func print() {\n        for pair in buckets {\n            if pair == nil {\n                Swift.print(\"null\")\n            } else if pair == TOMBSTONE {\n                Swift.print(\"TOMBSTONE\")\n            } else {\n                Swift.print(\"\\(pair!.key) -&gt; \\(pair!.val)\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.js<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    #size; // \u952e\u503c\u5bf9\u6570\u91cf\n    #capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    #loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    #extendRatio; // \u6269\u5bb9\u500d\u6570\n    #buckets; // \u6876\u6570\u7ec4\n    #TOMBSTONE; // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\n        this.#capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n        this.#loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        this.#extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n        this.#buckets = Array(this.#capacity).fill(null); // \u6876\u6570\u7ec4\n        this.#TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    #findBucket(key) {\n        let index = this.#hashFunc(key);\n        let firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (this.#buckets[index] !== null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (this.#buckets[index].key === key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone !== -1) {\n                    this.#buckets[firstTombstone] = this.#buckets[index];\n                    this.#buckets[index] = this.#TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.#buckets[index] === this.#TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % this.#capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            return this.#buckets[index].val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key, val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        this.#buckets[index] = new Pair(key, val);\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index] = this.#TOMBSTONE;\n            this.#size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    #extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = Array(this.#capacity).fill(null);\n        this.#size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.#TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print() {\n        for (const pair of this.#buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.#TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.ts<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    private size: number; // \u952e\u503c\u5bf9\u6570\u91cf\n    private capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    private loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    private extendRatio: number; // \u6269\u5bb9\u500d\u6570\n    private buckets: Array&lt;Pair | null&gt;; // \u6876\u6570\u7ec4\n    private TOMBSTONE: Pair; // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\n        this.capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n        this.loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        this.extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n        this.buckets = Array(this.capacity).fill(null); // \u6876\u6570\u7ec4\n        this.TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private hashFunc(key: number): number {\n        return key % this.capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    private loadFactor(): number {\n        return this.size / this.capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    private findBucket(key: number): number {\n        let index = this.hashFunc(key);\n        let firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (this.buckets[index] !== null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (this.buckets[index]!.key === key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone !== -1) {\n                    this.buckets[firstTombstone] = this.buckets[index];\n                    this.buckets[index] = this.TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.buckets[index] === this.TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % this.capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key: number): string | null {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            return this.buckets[index]!.val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key: number, val: string): void {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.loadFactor() &gt; this.loadThres) {\n            this.extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index]!.val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        this.buckets[index] = new Pair(key, val);\n        this.size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key: number): void {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index] = this.TOMBSTONE;\n            this.size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    private extend(): void {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.capacity *= this.extendRatio;\n        this.buckets = Array(this.capacity).fill(null);\n        this.size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print(): void {\n        for (const pair of this.buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.dart<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n  late int _size; // \u952e\u503c\u5bf9\u6570\u91cf\n  int _capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n  double _loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n  int _extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n  late List&lt;Pair?&gt; _buckets; // \u6876\u6570\u7ec4\n  Pair _TOMBSTONE = Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  HashMapOpenAddressing() {\n    _size = 0;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n  }\n\n  /* \u54c8\u5e0c\u51fd\u6570 */\n  int hashFunc(int key) {\n    return key % _capacity;\n  }\n\n  /* \u8d1f\u8f7d\u56e0\u5b50 */\n  double loadFactor() {\n    return _size / _capacity;\n  }\n\n  /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n  int findBucket(int key) {\n    int index = hashFunc(key);\n    int firstTombstone = -1;\n    // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n    while (_buckets[index] != null) {\n      // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n      if (_buckets[index]!.key == key) {\n        // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n        if (firstTombstone != -1) {\n          _buckets[firstTombstone] = _buckets[index];\n          _buckets[index] = _TOMBSTONE;\n          return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n        }\n        return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n      }\n      // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n      if (firstTombstone == -1 &amp;&amp; _buckets[index] == _TOMBSTONE) {\n        firstTombstone = index;\n      }\n      // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n      index = (index + 1) % _capacity;\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n    return firstTombstone == -1 ? index : firstTombstone;\n  }\n\n  /* \u67e5\u8be2\u64cd\u4f5c */\n  String? get(int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      return _buckets[index]!.val;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n    return null;\n  }\n\n  /* \u6dfb\u52a0\u64cd\u4f5c */\n  void put(int key, String val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor() &gt; _loadThres) {\n      extend();\n    }\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index]!.val = val;\n      return;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n    _buckets[index] = new Pair(key, val);\n    _size++;\n  }\n\n  /* \u5220\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index] = _TOMBSTONE;\n      _size--;\n    }\n  }\n\n  /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n  void extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    List&lt;Pair?&gt; bucketsTmp = _buckets;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    _capacity *= _extendRatio;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n    _size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (Pair? pair in bucketsTmp) {\n      if (pair != null &amp;&amp; pair != _TOMBSTONE) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* \u6253\u5370\u54c8\u5e0c\u8868 */\n  void printHashMap() {\n    for (Pair? pair in _buckets) {\n      if (pair == null) {\n        print(\"null\");\n      } else if (pair == _TOMBSTONE) {\n        print(\"TOMBSTONE\");\n      } else {\n        print(\"${pair.key} -&gt; ${pair.val}\");\n      }\n    }\n  }\n}\n</code></pre> hash_map_open_addressing.rs<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapOpenAddressing {\n    size: usize,                // \u952e\u503c\u5bf9\u6570\u91cf\n    capacity: usize,            // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    load_thres: f64,            // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    extend_ratio: usize,        // \u6269\u5bb9\u500d\u6570\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;, // \u6876\u6570\u7ec4\n    TOMBSTONE: Option&lt;Pair&gt;,    // \u5220\u9664\u6807\u8bb0\n}\n\nimpl HashMapOpenAddressing {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![None; 4],\n            TOMBSTONE: Some(Pair {\n                key: -1,\n                val: \"-1\".to_string(),\n            }),\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        (key % self.capacity as i32) as usize\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fn load_factor(&amp;self) -&gt; f64 {\n        self.size as f64 / self.capacity as f64\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    fn find_bucket(&amp;mut self, key: i32) -&gt; usize {\n        let mut index = self.hash_func(key);\n        let mut first_tombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while self.buckets[index].is_some() {\n            // \u82e5\u9047\u5230 key\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if self.buckets[index].as_ref().unwrap().key == key {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u5efa\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n                if first_tombstone != -1 {\n                    self.buckets[first_tombstone as usize] = self.buckets[index].take();\n                    self.buckets[index] = self.TOMBSTONE.clone();\n                    return first_tombstone as usize; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if first_tombstone == -1 &amp;&amp; self.buckets[index] == self.TOMBSTONE {\n                first_tombstone = index as i32;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % self.capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        if first_tombstone == -1 {\n            index\n        } else {\n            first_tombstone as usize\n        }\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fn get(&amp;mut self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            return self.buckets[index].as_ref().map(|pair| &amp;pair.val as &amp;str);\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        None\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index].as_mut().unwrap().val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        self.buckets[index] = Some(Pair { key, val });\n        self.size += 1;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fn remove(&amp;mut self, key: i32) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index] = self.TOMBSTONE.clone();\n            self.size -= 1;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fn extend(&amp;mut self) {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let buckets_tmp = self.buckets.clone();\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![None; self.capacity];\n        self.size = 0;\n\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for pair in buckets_tmp {\n            if pair.is_none() || pair == self.TOMBSTONE {\n                continue;\n            }\n            let pair = pair.unwrap();\n\n            self.put(pair.key, pair.val);\n        }\n    }\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fn print(&amp;self) {\n        for pair in &amp;self.buckets {\n            if pair.is_none() {\n                println!(\"null\");\n            } else if pair == &amp;self.TOMBSTONE {\n                println!(\"TOMBSTONE\");\n            } else {\n                let pair = pair.as_ref().unwrap();\n                println!(\"{} -&gt; {}\", pair.key, pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.c<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\n    int size;         // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity;     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio;  // \u6269\u5bb9\u500d\u6570\n    Pair **buckets;   // \u6876\u6570\u7ec4\n    Pair *TOMBSTONE;  // \u5220\u9664\u6807\u8bb0\n} HashMapOpenAddressing;\n\n/* \u6784\u9020\u51fd\u6570 */\nHashMapOpenAddressing *newHashMapOpenAddressing() {\n    HashMapOpenAddressing *hashMap = (HashMapOpenAddressing *)malloc(sizeof(HashMapOpenAddressing));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Pair **)malloc(sizeof(Pair *) * hashMap-&gt;capacity);\n    hashMap-&gt;TOMBSTONE = (Pair *)malloc(sizeof(Pair));\n    hashMap-&gt;TOMBSTONE-&gt;key = -1;\n    hashMap-&gt;TOMBSTONE-&gt;val = \"-1\";\n\n    return hashMap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapOpenAddressing(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapOpenAddressing *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapOpenAddressing *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nint findBucket(HashMapOpenAddressing *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    int firstTombstone = -1;\n    // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n    while (hashMap-&gt;buckets[index] != NULL) {\n        // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        if (hashMap-&gt;buckets[index]-&gt;key == key) {\n            // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n            if (firstTombstone != -1) {\n                hashMap-&gt;buckets[firstTombstone] = hashMap-&gt;buckets[index];\n                hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n                return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n            }\n            return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n        }\n        // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n        if (firstTombstone == -1 &amp;&amp; hashMap-&gt;buckets[index] == hashMap-&gt;TOMBSTONE) {\n            firstTombstone = index;\n        }\n        // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n        index = (index + 1) % hashMap-&gt;capacity;\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n    return firstTombstone == -1 ? index : firstTombstone;\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapOpenAddressing *hashMap, int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        return hashMap-&gt;buckets[index]-&gt;val;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n    return \"\";\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapOpenAddressing *hashMap, int key, char *val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        free(hashMap-&gt;buckets[index]-&gt;val);\n        hashMap-&gt;buckets[index]-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n        strcpy(hashMap-&gt;buckets[index]-&gt;val, val);\n        hashMap-&gt;buckets[index]-&gt;val[strlen(val)] = '\\0';\n        return;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n    Pair *pair = (Pair *)malloc(sizeof(Pair));\n    pair-&gt;key = key;\n    pair-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n    strcpy(pair-&gt;val, val);\n    pair-&gt;val[strlen(val)] = '\\0';\n\n    hashMap-&gt;buckets[index] = pair;\n    hashMap-&gt;size++;\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapOpenAddressing *hashMap, int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        Pair *pair = hashMap-&gt;buckets[index];\n        free(pair-&gt;val);\n        free(pair);\n        hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n        hashMap-&gt;size--;\n    }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapOpenAddressing *hashMap) {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    Pair **bucketsTmp = hashMap-&gt;buckets;\n    int oldCapacity = hashMap-&gt;capacity;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Pair **)malloc(sizeof(Pair *) * hashMap-&gt;capacity);\n    hashMap-&gt;size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Pair *pair = bucketsTmp[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            put(hashMap, pair-&gt;key, pair-&gt;val);\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(bucketsTmp);\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair == NULL) {\n            printf(\"NULL\\n\");\n        } else if (pair == hashMap-&gt;TOMBSTONE) {\n            printf(\"TOMBSTONE\\n\");\n        } else {\n            printf(\"%d -&gt; %s\\n\", pair-&gt;key, pair-&gt;val);\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.kt<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    private var size: Int = 0 // \u952e\u503c\u5bf9\u6570\u91cf\n    private var capacity = 4 // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    private val loadThres: Double = 2.0 / 3.0 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    private val extendRatio = 2 // \u6269\u5bb9\u500d\u6570\n    private var buckets: Array&lt;Pair?&gt; // \u6876\u6570\u7ec4\n    private val TOMBSTONE = Pair(-1, \"-1\") // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init {\n        buckets = arrayOfNulls(capacity)\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    fun findBucket(key: Int): Int {\n        var index = hashFunc(key)\n        var firstTombstone = -1\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if (buckets[index]?.key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\u5904\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return if (firstTombstone == -1) index else firstTombstone\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]?.value\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]!!.value = value\n            return\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = Pair(key, value)\n        size++\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        val index = findBucket(key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE\n            size--\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fun extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        val bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        buckets = arrayOfNulls(capacity)\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair.value)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (pair in buckets) {\n            if (pair == null) {\n                println(\"null\")\n            } else if (pair == TOMBSTONE) {\n                println(\"TOMESTOME\")\n            } else {\n                println(\"${pair.key} -&gt; ${pair.value}\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.rb<pre><code>[class]{HashMapOpenAddressing}-[func]{}\n</code></pre> hash_map_open_addressing.zig<pre><code>[class]{HashMapOpenAddressing}-[func]{}\n</code></pre>"},{"location":"chapter_hashing/hash_collision/#2","title":"2. \u00a0 \u5e73\u65b9\u63a2\u6d4b","text":"<p>\u5e73\u65b9\u63a2\u6d4b\u4e0e\u7ebf\u6027\u63a2\u6d4b\u7c7b\u4f3c\uff0c\u90fd\u662f\u5f00\u653e\u5bfb\u5740\u7684\u5e38\u89c1\u7b56\u7565\u4e4b\u4e00\u3002\u5f53\u53d1\u751f\u51b2\u7a81\u65f6\uff0c\u5e73\u65b9\u63a2\u6d4b\u4e0d\u662f\u7b80\u5355\u5730\u8df3\u8fc7\u4e00\u4e2a\u56fa\u5b9a\u7684\u6b65\u6570\uff0c\u800c\u662f\u8df3\u8fc7\u201c\u63a2\u6d4b\u6b21\u6570\u7684\u5e73\u65b9\u201d\u7684\u6b65\u6570\uff0c\u5373 \\(1, 4, 9, \\dots\\) \u6b65\u3002</p> <p>\u5e73\u65b9\u63a2\u6d4b\u4e3b\u8981\u5177\u6709\u4ee5\u4e0b\u4f18\u52bf\u3002</p> <ul> <li>\u5e73\u65b9\u63a2\u6d4b\u901a\u8fc7\u8df3\u8fc7\u63a2\u6d4b\u6b21\u6570\u5e73\u65b9\u7684\u8ddd\u79bb\uff0c\u8bd5\u56fe\u7f13\u89e3\u7ebf\u6027\u63a2\u6d4b\u7684\u805a\u96c6\u6548\u5e94\u3002</li> <li>\u5e73\u65b9\u63a2\u6d4b\u4f1a\u8df3\u8fc7\u66f4\u5927\u7684\u8ddd\u79bb\u6765\u5bfb\u627e\u7a7a\u4f4d\u7f6e\uff0c\u6709\u52a9\u4e8e\u6570\u636e\u5206\u5e03\u5f97\u66f4\u52a0\u5747\u5300\u3002</li> </ul> <p>\u7136\u800c\uff0c\u5e73\u65b9\u63a2\u6d4b\u5e76\u4e0d\u662f\u5b8c\u7f8e\u7684\u3002</p> <ul> <li>\u4ecd\u7136\u5b58\u5728\u805a\u96c6\u73b0\u8c61\uff0c\u5373\u67d0\u4e9b\u4f4d\u7f6e\u6bd4\u5176\u4ed6\u4f4d\u7f6e\u66f4\u5bb9\u6613\u88ab\u5360\u7528\u3002</li> <li>\u7531\u4e8e\u5e73\u65b9\u7684\u589e\u957f\uff0c\u5e73\u65b9\u63a2\u6d4b\u53ef\u80fd\u4e0d\u4f1a\u63a2\u6d4b\u6574\u4e2a\u54c8\u5e0c\u8868\uff0c\u8fd9\u610f\u5473\u7740\u5373\u4f7f\u54c8\u5e0c\u8868\u4e2d\u6709\u7a7a\u6876\uff0c\u5e73\u65b9\u63a2\u6d4b\u4e5f\u53ef\u80fd\u65e0\u6cd5\u8bbf\u95ee\u5230\u5b83\u3002</li> </ul>"},{"location":"chapter_hashing/hash_collision/#3","title":"3. \u00a0 \u591a\u6b21\u54c8\u5e0c","text":"<p>\u987e\u540d\u601d\u4e49\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\)\u3001\\(f_2(x)\\)\u3001\\(f_3(x)\\)\u3001\\(\\dots\\) \u8fdb\u884c\u63a2\u6d4b\u3002</p> <ul> <li>\u63d2\u5165\u5143\u7d20\uff1a\u82e5\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\) \u51fa\u73b0\u51b2\u7a81\uff0c\u5219\u5c1d\u8bd5 \\(f_2(x)\\) \uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u5230\u627e\u5230\u7a7a\u4f4d\u540e\u63d2\u5165\u5143\u7d20\u3002</li> <li>\u67e5\u627e\u5143\u7d20\uff1a\u5728\u76f8\u540c\u7684\u54c8\u5e0c\u51fd\u6570\u987a\u5e8f\u4e0b\u8fdb\u884c\u67e5\u627e\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u65f6\u8fd4\u56de\uff1b\u82e5\u9047\u5230\u7a7a\u4f4d\u6216\u5df2\u5c1d\u8bd5\u6240\u6709\u54c8\u5e0c\u51fd\u6570\uff0c\u8bf4\u660e\u54c8\u5e0c\u8868\u4e2d\u4e0d\u5b58\u5728\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de <code>None</code> \u3002</li> </ul> <p>\u4e0e\u7ebf\u6027\u63a2\u6d4b\u76f8\u6bd4\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u4f1a\u5e26\u6765\u989d\u5916\u7684\u8ba1\u7b97\u91cf\u3002</p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u548c\u591a\u6b21\u54c8\u5e0c\uff09\u54c8\u5e0c\u8868\u90fd\u5b58\u5728\u201c\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u201d\u7684\u95ee\u9898\u3002</p>"},{"location":"chapter_hashing/hash_collision/#623","title":"6.2.3 \u00a0 \u7f16\u7a0b\u8bed\u8a00\u7684\u9009\u62e9","text":"<p>\u5404\u79cd\u7f16\u7a0b\u8bed\u8a00\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u54c8\u5e0c\u8868\u5b9e\u73b0\u7b56\u7565\uff0c\u4e0b\u9762\u4e3e\u51e0\u4e2a\u4f8b\u5b50\u3002</p> <ul> <li>Python \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002\u5b57\u5178 <code>dict</code> \u4f7f\u7528\u4f2a\u968f\u673a\u6570\u8fdb\u884c\u63a2\u6d4b\u3002</li> <li>Java \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002\u81ea JDK 1.8 \u4ee5\u6765\uff0c\u5f53 <code>HashMap</code> \u5185\u6570\u7ec4\u957f\u5ea6\u8fbe\u5230 64 \u4e14\u94fe\u8868\u957f\u5ea6\u8fbe\u5230 8 \u65f6\uff0c\u94fe\u8868\u4f1a\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u4ee5\u63d0\u5347\u67e5\u627e\u6027\u80fd\u3002</li> <li>Go \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002Go \u89c4\u5b9a\u6bcf\u4e2a\u6876\u6700\u591a\u5b58\u50a8 8 \u4e2a\u952e\u503c\u5bf9\uff0c\u8d85\u51fa\u5bb9\u91cf\u5219\u8fde\u63a5\u4e00\u4e2a\u6ea2\u51fa\u6876\uff1b\u5f53\u6ea2\u51fa\u6876\u8fc7\u591a\u65f6\uff0c\u4f1a\u6267\u884c\u4e00\u6b21\u7279\u6b8a\u7684\u7b49\u91cf\u6269\u5bb9\u64cd\u4f5c\uff0c\u4ee5\u786e\u4fdd\u6027\u80fd\u3002</li> </ul>"},{"location":"chapter_hashing/hash_map/","title":"6.1 \u00a0 \u54c8\u5e0c\u8868","text":"<p>\u54c8\u5e0c\u8868\uff08hash table\uff09\uff0c\u53c8\u79f0\u6563\u5217\u8868\uff0c\u5b83\u901a\u8fc7\u5efa\u7acb\u952e <code>key</code> \u4e0e\u503c <code>value</code> \u4e4b\u95f4\u7684\u6620\u5c04\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u8be2\u3002\u5177\u4f53\u800c\u8a00\uff0c\u6211\u4eec\u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u4e00\u4e2a\u952e <code>key</code> \uff0c\u5219\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u5bf9\u5e94\u7684\u503c <code>value</code> \u3002</p> <p>\u5982\u56fe 6-1 \u6240\u793a\uff0c\u7ed9\u5b9a \\(n\\) \u4e2a\u5b66\u751f\uff0c\u6bcf\u4e2a\u5b66\u751f\u90fd\u6709\u201c\u59d3\u540d\u201d\u548c\u201c\u5b66\u53f7\u201d\u4e24\u9879\u6570\u636e\u3002\u5047\u5982\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u201c\u8f93\u5165\u4e00\u4e2a\u5b66\u53f7\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u59d3\u540d\u201d\u7684\u67e5\u8be2\u529f\u80fd\uff0c\u5219\u53ef\u4ee5\u91c7\u7528\u56fe 6-1 \u6240\u793a\u7684\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002</p> <p></p> <p> \u56fe 6-1 \u00a0 \u54c8\u5e0c\u8868\u7684\u62bd\u8c61\u8868\u793a </p> <p>\u9664\u54c8\u5e0c\u8868\u5916\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u4e5f\u53ef\u4ee5\u5b9e\u73b0\u67e5\u8be2\u529f\u80fd\uff0c\u5b83\u4eec\u7684\u6548\u7387\u5bf9\u6bd4\u5982\u8868 6-1 \u6240\u793a\u3002</p> <ul> <li>\u6dfb\u52a0\u5143\u7d20\uff1a\u4ec5\u9700\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u6570\u7ec4\uff08\u94fe\u8868\uff09\u7684\u5c3e\u90e8\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002</li> <li>\u67e5\u8be2\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\uff08\u94fe\u8868\uff09\u662f\u4e71\u5e8f\u7684\uff0c\u56e0\u6b64\u9700\u8981\u904d\u5386\u5176\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> <li>\u5220\u9664\u5143\u7d20\uff1a\u9700\u8981\u5148\u67e5\u8be2\u5230\u5143\u7d20\uff0c\u518d\u4ece\u6570\u7ec4\uff08\u94fe\u8868\uff09\u4e2d\u5220\u9664\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> </ul> <p> \u8868 6-1 \u00a0 \u5143\u7d20\u67e5\u8be2\u6548\u7387\u5bf9\u6bd4 </p> \u6570\u7ec4 \u94fe\u8868 \u54c8\u5e0c\u8868 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u5728\u54c8\u5e0c\u8868\u4e2d\u8fdb\u884c\u589e\u5220\u67e5\u6539\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f \\(O(1)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</p>"},{"location":"chapter_hashing/hash_map/#611","title":"6.1.1 \u00a0 \u54c8\u5e0c\u8868\u5e38\u7528\u64cd\u4f5c","text":"<p>\u54c8\u5e0c\u8868\u7684\u5e38\u89c1\u64cd\u4f5c\u5305\u62ec\uff1a\u521d\u59cb\u5316\u3001\u67e5\u8be2\u64cd\u4f5c\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u548c\u5220\u9664\u952e\u503c\u5bf9\u7b49\uff0c\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map.py<pre><code># \u521d\u59cb\u5316\u54c8\u5e0c\u8868\nhmap: dict = {}\n\n# \u6dfb\u52a0\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u5570\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9e2d\"\n\n# \u67e5\u8be2\u64cd\u4f5c\n# \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname: str = hmap[15937]\n\n# \u5220\u9664\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nhmap.pop(10583)\n</code></pre> hash_map.cpp<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nunordered_map&lt;int, string&gt; map;\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.erase(10583);\n</code></pre> hash_map.java<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.put(12836, \"\u5c0f\u54c8\");   \nmap.put(15937, \"\u5c0f\u5570\");   \nmap.put(16750, \"\u5c0f\u7b97\");   \nmap.put(13276, \"\u5c0f\u6cd5\");\nmap.put(10583, \"\u5c0f\u9e2d\");\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map.get(15937);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n</code></pre> hash_map.cs<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nDictionary&lt;int, string&gt; map = new() {\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    // \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\n    { 12836, \"\u5c0f\u54c8\" },\n    { 15937, \"\u5c0f\u5570\" },\n    { 16750, \"\u5c0f\u7b97\" },\n    { 13276, \"\u5c0f\u6cd5\" },\n    { 10583, \"\u5c0f\u9e2d\" }\n};\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.Remove(10583);\n</code></pre> hash_map_test.go<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nhmap := make(map[int]string)\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u5570\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9e2d\"\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname := hmap[15937]\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\ndelete(hmap, 10583)\n</code></pre> hash_map.swift<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nvar map: [Int: String] = [:]\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u5570\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9e2d\"\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map[15937]!\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.removeValue(forKey: 10583)\n</code></pre> hash_map.js<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\n</code></pre> hash_map.ts<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map&lt;number, string&gt;();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\nconsole.info('\\n\u6dfb\u52a0\u5b8c\u6210\u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -&gt; Value');\nconsole.info(map);\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\nconsole.info('\\n\u8f93\u5165\u5b66\u53f7 15937 \uff0c\u67e5\u8be2\u5230\u59d3\u540d ' + name);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\nconsole.info('\\n\u5220\u9664 10583 \u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -&gt; Value');\nconsole.info(map);\n</code></pre> hash_map.dart<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap&lt;int, String&gt; map = {};\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map[15937];\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n</code></pre> hash_map.rs<pre><code>use std::collections::HashMap;\n\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nlet mut map: HashMap&lt;i32, String&gt; = HashMap::new();\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.insert(12836, \"\u5c0f\u54c8\".to_string());\nmap.insert(15937, \"\u5c0f\u5570\".to_string());\nmap.insert(16750, \"\u5c0f\u7b97\".to_string());\nmap.insert(13279, \"\u5c0f\u6cd5\".to_string());\nmap.insert(10583, \"\u5c0f\u9e2d\".to_string());\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet _name: Option&lt;&amp;String&gt; = map.get(&amp;15937);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nlet _removed_value: Option&lt;String&gt; = map.remove(&amp;10583);\n</code></pre> hash_map.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n</code></pre> hash_map.kt<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nval map = HashMap&lt;Int,String&gt;()\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u5570\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9e2d\"\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nval name = map[15937]\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583)\n</code></pre> hash_map.rb<pre><code>\n</code></pre> hash_map.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u54c8\u5e0c\u8868\u6709\u4e09\u79cd\u5e38\u7528\u7684\u904d\u5386\u65b9\u5f0f\uff1a\u904d\u5386\u952e\u503c\u5bf9\u3001\u904d\u5386\u952e\u548c\u904d\u5386\u503c\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig hash_map.py<pre><code># \u904d\u5386\u54c8\u5e0c\u8868\n# \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor key, value in hmap.items():\n    print(key, \"-&gt;\", value)\n# \u5355\u72ec\u904d\u5386\u952e key\nfor key in hmap.keys():\n    print(key)\n# \u5355\u72ec\u904d\u5386\u503c value\nfor value in hmap.values():\n    print(value)\n</code></pre> hash_map.cpp<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor (auto kv: map) {\n    cout &lt;&lt; kv.first &lt;&lt; \" -&gt; \" &lt;&lt; kv.second &lt;&lt; endl;\n}\n// \u4f7f\u7528\u8fed\u4ee3\u5668\u904d\u5386 key-&gt;value\nfor (auto iter = map.begin(); iter != map.end(); iter++) {\n    cout &lt;&lt; iter-&gt;first &lt;&lt; \"-&gt;\" &lt;&lt; iter-&gt;second &lt;&lt; endl;\n}\n</code></pre> hash_map.java<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor (Map.Entry &lt;Integer, String&gt; kv: map.entrySet()) {\n    System.out.println(kv.getKey() + \" -&gt; \" + kv.getValue());\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (int key: map.keySet()) {\n    System.out.println(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (String val: map.values()) {\n    System.out.println(val);\n}\n</code></pre> hash_map.cs<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nforeach (var kv in map) {\n    Console.WriteLine(kv.Key + \" -&gt; \" + kv.Value);\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nforeach (int key in map.Keys) {\n    Console.WriteLine(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nforeach (string val in map.Values) {\n    Console.WriteLine(val);\n}\n</code></pre> hash_map_test.go<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor key, value := range hmap {\n    fmt.Println(key, \"-&gt;\", value)\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor key := range hmap {\n    fmt.Println(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor _, value := range hmap {\n    fmt.Println(value)\n}\n</code></pre> hash_map.swift<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nfor (key, value) in map {\n    print(\"\\(key) -&gt; \\(value)\")\n}\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys {\n    print(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values {\n    print(value)\n}\n</code></pre> hash_map.js<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.ts<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.dart<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nmap.forEach((key, value) {\n  print('$key -&gt; $value');\n});\n\n// \u5355\u72ec\u904d\u5386\u952e Key\nmap.keys.forEach((key) {\n  print(key);\n});\n\n// \u5355\u72ec\u904d\u5386\u503c Value\nmap.values.forEach((value) {\n  print(value);\n});\n</code></pre> hash_map.rs<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nfor (key, value) in &amp;map {\n    println!(\"{key} -&gt; {value}\");\n}\n\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys() {\n    println!(\"{key}\"); \n}\n\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values() {\n    println!(\"{value}\");\n}\n</code></pre> hash_map.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n</code></pre> hash_map.kt<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor ((key, value) in map) {\n    println(\"$key -&gt; $value\")\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (key in map.keys) {\n    println(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (_val in map.values) {\n    println(_val)\n}\n</code></pre> hash_map.rb<pre><code>\n</code></pre> hash_map.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_hashing/hash_map/#612","title":"6.1.2 \u00a0 \u54c8\u5e0c\u8868\u7b80\u5355\u5b9e\u73b0","text":"<p>\u6211\u4eec\u5148\u8003\u8651\u6700\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u4ec5\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u5b9e\u73b0\u54c8\u5e0c\u8868\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5c06\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u7a7a\u4f4d\u79f0\u4e3a\u6876\uff08bucket\uff09\uff0c\u6bcf\u4e2a\u6876\u53ef\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u56e0\u6b64\uff0c\u67e5\u8be2\u64cd\u4f5c\u5c31\u662f\u627e\u5230 <code>key</code> \u5bf9\u5e94\u7684\u6876\uff0c\u5e76\u5728\u6876\u4e2d\u83b7\u53d6 <code>value</code> \u3002</p> <p>\u90a3\u4e48\uff0c\u5982\u4f55\u57fa\u4e8e <code>key</code> \u5b9a\u4f4d\u5bf9\u5e94\u7684\u6876\u5462\uff1f\u8fd9\u662f\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\uff08hash function\uff09\u5b9e\u73b0\u7684\u3002\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u8f93\u5165\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u8f93\u51fa\u7a7a\u95f4\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8f93\u5165\u7a7a\u95f4\u662f\u6240\u6709 <code>key</code> \uff0c\u8f93\u51fa\u7a7a\u95f4\u662f\u6240\u6709\u6876\uff08\u6570\u7ec4\u7d22\u5f15\uff09\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u8f93\u5165\u4e00\u4e2a <code>key</code> \uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u8be5 <code>key</code> \u5bf9\u5e94\u7684\u952e\u503c\u5bf9\u5728\u6570\u7ec4\u4e2d\u7684\u5b58\u50a8\u4f4d\u7f6e\u3002</p> <p>\u8f93\u5165\u4e00\u4e2a <code>key</code> \uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u8fc7\u7a0b\u5206\u4e3a\u4ee5\u4e0b\u4e24\u6b65\u3002</p> <ol> <li>\u901a\u8fc7\u67d0\u79cd\u54c8\u5e0c\u7b97\u6cd5 <code>hash()</code> \u8ba1\u7b97\u5f97\u5230\u54c8\u5e0c\u503c\u3002</li> <li>\u5c06\u54c8\u5e0c\u503c\u5bf9\u6876\u6570\u91cf\uff08\u6570\u7ec4\u957f\u5ea6\uff09<code>capacity</code> \u53d6\u6a21\uff0c\u4ece\u800c\u83b7\u53d6\u8be5 <code>key</code> \u5bf9\u5e94\u7684\u6570\u7ec4\u7d22\u5f15 <code>index</code> \u3002</li> </ol> <pre><code>index = hash(key) % capacity\n</code></pre> <p>\u968f\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5229\u7528 <code>index</code> \u5728\u54c8\u5e0c\u8868\u4e2d\u8bbf\u95ee\u5bf9\u5e94\u7684\u6876\uff0c\u4ece\u800c\u83b7\u53d6 <code>value</code> \u3002</p> <p>\u8bbe\u6570\u7ec4\u957f\u5ea6 <code>capacity = 100</code>\u3001\u54c8\u5e0c\u7b97\u6cd5 <code>hash(key) = key</code> \uff0c\u6613\u5f97\u54c8\u5e0c\u51fd\u6570\u4e3a <code>key % 100</code> \u3002\u56fe 6-2 \u4ee5 <code>key</code> \u5b66\u53f7\u548c <code>value</code> \u59d3\u540d\u4e3a\u4f8b\uff0c\u5c55\u793a\u4e86\u54c8\u5e0c\u51fd\u6570\u7684\u5de5\u4f5c\u539f\u7406\u3002</p> <p></p> <p> \u56fe 6-2 \u00a0 \u54c8\u5e0c\u51fd\u6570\u5de5\u4f5c\u539f\u7406 </p> <p>\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u4e2a\u7b80\u5355\u54c8\u5e0c\u8868\u3002\u5176\u4e2d\uff0c\u6211\u4eec\u5c06 <code>key</code> \u548c <code>value</code> \u5c01\u88c5\u6210\u4e00\u4e2a\u7c7b <code>Pair</code> \uff0c\u4ee5\u8868\u793a\u952e\u503c\u5bf9\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_hash_map.py<pre><code>class Pair:\n    \"\"\"\u952e\u503c\u5bf9\"\"\"\n\n    def __init__(self, key: int, val: str):\n        self.key = key\n        self.val = val\n\nclass ArrayHashMap:\n    \"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        # \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        self.buckets: list[Pair | None] = [None] * 100\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n        index = key % 100\n        return index\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n        index: int = self.hash_func(key)\n        pair: Pair = self.buckets[index]\n        if pair is None:\n            return None\n        return pair.val\n\n    def put(self, key: int, val: str):\n        \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n        pair = Pair(key, val)\n        index: int = self.hash_func(key)\n        self.buckets[index] = pair\n\n    def remove(self, key: int):\n        \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n        index: int = self.hash_func(key)\n        # \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\n        self.buckets[index] = None\n\n    def entry_set(self) -&gt; list[Pair]:\n        \"\"\"\u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\"\"\"\n        result: list[Pair] = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair)\n        return result\n\n    def key_set(self) -&gt; list[int]:\n        \"\"\"\u83b7\u53d6\u6240\u6709\u952e\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.key)\n        return result\n\n    def value_set(self) -&gt; list[str]:\n        \"\"\"\u83b7\u53d6\u6240\u6709\u503c\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.val)\n        return result\n\n    def print(self):\n        \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n        for pair in self.buckets:\n            if pair is not None:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> array_hash_map.cpp<pre><code>/* \u952e\u503c\u5bf9 */\nstruct Pair {\n  public:\n    int key;\n    string val;\n    Pair(int key, string val) {\n        this-&gt;key = key;\n        this-&gt;val = val;\n    }\n};\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n  private:\n    vector&lt;Pair *&gt; buckets;\n\n  public:\n    ArrayHashMap() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = vector&lt;Pair *&gt;(100);\n    }\n\n    ~ArrayHashMap() {\n        // \u91ca\u653e\u5185\u5b58\n        for (const auto &amp;bucket : buckets) {\n            delete bucket;\n        }\n        buckets.clear();\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    string get(int key) {\n        int index = hashFunc(key);\n        Pair *pair = buckets[index];\n        if (pair == nullptr)\n            return \"\";\n        return pair-&gt;val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, string val) {\n        Pair *pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        // \u91ca\u653e\u5185\u5b58\u5e76\u7f6e\u4e3a nullptr\n        delete buckets[index];\n        buckets[index] = nullptr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    vector&lt;Pair *&gt; pairSet() {\n        vector&lt;Pair *&gt; pairSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                pairSet.push_back(pair);\n            }\n        }\n        return pairSet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    vector&lt;int&gt; keySet() {\n        vector&lt;int&gt; keySet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                keySet.push_back(pair-&gt;key);\n            }\n        }\n        return keySet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    vector&lt;string&gt; valueSet() {\n        vector&lt;string&gt; valueSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                valueSet.push_back(pair-&gt;val);\n            }\n        }\n        return valueSet;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (Pair *kv : pairSet()) {\n            cout &lt;&lt; kv-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; kv-&gt;val &lt;&lt; endl;\n        }\n    }\n};\n</code></pre> array_hash_map.java<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair {\n    public int key;\n    public String val;\n\n    public Pair(int key, String val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private List&lt;Pair&gt; buckets;\n\n    public ArrayHashMap() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = new ArrayList&lt;&gt;();\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.add(null);\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public String get(int key) {\n        int index = hashFunc(key);\n        Pair pair = buckets.get(index);\n        if (pair == null)\n            return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void put(int key, String val) {\n        Pair pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets.set(index, pair);\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void remove(int key) {\n        int index = hashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets.set(index, null);\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    public List&lt;Pair&gt; pairSet() {\n        List&lt;Pair&gt; pairSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                pairSet.add(pair);\n        }\n        return pairSet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    public List&lt;Integer&gt; keySet() {\n        List&lt;Integer&gt; keySet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                keySet.add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    public List&lt;String&gt; valueSet() {\n        List&lt;String&gt; valueSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                valueSet.add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void print() {\n        for (Pair kv : pairSet()) {\n            System.out.println(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.cs<pre><code>/* \u952e\u503c\u5bf9 int-&gt;string */\nclass Pair(int key, string val) {\n    public int key = key;\n    public string val = val;\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    List&lt;Pair?&gt; buckets;\n    public ArrayHashMap() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = [];\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.Add(null);\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int HashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        Pair? pair = buckets[index];\n        if (pair == null) return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        Pair pair = new(key, val);\n        int index = HashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = null;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    public List&lt;Pair&gt; PairSet() {\n        List&lt;Pair&gt; pairSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                pairSet.Add(pair);\n        }\n        return pairSet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    public List&lt;int&gt; KeySet() {\n        List&lt;int&gt; keySet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                keySet.Add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    public List&lt;string&gt; ValueSet() {\n        List&lt;string&gt; valueSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                valueSet.Add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void Print() {\n        foreach (Pair kv in PairSet()) {\n            Console.WriteLine(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.go<pre><code>/* \u952e\u503c\u5bf9 */\ntype pair struct {\n    key int\n    val string\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntype arrayHashMap struct {\n    buckets []*pair\n}\n\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nfunc newArrayHashMap() *arrayHashMap {\n    // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n    buckets := make([]*pair, 100)\n    return &amp;arrayHashMap{buckets: buckets}\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (a *arrayHashMap) hashFunc(key int) int {\n    index := key % 100\n    return index\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (a *arrayHashMap) get(key int) string {\n    index := a.hashFunc(key)\n    pair := a.buckets[index]\n    if pair == nil {\n        return \"Not Found\"\n    }\n    return pair.val\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (a *arrayHashMap) put(key int, val string) {\n    pair := &amp;pair{key: key, val: val}\n    index := a.hashFunc(key)\n    a.buckets[index] = pair\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (a *arrayHashMap) remove(key int) {\n    index := a.hashFunc(key)\n    // \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\n    a.buckets[index] = nil\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e\u5bf9 */\nfunc (a *arrayHashMap) pairSet() []*pair {\n    var pairs []*pair\n    for _, pair := range a.buckets {\n        if pair != nil {\n            pairs = append(pairs, pair)\n        }\n    }\n    return pairs\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc (a *arrayHashMap) keySet() []int {\n    var keys []int\n    for _, pair := range a.buckets {\n        if pair != nil {\n            keys = append(keys, pair.key)\n        }\n    }\n    return keys\n}\n\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc (a *arrayHashMap) valueSet() []string {\n    var values []string\n    for _, pair := range a.buckets {\n        if pair != nil {\n            values = append(values, pair.val)\n        }\n    }\n    return values\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (a *arrayHashMap) print() {\n    for _, pair := range a.buckets {\n        if pair != nil {\n            fmt.Println(pair.key, \"-&gt;\", pair.val)\n        }\n    }\n}\n</code></pre> array_hash_map.swift<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair: Equatable {\n    public var key: Int\n    public var val: String\n\n    public init(key: Int, val: String) {\n        self.key = key\n        self.val = val\n    }\n\n    public static func == (lhs: Pair, rhs: Pair) -&gt; Bool {\n        lhs.key == rhs.key &amp;&amp; lhs.val == rhs.val\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private var buckets: [Pair?]\n\n    init() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = Array(repeating: nil, count: 100)\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private func hashFunc(key: Int) -&gt; Int {\n        let index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let pair = buckets[index]\n        return pair?.val\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        let pair = Pair(key: key, val: val)\n        let index = hashFunc(key: key)\n        buckets[index] = pair\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        // \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = nil\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    func pairSet() -&gt; [Pair] {\n        buckets.compactMap { $0 }\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    func keySet() -&gt; [Int] {\n        buckets.compactMap { $0?.key }\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    func valueSet() -&gt; [String] {\n        buckets.compactMap { $0?.val }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    func print() {\n        for pair in pairSet() {\n            Swift.print(\"\\(pair.key) -&gt; \\(pair.val)\")\n        }\n    }\n}\n</code></pre> array_hash_map.js<pre><code>/* \u952e\u503c\u5bf9 Number -&gt; String */\nclass Pair {\n    constructor(key, val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    #buckets;\n    constructor() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        this.#buckets = new Array(100).fill(null);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key) {\n        return key % 100;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key) {\n        let index = this.#hashFunc(key);\n        let pair = this.#buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    set(key, val) {\n        let index = this.#hashFunc(key);\n        this.#buckets[index] = new Pair(key, val);\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    delete(key) {\n        let index = this.#hashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        this.#buckets[index] = null;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    entries() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    keys() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    values() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.ts<pre><code>/* \u952e\u503c\u5bf9 Number -&gt; String */\nclass Pair {\n    public key: number;\n    public val: string;\n\n    constructor(key: number, val: string) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private readonly buckets: (Pair | null)[];\n\n    constructor() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        this.buckets = new Array(100).fill(null);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private hashFunc(key: number): number {\n        return key % 100;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public get(key: number): string | null {\n        let index = this.hashFunc(key);\n        let pair = this.buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public set(key: number, val: string) {\n        let index = this.hashFunc(key);\n        this.buckets[index] = new Pair(key, val);\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public delete(key: number) {\n        let index = this.hashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        this.buckets[index] = null;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    public entries(): (Pair | null)[] {\n        let arr: (Pair | null)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    public keys(): (number | undefined)[] {\n        let arr: (number | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    public values(): (string | undefined)[] {\n        let arr: (string | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.dart<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair {\n  int key;\n  String val;\n  Pair(this.key, this.val);\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n  late List&lt;Pair?&gt; _buckets;\n\n  ArrayHashMap() {\n    // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n    _buckets = List.filled(100, null);\n  }\n\n  /* \u54c8\u5e0c\u51fd\u6570 */\n  int _hashFunc(int key) {\n    final int index = key % 100;\n    return index;\n  }\n\n  /* \u67e5\u8be2\u64cd\u4f5c */\n  String? get(int key) {\n    final int index = _hashFunc(key);\n    final Pair? pair = _buckets[index];\n    if (pair == null) {\n      return null;\n    }\n    return pair.val;\n  }\n\n  /* \u6dfb\u52a0\u64cd\u4f5c */\n  void put(int key, String val) {\n    final Pair pair = Pair(key, val);\n    final int index = _hashFunc(key);\n    _buckets[index] = pair;\n  }\n\n  /* \u5220\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    final int index = _hashFunc(key);\n    _buckets[index] = null;\n  }\n\n  /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n  List&lt;Pair&gt; pairSet() {\n    List&lt;Pair&gt; pairSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        pairSet.add(pair);\n      }\n    }\n    return pairSet;\n  }\n\n  /* \u83b7\u53d6\u6240\u6709\u952e */\n  List&lt;int&gt; keySet() {\n    List&lt;int&gt; keySet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        keySet.add(pair.key);\n      }\n    }\n    return keySet;\n  }\n\n  /* \u83b7\u53d6\u6240\u6709\u503c */\n  List&lt;String&gt; values() {\n    List&lt;String&gt; valueSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        valueSet.add(pair.val);\n      }\n    }\n    return valueSet;\n  }\n\n  /* \u6253\u5370\u54c8\u5e0c\u8868 */\n  void printHashMap() {\n    for (final Pair kv in pairSet()) {\n      print(\"${kv.key} -&gt; ${kv.val}\");\n    }\n  }\n}\n</code></pre> array_hash_map.rs<pre><code>/* \u952e\u503c\u5bf9 */\n#[derive(Debug, Clone, PartialEq)]\npub struct Pair {\n    pub key: i32,\n    pub val: String,\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\npub struct ArrayHashMap {\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;,\n}\n\nimpl ArrayHashMap {\n    pub fn new() -&gt; ArrayHashMap {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        Self {\n            buckets: vec![None; 100],\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % 100\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    pub fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;String&gt; {\n        let index = self.hash_func(key);\n        self.buckets[index].as_ref().map(|pair| &amp;pair.val)\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    pub fn put(&amp;mut self, key: i32, val: &amp;str) {\n        let index = self.hash_func(key);\n        self.buckets[index] = Some(Pair {\n            key,\n            val: val.to_string(),\n        });\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    pub fn remove(&amp;mut self, key: i32) {\n        let index = self.hash_func(key);\n        // \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\n        self.buckets[index] = None;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    pub fn entry_set(&amp;self) -&gt; Vec&lt;&amp;Pair&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref())\n            .collect()\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    pub fn key_set(&amp;self) -&gt; Vec&lt;&amp;i32&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.key))\n            .collect()\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    pub fn value_set(&amp;self) -&gt; Vec&lt;&amp;String&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.val))\n            .collect()\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    pub fn print(&amp;self) {\n        for pair in self.entry_set() {\n            println!(\"{} -&gt; {}\", pair.key, pair.val);\n        }\n    }\n}\n</code></pre> array_hash_map.c<pre><code>/* \u952e\u503c\u5bf9 int-&gt;string */\ntypedef struct {\n    int key;\n    char *val;\n} Pair;\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntypedef struct {\n    Pair *buckets[HASHTABLE_CAPACITY];\n} ArrayHashMap;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayHashMap *newArrayHashMap() {\n    ArrayHashMap *hmap = malloc(sizeof(ArrayHashMap));\n    return hmap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayHashMap(ArrayHashMap *hmap) {\n    for (int i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            free(hmap-&gt;buckets[i]-&gt;val);\n            free(hmap-&gt;buckets[i]);\n        }\n    }\n    free(hmap);\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(ArrayHashMap *hmap, const int key, const char *val) {\n    Pair *Pair = malloc(sizeof(Pair));\n    Pair-&gt;key = key;\n    Pair-&gt;val = malloc(strlen(val) + 1);\n    strcpy(Pair-&gt;val, val);\n\n    int index = hashFunc(key);\n    hmap-&gt;buckets[index] = Pair;\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(ArrayHashMap *hmap, const int key) {\n    int index = hashFunc(key);\n    free(hmap-&gt;buckets[index]-&gt;val);\n    free(hmap-&gt;buckets[index]);\n    hmap-&gt;buckets[index] = NULL;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nvoid pairSet(ArrayHashMap *hmap, MapSet *set) {\n    Pair *entries;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    entries = malloc(sizeof(Pair) * total);\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            entries[index].key = hmap-&gt;buckets[i]-&gt;key;\n            entries[index].val = malloc(strlen(hmap-&gt;buckets[i]-&gt;val) + 1);\n            strcpy(entries[index].val, hmap-&gt;buckets[i]-&gt;val);\n            index++;\n        }\n    }\n    set-&gt;set = entries;\n    set-&gt;len = total;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e */\nvoid keySet(ArrayHashMap *hmap, MapSet *set) {\n    int *keys;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    keys = malloc(total * sizeof(int));\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            keys[index] = hmap-&gt;buckets[i]-&gt;key;\n            index++;\n        }\n    }\n    set-&gt;set = keys;\n    set-&gt;len = total;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvoid valueSet(ArrayHashMap *hmap, MapSet *set) {\n    char **vals;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    vals = malloc(total * sizeof(char *));\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            vals[index] = hmap-&gt;buckets[i]-&gt;val;\n            index++;\n        }\n    }\n    set-&gt;set = vals;\n    set-&gt;len = total;\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(ArrayHashMap *hmap) {\n    int i;\n    MapSet set;\n    pairSet(hmap, &amp;set);\n    Pair *entries = (Pair *)set.set;\n    for (i = 0; i &lt; set.len; i++) {\n        printf(\"%d -&gt; %s\\n\", entries[i].key, entries[i].val);\n    }\n    free(set.set);\n}\n</code></pre> array_hash_map.kt<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair(\n    var key: Int,\n    var value: String\n)\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private val buckets = arrayOfNulls&lt;Pair&gt;(100)\n\n    init {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        for (i in 0..&lt;100) {\n            buckets[i] = null\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        val index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val pair = buckets[index] ?: return null\n        return pair.value\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        val pair = Pair(key, value)\n        val index = hashFunc(key)\n        buckets[index] = pair\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = null\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    fun pairSet(): MutableList&lt;Pair&gt; {\n        val pairSet = ArrayList&lt;Pair&gt;()\n        for (pair in buckets) {\n            if (pair != null) pairSet.add(pair)\n        }\n        return pairSet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    fun keySet(): MutableList&lt;Int&gt; {\n        val keySet = ArrayList&lt;Int&gt;()\n        for (pair in buckets) {\n            if (pair != null) keySet.add(pair.key)\n        }\n        return keySet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    fun valueSet(): MutableList&lt;String&gt; {\n        val valueSet = ArrayList&lt;String&gt;()\n        for (pair in buckets) {\n            pair?.let { valueSet.add(it.value) }\n        }\n        return valueSet\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (kv in pairSet()) {\n            val key = kv.key\n            val value = kv.value\n            println(\"${key}-&gt;${value}\")\n        }\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private val buckets = arrayOfNulls&lt;Pair&gt;(100)\n\n    init {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        for (i in 0..&lt;100) {\n            buckets[i] = null\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fun hashFunc(key: Int): Int {\n        val index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val pair = buckets[index] ?: return null\n        return pair.value\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fun put(key: Int, value: String) {\n        val pair = Pair(key, value)\n        val index = hashFunc(key)\n        buckets[index] = pair\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = null\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    fun pairSet(): MutableList&lt;Pair&gt; {\n        val pairSet = ArrayList&lt;Pair&gt;()\n        for (pair in buckets) {\n            if (pair != null) pairSet.add(pair)\n        }\n        return pairSet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    fun keySet(): MutableList&lt;Int&gt; {\n        val keySet = ArrayList&lt;Int&gt;()\n        for (pair in buckets) {\n            if (pair != null) keySet.add(pair.key)\n        }\n        return keySet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    fun valueSet(): MutableList&lt;String&gt; {\n        val valueSet = ArrayList&lt;String&gt;()\n        for (pair in buckets) {\n            pair?.let { valueSet.add(it.value) }\n        }\n        return valueSet\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fun print() {\n        for (kv in pairSet()) {\n            val key = kv.key\n            val value = kv.value\n            println(\"${key}-&gt;${value}\")\n        }\n    }\n}\n</code></pre> array_hash_map.rb<pre><code>[class]{Pair}-[func]{}\n\n[class]{ArrayHashMap}-[func]{}\n</code></pre> array_hash_map.zig<pre><code>// \u952e\u503c\u5bf9\nconst Pair = struct {\n    key: usize = undefined,\n    val: []const u8 = undefined,\n\n   pub fn init(key: usize, val: []const u8) Pair {\n        return Pair {\n            .key = key,\n            .val = val,\n        };\n    }\n};\n\n// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\nfn ArrayHashMap(comptime T: type) type {\n    return struct {\n        bucket: ?std.ArrayList(?T) = null,\n        mem_allocator: std.mem.Allocator = undefined,\n\n        const Self = @This();\n\n        // \u6784\u9020\u51fd\u6570\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            self.mem_allocator = allocator;\n            // \u521d\u59cb\u5316\u4e00\u4e2a\u957f\u5ea6\u4e3a 100 \u7684\u6876\uff08\u6570\u7ec4\uff09\n            self.bucket = std.ArrayList(?T).init(self.mem_allocator);\n            var i: i32 = 0;\n            while (i &lt; 100) : (i += 1) {\n                try self.bucket.?.append(null);\n            }\n        }\n\n        // \u6790\u6784\u51fd\u6570\n        pub fn deinit(self: *Self) void {\n            if (self.bucket != null) self.bucket.?.deinit();\n        }\n\n        // \u54c8\u5e0c\u51fd\u6570\n        fn hashFunc(key: usize) usize {\n            var index = key % 100;\n            return index;\n        }\n\n        // \u67e5\u8be2\u64cd\u4f5c\n        pub fn get(self: *Self, key: usize) []const u8 {\n            var index = hashFunc(key);\n            var pair = self.bucket.?.items[index];\n            return pair.?.val;\n        }\n\n        // \u6dfb\u52a0\u64cd\u4f5c\n        pub fn put(self: *Self, key: usize, val: []const u8) !void {\n            var pair = Pair.init(key, val);\n            var index = hashFunc(key);\n            self.bucket.?.items[index] = pair;\n        }\n\n        // \u5220\u9664\u64cd\u4f5c\n        pub fn remove(self: *Self, key: usize) !void {\n            var index = hashFunc(key);\n            // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n            self.bucket.?.items[index] = null;\n        }       \n\n        // \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\n        pub fn pairSet(self: *Self) !std.ArrayList(T) {\n            var entry_set = std.ArrayList(T).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try entry_set.append(item.?);\n            }\n            return entry_set;\n        }  \n\n        // \u83b7\u53d6\u6240\u6709\u952e\n        pub fn keySet(self: *Self) !std.ArrayList(usize) {\n            var key_set = std.ArrayList(usize).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try key_set.append(item.?.key);\n            }\n            return key_set;\n        }  \n\n        // \u83b7\u53d6\u6240\u6709\u503c\n        pub fn valueSet(self: *Self) !std.ArrayList([]const u8) {\n            var value_set = std.ArrayList([]const u8).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try value_set.append(item.?.val);\n            }\n            return value_set;\n        }\n\n        // \u6253\u5370\u54c8\u5e0c\u8868\n        pub fn print(self: *Self) !void {\n            var entry_set = try self.pairSet();\n            defer entry_set.deinit();\n            for (entry_set.items) |item| {\n                std.debug.print(\"{} -&gt; {s}\\n\", .{item.key, item.val});\n            }\n        }\n    };\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_hashing/hash_map/#613","title":"6.1.3 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u4e0e\u6269\u5bb9","text":"<p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u6240\u6709 <code>key</code> \u6784\u6210\u7684\u8f93\u5165\u7a7a\u95f4\u6620\u5c04\u5230\u6570\u7ec4\u6240\u6709\u7d22\u5f15\u6784\u6210\u7684\u8f93\u51fa\u7a7a\u95f4\uff0c\u800c\u8f93\u5165\u7a7a\u95f4\u5f80\u5f80\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7406\u8bba\u4e0a\u4e00\u5b9a\u5b58\u5728\u201c\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u76f8\u540c\u8f93\u51fa\u201d\u7684\u60c5\u51b5\u3002</p> <p>\u5bf9\u4e8e\u4e0a\u8ff0\u793a\u4f8b\u4e2d\u7684\u54c8\u5e0c\u51fd\u6570\uff0c\u5f53\u8f93\u5165\u7684 <code>key</code> \u540e\u4e24\u4f4d\u76f8\u540c\u65f6\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u51fa\u7ed3\u679c\u4e5f\u76f8\u540c\u3002\u4f8b\u5982\uff0c\u67e5\u8be2\u5b66\u53f7\u4e3a 12836 \u548c 20336 \u7684\u4e24\u4e2a\u5b66\u751f\u65f6\uff0c\u6211\u4eec\u5f97\u5230\uff1a</p> <pre><code>12836 % 100 = 36\n20336 % 100 = 36\n</code></pre> <p>\u5982\u56fe 6-3 \u6240\u793a\uff0c\u4e24\u4e2a\u5b66\u53f7\u6307\u5411\u4e86\u540c\u4e00\u4e2a\u59d3\u540d\uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002\u6211\u4eec\u5c06\u8fd9\u79cd\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u540c\u4e00\u8f93\u51fa\u7684\u60c5\u51b5\u79f0\u4e3a\u54c8\u5e0c\u51b2\u7a81\uff08hash collision\uff09\u3002</p> <p></p> <p> \u56fe 6-3 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u793a\u4f8b </p> <p>\u5bb9\u6613\u60f3\u5230\uff0c\u54c8\u5e0c\u8868\u5bb9\u91cf \\(n\\) \u8d8a\u5927\uff0c\u591a\u4e2a <code>key</code> \u88ab\u5206\u914d\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\uff0c\u51b2\u7a81\u5c31\u8d8a\u5c11\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002</p> <p>\u5982\u56fe 6-4 \u6240\u793a\uff0c\u6269\u5bb9\u524d\u952e\u503c\u5bf9 <code>(136, A)</code> \u548c <code>(236, D)</code> \u53d1\u751f\u51b2\u7a81\uff0c\u6269\u5bb9\u540e\u51b2\u7a81\u6d88\u5931\u3002</p> <p></p> <p> \u56fe 6-4 \u00a0 \u54c8\u5e0c\u8868\u6269\u5bb9 </p> <p>\u7c7b\u4f3c\u4e8e\u6570\u7ec4\u6269\u5bb9\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u5c06\u6240\u6709\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u8fc1\u79fb\u81f3\u65b0\u54c8\u5e0c\u8868\uff0c\u975e\u5e38\u8017\u65f6\uff1b\u5e76\u4e14\u7531\u4e8e\u54c8\u5e0c\u8868\u5bb9\u91cf <code>capacity</code> \u6539\u53d8\uff0c\u6211\u4eec\u9700\u8981\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u6765\u91cd\u65b0\u8ba1\u7b97\u6240\u6709\u952e\u503c\u5bf9\u7684\u5b58\u50a8\u4f4d\u7f6e\uff0c\u8fd9\u8fdb\u4e00\u6b65\u589e\u52a0\u4e86\u6269\u5bb9\u8fc7\u7a0b\u7684\u8ba1\u7b97\u5f00\u9500\u3002\u4e3a\u6b64\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u9884\u7559\u8db3\u591f\u5927\u7684\u54c8\u5e0c\u8868\u5bb9\u91cf\uff0c\u9632\u6b62\u9891\u7e41\u6269\u5bb9\u3002</p> <p>\u8d1f\u8f7d\u56e0\u5b50\uff08load factor\uff09\u662f\u54c8\u5e0c\u8868\u7684\u4e00\u4e2a\u91cd\u8981\u6982\u5ff5\uff0c\u5176\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u7684\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u7528\u4e8e\u8861\u91cf\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u4e5f\u5e38\u4f5c\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u89e6\u53d1\u6761\u4ef6\u3002\u4f8b\u5982\u5728 Java \u4e2d\uff0c\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(0.75\\) \u65f6\uff0c\u7cfb\u7edf\u4f1a\u5c06\u54c8\u5e0c\u8868\u6269\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002</p>"},{"location":"chapter_hashing/summary/","title":"6.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_hashing/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u8f93\u5165 <code>key</code> \uff0c\u54c8\u5e0c\u8868\u80fd\u591f\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u67e5\u8be2\u5230 <code>value</code> \uff0c\u6548\u7387\u975e\u5e38\u9ad8\u3002</li> <li>\u5e38\u89c1\u7684\u54c8\u5e0c\u8868\u64cd\u4f5c\u5305\u62ec\u67e5\u8be2\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u3001\u5220\u9664\u952e\u503c\u5bf9\u548c\u904d\u5386\u54c8\u5e0c\u8868\u7b49\u3002</li> <li>\u54c8\u5e0c\u51fd\u6570\u5c06 <code>key</code> \u6620\u5c04\u4e3a\u6570\u7ec4\u7d22\u5f15\uff0c\u4ece\u800c\u8bbf\u95ee\u5bf9\u5e94\u6876\u5e76\u83b7\u53d6 <code>value</code> \u3002</li> <li>\u4e24\u4e2a\u4e0d\u540c\u7684 <code>key</code> \u53ef\u80fd\u5728\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u540e\u5f97\u5230\u76f8\u540c\u7684\u6570\u7ec4\u7d22\u5f15\uff0c\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u51fa\u9519\uff0c\u8fd9\u79cd\u73b0\u8c61\u88ab\u79f0\u4e3a\u54c8\u5e0c\u51b2\u7a81\u3002</li> <li>\u54c8\u5e0c\u8868\u5bb9\u91cf\u8d8a\u5927\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002\u56e0\u6b64\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\u3002\u4e0e\u6570\u7ec4\u6269\u5bb9\u7c7b\u4f3c\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u64cd\u4f5c\u7684\u5f00\u9500\u5f88\u5927\u3002</li> <li>\u8d1f\u8f7d\u56e0\u5b50\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u4e2d\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u53cd\u6620\u4e86\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u5e38\u7528\u4f5c\u89e6\u53d1\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u6761\u4ef6\u3002</li> <li>\u94fe\u5f0f\u5730\u5740\u901a\u8fc7\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u5316\u4e3a\u94fe\u8868\uff0c\u5c06\u6240\u6709\u51b2\u7a81\u5143\u7d20\u5b58\u50a8\u5728\u540c\u4e00\u4e2a\u94fe\u8868\u4e2d\u3002\u7136\u800c\uff0c\u94fe\u8868\u8fc7\u957f\u4f1a\u964d\u4f4e\u67e5\u8be2\u6548\u7387\uff0c\u53ef\u4ee5\u901a\u8fc7\u8fdb\u4e00\u6b65\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6765\u63d0\u9ad8\u6548\u7387\u3002</li> <li>\u5f00\u653e\u5bfb\u5740\u901a\u8fc7\u591a\u6b21\u63a2\u6d4b\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\u3002\u7ebf\u6027\u63a2\u6d4b\u4f7f\u7528\u56fa\u5b9a\u6b65\u957f\uff0c\u7f3a\u70b9\u662f\u4e0d\u80fd\u5220\u9664\u5143\u7d20\uff0c\u4e14\u5bb9\u6613\u4ea7\u751f\u805a\u96c6\u3002\u591a\u6b21\u54c8\u5e0c\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u8fdb\u884c\u63a2\u6d4b\uff0c\u76f8\u8f83\u7ebf\u6027\u63a2\u6d4b\u66f4\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u589e\u52a0\u4e86\u8ba1\u7b97\u91cf\u3002</li> <li>\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u54c8\u5e0c\u8868\u5b9e\u73b0\u3002\u4f8b\u5982\uff0cJava \u7684 <code>HashMap</code> \u4f7f\u7528\u94fe\u5f0f\u5730\u5740\uff0c\u800c Python \u7684 <code>Dict</code> \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002</li> <li>\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5e0c\u671b\u54c8\u5e0c\u7b97\u6cd5\u5177\u6709\u786e\u5b9a\u6027\u3001\u9ad8\u6548\u7387\u548c\u5747\u5300\u5206\u5e03\u7684\u7279\u70b9\u3002\u5728\u5bc6\u7801\u5b66\u4e2d\uff0c\u54c8\u5e0c\u7b97\u6cd5\u8fd8\u5e94\u8be5\u5177\u5907\u6297\u78b0\u649e\u6027\u548c\u96ea\u5d29\u6548\u5e94\u3002</li> <li>\u54c8\u5e0c\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u5747\u5300\u5206\u5e03\uff0c\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002</li> <li>\u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5\u5305\u62ec MD5\u3001SHA-1\u3001SHA-2 \u548c SHA-3 \u7b49\u3002MD5 \u5e38\u7528\u4e8e\u6821\u9a8c\u6587\u4ef6\u5b8c\u6574\u6027\uff0cSHA-2 \u5e38\u7528\u4e8e\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u3002</li> <li>\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u662f\u53ef\u54c8\u5e0c\u7684\u3002</li> </ul>"},{"location":"chapter_hashing/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u662f \\(O(n)\\) \uff1f</p> <p>\u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u9000\u5316\u81f3 \\(O(n)\\) \u3002\u5f53\u54c8\u5e0c\u51fd\u6570\u8bbe\u8ba1\u5f97\u6bd4\u8f83\u597d\u3001\u5bb9\u91cf\u8bbe\u7f6e\u6bd4\u8f83\u5408\u7406\u3001\u51b2\u7a81\u6bd4\u8f83\u5e73\u5747\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u6211\u4eec\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u54c8\u5e0c\u8868\u65f6\uff0c\u901a\u5e38\u8ba4\u4e3a\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002</p> <p>Q\uff1a\u4e3a\u4ec0\u4e48\u4e0d\u4f7f\u7528\u54c8\u5e0c\u51fd\u6570 \\(f(x) = x\\) \u5462\uff1f\u8fd9\u6837\u5c31\u4e0d\u4f1a\u6709\u51b2\u7a81\u4e86\u3002</p> <p>\u5728 \\(f(x) = x\\) \u54c8\u5e0c\u51fd\u6570\u4e0b\uff0c\u6bcf\u4e2a\u5143\u7d20\u5bf9\u5e94\u552f\u4e00\u7684\u6876\u7d22\u5f15\uff0c\u8fd9\u4e0e\u6570\u7ec4\u7b49\u4ef7\u3002\u7136\u800c\uff0c\u8f93\u5165\u7a7a\u95f4\u901a\u5e38\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff08\u6570\u7ec4\u957f\u5ea6\uff09\uff0c\u56e0\u6b64\u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u54c8\u5e0c\u8868\u7684\u76ee\u6807\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u72b6\u6001\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u7a7a\u95f4\uff0c\u5e76\u63d0\u4f9b \\(O(1)\\) \u7684\u67e5\u8be2\u6548\u7387\u3002</p> <p>Q\uff1a\u54c8\u5e0c\u8868\u5e95\u5c42\u5b9e\u73b0\u662f\u6570\u7ec4\u3001\u94fe\u8868\u3001\u4e8c\u53c9\u6811\uff0c\u4f46\u4e3a\u4ec0\u4e48\u6548\u7387\u53ef\u4ee5\u6bd4\u5b83\u4eec\u66f4\u9ad8\u5462\uff1f</p> <p>\u9996\u5148\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\uff0c\u4f46\u7a7a\u95f4\u6548\u7387\u53d8\u4f4e\u4e86\u3002\u54c8\u5e0c\u8868\u6709\u76f8\u5f53\u4e00\u90e8\u5206\u5185\u5b58\u672a\u4f7f\u7528\u3002</p> <p>\u5176\u6b21\uff0c\u53ea\u662f\u5728\u7279\u5b9a\u4f7f\u7528\u573a\u666f\u4e0b\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\u4e86\u3002\u5982\u679c\u4e00\u4e2a\u529f\u80fd\u80fd\u591f\u5728\u76f8\u540c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u4f7f\u7528\u6570\u7ec4\u6216\u94fe\u8868\u5b9e\u73b0\uff0c\u90a3\u4e48\u901a\u5e38\u6bd4\u54c8\u5e0c\u8868\u66f4\u5feb\u3002\u8fd9\u662f\u56e0\u4e3a\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u9700\u8981\u5f00\u9500\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u6570\u9879\u66f4\u5927\u3002</p> <p>\u6700\u540e\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u80fd\u53d1\u751f\u52a3\u5316\u3002\u4f8b\u5982\u5728\u94fe\u5f0f\u5730\u5740\u4e2d\uff0c\u6211\u4eec\u91c7\u53d6\u5728\u94fe\u8868\u6216\u7ea2\u9ed1\u6811\u4e2d\u6267\u884c\u67e5\u627e\u64cd\u4f5c\uff0c\u4ecd\u7136\u6709\u9000\u5316\u81f3 \\(O(n)\\) \u65f6\u95f4\u7684\u98ce\u9669\u3002</p> <p>Q\uff1a\u591a\u6b21\u54c8\u5e0c\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\u5417\uff1f\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u8fd8\u80fd\u518d\u6b21\u4f7f\u7528\u5417\uff1f</p> <p>\u591a\u6b21\u54c8\u5e0c\u662f\u5f00\u653e\u5bfb\u5740\u7684\u4e00\u79cd\uff0c\u5f00\u653e\u5bfb\u5740\u6cd5\u90fd\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\uff0c\u9700\u8981\u901a\u8fc7\u6807\u8bb0\u5220\u9664\u3002\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u53ef\u4ee5\u518d\u6b21\u4f7f\u7528\u3002\u5f53\u5c06\u65b0\u5143\u7d20\u63d2\u5165\u54c8\u5e0c\u8868\uff0c\u5e76\u4e14\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u4f4d\u7f6e\u65f6\uff0c\u8be5\u4f4d\u7f6e\u53ef\u4ee5\u88ab\u65b0\u5143\u7d20\u4f7f\u7528\u3002\u8fd9\u6837\u505a\u65e2\u80fd\u4fdd\u6301\u54c8\u5e0c\u8868\u7684\u63a2\u6d4b\u5e8f\u5217\u4e0d\u53d8\uff0c\u53c8\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u7684\u7a7a\u95f4\u4f7f\u7528\u7387\u3002</p> <p>Q\uff1a\u4e3a\u4ec0\u4e48\u5728\u7ebf\u6027\u63a2\u6d4b\u4e2d\uff0c\u67e5\u627e\u5143\u7d20\u7684\u65f6\u5019\u4f1a\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u5462\uff1f</p> <p>\u67e5\u627e\u7684\u65f6\u5019\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u5bf9\u5e94\u7684\u6876\u548c\u952e\u503c\u5bf9\uff0c\u53d1\u73b0 <code>key</code> \u4e0d\u5339\u914d\uff0c\u8fd9\u5c31\u4ee3\u8868\u6709\u54c8\u5e0c\u51b2\u7a81\u3002\u56e0\u6b64\uff0c\u7ebf\u6027\u63a2\u6d4b\u6cd5\u4f1a\u6839\u636e\u9884\u5148\u8bbe\u5b9a\u7684\u6b65\u957f\u4f9d\u6b21\u5411\u4e0b\u67e5\u627e\uff0c\u76f4\u81f3\u627e\u5230\u6b63\u786e\u7684\u952e\u503c\u5bf9\u6216\u65e0\u6cd5\u627e\u5230\u8df3\u51fa\u4e3a\u6b62\u3002</p> <p>Q\uff1a\u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u6269\u5bb9\u80fd\u591f\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\uff1f</p> <p>\u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d6\u6a21\uff08\u53d6\u4f59\uff09\uff0c\u8ba9\u8f93\u51fa\u503c\u843d\u5728\u6570\u7ec4\u7d22\u5f15\u8303\u56f4\u5185\uff1b\u5728\u6269\u5bb9\u540e\uff0c\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d1\u751f\u53d8\u5316\uff0c\u800c <code>key</code> \u5bf9\u5e94\u7684\u7d22\u5f15\u4e5f\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002\u539f\u5148\u843d\u5728\u540c\u4e00\u4e2a\u6876\u7684\u591a\u4e2a <code>key</code> \uff0c\u5728\u6269\u5bb9\u540e\u53ef\u80fd\u4f1a\u88ab\u5206\u914d\u5230\u591a\u4e2a\u6876\u4e2d\uff0c\u4ece\u800c\u5b9e\u73b0\u54c8\u5e0c\u51b2\u7a81\u7684\u7f13\u89e3\u3002</p>"},{"location":"chapter_heap/","title":"\u7b2c 8 \u7ae0 \u00a0 \u5806","text":"<p>Abstract</p> <p>\u5806\u5c31\u50cf\u662f\u5c71\u5cb3\u5cf0\u5ce6\uff0c\u5c42\u53e0\u8d77\u4f0f\u3001\u5f62\u6001\u5404\u5f02\u3002</p> <p>\u5ea7\u5ea7\u5c71\u5cf0\u9ad8\u4f4e\u9519\u843d\uff0c\u800c\u6700\u9ad8\u7684\u5c71\u5cf0\u603b\u662f\u6700\u5148\u6620\u5165\u773c\u5e18\u3002</p>"},{"location":"chapter_heap/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>8.1 \u00a0 \u5806</li> <li>8.2 \u00a0 \u5efa\u5806\u64cd\u4f5c</li> <li>8.3 \u00a0 Top-k \u95ee\u9898</li> <li>8.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_heap/build_heap/","title":"8.2 \u00a0 \u5efa\u5806\u64cd\u4f5c","text":"<p>\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u4f7f\u7528\u4e00\u4e2a\u5217\u8868\u7684\u6240\u6709\u5143\u7d20\u6765\u6784\u5efa\u4e00\u4e2a\u5806\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u88ab\u79f0\u4e3a\u201c\u5efa\u5806\u64cd\u4f5c\u201d\u3002</p>"},{"location":"chapter_heap/build_heap/#821","title":"8.2.1 \u00a0 \u501f\u52a9\u5165\u5806\u64cd\u4f5c\u5b9e\u73b0","text":"<p>\u6211\u4eec\u9996\u5148\u521b\u5efa\u4e00\u4e2a\u7a7a\u5806\uff0c\u7136\u540e\u904d\u5386\u5217\u8868\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e2a\u5143\u7d20\u6267\u884c\u201c\u5165\u5806\u64cd\u4f5c\u201d\uff0c\u5373\u5148\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u5806\u7684\u5c3e\u90e8\uff0c\u518d\u5bf9\u8be5\u5143\u7d20\u6267\u884c\u201c\u4ece\u5e95\u81f3\u9876\u201d\u5806\u5316\u3002</p> <p>\u6bcf\u5f53\u4e00\u4e2a\u5143\u7d20\u5165\u5806\uff0c\u5806\u7684\u957f\u5ea6\u5c31\u52a0\u4e00\u3002\u7531\u4e8e\u8282\u70b9\u662f\u4ece\u9876\u5230\u5e95\u4f9d\u6b21\u88ab\u6dfb\u52a0\u8fdb\u4e8c\u53c9\u6811\u7684\uff0c\u56e0\u6b64\u5806\u662f\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u6784\u5efa\u7684\u3002</p> <p>\u8bbe\u5143\u7d20\u6570\u91cf\u4e3a \\(n\\) \uff0c\u6bcf\u4e2a\u5143\u7d20\u7684\u5165\u5806\u64cd\u4f5c\u4f7f\u7528 \\(O(\\log{n})\\) \u65f6\u95f4\uff0c\u56e0\u6b64\u8be5\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p>"},{"location":"chapter_heap/build_heap/#822","title":"8.2.2 \u00a0 \u901a\u8fc7\u904d\u5386\u5806\u5316\u5b9e\u73b0","text":"<p>\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9e\u73b0\u4e00\u79cd\u66f4\u4e3a\u9ad8\u6548\u7684\u5efa\u5806\u65b9\u6cd5\uff0c\u5171\u5206\u4e3a\u4e24\u6b65\u3002</p> <ol> <li>\u5c06\u5217\u8868\u6240\u6709\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u5730\u6dfb\u52a0\u5230\u5806\u4e2d\uff0c\u6b64\u65f6\u5806\u7684\u6027\u8d28\u5c1a\u672a\u5f97\u5230\u6ee1\u8db3\u3002</li> <li>\u5012\u5e8f\u904d\u5386\u5806\uff08\u5c42\u5e8f\u904d\u5386\u7684\u5012\u5e8f\uff09\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e2a\u975e\u53f6\u8282\u70b9\u6267\u884c\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u3002</li> </ol> <p>\u6bcf\u5f53\u5806\u5316\u4e00\u4e2a\u8282\u70b9\u540e\uff0c\u4ee5\u8be5\u8282\u70b9\u4e3a\u6839\u8282\u70b9\u7684\u5b50\u6811\u5c31\u5f62\u6210\u4e00\u4e2a\u5408\u6cd5\u7684\u5b50\u5806\u3002\u800c\u7531\u4e8e\u662f\u5012\u5e8f\u904d\u5386\uff0c\u56e0\u6b64\u5806\u662f\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u6784\u5efa\u7684\u3002</p> <p>\u4e4b\u6240\u4ee5\u9009\u62e9\u5012\u5e8f\u904d\u5386\uff0c\u662f\u56e0\u4e3a\u8fd9\u6837\u80fd\u591f\u4fdd\u8bc1\u5f53\u524d\u8282\u70b9\u4e4b\u4e0b\u7684\u5b50\u6811\u5df2\u7ecf\u662f\u5408\u6cd5\u7684\u5b50\u5806\uff0c\u8fd9\u6837\u5806\u5316\u5f53\u524d\u8282\u70b9\u624d\u662f\u6709\u6548\u7684\u3002</p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u7531\u4e8e\u53f6\u8282\u70b9\u6ca1\u6709\u5b50\u8282\u70b9\uff0c\u56e0\u6b64\u5b83\u4eec\u5929\u7136\u5c31\u662f\u5408\u6cd5\u7684\u5b50\u5806\uff0c\u65e0\u987b\u5806\u5316\u3002\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u6700\u540e\u4e00\u4e2a\u975e\u53f6\u8282\u70b9\u662f\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u7684\u7236\u8282\u70b9\uff0c\u6211\u4eec\u4ece\u5b83\u5f00\u59cb\u5012\u5e8f\u904d\u5386\u5e76\u6267\u884c\u5806\u5316\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def __init__(self, nums: list[int]):\n    \"\"\"\u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\"\"\"\n    # \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    self.max_heap = nums\n    # \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in range(self.parent(self.size() - 1), -1, -1):\n        self.sift_down(i)\n</code></pre> my_heap.cpp<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(vector&lt;int&gt; nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = nums;\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List&lt;Integer&gt; nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = new ArrayList&lt;&gt;(nums);\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(IEnumerable&lt;int&gt; nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = new List&lt;int&gt;(nums);\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    var size = Parent(this.Size() - 1);\n    for (int i = size; i &gt;= 0; i--) {\n        SiftDown(i);\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nfunc newMaxHeap(nums []any) *maxHeap {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    h := &amp;maxHeap{data: nums}\n    for i := h.parent(len(h.data) - 1); i &gt;= 0; i-- {\n        // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n        h.siftDown(i)\n    }\n    return h\n}\n</code></pre> my_heap.swift<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\ninit(nums: [Int]) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = nums\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in (0 ... parent(i: size() - 1)).reversed() {\n        siftDown(i: i)\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    this.#maxHeap = nums === undefined ? [] : [...nums];\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = this.#parent(this.size() - 1); i &gt;= 0; i--) {\n        this.#siftDown(i);\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums?: number[]) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    this.maxHeap = nums === undefined ? [] : [...nums];\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = this.parent(this.size() - 1); i &gt;= 0; i--) {\n        this.siftDown(i);\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List&lt;int&gt; nums) {\n  // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n  _maxHeap = nums;\n  // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n  for (int i = _parent(size() - 1); i &gt;= 0; i--) {\n    siftDown(i);\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nfn new(nums: Vec&lt;i32&gt;) -&gt; Self {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    let mut heap = MaxHeap { max_heap: nums };\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in (0..=Self::parent(heap.size() - 1)).rev() {\n        heap.sift_down(i);\n    }\n    heap\n}\n</code></pre> my_heap.c<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nMaxHeap *newMaxHeap(int nums[], int size) {\n    // \u6240\u6709\u5143\u7d20\u5165\u5806\n    MaxHeap *maxHeap = (MaxHeap *)malloc(sizeof(MaxHeap));\n    maxHeap-&gt;size = size;\n    memcpy(maxHeap-&gt;data, nums, size * sizeof(int));\n    for (int i = parent(maxHeap, size - 1); i &gt;= 0; i--) {\n        // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n        siftDown(maxHeap, i);\n    }\n    return maxHeap;\n}\n</code></pre> my_heap.kt<pre><code>/* \u5927\u9876\u5806 */\nclass MaxHeap(nums: List&lt;Int&gt;?) {\n    // \u4f7f\u7528\u5217\u8868\u800c\u975e\u6570\u7ec4\uff0c\u8fd9\u6837\u65e0\u987b\u8003\u8651\u6269\u5bb9\u95ee\u9898\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    private val maxHeap = ArrayList(nums!!)\n\n    /* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\n    init {\n        // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n        for (i in parent(size() - 1) downTo 0) {\n            siftDown(i)\n        }\n    }\n\n    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    private fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    private fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    private fun parent(i: Int): Int {\n        return (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u4ea4\u6362\u5143\u7d20 */\n    private fun swap(i: Int, j: Int) {\n        maxHeap[i] = maxHeap[j].also { maxHeap[j] = maxHeap[i] }\n    }\n\n    /* \u83b7\u53d6\u5806\u5927\u5c0f */\n    fun size(): Int {\n        return maxHeap.size\n    }\n\n    /* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        /* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\n        return size() == 0\n    }\n\n    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\n    fun peek(): Int {\n        return maxHeap[0]\n    }\n\n    /* \u5143\u7d20\u5165\u5806 */\n    fun push(value: Int) {\n        // \u6dfb\u52a0\u8282\u70b9\n        maxHeap.add(value)\n        // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n        siftUp(size() - 1)\n    }\n\n    /* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n    private fun siftUp(it: Int) {\n        // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n        var i = it\n        while (true) {\n            // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n            val p = parent(i)\n            // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n            if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n            // \u4ea4\u6362\u4e24\u8282\u70b9\n            swap(i, p)\n            // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n            i = p\n        }\n    }\n\n    /* \u5143\u7d20\u51fa\u5806 */\n    fun pop(): Int {\n        // \u5224\u7a7a\u5904\u7406\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        swap(0, size() - 1)\n        // \u5220\u9664\u8282\u70b9\n        val value = maxHeap.removeAt(size() - 1)\n        // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n        siftDown(0)\n        // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n        return value\n    }\n\n    /* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n    private fun siftDown(it: Int) {\n        // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n        var i = it\n        while (true) {\n            // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n            val l = left(i)\n            val r = right(i)\n            var ma = i\n            if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n            if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n            // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n            if (ma == i) break\n            // \u4ea4\u6362\u4e24\u8282\u70b9\n            swap(i, ma)\n            // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n            i = ma\n        }\n    }\n\n    /* \u6253\u5370\u5806\uff08\u4e8c\u53c9\u6811\uff09 */\n    fun print() {\n        val queue = PriorityQueue { a: Int, b: Int -&gt; b - a }\n        queue.addAll(maxHeap)\n        printHeap(queue)\n    }\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{__init__}\n</code></pre> my_heap.zig<pre><code>// \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\nfn init(self: *Self, allocator: std.mem.Allocator, nums: []const T) !void {\n    if (self.max_heap != null) return;\n    self.max_heap = std.ArrayList(T).init(allocator);\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    try self.max_heap.?.appendSlice(nums);\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    var i: usize = parent(self.size() - 1) + 1;\n    while (i &gt; 0) : (i -= 1) {\n        try self.siftDown(i - 1);\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_heap/build_heap/#823","title":"8.2.3 \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>\u4e0b\u9762\uff0c\u6211\u4eec\u6765\u5c1d\u8bd5\u63a8\u7b97\u7b2c\u4e8c\u79cd\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <ul> <li>\u5047\u8bbe\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u5219\u53f6\u8282\u70b9\u6570\u91cf\u4e3a \\((n + 1) / 2\\) \uff0c\u5176\u4e2d \\(/\\) \u4e3a\u5411\u4e0b\u6574\u9664\u3002\u56e0\u6b64\u9700\u8981\u5806\u5316\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\((n - 1) / 2\\) \u3002</li> <li>\u5728\u4ece\u9876\u81f3\u5e95\u5806\u5316\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e2a\u8282\u70b9\u6700\u591a\u5806\u5316\u5230\u53f6\u8282\u70b9\uff0c\u56e0\u6b64\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u4e3a\u4e8c\u53c9\u6811\u9ad8\u5ea6 \\(\\log n\\) \u3002</li> </ul> <p>\u5c06\u4e0a\u8ff0\u4e24\u8005\u76f8\u4e58\uff0c\u53ef\u5f97\u5230\u5efa\u5806\u8fc7\u7a0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u4f46\u8fd9\u4e2a\u4f30\u7b97\u7ed3\u679c\u5e76\u4e0d\u51c6\u786e\uff0c\u56e0\u4e3a\u6211\u4eec\u6ca1\u6709\u8003\u8651\u5230\u4e8c\u53c9\u6811\u5e95\u5c42\u8282\u70b9\u6570\u91cf\u8fdc\u591a\u4e8e\u9876\u5c42\u8282\u70b9\u7684\u6027\u8d28\u3002</p> <p>\u63a5\u4e0b\u6765\u6211\u4eec\u6765\u8fdb\u884c\u66f4\u4e3a\u51c6\u786e\u7684\u8ba1\u7b97\u3002\u4e3a\u4e86\u964d\u4f4e\u8ba1\u7b97\u96be\u5ea6\uff0c\u5047\u8bbe\u7ed9\u5b9a\u4e00\u4e2a\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \u3001\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u201c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u201d\uff0c\u8be5\u5047\u8bbe\u4e0d\u4f1a\u5f71\u54cd\u8ba1\u7b97\u7ed3\u679c\u7684\u6b63\u786e\u6027\u3002</p> <p></p> <p> \u56fe 8-5 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u5404\u5c42\u8282\u70b9\u6570\u91cf </p> <p>\u5982\u56fe 8-5 \u6240\u793a\uff0c\u8282\u70b9\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u7684\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u7b49\u4e8e\u8be5\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u800c\u8be5\u8ddd\u79bb\u6b63\u662f\u201c\u8282\u70b9\u9ad8\u5ea6\u201d\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5bf9\u5404\u5c42\u7684\u201c\u8282\u70b9\u6570\u91cf \\(\\times\\) \u8282\u70b9\u9ad8\u5ea6\u201d\u6c42\u548c\uff0c\u5f97\u5230\u6240\u6709\u8282\u70b9\u7684\u5806\u5316\u8fed\u4ee3\u6b21\u6570\u7684\u603b\u548c\u3002</p> \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{(h-1)}\\times1 \\] <p>\u5316\u7b80\u4e0a\u5f0f\u9700\u8981\u501f\u52a9\u4e2d\u5b66\u7684\u6570\u5217\u77e5\u8bc6\uff0c\u5148\u5c06 \\(T(h)\\) \u4e58\u4ee5 \\(2\\) \uff0c\u5f97\u5230\uff1a</p> \\[ \\begin{aligned} T(h) &amp; = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{h-1}\\times1 \\newline 2 T(h) &amp; = 2^1h + 2^2(h-1) + 2^3(h-2) + \\dots + 2^{h}\\times1 \\newline \\end{aligned} \\] <p>\u4f7f\u7528\u9519\u4f4d\u76f8\u51cf\u6cd5\uff0c\u7528\u4e0b\u5f0f \\(2 T(h)\\) \u51cf\u53bb\u4e0a\u5f0f \\(T(h)\\) \uff0c\u53ef\u5f97\uff1a</p> \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\dots + 2^{h-1} + 2^h \\] <p>\u89c2\u5bdf\u4e0a\u5f0f\uff0c\u53d1\u73b0 \\(T(h)\\) \u662f\u4e00\u4e2a\u7b49\u6bd4\u6570\u5217\uff0c\u53ef\u76f4\u63a5\u4f7f\u7528\u6c42\u548c\u516c\u5f0f\uff0c\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\uff1a</p> \\[ \\begin{aligned} T(h) &amp; = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline &amp; = 2^{h+1} - h - 2 \\newline &amp; = O(2^h) \\end{aligned} \\] <p>\u8fdb\u4e00\u6b65\uff0c\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n = 2^{h+1} - 1\\) \uff0c\u6613\u5f97\u590d\u6742\u5ea6\u4e3a \\(O(2^h) = O(n)\\) \u3002\u4ee5\u4e0a\u63a8\u7b97\u8868\u660e\uff0c\u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</p>"},{"location":"chapter_heap/heap/","title":"8.1 \u00a0 \u5806","text":"<p>\u5806\uff08heap\uff09\u662f\u4e00\u79cd\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u4e3b\u8981\u53ef\u5206\u4e3a\u4e24\u79cd\u7c7b\u578b\uff0c\u5982\u56fe 8-1 \u6240\u793a\u3002</p> <ul> <li>\u5c0f\u9876\u5806\uff08min heap\uff09\uff1a\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\leq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\u3002</li> <li>\u5927\u9876\u5806\uff08max heap\uff09\uff1a\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\geq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\u3002</li> </ul> <p></p> <p> \u56fe 8-1 \u00a0 \u5c0f\u9876\u5806\u4e0e\u5927\u9876\u5806 </p> <p>\u5806\u4f5c\u4e3a\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u6700\u5e95\u5c42\u8282\u70b9\u9760\u5de6\u586b\u5145\uff0c\u5176\u4ed6\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u3002</li> <li>\u6211\u4eec\u5c06\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\u79f0\u4e3a\u201c\u5806\u9876\u201d\uff0c\u5c06\u5e95\u5c42\u6700\u9760\u53f3\u7684\u8282\u70b9\u79f0\u4e3a\u201c\u5806\u5e95\u201d\u3002</li> <li>\u5bf9\u4e8e\u5927\u9876\u5806\uff08\u5c0f\u9876\u5806\uff09\uff0c\u5806\u9876\u5143\u7d20\uff08\u6839\u8282\u70b9\uff09\u7684\u503c\u662f\u6700\u5927\uff08\u6700\u5c0f\uff09\u7684\u3002</li> </ul>"},{"location":"chapter_heap/heap/#811","title":"8.1.1 \u00a0 \u5806\u7684\u5e38\u7528\u64cd\u4f5c","text":"<p>\u9700\u8981\u6307\u51fa\u7684\u662f\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u662f\u4f18\u5148\u961f\u5217\uff08priority queue\uff09\uff0c\u8fd9\u662f\u4e00\u79cd\u62bd\u8c61\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5b9a\u4e49\u4e3a\u5177\u6709\u4f18\u5148\u7ea7\u6392\u5e8f\u7684\u961f\u5217\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u5806\u901a\u5e38\u7528\u4e8e\u5b9e\u73b0\u4f18\u5148\u961f\u5217\uff0c\u5927\u9876\u5806\u76f8\u5f53\u4e8e\u5143\u7d20\u6309\u4ece\u5927\u5230\u5c0f\u7684\u987a\u5e8f\u51fa\u961f\u7684\u4f18\u5148\u961f\u5217\u3002\u4ece\u4f7f\u7528\u89d2\u5ea6\u6765\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u201c\u4f18\u5148\u961f\u5217\u201d\u548c\u201c\u5806\u201d\u770b\u4f5c\u7b49\u4ef7\u7684\u6570\u636e\u7ed3\u6784\u3002\u56e0\u6b64\uff0c\u672c\u4e66\u5bf9\u4e24\u8005\u4e0d\u505a\u7279\u522b\u533a\u5206\uff0c\u7edf\u4e00\u79f0\u4f5c\u201c\u5806\u201d\u3002</p> <p>\u5806\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u8868 8-1 \uff0c\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002</p> <p> \u8868 8-1 \u00a0 \u5806\u7684\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 <code>push()</code> \u5143\u7d20\u5165\u5806 \\(O(\\log n)\\) <code>pop()</code> \u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) <code>peek()</code> \u8bbf\u95ee\u5806\u9876\u5143\u7d20\uff08\u5bf9\u4e8e\u5927 / \u5c0f\u9876\u5806\u5206\u522b\u4e3a\u6700\u5927 / \u5c0f\u503c\uff09 \\(O(1)\\) <code>size()</code> \u83b7\u53d6\u5806\u7684\u5143\u7d20\u6570\u91cf \\(O(1)\\) <code>isEmpty()</code> \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a \\(O(1)\\) <p>\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u5806\u7c7b\uff08\u6216\u4f18\u5148\u961f\u5217\u7c7b\uff09\u3002</p> <p>\u7c7b\u4f3c\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e2d\u7684\u201c\u4ece\u5c0f\u5230\u5927\u6392\u5217\u201d\u548c\u201c\u4ece\u5927\u5230\u5c0f\u6392\u5217\u201d\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u8bbe\u7f6e\u4e00\u4e2a <code>flag</code> \u6216\u4fee\u6539 <code>Comparator</code> \u5b9e\u73b0\u201c\u5c0f\u9876\u5806\u201d\u4e0e\u201c\u5927\u9876\u5806\u201d\u4e4b\u95f4\u7684\u8f6c\u6362\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig heap.py<pre><code># \u521d\u59cb\u5316\u5c0f\u9876\u5806\nmin_heap, flag = [], 1\n# \u521d\u59cb\u5316\u5927\u9876\u5806\nmax_heap, flag = [], -1\n\n# Python \u7684 heapq \u6a21\u5757\u9ed8\u8ba4\u5b9e\u73b0\u5c0f\u9876\u5806\n# \u8003\u8651\u5c06\u201c\u5143\u7d20\u53d6\u8d1f\u201d\u540e\u518d\u5165\u5806\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5c06\u5927\u5c0f\u5173\u7cfb\u98a0\u5012\uff0c\u4ece\u800c\u5b9e\u73b0\u5927\u9876\u5806\n# \u5728\u672c\u793a\u4f8b\u4e2d\uff0cflag = 1 \u65f6\u5bf9\u5e94\u5c0f\u9876\u5806\uff0cflag = -1 \u65f6\u5bf9\u5e94\u5927\u9876\u5806\n\n# \u5143\u7d20\u5165\u5806\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n\n# \u83b7\u53d6\u5806\u9876\u5143\u7d20\npeek: int = flag * max_heap[0] # 5\n\n# \u5806\u9876\u5143\u7d20\u51fa\u5806\n# \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n\n# \u83b7\u53d6\u5806\u5927\u5c0f\nsize: int = len(max_heap)\n\n# \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = not max_heap\n\n# \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\nmin_heap: list[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n</code></pre> heap.cpp<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap;\n// \u521d\u59cb\u5316\u5927\u9876\u5806\npriority_queue&lt;int, vector&lt;int&gt;, less&lt;int&gt;&gt; maxHeap;\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.top(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.empty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nvector&lt;int&gt; input{1, 3, 2, 5, 4};\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap(input.begin(), input.end());\n</code></pre> heap.java<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nQueue&lt;Integer&gt; minHeap = new PriorityQueue&lt;&gt;();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nQueue&lt;Integer&gt; maxHeap = new PriorityQueue&lt;&gt;((a, b) -&gt; b - a);\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.peek(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll(); // 5\npeek = maxHeap.poll(); // 4\npeek = maxHeap.poll(); // 3\npeek = maxHeap.poll(); // 2\npeek = maxHeap.poll(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = maxHeap.isEmpty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue&lt;&gt;(Arrays.asList(1, 3, 2, 5, 4));\n</code></pre> heap.cs<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nPriorityQueue&lt;int, int&gt; minHeap = new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nPriorityQueue&lt;int, int&gt; maxHeap = new(Comparer&lt;int&gt;.Create((x, y) =&gt; y - x));\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.Peek();//5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.Count;\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.Count == 0;\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue&lt;int, int&gt;([(1, 1), (3, 3), (2, 2), (5, 5), (4, 4)]);\n</code></pre> heap.go<pre><code>// Go \u8bed\u8a00\u4e2d\u53ef\u4ee5\u901a\u8fc7\u5b9e\u73b0 heap.Interface \u6765\u6784\u5efa\u6574\u6570\u5927\u9876\u5806\n// \u5b9e\u73b0 heap.Interface \u9700\u8981\u540c\u65f6\u5b9e\u73b0 sort.Interface\ntype intHeap []any\n\n// Push heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u63a8\u5165\u5143\u7d20\u5230\u5806\nfunc (h *intHeap) Push(x any) {\n    // Push \u548c Pop \u4f7f\u7528 pointer receiver \u4f5c\u4e3a\u53c2\u6570\n    // \u56e0\u4e3a\u5b83\u4eec\u4e0d\u4ec5\u4f1a\u5bf9\u5207\u7247\u7684\u5185\u5bb9\u8fdb\u884c\u8c03\u6574\uff0c\u8fd8\u4f1a\u4fee\u6539\u5207\u7247\u7684\u957f\u5ea6\u3002\n    *h = append(*h, x.(int))\n}\n\n// Pop heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u5f39\u51fa\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Pop() any {\n    // \u5f85\u51fa\u5806\u5143\u7d20\u5b58\u653e\u5728\u6700\u540e\n    last := (*h)[len(*h)-1]\n    *h = (*h)[:len(*h)-1]\n    return last\n}\n\n// Len sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Len() int {\n    return len(*h)\n}\n\n// Less sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Less(i, j int) bool {\n    // \u5982\u679c\u5b9e\u73b0\u5c0f\u9876\u5806\uff0c\u5219\u9700\u8981\u8c03\u6574\u4e3a\u5c0f\u4e8e\u53f7\n    return (*h)[i].(int) &gt; (*h)[j].(int)\n}\n\n// Swap sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Swap(i, j int) {\n    (*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n\n// Top \u83b7\u53d6\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Top() any {\n    return (*h)[0]\n}\n\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n    /* \u521d\u59cb\u5316\u5806 */\n    // \u521d\u59cb\u5316\u5927\u9876\u5806\n    maxHeap := &amp;intHeap{}\n    heap.Init(maxHeap)\n    /* \u5143\u7d20\u5165\u5806 */\n    // \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u6dfb\u52a0\u5143\u7d20\n    heap.Push(maxHeap, 1)\n    heap.Push(maxHeap, 3)\n    heap.Push(maxHeap, 2)\n    heap.Push(maxHeap, 4)\n    heap.Push(maxHeap, 5)\n\n    /* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\n    top := maxHeap.Top()\n    fmt.Printf(\"\u5806\u9876\u5143\u7d20\u4e3a %d\\n\", top)\n\n    /* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n    // \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u79fb\u9664\u5143\u7d20\n    heap.Pop(maxHeap) // 5\n    heap.Pop(maxHeap) // 4\n    heap.Pop(maxHeap) // 3\n    heap.Pop(maxHeap) // 2\n    heap.Pop(maxHeap) // 1\n\n    /* \u83b7\u53d6\u5806\u5927\u5c0f */\n    size := len(*maxHeap)\n    fmt.Printf(\"\u5806\u5143\u7d20\u6570\u91cf\u4e3a %d\\n\", size)\n\n    /* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\n    isEmpty := len(*maxHeap) == 0\n    fmt.Printf(\"\u5806\u662f\u5426\u4e3a\u7a7a %t\\n\", isEmpty)\n}\n</code></pre> heap.swift<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// Swift \u7684 Heap \u7c7b\u578b\u540c\u65f6\u652f\u6301\u6700\u5927\u5806\u548c\u6700\u5c0f\u5806\uff0c\u4e14\u9700\u8981\u5f15\u5165 swift-collections\nvar heap = Heap&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u5806 */\nheap.insert(1)\nheap.insert(3)\nheap.insert(2)\nheap.insert(5)\nheap.insert(4)\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nvar peek = heap.max()!\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\npeek = heap.removeMax() // 5\npeek = heap.removeMax() // 4\npeek = heap.removeMax() // 3\npeek = heap.removeMax() // 2\npeek = heap.removeMax() // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nlet size = heap.count\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = heap.isEmpty\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nlet heap2 = Heap([1, 3, 2, 5, 4])\n</code></pre> heap.js<pre><code>// JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.ts<pre><code>// TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.dart<pre><code>// Dart \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.rs<pre><code>use std::collections::BinaryHeap;\nuse std::cmp::Reverse;\n\n/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nlet mut min_heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\nlet mut max_heap = BinaryHeap::new();\n\n/* \u5143\u7d20\u5165\u5806 */\nmax_heap.push(1);\nmax_heap.push(3);\nmax_heap.push(2);\nmax_heap.push(5);\nmax_heap.push(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nlet peek = max_heap.peek().unwrap();  // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nlet peek = max_heap.pop().unwrap();   // 5\nlet peek = max_heap.pop().unwrap();   // 4\nlet peek = max_heap.pop().unwrap();   // 3\nlet peek = max_heap.pop().unwrap();   // 2\nlet peek = max_heap.pop().unwrap();   // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nlet size = max_heap.len();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = max_heap.is_empty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nlet min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]);\n</code></pre> heap.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.kt<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nvar minHeap = PriorityQueue&lt;Int&gt;()\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nval maxHeap = PriorityQueue { a: Int, b: Int -&gt; b - a }\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1)\nmaxHeap.offer(3)\nmaxHeap.offer(2)\nmaxHeap.offer(5)\nmaxHeap.offer(4)\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nvar peek = maxHeap.peek() // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll() // 5\npeek = maxHeap.poll() // 4\npeek = maxHeap.poll() // 3\npeek = maxHeap.poll() // 2\npeek = maxHeap.poll() // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nval size = maxHeap.size\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nval isEmpty = maxHeap.isEmpty()\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = PriorityQueue(mutableListOf(1, 3, 2, 5, 4))\n</code></pre> heap.rb<pre><code>\n</code></pre> heap.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_heap/heap/#812","title":"8.1.2 \u00a0 \u5806\u7684\u5b9e\u73b0","text":"<p>\u4e0b\u6587\u5b9e\u73b0\u7684\u662f\u5927\u9876\u5806\u3002\u82e5\u8981\u5c06\u5176\u8f6c\u6362\u4e3a\u5c0f\u9876\u5806\uff0c\u53ea\u9700\u5c06\u6240\u6709\u5927\u5c0f\u903b\u8f91\u5224\u65ad\u53d6\u9006\uff08\u4f8b\u5982\uff0c\u5c06 \\(\\geq\\) \u66ff\u6362\u4e3a \\(\\leq\\) \uff09\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002</p>"},{"location":"chapter_heap/heap/#1","title":"1. \u00a0 \u5806\u7684\u5b58\u50a8\u4e0e\u8868\u793a","text":"<p>\u201c\u4e8c\u53c9\u6811\u201d\u7ae0\u8282\u8bb2\u8fc7\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u7531\u4e8e\u5806\u6b63\u662f\u4e00\u79cd\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u56e0\u6b64\u6211\u4eec\u5c06\u91c7\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002</p> <p>\u5f53\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u4e8c\u53c9\u6811\u65f6\uff0c\u5143\u7d20\u4ee3\u8868\u8282\u70b9\u503c\uff0c\u7d22\u5f15\u4ee3\u8868\u8282\u70b9\u5728\u4e8c\u53c9\u6811\u4e2d\u7684\u4f4d\u7f6e\u3002\u8282\u70b9\u6307\u9488\u901a\u8fc7\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u6765\u5b9e\u73b0\u3002</p> <p>\u5982\u56fe 8-2 \u6240\u793a\uff0c\u7ed9\u5b9a\u7d22\u5f15 \\(i\\) \uff0c\u5176\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\u4e3a \\(2i + 2\\) \uff0c\u7236\u8282\u70b9\u7684\u7d22\u5f15\u4e3a \\((i - 1) / 2\\)\uff08\u5411\u4e0b\u6574\u9664\uff09\u3002\u5f53\u7d22\u5f15\u8d8a\u754c\u65f6\uff0c\u8868\u793a\u7a7a\u8282\u70b9\u6216\u8282\u70b9\u4e0d\u5b58\u5728\u3002</p> <p></p> <p> \u56fe 8-2 \u00a0 \u5806\u7684\u8868\u793a\u4e0e\u5b58\u50a8 </p> <p>\u6211\u4eec\u53ef\u4ee5\u5c06\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def left(self, i: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n    return 2 * i + 1\n\ndef right(self, i: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n    return 2 * i + 2\n\ndef parent(self, i: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n    return (i - 1) // 2  # \u5411\u4e0b\u6574\u9664\n</code></pre> my_heap.cpp<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.java<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.cs<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint Left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint Right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint Parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.go<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) left(i int) int {\n    return 2*i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) right(i int) int {\n    return 2*i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (h *maxHeap) parent(i int) int {\n    // \u5411\u4e0b\u6574\u9664\n    return (i - 1) / 2\n}\n</code></pre> my_heap.swift<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc left(i: Int) -&gt; Int {\n    2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc right(i: Int) -&gt; Int {\n    2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc parent(i: Int) -&gt; Int {\n    (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.js<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n#left(i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n#right(i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n#parent(i) {\n    return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.ts<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nleft(i: number): number {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nright(i: number): number {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nparent(i: number): number {\n    return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.dart<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint _left(int i) {\n  return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint _right(int i) {\n  return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint _parent(int i) {\n  return (i - 1) ~/ 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.rs<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfn left(i: usize) -&gt; usize {\n    2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfn right(i: usize) -&gt; usize {\n    2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfn parent(i: usize) -&gt; usize {\n    (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.c<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint left(MaxHeap *maxHeap, int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint right(MaxHeap *maxHeap, int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint parent(MaxHeap *maxHeap, int i) {\n    return (i - 1) / 2;\n}\n</code></pre> my_heap.kt<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfun left(i: Int): Int {\n    return 2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfun right(i: Int): Int {\n    return 2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfun parent(i: Int): Int {\n    return (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{left}\n\n[class]{MaxHeap}-[func]{right}\n\n[class]{MaxHeap}-[func]{parent}\n</code></pre> my_heap.zig<pre><code>// \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\nfn left(i: usize) usize {\n    return 2 * i + 1;\n}\n\n// \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\nfn right(i: usize) usize {\n    return 2 * i + 2;\n}\n\n// \u83b7\u53d6\u7236\u8282\u70b9\u7684\u7d22\u5f15\nfn parent(i: usize) usize {\n    // return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n    return @divFloor(i - 1, 2);\n}\n</code></pre>"},{"location":"chapter_heap/heap/#2","title":"2. \u00a0 \u8bbf\u95ee\u5806\u9876\u5143\u7d20","text":"<p>\u5806\u9876\u5143\u7d20\u5373\u4e3a\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u4e5f\u5c31\u662f\u5217\u8868\u7684\u9996\u4e2a\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def peek(self) -&gt; int:\n    \"\"\"\u8bbf\u95ee\u5806\u9876\u5143\u7d20\"\"\"\n    return self.max_heap[0]\n</code></pre> my_heap.cpp<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.java<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n    return maxHeap.get(0);\n}\n</code></pre> my_heap.cs<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint Peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.go<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc (h *maxHeap) peek() any {\n    return h.data[0]\n}\n</code></pre> my_heap.swift<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc peek() -&gt; Int {\n    maxHeap[0]\n}\n</code></pre> my_heap.js<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek() {\n    return this.#maxHeap[0];\n}\n</code></pre> my_heap.ts<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek(): number {\n    return this.maxHeap[0];\n}\n</code></pre> my_heap.dart<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n  return _maxHeap[0];\n}\n</code></pre> my_heap.rs<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfn peek(&amp;self) -&gt; Option&lt;i32&gt; {\n    self.max_heap.first().copied()\n}\n</code></pre> my_heap.c<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek(MaxHeap *maxHeap) {\n    return maxHeap-&gt;data[0];\n}\n</code></pre> my_heap.kt<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfun peek(): Int {\n    return maxHeap[0]\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{peek}\n</code></pre> my_heap.zig<pre><code>// \u8bbf\u95ee\u5806\u9876\u5143\u7d20\nfn peek(self: *Self) T {\n    return self.max_heap.?.items[0];\n}  \n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_heap/heap/#3","title":"3. \u00a0 \u5143\u7d20\u5165\u5806","text":"<p>\u7ed9\u5b9a\u5143\u7d20 <code>val</code> \uff0c\u6211\u4eec\u9996\u5148\u5c06\u5176\u6dfb\u52a0\u5230\u5806\u5e95\u3002\u6dfb\u52a0\u4e4b\u540e\uff0c\u7531\u4e8e <code>val</code> \u53ef\u80fd\u5927\u4e8e\u5806\u4e2d\u5176\u4ed6\u5143\u7d20\uff0c\u5806\u7684\u6210\u7acb\u6761\u4ef6\u53ef\u80fd\u5df2\u88ab\u7834\u574f\uff0c\u56e0\u6b64\u9700\u8981\u4fee\u590d\u4ece\u63d2\u5165\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u8fd9\u4e2a\u64cd\u4f5c\u88ab\u79f0\u4e3a\u5806\u5316\uff08heapify\uff09\u3002</p> <p>\u8003\u8651\u4ece\u5165\u5806\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u6267\u884c\u5806\u5316\u3002\u5982\u56fe 8-3 \u6240\u793a\uff0c\u6211\u4eec\u6bd4\u8f83\u63d2\u5165\u8282\u70b9\u4e0e\u5176\u7236\u8282\u70b9\u7684\u503c\uff0c\u5982\u679c\u63d2\u5165\u8282\u70b9\u66f4\u5927\uff0c\u5219\u5c06\u5b83\u4eec\u4ea4\u6362\u3002\u7136\u540e\u7ee7\u7eed\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u4ece\u5e95\u81f3\u9876\u4fee\u590d\u5806\u4e2d\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u76f4\u81f3\u8d8a\u8fc7\u6839\u8282\u70b9\u6216\u9047\u5230\u65e0\u987b\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 8-3 \u00a0 \u5143\u7d20\u5165\u5806\u6b65\u9aa4 </p> <p>\u8bbe\u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u6811\u7684\u9ad8\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u7531\u6b64\u53ef\u77e5\uff0c\u5806\u5316\u64cd\u4f5c\u7684\u5faa\u73af\u8f6e\u6570\u6700\u591a\u4e3a \\(O(\\log n)\\) \uff0c\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def push(self, val: int):\n    \"\"\"\u5143\u7d20\u5165\u5806\"\"\"\n    # \u6dfb\u52a0\u8282\u70b9\n    self.max_heap.append(val)\n    # \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    self.sift_up(self.size() - 1)\n\ndef sift_up(self, i: int):\n    \"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\"\"\"\n    while True:\n        # \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        p = self.parent(i)\n        # \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if p &lt; 0 or self.max_heap[i] &lt;= self.max_heap[p]:\n            break\n        # \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, p)\n        # \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.push_back(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(maxHeap[i], maxHeap[p]);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.add(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap.get(i) &lt;= maxHeap.get(p))\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid Push(int val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.Add(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    SiftUp(Size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid SiftUp(int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = Parent(i);\n        // \u82e5\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        Swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunc (h *maxHeap) push(val any) {\n    // \u6dfb\u52a0\u8282\u70b9\n    h.data = append(h.data, val)\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    h.siftUp(len(h.data) - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc (h *maxHeap) siftUp(i int) {\n    for true {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        p := h.parent(i)\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if p &lt; 0 || h.data[i].(int) &lt;= h.data[p].(int) {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        h.swap(i, p)\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunc push(val: Int) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.append(val)\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(i: size() - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc siftUp(i: Int) {\n    var i = i\n    while true {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        let p = parent(i: i)\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if p &lt; 0 || maxHeap[i] &lt;= maxHeap[p] {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i: i, j: p)\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u5165\u5806 */\npush(val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    this.#maxHeap.push(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    this.#siftUp(this.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n#siftUp(i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        const p = this.#parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || this.#maxHeap[i] &lt;= this.#maxHeap[p]) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.#swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u5165\u5806 */\npush(val: number): void {\n    // \u6dfb\u52a0\u8282\u70b9\n    this.maxHeap.push(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    this.siftUp(this.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nsiftUp(i: number): void {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        const p = this.parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || this.maxHeap[i] &lt;= this.maxHeap[p]) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n  // \u6dfb\u52a0\u8282\u70b9\n  _maxHeap.add(val);\n  // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n  siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n  while (true) {\n    // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n    int p = _parent(i);\n    // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n    if (p &lt; 0 || _maxHeap[i] &lt;= _maxHeap[p]) {\n      break;\n    }\n    // \u4ea4\u6362\u4e24\u8282\u70b9\n    _swap(i, p);\n    // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n    i = p;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfn push(&amp;mut self, val: i32) {\n    // \u6dfb\u52a0\u8282\u70b9\n    self.max_heap.push(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    self.sift_up(self.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfn sift_up(&amp;mut self, mut i: usize) {\n    loop {\n        // \u8282\u70b9 i \u5df2\u7ecf\u662f\u5806\u9876\u8282\u70b9\u4e86\uff0c\u7ed3\u675f\u5806\u5316\n        if i == 0 {\n            break;\n        }\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        let p = Self::parent(i);\n        // \u5f53\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if self.max_heap[i] &lt;= self.max_heap[p] {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(MaxHeap *maxHeap, int val) {\n    // \u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c\u4e0d\u5e94\u8be5\u6dfb\u52a0\u8fd9\u4e48\u591a\u8282\u70b9\n    if (maxHeap-&gt;size == MAX_SIZE) {\n        printf(\"heap is full!\");\n        return;\n    }\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap-&gt;data[maxHeap-&gt;size] = val;\n    maxHeap-&gt;size++;\n\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(maxHeap, maxHeap-&gt;size - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = parent(maxHeap, i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap-&gt;data[i] &lt;= maxHeap-&gt;data[p]) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(maxHeap, i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfun push(value: Int) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.add(value)\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(size() - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfun siftUp(it: Int) {\n    // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n    var i = it\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        val p = parent(i)\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, p)\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{push}\n\n[class]{MaxHeap}-[func]{sift_up}\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u5165\u5806\nfn push(self: *Self, val: T) !void {\n    // \u6dfb\u52a0\u8282\u70b9\n    try self.max_heap.?.append(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    try self.siftUp(self.size() - 1);\n}  \n\n// \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\nfn siftUp(self: *Self, i_: usize) !void {\n    var i = i_;\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        var p = parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 or self.max_heap.?.items[i] &lt;= self.max_heap.?.items[p]) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        try self.swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_heap/heap/#4","title":"4. \u00a0 \u5806\u9876\u5143\u7d20\u51fa\u5806","text":"<p>\u5806\u9876\u5143\u7d20\u662f\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u5373\u5217\u8868\u9996\u5143\u7d20\u3002\u5982\u679c\u6211\u4eec\u76f4\u63a5\u4ece\u5217\u8868\u4e2d\u5220\u9664\u9996\u5143\u7d20\uff0c\u90a3\u4e48\u4e8c\u53c9\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u7d22\u5f15\u90fd\u4f1a\u53d1\u751f\u53d8\u5316\uff0c\u8fd9\u5c06\u4f7f\u5f97\u540e\u7eed\u4f7f\u7528\u5806\u5316\u8fdb\u884c\u4fee\u590d\u53d8\u5f97\u56f0\u96be\u3002\u4e3a\u4e86\u5c3d\u91cf\u51cf\u5c11\u5143\u7d20\u7d22\u5f15\u7684\u53d8\u52a8\uff0c\u6211\u4eec\u91c7\u7528\u4ee5\u4e0b\u64cd\u4f5c\u6b65\u9aa4\u3002</p> <ol> <li>\u4ea4\u6362\u5806\u9876\u5143\u7d20\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff09\u3002</li> <li>\u4ea4\u6362\u5b8c\u6210\u540e\uff0c\u5c06\u5806\u5e95\u4ece\u5217\u8868\u4e2d\u5220\u9664\uff08\u6ce8\u610f\uff0c\u7531\u4e8e\u5df2\u7ecf\u4ea4\u6362\uff0c\u56e0\u6b64\u5b9e\u9645\u4e0a\u5220\u9664\u7684\u662f\u539f\u6765\u7684\u5806\u9876\u5143\u7d20\uff09\u3002</li> <li>\u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u6267\u884c\u5806\u5316\u3002</li> </ol> <p>\u5982\u56fe 8-4 \u6240\u793a\uff0c\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u7684\u64cd\u4f5c\u65b9\u5411\u4e0e\u201c\u4ece\u5e95\u81f3\u9876\u5806\u5316\u201d\u76f8\u53cd\uff0c\u6211\u4eec\u5c06\u6839\u8282\u70b9\u7684\u503c\u4e0e\u5176\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u503c\u8fdb\u884c\u6bd4\u8f83\uff0c\u5c06\u6700\u5927\u7684\u5b50\u8282\u70b9\u4e0e\u6839\u8282\u70b9\u4ea4\u6362\u3002\u7136\u540e\u5faa\u73af\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\u6216\u9047\u5230\u65e0\u987b\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 8-4 \u00a0 \u5806\u9876\u5143\u7d20\u51fa\u5806\u6b65\u9aa4 </p> <p>\u4e0e\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u76f8\u4f3c\uff0c\u5806\u9876\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4e3a \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig my_heap.py<pre><code>def pop(self) -&gt; int:\n    \"\"\"\u5143\u7d20\u51fa\u5806\"\"\"\n    # \u5224\u7a7a\u5904\u7406\n    if self.is_empty():\n        raise IndexError(\"\u5806\u4e3a\u7a7a\")\n    # \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    self.swap(0, self.size() - 1)\n    # \u5220\u9664\u8282\u70b9\n    val = self.max_heap.pop()\n    # \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    self.sift_down(0)\n    # \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val\n\ndef sift_down(self, i: int):\n    \"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\n    while True:\n        # \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        l, r, ma = self.left(i), self.right(i), i\n        if l &lt; self.size() and self.max_heap[l] &gt; self.max_heap[ma]:\n            ma = l\n        if r &lt; self.size() and self.max_heap[r] &gt; self.max_heap[ma]:\n            ma = r\n        # \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i:\n            break\n        # \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, ma)\n        # \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nvoid pop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty()) {\n        throw out_of_range(\"\u5806\u4e3a\u7a7a\");\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(maxHeap[0], maxHeap[size() - 1]);\n    // \u5220\u9664\u8282\u70b9\n    maxHeap.pop_back();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(0);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        swap(maxHeap[i], maxHeap[ma]);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty())\n        throw new IndexOutOfBoundsException();\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(0, size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    int val = maxHeap.remove(size() - 1);\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap.get(l) &gt; maxHeap.get(ma))\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap.get(r) &gt; maxHeap.get(ma))\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint Pop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (IsEmpty())\n        throw new IndexOutOfRangeException();\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    Swap(0, Size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    int val = maxHeap.Last();\n    maxHeap.RemoveAt(Size() - 1);\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    SiftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid SiftDown(int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = Left(i), r = Right(i), ma = i;\n        if (l &lt; Size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; Size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // \u82e5\u201c\u8282\u70b9 i \u6700\u5927\u201d\u6216\u201c\u8d8a\u8fc7\u53f6\u8282\u70b9\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\n        if (ma == i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        Swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfunc (h *maxHeap) pop() any {\n    // \u5224\u7a7a\u5904\u7406\n    if h.isEmpty() {\n        fmt.Println(\"error\")\n        return nil\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    h.swap(0, h.size()-1)\n    // \u5220\u9664\u8282\u70b9\n    val := h.data[len(h.data)-1]\n    h.data = h.data[:len(h.data)-1]\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    h.siftDown(0)\n\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc (h *maxHeap) siftDown(i int) {\n    for true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\n        l, r, max := h.left(i), h.right(i), i\n        if l &lt; h.size() &amp;&amp; h.data[l].(int) &gt; h.data[max].(int) {\n            max = l\n        }\n        if r &lt; h.size() &amp;&amp; h.data[r].(int) &gt; h.data[max].(int) {\n            max = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if max == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        h.swap(i, max)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = max\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfunc pop() -&gt; Int {\n    // \u5224\u7a7a\u5904\u7406\n    if isEmpty() {\n        fatalError(\"\u5806\u4e3a\u7a7a\")\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(i: 0, j: size() - 1)\n    // \u5220\u9664\u8282\u70b9\n    let val = maxHeap.remove(at: size() - 1)\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(i: 0)\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(i: Int) {\n    var i = i\n    while true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = left(i: i)\n        let r = right(i: i)\n        var ma = i\n        if l &lt; size(), maxHeap[l] &gt; maxHeap[ma] {\n            ma = l\n        }\n        if r &lt; size(), maxHeap[r] &gt; maxHeap[ma] {\n            ma = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i: i, j: ma)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u51fa\u5806 */\npop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (this.isEmpty()) throw new Error('\u5806\u4e3a\u7a7a');\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    this.#swap(0, this.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    const val = this.#maxHeap.pop();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    this.#siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n#siftDown(i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        const l = this.#left(i),\n            r = this.#right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.#maxHeap[l] &gt; this.#maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.#maxHeap[r] &gt; this.#maxHeap[ma]) ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.#swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u51fa\u5806 */\npop(): number {\n    // \u5224\u7a7a\u5904\u7406\n    if (this.isEmpty()) throw new RangeError('Heap is empty.');\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    this.swap(0, this.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    const val = this.maxHeap.pop();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    this.siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nsiftDown(i: number): void {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        const l = this.left(i),\n            r = this.right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.maxHeap[l] &gt; this.maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.maxHeap[r] &gt; this.maxHeap[ma]) ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n  // \u5224\u7a7a\u5904\u7406\n  if (isEmpty()) throw Exception('\u5806\u4e3a\u7a7a');\n  // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n  _swap(0, size() - 1);\n  // \u5220\u9664\u8282\u70b9\n  int val = _maxHeap.removeLast();\n  // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n  siftDown(0);\n  // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n  return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n  while (true) {\n    // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n    int l = _left(i);\n    int r = _right(i);\n    int ma = i;\n    if (l &lt; size() &amp;&amp; _maxHeap[l] &gt; _maxHeap[ma]) ma = l;\n    if (r &lt; size() &amp;&amp; _maxHeap[r] &gt; _maxHeap[ma]) ma = r;\n    // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n    if (ma == i) break;\n    // \u4ea4\u6362\u4e24\u8282\u70b9\n    _swap(i, ma);\n    // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n    i = ma;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfn pop(&amp;mut self) -&gt; i32 {\n    // \u5224\u7a7a\u5904\u7406\n    if self.is_empty() {\n        panic!(\"index out of bounds\");\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    self.swap(0, self.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    let val = self.max_heap.remove(self.size() - 1);\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    self.sift_down(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfn sift_down(&amp;mut self, mut i: usize) {\n    loop {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let (l, r, mut ma) = (Self::left(i), Self::right(i), i);\n        if l &lt; self.size() &amp;&amp; self.max_heap[l] &gt; self.max_heap[ma] {\n            ma = l;\n        }\n        if r &lt; self.size() &amp;&amp; self.max_heap[r] &gt; self.max_heap[ma] {\n            ma = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop(MaxHeap *maxHeap) {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty(maxHeap)) {\n        printf(\"heap is empty!\");\n        return INT_MAX;\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(maxHeap, 0, size(maxHeap) - 1);\n    // \u5220\u9664\u8282\u70b9\n    int val = maxHeap-&gt;data[maxHeap-&gt;size - 1];\n    maxHeap-&gt;size--;\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(maxHeap, 0);\n\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\n        int l = left(maxHeap, i);\n        int r = right(maxHeap, i);\n        int max = i;\n        if (l &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[l] &gt; maxHeap-&gt;data[max]) {\n            max = l;\n        }\n        if (r &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[r] &gt; maxHeap-&gt;data[max]) {\n            max = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (max == i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(maxHeap, i, max);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = max;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfun pop(): Int {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty()) throw IndexOutOfBoundsException()\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(0, size() - 1)\n    // \u5220\u9664\u8282\u70b9\n    val value = maxHeap.removeAt(size() - 1)\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(0)\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return value\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfun siftDown(it: Int) {\n    // Kotlin\u7684\u51fd\u6570\u53c2\u6570\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u521b\u5efa\u4e34\u65f6\u53d8\u91cf\n    var i = it\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        val l = left(i)\n        val r = right(i)\n        var ma = i\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) break\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, ma)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n</code></pre> my_heap.rb<pre><code>[class]{MaxHeap}-[func]{pop}\n\n[class]{MaxHeap}-[func]{sift_down}\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u51fa\u5806\nfn pop(self: *Self) !T {\n    // \u5224\u65ad\u5904\u7406\n    if (self.isEmpty()) unreachable;\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    try self.swap(0, self.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    var val = self.max_heap.?.pop();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    try self.siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n} \n\n// \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\nfn siftDown(self: *Self, i_: usize) !void {\n    var i = i_;\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        var l = left(i);\n        var r = right(i);\n        var ma = i;\n        if (l &lt; self.size() and self.max_heap.?.items[l] &gt; self.max_heap.?.items[ma]) ma = l;\n        if (r &lt; self.size() and self.max_heap.?.items[r] &gt; self.max_heap.?.items[ma]) ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        try self.swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_heap/heap/#813","title":"8.1.3 \u00a0 \u5806\u7684\u5e38\u89c1\u5e94\u7528","text":"<ul> <li>\u4f18\u5148\u961f\u5217\uff1a\u5806\u901a\u5e38\u4f5c\u4e3a\u5b9e\u73b0\u4f18\u5148\u961f\u5217\u7684\u9996\u9009\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \uff0c\u800c\u5efa\u961f\u64cd\u4f5c\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e9b\u64cd\u4f5c\u90fd\u975e\u5e38\u9ad8\u6548\u3002</li> <li>\u5806\u6392\u5e8f\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u5b83\u4eec\u5efa\u7acb\u4e00\u4e2a\u5806\uff0c\u7136\u540e\u4e0d\u65ad\u5730\u6267\u884c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\uff0c\u4ece\u800c\u5f97\u5230\u6709\u5e8f\u6570\u636e\u3002\u7136\u800c\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u4f7f\u7528\u4e00\u79cd\u66f4\u4f18\u96c5\u7684\u65b9\u5f0f\u5b9e\u73b0\u5806\u6392\u5e8f\uff0c\u8be6\u89c1\u201c\u5806\u6392\u5e8f\u201d\u7ae0\u8282\u3002</li> <li>\u83b7\u53d6\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff1a\u8fd9\u662f\u4e00\u4e2a\u7ecf\u5178\u7684\u7b97\u6cd5\u95ee\u9898\uff0c\u540c\u65f6\u4e5f\u662f\u4e00\u79cd\u5178\u578b\u5e94\u7528\uff0c\u4f8b\u5982\u9009\u62e9\u70ed\u5ea6\u524d 10 \u7684\u65b0\u95fb\u4f5c\u4e3a\u5fae\u535a\u70ed\u641c\uff0c\u9009\u53d6\u9500\u91cf\u524d 10 \u7684\u5546\u54c1\u7b49\u3002</li> </ul>"},{"location":"chapter_heap/summary/","title":"8.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_heap/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u5806\u662f\u4e00\u68f5\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u6839\u636e\u6210\u7acb\u6761\u4ef6\u53ef\u5206\u4e3a\u5927\u9876\u5806\u548c\u5c0f\u9876\u5806\u3002\u5927\uff08\u5c0f\uff09\u9876\u5806\u7684\u5806\u9876\u5143\u7d20\u662f\u6700\u5927\uff08\u5c0f\uff09\u7684\u3002</li> <li>\u4f18\u5148\u961f\u5217\u7684\u5b9a\u4e49\u662f\u5177\u6709\u51fa\u961f\u4f18\u5148\u7ea7\u7684\u961f\u5217\uff0c\u901a\u5e38\u4f7f\u7528\u5806\u6765\u5b9e\u73b0\u3002</li> <li>\u5806\u7684\u5e38\u7528\u64cd\u4f5c\u53ca\u5176\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5305\u62ec\uff1a\u5143\u7d20\u5165\u5806 \\(O(\\log n)\\)\u3001\u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) \u548c\u8bbf\u95ee\u5806\u9876\u5143\u7d20 \\(O(1)\\) \u7b49\u3002</li> <li>\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u8868\u793a\uff0c\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002</li> <li>\u5806\u5316\u64cd\u4f5c\u7528\u4e8e\u7ef4\u62a4\u5806\u7684\u6027\u8d28\uff0c\u5728\u5165\u5806\u548c\u51fa\u5806\u64cd\u4f5c\u4e2d\u90fd\u4f1a\u7528\u5230\u3002</li> <li>\u8f93\u5165 \\(n\\) \u4e2a\u5143\u7d20\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</li> <li>Top-k \u662f\u4e00\u4e2a\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\uff0c\u53ef\u4ee5\u4f7f\u7528\u5806\u6570\u636e\u7ed3\u6784\u9ad8\u6548\u89e3\u51b3\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002</li> </ul>"},{"location":"chapter_heap/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u6570\u636e\u7ed3\u6784\u7684\u201c\u5806\u201d\u4e0e\u5185\u5b58\u7ba1\u7406\u7684\u201c\u5806\u201d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\u5417\uff1f</p> <p>\u4e24\u8005\u4e0d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\uff0c\u53ea\u662f\u78b0\u5de7\u90fd\u53eb\u201c\u5806\u201d\u3002\u8ba1\u7b97\u673a\u7cfb\u7edf\u5185\u5b58\u4e2d\u7684\u5806\u662f\u52a8\u6001\u5185\u5b58\u5206\u914d\u7684\u4e00\u90e8\u5206\uff0c\u7a0b\u5e8f\u5728\u8fd0\u884c\u65f6\u53ef\u4ee5\u4f7f\u7528\u5b83\u6765\u5b58\u50a8\u6570\u636e\u3002\u7a0b\u5e8f\u53ef\u4ee5\u8bf7\u6c42\u4e00\u5b9a\u91cf\u7684\u5806\u5185\u5b58\uff0c\u7528\u4e8e\u5b58\u50a8\u5982\u5bf9\u8c61\u548c\u6570\u7ec4\u7b49\u590d\u6742\u7ed3\u6784\u3002\u5f53\u8fd9\u4e9b\u6570\u636e\u4e0d\u518d\u9700\u8981\u65f6\uff0c\u7a0b\u5e8f\u9700\u8981\u91ca\u653e\u8fd9\u4e9b\u5185\u5b58\uff0c\u4ee5\u9632\u6b62\u5185\u5b58\u6cc4\u6f0f\u3002\u76f8\u8f83\u4e8e\u6808\u5185\u5b58\uff0c\u5806\u5185\u5b58\u7684\u7ba1\u7406\u548c\u4f7f\u7528\u9700\u8981\u66f4\u8c28\u614e\uff0c\u4f7f\u7528\u4e0d\u5f53\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5185\u5b58\u6cc4\u6f0f\u548c\u91ce\u6307\u9488\u7b49\u95ee\u9898\u3002</p>"},{"location":"chapter_heap/top_k/","title":"8.3 \u00a0 Top-k \u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u65e0\u5e8f\u6570\u7ec4 <code>nums</code> \uff0c\u8bf7\u8fd4\u56de\u6570\u7ec4\u4e2d\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u3002</p> <p>\u5bf9\u4e8e\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u5148\u4ecb\u7ecd\u4e24\u79cd\u601d\u8def\u6bd4\u8f83\u76f4\u63a5\u7684\u89e3\u6cd5\uff0c\u518d\u4ecb\u7ecd\u6548\u7387\u66f4\u9ad8\u7684\u5806\u89e3\u6cd5\u3002</p>"},{"location":"chapter_heap/top_k/#831","title":"8.3.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u904d\u5386\u9009\u62e9","text":"<p>\u6211\u4eec\u53ef\u4ee5\u8fdb\u884c\u56fe 8-6 \u6240\u793a\u7684 \\(k\\) \u8f6e\u904d\u5386\uff0c\u5206\u522b\u5728\u6bcf\u8f6e\u4e2d\u63d0\u53d6\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(\\dots\\)\u3001\\(k\\) \u5927\u7684\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nk)\\) \u3002</p> <p>\u6b64\u65b9\u6cd5\u53ea\u9002\u7528\u4e8e \\(k \\ll n\\) \u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u5f53 \\(k\\) \u4e0e \\(n\\) \u6bd4\u8f83\u63a5\u8fd1\u65f6\uff0c\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411\u4e8e \\(O(n^2)\\) \uff0c\u975e\u5e38\u8017\u65f6\u3002</p> <p></p> <p> \u56fe 8-6 \u00a0 \u904d\u5386\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 </p> <p>Tip</p> <p>\u5f53 \\(k = n\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5b8c\u6574\u7684\u6709\u5e8f\u5e8f\u5217\uff0c\u6b64\u65f6\u7b49\u4ef7\u4e8e\u201c\u9009\u62e9\u6392\u5e8f\u201d\u7b97\u6cd5\u3002</p>"},{"location":"chapter_heap/top_k/#832","title":"8.3.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u6392\u5e8f","text":"<p>\u5982\u56fe 8-7 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u5bf9\u6570\u7ec4 <code>nums</code> \u8fdb\u884c\u6392\u5e8f\uff0c\u518d\u8fd4\u56de\u6700\u53f3\u8fb9\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p> <p>\u663e\u7136\uff0c\u8be5\u65b9\u6cd5\u201c\u8d85\u989d\u201d\u5b8c\u6210\u4efb\u52a1\u4e86\uff0c\u56e0\u4e3a\u6211\u4eec\u53ea\u9700\u627e\u51fa\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u5373\u53ef\uff0c\u800c\u4e0d\u9700\u8981\u6392\u5e8f\u5176\u4ed6\u5143\u7d20\u3002</p> <p></p> <p> \u56fe 8-7 \u00a0 \u6392\u5e8f\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 </p>"},{"location":"chapter_heap/top_k/#833","title":"8.3.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u5806","text":"<p>\u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u5806\u66f4\u52a0\u9ad8\u6548\u5730\u89e3\u51b3 Top-k \u95ee\u9898\uff0c\u6d41\u7a0b\u5982\u56fe 8-8 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5176\u5806\u9876\u5143\u7d20\u6700\u5c0f\u3002</li> <li>\u5148\u5c06\u6570\u7ec4\u7684\u524d \\(k\\) \u4e2a\u5143\u7d20\u4f9d\u6b21\u5165\u5806\u3002</li> <li>\u4ece\u7b2c \\(k + 1\\) \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\uff0c\u5e76\u5c06\u5f53\u524d\u5143\u7d20\u5165\u5806\u3002</li> <li>\u904d\u5386\u5b8c\u6210\u540e\uff0c\u5806\u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 8-8 \u00a0 \u57fa\u4e8e\u5806\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 </p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig top_k.py<pre><code>def top_k_heap(nums: list[int], k: int) -&gt; list[int]:\n    \"\"\"\u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\"\"\"\n    # \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    heap = []\n    # \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for i in range(k):\n        heapq.heappush(heap, nums[i])\n    # \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for i in range(k, len(nums)):\n        # \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if nums[i] &gt; heap[0]:\n            heapq.heappop(heap)\n            heapq.heappush(heap, nums[i])\n    return heap\n</code></pre> top_k.cpp<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; topKHeap(vector&lt;int&gt; &amp;nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    priority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; heap;\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        heap.push(nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; nums.size(); i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.top()) {\n            heap.pop();\n            heap.push(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.java<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nQueue&lt;Integer&gt; topKHeap(int[] nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    Queue&lt;Integer&gt; heap = new PriorityQueue&lt;Integer&gt;();\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        heap.offer(nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; nums.length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll();\n            heap.offer(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.cs<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nPriorityQueue&lt;int, int&gt; TopKHeap(int[] nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    PriorityQueue&lt;int, int&gt; heap = new();\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        heap.Enqueue(nums[i], nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; nums.Length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.Peek()) {\n            heap.Dequeue();\n            heap.Enqueue(nums[i], nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.go<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums []int, k int) *minHeap {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    h := &amp;minHeap{}\n    heap.Init(h)\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for i := 0; i &lt; k; i++ {\n        heap.Push(h, nums[i])\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for i := k; i &lt; len(nums); i++ {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if nums[i] &gt; h.Top().(int) {\n            heap.Pop(h)\n            heap.Push(h, nums[i])\n        }\n    }\n    return h\n}\n</code></pre> top_k.swift<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums: [Int], k: Int) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5e76\u5c06\u524d k \u4e2a\u5143\u7d20\u5efa\u5806\n    var heap = Heap(nums.prefix(k))\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for i in nums.indices.dropFirst(k) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if nums[i] &gt; heap.min()! {\n            _ = heap.removeMin()\n            heap.insert(nums[i])\n        }\n    }\n    return heap.unordered\n}\n</code></pre> top_k.js<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap, val) {\n    // \u5143\u7d20\u53d6\u53cd\n    maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.pop();\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return maxHeap.getMaxHeap().map((num) =&gt; -num);\n}\n\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums, k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n    const maxHeap = new MaxHeap([]);\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (let i = k; i &lt; nums.length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.ts<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap: MaxHeap, val: number): void {\n    // \u5143\u7d20\u53d6\u53cd\n    maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap: MaxHeap): number {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.pop();\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap: MaxHeap): number {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap: MaxHeap): number[] {\n    // \u5143\u7d20\u53d6\u53cd\n    return maxHeap.getMaxHeap().map((num: number) =&gt; -num);\n}\n\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums: number[], k: number): number[] {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n    const maxHeap = new MaxHeap([]);\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (let i = k; i &lt; nums.length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.dart<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nMinHeap topKHeap(List&lt;int&gt; nums, int k) {\n  // \u521d\u59cb\u5316\u5c0f\u9876\u5806\uff0c\u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n  MinHeap heap = MinHeap(nums.sublist(0, k));\n  // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n  for (int i = k; i &lt; nums.length; i++) {\n    // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n    if (nums[i] &gt; heap.peek()) {\n      heap.pop();\n      heap.push(nums[i]);\n    }\n  }\n  return heap;\n}\n</code></pre> top_k.rs<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfn top_k_heap(nums: Vec&lt;i32&gt;, k: usize) -&gt; BinaryHeap&lt;Reverse&lt;i32&gt;&gt; {\n    // BinaryHeap \u662f\u5927\u9876\u5806\uff0c\u4f7f\u7528 Reverse \u5c06\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u5b9e\u73b0\u5c0f\u9876\u5806\n    let mut heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for &amp;num in nums.iter().take(k) {\n        heap.push(Reverse(num));\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for &amp;num in nums.iter().skip(k) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if num &gt; heap.peek().unwrap().0 {\n            heap.pop();\n            heap.push(Reverse(num));\n        }\n    }\n    heap\n}\n</code></pre> top_k.c<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid pushMinHeap(MaxHeap *maxHeap, int val) {\n    // \u5143\u7d20\u53d6\u53cd\n    push(maxHeap, -val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nint popMinHeap(MaxHeap *maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -pop(maxHeap);\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peekMinHeap(MaxHeap *maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -peek(maxHeap);\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n// \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\u7684\u51fd\u6570\nint *topKHeap(int *nums, int sizeNums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n    int *empty = (int *)malloc(0);\n    MaxHeap *maxHeap = newMaxHeap(empty, 0);\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; sizeNums; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    int *res = getMinHeap(maxHeap);\n    // \u91ca\u653e\u5185\u5b58\n    delMaxHeap(maxHeap);\n    return res;\n}\n</code></pre> top_k.kt<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfun topKHeap(nums: IntArray, k: Int): Queue&lt;Int&gt; {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    val heap = PriorityQueue&lt;Int&gt;()\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (i in 0..&lt;k) {\n        heap.offer(nums[i])\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (i in k..&lt;nums.size) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll()\n            heap.offer(nums[i])\n        }\n    }\n    return heap\n}\n</code></pre> top_k.rb<pre><code>[class]{}-[func]{top_k_heap}\n</code></pre> top_k.zig<pre><code>[class]{}-[func]{topKHeap}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u603b\u5171\u6267\u884c\u4e86 \\(n\\) \u8f6e\u5165\u5806\u548c\u51fa\u5806\uff0c\u5806\u7684\u6700\u5927\u957f\u5ea6\u4e3a \\(k\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002\u8be5\u65b9\u6cd5\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u5f53 \\(k\\) \u8f83\u5c0f\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \uff1b\u5f53 \\(k\\) \u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(O(n \\log n)\\) \u3002</p> <p>\u53e6\u5916\uff0c\u8be5\u65b9\u6cd5\u9002\u7528\u4e8e\u52a8\u6001\u6570\u636e\u6d41\u7684\u4f7f\u7528\u573a\u666f\u3002\u5728\u4e0d\u65ad\u52a0\u5165\u6570\u636e\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u6301\u7eed\u7ef4\u62a4\u5806\u5185\u7684\u5143\u7d20\uff0c\u4ece\u800c\u5b9e\u73b0\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u7684\u52a8\u6001\u66f4\u65b0\u3002</p>"},{"location":"chapter_hello_algo/","title":"\u5e8f","text":"<p>\u51e0\u5e74\u524d\uff0c\u6211\u5728\u529b\u6263\u4e0a\u5206\u4eab\u4e86\u201c\u5251\u6307 Offer\u201d\u7cfb\u5217\u9898\u89e3\uff0c\u53d7\u5230\u4e86\u8bb8\u591a\u8bfb\u8005\u7684\u9f13\u52b1\u548c\u652f\u6301\u3002\u5728\u4e0e\u8bfb\u8005\u4ea4\u6d41\u671f\u95f4\uff0c\u6211\u6700\u5e38\u88ab\u95ee\u7684\u4e00\u4e2a\u95ee\u9898\u662f\u201c\u5982\u4f55\u5165\u95e8\u7b97\u6cd5\u201d\u3002\u9010\u6e10\u5730\uff0c\u6211\u5bf9\u8fd9\u4e2a\u95ee\u9898\u4ea7\u751f\u4e86\u6d53\u539a\u7684\u5174\u8da3\u3002</p> <p>\u4e24\u773c\u4e00\u62b9\u9ed1\u5730\u5237\u9898\u4f3c\u4e4e\u662f\u6700\u53d7\u6b22\u8fce\u7684\u65b9\u6cd5\uff0c\u7b80\u5355\u3001\u76f4\u63a5\u4e14\u6709\u6548\u3002\u7136\u800c\u5237\u9898\u5c31\u5982\u540c\u73a9\u201c\u626b\u96f7\u201d\u6e38\u620f\uff0c\u81ea\u5b66\u80fd\u529b\u5f3a\u7684\u4eba\u80fd\u591f\u987a\u5229\u5c06\u5730\u96f7\u9010\u4e2a\u6392\u6389\uff0c\u800c\u57fa\u7840\u4e0d\u8db3\u7684\u4eba\u5f88\u53ef\u80fd\u88ab\u70b8\u5f97\u6ee1\u5934\u662f\u5305\uff0c\u5e76\u5728\u632b\u6298\u4e2d\u6b65\u6b65\u9000\u7f29\u3002\u901a\u8bfb\u6559\u6750\u4e5f\u662f\u4e00\u79cd\u5e38\u89c1\u505a\u6cd5\uff0c\u4f46\u5bf9\u4e8e\u9762\u5411\u6c42\u804c\u7684\u4eba\u6765\u8bf4\uff0c\u6bd5\u4e1a\u8bba\u6587\u3001\u6295\u9012\u7b80\u5386\u3001\u51c6\u5907\u7b14\u8bd5\u548c\u9762\u8bd5\u5df2\u7ecf\u6d88\u8017\u4e86\u5927\u90e8\u5206\u7cbe\u529b\uff0c\u5543\u539a\u91cd\u7684\u4e66\u5f80\u5f80\u53d8\u6210\u4e86\u4e00\u9879\u8270\u5de8\u7684\u6311\u6218\u3002</p> <p>\u5982\u679c\u4f60\u4e5f\u9762\u4e34\u7c7b\u4f3c\u7684\u56f0\u6270\uff0c\u90a3\u4e48\u5f88\u5e78\u8fd0\u8fd9\u672c\u4e66\u201c\u627e\u201d\u5230\u4e86\u4f60\u3002\u672c\u4e66\u662f\u6211\u5bf9\u8fd9\u4e2a\u95ee\u9898\u7ed9\u51fa\u7684\u7b54\u6848\uff0c\u5373\u4f7f\u4e0d\u662f\u6700\u4f18\u89e3\uff0c\u4e5f\u81f3\u5c11\u662f\u4e00\u6b21\u79ef\u6781\u7684\u5c1d\u8bd5\u3002\u672c\u4e66\u867d\u7136\u4e0d\u8db3\u4ee5\u8ba9\u4f60\u76f4\u63a5\u62ff\u5230 Offer\uff0c\u4f46\u4f1a\u5f15\u5bfc\u4f60\u63a2\u7d22\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u201c\u77e5\u8bc6\u5730\u56fe\u201d\uff0c\u5e26\u4f60\u4e86\u89e3\u4e0d\u540c\u201c\u5730\u96f7\u201d\u7684\u5f62\u72b6\u3001\u5927\u5c0f\u548c\u5206\u5e03\u4f4d\u7f6e\uff0c\u8ba9\u4f60\u638c\u63e1\u5404\u79cd\u201c\u6392\u96f7\u65b9\u6cd5\u201d\u3002\u6709\u4e86\u8fd9\u4e9b\u672c\u9886\uff0c\u76f8\u4fe1\u4f60\u53ef\u4ee5\u66f4\u52a0\u81ea\u5982\u5730\u5237\u9898\u548c\u9605\u8bfb\u6587\u732e\uff0c\u9010\u6b65\u6784\u5efa\u8d77\u5b8c\u6574\u7684\u77e5\u8bc6\u4f53\u7cfb\u3002</p> <p>\u6211\u6df1\u6df1\u8d5e\u540c\u8d39\u66fc\u6559\u6388\u6240\u8a00\uff1a\u201cKnowledge isn't free. You have to pay attention.\u201d\u4ece\u8fd9\u4e2a\u610f\u4e49\u4e0a\u770b\uff0c\u8fd9\u672c\u4e66\u5e76\u975e\u5b8c\u5168\u201c\u514d\u8d39\u201d\u3002\u4e3a\u4e86\u4e0d\u8f9c\u8d1f\u4f60\u4e3a\u672c\u4e66\u6240\u4ed8\u51fa\u7684\u5b9d\u8d35\u201c\u6ce8\u610f\u529b\u201d\uff0c\u6211\u4f1a\u7aed\u5c3d\u6240\u80fd\uff0c\u6295\u5165\u6700\u5927\u7684\u201c\u6ce8\u610f\u529b\u201d\u6765\u5b8c\u6210\u672c\u4e66\u7684\u521b\u4f5c\u3002</p> <p>\u672c\u4eba\u81ea\u77e5\u5b66\u758f\u624d\u6d45\uff0c\u4e66\u4e2d\u5185\u5bb9\u867d\u7136\u5df2\u7ecf\u8fc7\u4e00\u6bb5\u65f6\u95f4\u7684\u6253\u78e8\uff0c\u4f46\u4e00\u5b9a\u4ecd\u6709\u8bb8\u591a\u9519\u8bef\uff0c\u6073\u8bf7\u5404\u4f4d\u8001\u5e08\u548c\u540c\u5b66\u6279\u8bc4\u6307\u6b63\u3002</p> <p></p> Hello\uff0c\u7b97\u6cd5\uff01 <p>\u8ba1\u7b97\u673a\u7684\u51fa\u73b0\u7ed9\u4e16\u754c\u5e26\u6765\u4e86\u5de8\u5927\u53d8\u9769\uff0c\u5b83\u51ed\u501f\u9ad8\u901f\u7684\u8ba1\u7b97\u80fd\u529b\u548c\u51fa\u8272\u7684\u53ef\u7f16\u7a0b\u6027\uff0c\u6210\u4e3a\u4e86\u6267\u884c\u7b97\u6cd5\u4e0e\u5904\u7406\u6570\u636e\u7684\u7406\u60f3\u5a92\u4ecb\u3002\u65e0\u8bba\u662f\u7535\u5b50\u6e38\u620f\u7684\u903c\u771f\u753b\u9762\u3001\u81ea\u52a8\u9a7e\u9a76\u7684\u667a\u80fd\u51b3\u7b56\uff0c\u8fd8\u662f AlphaGo \u7684\u7cbe\u5f69\u68cb\u5c40\u3001ChatGPT \u7684\u81ea\u7136\u4ea4\u4e92\uff0c\u8fd9\u4e9b\u5e94\u7528\u90fd\u662f\u7b97\u6cd5\u5728\u8ba1\u7b97\u673a\u4e0a\u7684\u7cbe\u5999\u6f14\u7ece\u3002</p> <p>\u4e8b\u5b9e\u4e0a\uff0c\u5728\u8ba1\u7b97\u673a\u95ee\u4e16\u4e4b\u524d\uff0c\u7b97\u6cd5\u548c\u6570\u636e\u7ed3\u6784\u5c31\u5df2\u7ecf\u5b58\u5728\u4e8e\u4e16\u754c\u7684\u5404\u4e2a\u89d2\u843d\u3002\u65e9\u671f\u7684\u7b97\u6cd5\u76f8\u5bf9\u7b80\u5355\uff0c\u4f8b\u5982\u53e4\u4ee3\u7684\u8ba1\u6570\u65b9\u6cd5\u548c\u5de5\u5177\u5236\u4f5c\u6b65\u9aa4\u7b49\u3002\u968f\u7740\u6587\u660e\u7684\u8fdb\u6b65\uff0c\u7b97\u6cd5\u9010\u6e10\u53d8\u5f97\u66f4\u52a0\u7cbe\u7ec6\u548c\u590d\u6742\u3002\u4ece\u5de7\u593a\u5929\u5de5\u7684\u5320\u4eba\u6280\u827a\u3001\u5230\u89e3\u653e\u751f\u4ea7\u529b\u7684\u5de5\u4e1a\u4ea7\u54c1\u3001\u518d\u5230\u5b87\u5b99\u8fd0\u884c\u7684\u79d1\u5b66\u89c4\u5f8b\uff0c\u51e0\u4e4e\u6bcf\u4e00\u4ef6\u5e73\u51e1\u6216\u4ee4\u4eba\u60ca\u53f9\u7684\u4e8b\u7269\u80cc\u540e\uff0c\u90fd\u9690\u85cf\u7740\u7cbe\u5999\u7684\u7b97\u6cd5\u601d\u60f3\u3002</p> <p>\u540c\u6837\uff0c\u6570\u636e\u7ed3\u6784\u65e0\u5904\u4e0d\u5728\uff1a\u5927\u5230\u793e\u4f1a\u7f51\u7edc\uff0c\u5c0f\u5230\u5730\u94c1\u7ebf\u8def\uff0c\u8bb8\u591a\u7cfb\u7edf\u90fd\u53ef\u4ee5\u5efa\u6a21\u4e3a\u201c\u56fe\u201d\uff1b\u5927\u5230\u4e00\u4e2a\u56fd\u5bb6\uff0c\u5c0f\u5230\u4e00\u4e2a\u5bb6\u5ead\uff0c\u793e\u4f1a\u7684\u4e3b\u8981\u7ec4\u7ec7\u5f62\u5f0f\u5448\u73b0\u51fa\u201c\u6811\u201d\u7684\u7279\u5f81\uff1b\u51ac\u5929\u7684\u8863\u670d\u5c31\u50cf\u201c\u6808\u201d\uff0c\u6700\u5148\u7a7f\u4e0a\u7684\u6700\u540e\u624d\u80fd\u8131\u4e0b\uff1b\u7fbd\u6bdb\u7403\u7b52\u5219\u5982\u540c\u201c\u961f\u5217\u201d\uff0c\u4e00\u7aef\u653e\u5165\u3001\u53e6\u4e00\u7aef\u53d6\u51fa\uff1b\u5b57\u5178\u5c31\u50cf\u4e00\u4e2a\u201c\u54c8\u5e0c\u8868\u201d\uff0c\u80fd\u591f\u5feb\u901f\u67e5\u627e\u76ee\u6807\u8bcd\u6761\u3002</p> <p>\u672c\u4e66\u65e8\u5728\u901a\u8fc7\u6e05\u6670\u6613\u61c2\u7684\u52a8\u753b\u63d2\u56fe\u548c\u53ef\u8fd0\u884c\u7684\u4ee3\u7801\u793a\u4f8b\uff0c\u4f7f\u8bfb\u8005\u7406\u89e3\u7b97\u6cd5\u548c\u6570\u636e\u7ed3\u6784\u7684\u6838\u5fc3\u6982\u5ff5\uff0c\u5e76\u80fd\u591f\u901a\u8fc7\u7f16\u7a0b\u6765\u5b9e\u73b0\u5b83\u4eec\u3002\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u672c\u4e66\u81f4\u529b\u4e8e\u63ed\u793a\u7b97\u6cd5\u5728\u590d\u6742\u4e16\u754c\u4e2d\u7684\u751f\u52a8\u4f53\u73b0\uff0c\u5c55\u73b0\u7b97\u6cd5\u4e4b\u7f8e\u3002\u5e0c\u671b\u672c\u4e66\u80fd\u591f\u5e2e\u52a9\u5230\u4f60\uff01</p>"},{"location":"chapter_introduction/","title":"\u7b2c 1 \u7ae0 \u00a0 \u521d\u8bc6\u7b97\u6cd5","text":"<p>Abstract</p> <p>\u4e00\u4f4d\u5c11\u5973\u7fe9\u7fe9\u8d77\u821e\uff0c\u4e0e\u6570\u636e\u4ea4\u7ec7\u5728\u4e00\u8d77\uff0c\u88d9\u6446\u4e0a\u98d8\u626c\u7740\u7b97\u6cd5\u7684\u65cb\u5f8b\u3002</p> <p>\u5979\u9080\u8bf7\u4f60\u5171\u821e\uff0c\u8bf7\u7d27\u8ddf\u5979\u7684\u6b65\u4f10\uff0c\u8e0f\u5165\u5145\u6ee1\u903b\u8f91\u4e0e\u7f8e\u611f\u7684\u7b97\u6cd5\u4e16\u754c\u3002</p>"},{"location":"chapter_introduction/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>1.1 \u00a0 \u7b97\u6cd5\u65e0\u5904\u4e0d\u5728</li> <li>1.2 \u00a0 \u7b97\u6cd5\u662f\u4ec0\u4e48</li> <li>1.3 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_introduction/algorithms_are_everywhere/","title":"1.1 \u00a0 \u7b97\u6cd5\u65e0\u5904\u4e0d\u5728","text":"<p>\u5f53\u6211\u4eec\u542c\u5230\u201c\u7b97\u6cd5\u201d\u8fd9\u4e2a\u8bcd\u65f6\uff0c\u5f88\u81ea\u7136\u5730\u4f1a\u60f3\u5230\u6570\u5b66\u3002\u7136\u800c\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7b97\u6cd5\u5e76\u4e0d\u6d89\u53ca\u590d\u6742\u6570\u5b66\uff0c\u800c\u662f\u66f4\u591a\u5730\u4f9d\u8d56\u57fa\u672c\u903b\u8f91\uff0c\u8fd9\u4e9b\u903b\u8f91\u5728\u6211\u4eec\u7684\u65e5\u5e38\u751f\u6d3b\u4e2d\u5904\u5904\u53ef\u89c1\u3002</p> <p>\u5728\u6b63\u5f0f\u63a2\u8ba8\u7b97\u6cd5\u4e4b\u524d\uff0c\u6709\u4e00\u4e2a\u6709\u8da3\u7684\u4e8b\u5b9e\u503c\u5f97\u5206\u4eab\uff1a\u4f60\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u5e76\u4e60\u60ef\u5c06\u5b83\u4eec\u5e94\u7528\u5230\u65e5\u5e38\u751f\u6d3b\u4e2d\u4e86\u3002\u4e0b\u9762\u6211\u5c06\u4e3e\u51e0\u4e2a\u5177\u4f53\u7684\u4f8b\u5b50\u6765\u8bc1\u5b9e\u8fd9\u4e00\u70b9\u3002</p> <p>\u4f8b\u4e00\uff1a\u67e5\u5b57\u5178\u3002\u5728\u5b57\u5178\u91cc\uff0c\u6bcf\u4e2a\u6c49\u5b57\u90fd\u5bf9\u5e94\u4e00\u4e2a\u62fc\u97f3\uff0c\u800c\u5b57\u5178\u662f\u6309\u7167\u62fc\u97f3\u5b57\u6bcd\u987a\u5e8f\u6392\u5217\u7684\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u67e5\u627e\u4e00\u4e2a\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u5b57\uff0c\u901a\u5e38\u4f1a\u6309\u7167\u56fe 1-1 \u6240\u793a\u7684\u65b9\u5f0f\u5b9e\u73b0\u3002</p> <ol> <li>\u7ffb\u5f00\u5b57\u5178\u7ea6\u4e00\u534a\u7684\u9875\u6570\uff0c\u67e5\u770b\u8be5\u9875\u7684\u9996\u5b57\u6bcd\u662f\u4ec0\u4e48\uff0c\u5047\u8bbe\u9996\u5b57\u6bcd\u4e3a \\(m\\) \u3002</li> <li>\u7531\u4e8e\u5728\u62fc\u97f3\u5b57\u6bcd\u8868\u4e2d \\(r\\) \u4f4d\u4e8e \\(m\\) \u4e4b\u540e\uff0c\u6240\u4ee5\u6392\u9664\u5b57\u5178\u524d\u534a\u90e8\u5206\uff0c\u67e5\u627e\u8303\u56f4\u7f29\u5c0f\u5230\u540e\u534a\u90e8\u5206\u3002</li> <li>\u4e0d\u65ad\u91cd\u590d\u6b65\u9aa4 <code>1.</code> \u548c \u6b65\u9aa4 <code>2.</code> \uff0c\u76f4\u81f3\u627e\u5230\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u9875\u7801\u4e3a\u6b62\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 1-1 \u00a0 \u67e5\u5b57\u5178\u6b65\u9aa4 </p> <p>\u67e5\u5b57\u5178\u8fd9\u4e2a\u5c0f\u5b66\u751f\u5fc5\u5907\u6280\u80fd\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u8457\u540d\u7684\u201c\u4e8c\u5206\u67e5\u627e\u201d\u7b97\u6cd5\u3002\u4ece\u6570\u636e\u7ed3\u6784\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b57\u5178\u89c6\u4e3a\u4e00\u4e2a\u5df2\u6392\u5e8f\u7684\u201c\u6570\u7ec4\u201d\uff1b\u4ece\u7b97\u6cd5\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4e0a\u8ff0\u67e5\u5b57\u5178\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u770b\u4f5c\u201c\u4e8c\u5206\u67e5\u627e\u201d\u3002</p> <p>\u4f8b\u4e8c\uff1a\u6574\u7406\u6251\u514b\u3002\u6211\u4eec\u5728\u6253\u724c\u65f6\uff0c\u6bcf\u5c40\u90fd\u9700\u8981\u6574\u7406\u624b\u4e2d\u7684\u6251\u514b\u724c\uff0c\u4f7f\u5176\u4ece\u5c0f\u5230\u5927\u6392\u5217\uff0c\u5b9e\u73b0\u6d41\u7a0b\u5982\u56fe 1-2 \u6240\u793a\u3002</p> <ol> <li>\u5c06\u6251\u514b\u724c\u5212\u5206\u4e3a\u201c\u6709\u5e8f\u201d\u548c\u201c\u65e0\u5e8f\u201d\u4e24\u90e8\u5206\uff0c\u5e76\u5047\u8bbe\u521d\u59cb\u72b6\u6001\u4e0b\u6700\u5de6 1 \u5f20\u6251\u514b\u724c\u5df2\u7ecf\u6709\u5e8f\u3002</li> <li>\u5728\u65e0\u5e8f\u90e8\u5206\u62bd\u51fa\u4e00\u5f20\u6251\u514b\u724c\uff0c\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\uff1b\u5b8c\u6210\u540e\u6700\u5de6 2 \u5f20\u6251\u514b\u5df2\u7ecf\u6709\u5e8f\u3002</li> <li>\u4e0d\u65ad\u5faa\u73af\u6b65\u9aa4 <code>2.</code> \uff0c\u6bcf\u4e00\u8f6e\u5c06\u4e00\u5f20\u6251\u514b\u724c\u4ece\u65e0\u5e8f\u90e8\u5206\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\uff0c\u76f4\u81f3\u6240\u6709\u6251\u514b\u724c\u90fd\u6709\u5e8f\u3002</li> </ol> <p></p> <p> \u56fe 1-2 \u00a0 \u6251\u514b\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u4e0a\u8ff0\u6574\u7406\u6251\u514b\u724c\u7684\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u201c\u63d2\u5165\u6392\u5e8f\u201d\u7b97\u6cd5\uff0c\u5b83\u5728\u5904\u7406\u5c0f\u578b\u6570\u636e\u96c6\u65f6\u975e\u5e38\u9ad8\u6548\u3002\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u7684\u6392\u5e8f\u5e93\u51fd\u6570\u4e2d\u90fd\u6709\u63d2\u5165\u6392\u5e8f\u7684\u8eab\u5f71\u3002</p> <p>\u4f8b\u4e09\uff1a\u8d27\u5e01\u627e\u96f6\u3002\u5047\u8bbe\u6211\u4eec\u5728\u8d85\u5e02\u8d2d\u4e70\u4e86 \\(69\\) \u5143\u7684\u5546\u54c1\uff0c\u7ed9\u4e86\u6536\u94f6\u5458 \\(100\\) \u5143\uff0c\u5219\u6536\u94f6\u5458\u9700\u8981\u627e\u6211\u4eec \\(31\\) \u5143\u3002\u4ed6\u4f1a\u5f88\u81ea\u7136\u5730\u5b8c\u6210\u5982\u56fe 1-3 \u6240\u793a\u7684\u601d\u8003\u3002</p> <ol> <li>\u53ef\u9009\u9879\u662f\u6bd4 \\(31\\) \u5143\u9762\u503c\u66f4\u5c0f\u7684\u8d27\u5e01\uff0c\u5305\u62ec \\(1\\) \u5143\u3001\\(5\\) \u5143\u3001\\(10\\) \u5143\u3001\\(20\\) \u5143\u3002</li> <li>\u4ece\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(20\\) \u5143\uff0c\u5269\u4f59 \\(31 - 20 = 11\\) \u5143\u3002</li> <li>\u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(10\\) \u5143\uff0c\u5269\u4f59 \\(11 - 10 = 1\\) \u5143\u3002</li> <li>\u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(1\\) \u5143\uff0c\u5269\u4f59 \\(1 - 1 = 0\\) \u5143\u3002</li> <li>\u5b8c\u6210\u627e\u96f6\uff0c\u65b9\u6848\u4e3a \\(20 + 10 + 1 = 31\\) \u5143\u3002</li> </ol> <p></p> <p> \u56fe 1-3 \u00a0 \u8d27\u5e01\u627e\u96f6\u8fc7\u7a0b </p> <p>\u5728\u4ee5\u4e0a\u6b65\u9aa4\u4e2d\uff0c\u6211\u4eec\u6bcf\u4e00\u6b65\u90fd\u91c7\u53d6\u5f53\u524d\u770b\u6765\u6700\u597d\u7684\u9009\u62e9\uff08\u5c3d\u53ef\u80fd\u7528\u5927\u9762\u989d\u7684\u8d27\u5e01\uff09\uff0c\u6700\u7ec8\u5f97\u5230\u4e86\u53ef\u884c\u7684\u627e\u96f6\u65b9\u6848\u3002\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u201c\u8d2a\u5fc3\u201d\u7b97\u6cd5\u3002</p> <p>\u5c0f\u5230\u70f9\u996a\u4e00\u9053\u83dc\uff0c\u5927\u5230\u661f\u9645\u822a\u884c\uff0c\u51e0\u4e4e\u6240\u6709\u95ee\u9898\u7684\u89e3\u51b3\u90fd\u79bb\u4e0d\u5f00\u7b97\u6cd5\u3002\u8ba1\u7b97\u673a\u7684\u51fa\u73b0\u4f7f\u5f97\u6211\u4eec\u80fd\u591f\u901a\u8fc7\u7f16\u7a0b\u5c06\u6570\u636e\u7ed3\u6784\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\uff0c\u540c\u65f6\u7f16\u5199\u4ee3\u7801\u8c03\u7528 CPU \u548c GPU \u6267\u884c\u7b97\u6cd5\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u6211\u4eec\u5c31\u80fd\u628a\u751f\u6d3b\u4e2d\u7684\u95ee\u9898\u8f6c\u79fb\u5230\u8ba1\u7b97\u673a\u4e0a\uff0c\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u89e3\u51b3\u5404\u79cd\u590d\u6742\u95ee\u9898\u3002</p> <p>Tip</p> <p>\u5982\u679c\u4f60\u5bf9\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u3001\u6570\u7ec4\u548c\u4e8c\u5206\u67e5\u627e\u7b49\u6982\u5ff5\u4ecd\u611f\u5230\u4e00\u77e5\u534a\u89e3\uff0c\u8bf7\u7ee7\u7eed\u5f80\u4e0b\u9605\u8bfb\uff0c\u672c\u4e66\u5c06\u5f15\u5bfc\u4f60\u8fc8\u5165\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u77e5\u8bc6\u6bbf\u5802\u3002</p>"},{"location":"chapter_introduction/summary/","title":"1.3 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u7b97\u6cd5\u5728\u65e5\u5e38\u751f\u6d3b\u4e2d\u65e0\u5904\u4e0d\u5728\uff0c\u5e76\u4e0d\u662f\u9065\u4e0d\u53ef\u53ca\u7684\u9ad8\u6df1\u77e5\u8bc6\u3002\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u7528\u4ee5\u89e3\u51b3\u751f\u6d3b\u4e2d\u7684\u5927\u5c0f\u95ee\u9898\u3002</li> <li>\u67e5\u5b57\u5178\u7684\u539f\u7406\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u76f8\u4e00\u81f4\u3002\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u4f53\u73b0\u4e86\u5206\u800c\u6cbb\u4e4b\u7684\u91cd\u8981\u7b97\u6cd5\u601d\u60f3\u3002</li> <li>\u6574\u7406\u6251\u514b\u7684\u8fc7\u7a0b\u4e0e\u63d2\u5165\u6392\u5e8f\u7b97\u6cd5\u975e\u5e38\u7c7b\u4f3c\u3002\u63d2\u5165\u6392\u5e8f\u7b97\u6cd5\u9002\u5408\u6392\u5e8f\u5c0f\u578b\u6570\u636e\u96c6\u3002</li> <li>\u8d27\u5e01\u627e\u96f6\u7684\u6b65\u9aa4\u672c\u8d28\u4e0a\u662f\u8d2a\u5fc3\u7b97\u6cd5\uff0c\u6bcf\u4e00\u6b65\u90fd\u91c7\u53d6\u5f53\u524d\u770b\u6765\u6700\u597d\u7684\u9009\u62e9\u3002</li> <li>\u7b97\u6cd5\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u800c\u6570\u636e\u7ed3\u6784\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002</li> <li>\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7d27\u5bc6\u76f8\u8fde\u3002\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\uff0c\u800c\u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u4f5c\u7528\u7684\u821e\u53f0\u3002</li> <li>\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7c7b\u6bd4\u4e3a\u62fc\u88c5\u79ef\u6728\uff0c\u79ef\u6728\u4ee3\u8868\u6570\u636e\uff0c\u79ef\u6728\u7684\u5f62\u72b6\u548c\u8fde\u63a5\u65b9\u5f0f\u7b49\u4ee3\u8868\u6570\u636e\u7ed3\u6784\uff0c\u62fc\u88c5\u79ef\u6728\u7684\u6b65\u9aa4\u5219\u5bf9\u5e94\u7b97\u6cd5\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/","title":"1.2 \u00a0 \u7b97\u6cd5\u662f\u4ec0\u4e48","text":""},{"location":"chapter_introduction/what_is_dsa/#121","title":"1.2.1 \u00a0 \u7b97\u6cd5\u5b9a\u4e49","text":"<p>\u7b97\u6cd5\uff08algorithm\uff09\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u5b83\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u95ee\u9898\u662f\u660e\u786e\u7684\uff0c\u5305\u542b\u6e05\u6670\u7684\u8f93\u5165\u548c\u8f93\u51fa\u5b9a\u4e49\u3002</li> <li>\u5177\u6709\u53ef\u884c\u6027\uff0c\u80fd\u591f\u5728\u6709\u9650\u6b65\u9aa4\u3001\u65f6\u95f4\u548c\u5185\u5b58\u7a7a\u95f4\u4e0b\u5b8c\u6210\u3002</li> <li>\u5404\u6b65\u9aa4\u90fd\u6709\u786e\u5b9a\u7684\u542b\u4e49\uff0c\u5728\u76f8\u540c\u7684\u8f93\u5165\u548c\u8fd0\u884c\u6761\u4ef6\u4e0b\uff0c\u8f93\u51fa\u59cb\u7ec8\u76f8\u540c\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#122","title":"1.2.2 \u00a0 \u6570\u636e\u7ed3\u6784\u5b9a\u4e49","text":"<p>\u6570\u636e\u7ed3\u6784\uff08data structure\uff09\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\uff0c\u5177\u6709\u4ee5\u4e0b\u8bbe\u8ba1\u76ee\u6807\u3002</p> <ul> <li>\u7a7a\u95f4\u5360\u7528\u5c3d\u91cf\u5c11\uff0c\u4ee5\u8282\u7701\u8ba1\u7b97\u673a\u5185\u5b58\u3002</li> <li>\u6570\u636e\u64cd\u4f5c\u5c3d\u53ef\u80fd\u5feb\u901f\uff0c\u6db5\u76d6\u6570\u636e\u8bbf\u95ee\u3001\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66f4\u65b0\u7b49\u3002</li> <li>\u63d0\u4f9b\u7b80\u6d01\u7684\u6570\u636e\u8868\u793a\u548c\u903b\u8f91\u4fe1\u606f\uff0c\u4ee5\u4fbf\u7b97\u6cd5\u9ad8\u6548\u8fd0\u884c\u3002</li> </ul> <p>\u6570\u636e\u7ed3\u6784\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5145\u6ee1\u6743\u8861\u7684\u8fc7\u7a0b\u3002\u5982\u679c\u60f3\u5728\u67d0\u65b9\u9762\u53d6\u5f97\u63d0\u5347\uff0c\u5f80\u5f80\u9700\u8981\u5728\u53e6\u4e00\u65b9\u9762\u4f5c\u51fa\u59a5\u534f\u3002\u4e0b\u9762\u4e3e\u4e24\u4e2a\u4f8b\u5b50\u3002</p> <ul> <li>\u94fe\u8868\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5728\u6570\u636e\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u4e0a\u66f4\u52a0\u4fbf\u6377\uff0c\u4f46\u727a\u7272\u4e86\u6570\u636e\u8bbf\u95ee\u901f\u5ea6\u3002</li> <li>\u56fe\u76f8\u8f83\u4e8e\u94fe\u8868\uff0c\u63d0\u4f9b\u4e86\u66f4\u4e30\u5bcc\u7684\u903b\u8f91\u4fe1\u606f\uff0c\u4f46\u9700\u8981\u5360\u7528\u66f4\u5927\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#123","title":"1.2.3 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb","text":"<p>\u5982\u56fe 1-4 \u6240\u793a\uff0c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u9ad8\u5ea6\u76f8\u5173\u3001\u7d27\u5bc6\u7ed3\u5408\uff0c\u5177\u4f53\u8868\u73b0\u5728\u4ee5\u4e0b\u4e09\u4e2a\u65b9\u9762\u3002</p> <ul> <li>\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\u3002\u6570\u636e\u7ed3\u6784\u4e3a\u7b97\u6cd5\u63d0\u4f9b\u4e86\u7ed3\u6784\u5316\u5b58\u50a8\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u64cd\u4f5c\u6570\u636e\u7684\u65b9\u6cd5\u3002</li> <li>\u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u4f5c\u7528\u7684\u821e\u53f0\u3002\u6570\u636e\u7ed3\u6784\u672c\u8eab\u4ec5\u5b58\u50a8\u6570\u636e\u4fe1\u606f\uff0c\u7ed3\u5408\u7b97\u6cd5\u624d\u80fd\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u3002</li> <li>\u7b97\u6cd5\u901a\u5e38\u53ef\u4ee5\u57fa\u4e8e\u4e0d\u540c\u7684\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff0c\u4f46\u6267\u884c\u6548\u7387\u53ef\u80fd\u76f8\u5dee\u5f88\u5927\uff0c\u9009\u62e9\u5408\u9002\u7684\u6570\u636e\u7ed3\u6784\u662f\u5173\u952e\u3002</li> </ul> <p></p> <p> \u56fe 1-4 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb </p> <p>\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u72b9\u5982\u56fe 1-5 \u6240\u793a\u7684\u62fc\u88c5\u79ef\u6728\u3002\u4e00\u5957\u79ef\u6728\uff0c\u9664\u4e86\u5305\u542b\u8bb8\u591a\u96f6\u4ef6\u4e4b\u5916\uff0c\u8fd8\u9644\u6709\u8be6\u7ec6\u7684\u7ec4\u88c5\u8bf4\u660e\u4e66\u3002\u6211\u4eec\u6309\u7167\u8bf4\u660e\u4e66\u4e00\u6b65\u6b65\u64cd\u4f5c\uff0c\u5c31\u80fd\u7ec4\u88c5\u51fa\u7cbe\u7f8e\u7684\u79ef\u6728\u6a21\u578b\u3002</p> <p></p> <p> \u56fe 1-5 \u00a0 \u62fc\u88c5\u79ef\u6728 </p> <p>\u4e24\u8005\u7684\u8be6\u7ec6\u5bf9\u5e94\u5173\u7cfb\u5982\u8868 1-1 \u6240\u793a\u3002</p> <p> \u8868 1-1 \u00a0 \u5c06\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7c7b\u6bd4\u4e3a\u62fc\u88c5\u79ef\u6728 </p> \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5 \u62fc\u88c5\u79ef\u6728 \u8f93\u5165\u6570\u636e \u672a\u62fc\u88c5\u7684\u79ef\u6728 \u6570\u636e\u7ed3\u6784 \u79ef\u6728\u7ec4\u7ec7\u5f62\u5f0f\uff0c\u5305\u62ec\u5f62\u72b6\u3001\u5927\u5c0f\u3001\u8fde\u63a5\u65b9\u5f0f\u7b49 \u7b97\u6cd5 \u628a\u79ef\u6728\u62fc\u6210\u76ee\u6807\u5f62\u6001\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u6b65\u9aa4 \u8f93\u51fa\u6570\u636e \u79ef\u6728\u6a21\u578b <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u662f\u72ec\u7acb\u4e8e\u7f16\u7a0b\u8bed\u8a00\u7684\u3002\u6b63\u56e0\u5982\u6b64\uff0c\u672c\u4e66\u5f97\u4ee5\u63d0\u4f9b\u57fa\u4e8e\u591a\u79cd\u7f16\u7a0b\u8bed\u8a00\u7684\u5b9e\u73b0\u3002</p> <p>\u7ea6\u5b9a\u4fd7\u6210\u7684\u7b80\u79f0</p> <p>\u5728\u5b9e\u9645\u8ba8\u8bba\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u201c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u201d\u7b80\u79f0\u4e3a\u201c\u7b97\u6cd5\u201d\u3002\u6bd4\u5982\u4f17\u6240\u5468\u77e5\u7684 LeetCode \u7b97\u6cd5\u9898\u76ee\uff0c\u5b9e\u9645\u4e0a\u540c\u65f6\u8003\u67e5\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e24\u65b9\u9762\u7684\u77e5\u8bc6\u3002</p>"},{"location":"chapter_paperbook/","title":"\u7eb8\u8d28\u4e66","text":"<p>\u7ecf\u8fc7\u957f\u65f6\u95f4\u7684\u6253\u78e8\uff0c\u300aHello \u7b97\u6cd5\u300b\u7eb8\u8d28\u4e66\u7ec8\u4e8e\u53d1\u5e03\u4e86\uff01\u6b64\u65f6\u7684\u5fc3\u60c5\u53ef\u4ee5\u7528\u4e00\u53e5\u8bd7\u6765\u5f62\u5bb9\uff1a</p> <p>\u8ffd\u98ce\u8d76\u6708\u83ab\u505c\u7559\uff0c\u5e73\u829c\u5c3d\u5904\u662f\u6625\u5c71\u3002</p> <p></p> <p>\u4ee5\u4e0b\u89c6\u9891\u5c55\u793a\u4e86\u7eb8\u8d28\u4e66\uff0c\u5e76\u4e14\u5305\u542b\u6211\u7684\u4e00\u4e9b\u601d\u8003\uff1a</p> <ul> <li>\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u91cd\u8981\u6027\u3002</li> <li>\u4e3a\u4ec0\u4e48\u5728\u7eb8\u8d28\u4e66\u4e2d\u9009\u62e9 Python\u3002</li> <li>\u5bf9\u77e5\u8bc6\u5206\u4eab\u7684\u7406\u89e3\u3002</li> </ul> <p>\u65b0\u4eba UP \u4e3b\uff0c\u8bf7\u591a\u591a\u5173\u7167\u3001\u4e00\u952e\u4e09\u8fde\uff5e\u8c22\u8c22\uff01</p> <p>\u9644\u7eb8\u8d28\u4e66\u5feb\u7167\uff1a</p> <p></p> <p></p>"},{"location":"chapter_paperbook/#_2","title":"\u4f18\u52bf\u4e0e\u4e0d\u8db3","text":"<p>\u603b\u7ed3\u4e00\u4e0b\u7eb8\u8d28\u4e66\u53ef\u80fd\u4f1a\u7ed9\u5927\u5bb6\u5e26\u6765\u60ca\u559c\u7684\u5730\u65b9\uff1a</p> <ul> <li>\u91c7\u7528\u5168\u5f69\u5370\u5237\uff0c\u80fd\u591f\u539f\u6c41\u539f\u5473\u5730\u53d1\u6325\u51fa\u672c\u4e66\u201c\u52a8\u753b\u56fe\u89e3\u201d\u7684\u4f18\u52bf\u3002</li> <li>\u8003\u7a76\u7eb8\u5f20\u6750\u8d28\uff0c\u65e2\u4fdd\u8bc1\u8272\u5f69\u9ad8\u5ea6\u8fd8\u539f\uff0c\u4e5f\u4fdd\u7559\u7eb8\u8d28\u4e66\u7279\u6709\u7684\u8d28\u611f\u3002</li> <li>\u5728\u4e0d\u63d0\u5347\u5b9a\u4ef7\u7684\u524d\u63d0\u4e0b\uff0c\u9644\u8d60\u601d\u7ef4\u5bfc\u56fe\u6298\u9875\u3001\u4e66\u7b7e\u3002</li> <li>\u7eb8\u8d28\u4e66\u3001\u7f51\u9875\u7248\u3001PDF \u7248\u5185\u5bb9\u540c\u6b65\uff0c\u968f\u610f\u5207\u6362\u9605\u8bfb\u3002</li> </ul> <p>Tip</p> <p>\u7531\u4e8e\u7eb8\u8d28\u4e66\u548c\u7f51\u9875\u7248\u7684\u540c\u6b65\u6210\u672c\u8f83\u5927\uff0c\u56e0\u6b64\u53ef\u80fd\u4f1a\u6709\u4e00\u4e9b\u7ec6\u8282\u4e0a\u7684\u4e0d\u540c\uff0c\u8bf7\u60a8\u89c1\u8c05\uff01</p> <p>\u5f53\u7136\uff0c\u7eb8\u8d28\u4e66\u4e5f\u6709\u4e00\u4e9b\u503c\u5f97\u5927\u5bb6\u5165\u624b\u524d\u8003\u8651\u7684\u5730\u65b9\uff1a</p> <ul> <li>\u4f7f\u7528 Python \u8bed\u8a00\uff0c\u53ef\u80fd\u4e0d\u5339\u914d\u4f60\u7684\u4e3b\u8bed\u8a00\uff08\u4e5f\u8bb8\u53ef\u4ee5\u8d81\u6b64\u673a\u4f1a\u7ec3\u4e60 Python\uff09\u3002</li> <li>\u5168\u5f69\u5370\u5237\u867d\u7136\u5927\u5e45\u63d0\u5347\u4e86\u9605\u8bfb\u4f53\u9a8c\uff0c\u4f46\u4ef7\u683c\u4f1a\u6bd4\u9ed1\u767d\u5370\u5237\u9ad8\u4e00\u4e9b\u3002</li> </ul> <p>Tip</p> <p>\u201c\u5370\u5237\u8d28\u91cf\u201d\u548c\u201c\u4ef7\u683c\u201d\u5c31\u50cf\u7b97\u6cd5\u4e2d\u7684\u201c\u65f6\u95f4\u6548\u7387\u201d\u548c\u201c\u7a7a\u95f4\u6548\u7387\u201d\uff0c\u96be\u4ee5\u4e24\u5168\u3002\u800c\u6211\u8ba4\u4e3a\uff0c\u201c\u5370\u5237\u8d28\u91cf\u201d\u5bf9\u5e94\u7684\u662f\u201c\u65f6\u95f4\u6548\u7387\u201d\uff0c\u66f4\u5e94\u8be5\u88ab\u6ce8\u91cd\u3002</p>"},{"location":"chapter_paperbook/#_3","title":"\u8d2d\u4e70\u94fe\u63a5","text":"<p>\u5982\u679c\u4f60\u5bf9\u7eb8\u8d28\u4e66\u611f\u5174\u8da3\uff0c\u53ef\u4ee5\u8003\u8651\u5165\u624b\u4e00\u672c\u3002\u6211\u4eec\u4e3a\u5927\u5bb6\u4e89\u53d6\u5230\u4e86\u65b0\u4e66 5 \u6298\u4f18\u60e0\uff0c\u8bf7\u89c1\u6b64\u94fe\u63a5\u6216\u626b\u63cf\u4ee5\u4e0b\u4e8c\u7ef4\u7801\uff1a</p> <p></p>"},{"location":"chapter_paperbook/#_4","title":"\u5c3e\u8bb0","text":"<p>\u8d77\u521d\uff0c\u6211\u4f4e\u4f30\u4e86\u7eb8\u8d28\u4e66\u51fa\u7248\u7684\u5de5\u4f5c\u91cf\uff0c\u4ee5\u4e3a\u53ea\u8981\u7ef4\u62a4\u597d\u4e86\u5f00\u6e90\u9879\u76ee\uff0c\u7eb8\u8d28\u7248\u5c31\u53ef\u4ee5\u901a\u8fc7\u67d0\u4e9b\u81ea\u52a8\u5316\u624b\u6bb5\u751f\u6210\u51fa\u6765\u3002\u5b9e\u8df5\u8bc1\u660e\uff0c\u7eb8\u8d28\u4e66\u7684\u751f\u4ea7\u6d41\u7a0b\u4e0e\u5f00\u6e90\u9879\u76ee\u7684\u66f4\u65b0\u673a\u5236\u5b58\u5728\u5f88\u5927\u7684\u4e0d\u540c\uff0c\u4e24\u8005\u4e4b\u95f4\u7684\u8f6c\u5316\u9700\u8981\u505a\u8bb8\u591a\u989d\u5916\u5de5\u4f5c\u3002</p> <p>\u4e00\u672c\u4e66\u7684\u521d\u7a3f\u4e0e\u8fbe\u5230\u51fa\u7248\u6807\u51c6\u7684\u5b9a\u7a3f\u4e4b\u95f4\u4ecd\u6709\u8f83\u957f\u8ddd\u79bb\uff0c\u9700\u8981\u51fa\u7248\u793e\uff08\u7b56\u5212\u3001\u7f16\u8f91\u3001\u8bbe\u8ba1\u3001\u5e02\u573a\u7b49\uff09\u4e0e\u4f5c\u8005\u7684\u901a\u529b\u5408\u4f5c\u3001\u957f\u671f\u96d5\u7422\u3002\u5728\u6b64\u611f\u8c22\u56fe\u7075\u7b56\u5212\u7f16\u8f91\u738b\u519b\u82b1\u3001\u4ee5\u53ca\u4eba\u6c11\u90ae\u7535\u51fa\u7248\u793e\u548c\u56fe\u7075\u793e\u533a\u6bcf\u4f4d\u53c2\u4e0e\u672c\u4e66\u51fa\u7248\u6d41\u7a0b\u7684\u5de5\u4f5c\u4eba\u5458\uff01</p> <p>\u5e0c\u671b\u8fd9\u672c\u4e66\u80fd\u591f\u5e2e\u52a9\u5230\u4f60\uff01</p>"},{"location":"chapter_preface/","title":"\u7b2c 0 \u7ae0 \u00a0 \u524d\u8a00","text":"<p>Abstract</p> <p>\u7b97\u6cd5\u72b9\u5982\u7f8e\u5999\u7684\u4ea4\u54cd\u4e50\uff0c\u6bcf\u4e00\u884c\u4ee3\u7801\u90fd\u50cf\u97f5\u5f8b\u822c\u6d41\u6dcc\u3002</p> <p>\u613f\u8fd9\u672c\u4e66\u5728\u4f60\u7684\u8111\u6d77\u4e2d\u8f7b\u8f7b\u54cd\u8d77\uff0c\u7559\u4e0b\u72ec\u7279\u800c\u6df1\u523b\u7684\u65cb\u5f8b\u3002</p>"},{"location":"chapter_preface/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>0.1 \u00a0 \u5173\u4e8e\u672c\u4e66</li> <li>0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66</li> <li>0.3 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_preface/about_the_book/","title":"0.1 \u00a0 \u5173\u4e8e\u672c\u4e66","text":"<p>\u672c\u9879\u76ee\u65e8\u5728\u521b\u5efa\u4e00\u672c\u5f00\u6e90\u3001\u514d\u8d39\u3001\u5bf9\u65b0\u624b\u53cb\u597d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u6559\u7a0b\u3002</p> <ul> <li>\u5168\u4e66\u91c7\u7528\u52a8\u753b\u56fe\u89e3\uff0c\u7ed3\u6784\u5316\u5730\u8bb2\u89e3\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u5185\u5bb9\u6e05\u6670\u6613\u61c2\uff0c\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1\u3002</li> <li>\u7b97\u6cd5\u6e90\u4ee3\u7801\u7686\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u652f\u6301 Python\u3001C++\u3001Java\u3001C#\u3001Go\u3001Swift\u3001JavaScript\u3001TypeScript\u3001Dart\u3001Rust\u3001C \u548c Zig \u7b49\u8bed\u8a00\u3002</li> <li>\u9f13\u52b1\u8bfb\u8005\u5728\u7ebf\u4e0a\u7ae0\u8282\u8bc4\u8bba\u533a\u4e92\u5e2e\u4e92\u52a9\u3001\u5171\u540c\u8fdb\u6b65\uff0c\u63d0\u95ee\u4e0e\u8bc4\u8bba\u901a\u5e38\u53ef\u5728\u4e24\u65e5\u5185\u5f97\u5230\u56de\u590d\u3002</li> </ul>"},{"location":"chapter_preface/about_the_book/#011","title":"0.1.1 \u00a0 \u8bfb\u8005\u5bf9\u8c61","text":"<p>\u82e5\u4f60\u662f\u7b97\u6cd5\u521d\u5b66\u8005\uff0c\u4ece\u672a\u63a5\u89e6\u8fc7\u7b97\u6cd5\uff0c\u6216\u8005\u5df2\u7ecf\u6709\u4e00\u4e9b\u5237\u9898\u7ecf\u9a8c\uff0c\u5bf9\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6709\u6a21\u7cca\u7684\u8ba4\u8bc6\uff0c\u5728\u4f1a\u4e0e\u4e0d\u4f1a\u4e4b\u95f4\u53cd\u590d\u6a2a\u8df3\uff0c\u90a3\u4e48\u672c\u4e66\u6b63\u662f\u4e3a\u4f60\u91cf\u8eab\u5b9a\u5236\u7684\uff01</p> <p>\u5982\u679c\u4f60\u5df2\u7ecf\u79ef\u7d2f\u4e00\u5b9a\u7684\u5237\u9898\u91cf\uff0c\u719f\u6089\u5927\u90e8\u5206\u9898\u578b\uff0c\u90a3\u4e48\u672c\u4e66\u53ef\u52a9\u4f60\u56de\u987e\u4e0e\u68b3\u7406\u7b97\u6cd5\u77e5\u8bc6\u4f53\u7cfb\uff0c\u4ed3\u5e93\u6e90\u4ee3\u7801\u53ef\u4ee5\u5f53\u4f5c\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u6216\u201c\u7b97\u6cd5\u5b57\u5178\u201d\u6765\u4f7f\u7528\u3002</p> <p>\u82e5\u4f60\u662f\u7b97\u6cd5\u201c\u5927\u795e\u201d\uff0c\u6211\u4eec\u671f\u5f85\u6536\u5230\u4f60\u7684\u5b9d\u8d35\u5efa\u8bae\uff0c\u6216\u8005\u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c\u3002</p> <p>\u524d\u7f6e\u6761\u4ef6</p> <p>\u4f60\u9700\u8981\u81f3\u5c11\u5177\u5907\u4efb\u4e00\u8bed\u8a00\u7684\u7f16\u7a0b\u57fa\u7840\uff0c\u80fd\u591f\u9605\u8bfb\u548c\u7f16\u5199\u7b80\u5355\u4ee3\u7801\u3002</p>"},{"location":"chapter_preface/about_the_book/#012","title":"0.1.2 \u00a0 \u5185\u5bb9\u7ed3\u6784","text":"<p>\u672c\u4e66\u7684\u4e3b\u8981\u5185\u5bb9\u5982\u56fe 0-1 \u6240\u793a\u3002</p> <ul> <li>\u590d\u6742\u5ea6\u5206\u6790\uff1a\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u7684\u8bc4\u4ef7\u7ef4\u5ea6\u4e0e\u65b9\u6cd5\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u793a\u4f8b\u7b49\u3002</li> <li>\u6570\u636e\u7ed3\u6784\uff1a\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u6570\u636e\u7ed3\u6784\u7684\u5206\u7c7b\u65b9\u6cd5\u3002\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u7684\u5b9a\u4e49\u3001\u4f18\u7f3a\u70b9\u3001\u5e38\u7528\u64cd\u4f5c\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u5178\u578b\u5e94\u7528\u3001\u5b9e\u73b0\u65b9\u6cd5\u7b49\u3002</li> <li>\u7b97\u6cd5\uff1a\u641c\u7d22\u3001\u6392\u5e8f\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u8d2a\u5fc3\u7b49\u7b97\u6cd5\u7684\u5b9a\u4e49\u3001\u4f18\u7f3a\u70b9\u3001\u6548\u7387\u3001\u5e94\u7528\u573a\u666f\u3001\u89e3\u9898\u6b65\u9aa4\u548c\u793a\u4f8b\u95ee\u9898\u7b49\u3002</li> </ul> <p></p> <p> \u56fe 0-1 \u00a0 \u672c\u4e66\u4e3b\u8981\u5185\u5bb9 </p>"},{"location":"chapter_preface/about_the_book/#013","title":"0.1.3 \u00a0 \u81f4\u8c22","text":"<p>\u672c\u4e66\u5728\u5f00\u6e90\u793e\u533a\u4f17\u591a\u8d21\u732e\u8005\u7684\u5171\u540c\u52aa\u529b\u4e0b\u4e0d\u65ad\u5b8c\u5584\u3002\u611f\u8c22\u6bcf\u4e00\u4f4d\u6295\u5165\u65f6\u95f4\u4e0e\u7cbe\u529b\u7684\u64b0\u7a3f\u4eba\uff0c\u4ed6\u4eec\u662f\uff08\u6309\u7167 GitHub \u81ea\u52a8\u751f\u6210\u7684\u987a\u5e8f\uff09\uff1akrahets\u3001codingonion\u3001nuomi1\u3001Gonglja\u3001Reanon\u3001justin-tse\u3001danielsss\u3001hpstory\u3001S-N-O-R-L-A-X\u3001night-cruise\u3001msk397\u3001gvenusleo\u3001RiverTwilight\u3001gyt95\u3001zhuoqinyue\u3001Zuoxun\u3001Xia-Sang\u3001mingXta\u3001FangYuan33\u3001GN-Yu\u3001IsChristina\u3001xBLACKICEx\u3001guowei-gong\u3001Cathay-Chen\u3001mgisr\u3001JoseHung\u3001qualifier1024\u3001pengchzn\u3001Guanngxu\u3001longsizhuo\u3001L-Super\u3001what-is-me\u3001yuan0221\u3001lhxsm\u3001Slone123c\u3001WSL0809\u3001longranger2\u3001theNefelibatas\u3001xiongsp\u3001JeffersonHuang\u3001hongyun-robot\u3001K3v123\u3001yuelinxin\u3001a16su\u3001gaofer\u3001malone6\u3001Wonderdch\u3001xjr7670\u3001DullSword\u3001Horbin-Magician\u3001NI-SW\u3001reeswell\u3001XC-Zero\u3001XiaChuerwu\u3001yd-j\u3001iron-irax\u3001huawuque404\u3001MolDuM\u3001Nigh\u3001KorsChen\u3001foursevenlove\u300152coder\u3001bubble9um\u3001youshaoXG\u3001curly210102\u3001gltianwen\u3001fanchenggang\u3001Transmigration-zhou\u3001FloranceYeh\u3001FreddieLi\u3001ShiMaRing\u3001lipusheng\u3001Javesun99\u3001JackYang-hellobobo\u3001shanghai-Jerry\u30010130w\u3001Keynman\u3001psychelzh\u3001logan-qiu\u3001ZnYang2018\u3001MwumLi\u30011ch0\u3001Phoenix0415\u3001qingpeng9802\u3001Richard-Zhang1019\u3001QiLOL\u3001Suremotoo\u3001Turing-1024-Lee\u3001Evilrabbit520\u3001GaochaoZhu\u3001ZJKung\u3001linzeyan\u3001hezhizhen\u3001ZongYangL\u3001beintentional\u3001czruby\u3001coderlef\u3001dshlstarr\u3001szu17dmy\u3001fbigm\u3001gledfish\u3001hts0000\u3001boloboloda\u3001iStig\u3001jiaxianhua\u3001wenjianmin\u3001keshida\u3001kilikilikid\u3001lclc6\u3001lwbaptx\u3001liuxjerry\u3001lucaswangdev\u3001lyl625760\u3001chadyi\u3001noobcodemaker\u3001selear\u3001siqyka\u3001syd168\u30014yDX3906\u3001tao363\u3001wangwang105\u3001weibk\u3001yabo083\u3001yi427\u3001yishangzhang\u3001zhouLion\u3001baagod\u3001ElaBosak233\u3001xb534\u3001luluxia\u3001yanedie\u3001thomasq0\u3001YangXuanyi \u548c th1nk3r-ing \u3002</p> <p>\u672c\u4e66\u7684\u4ee3\u7801\u5ba1\u9605\u5de5\u4f5c\u7531 codingonion\u3001Gonglja\u3001gvenusleo\u3001hpstory\u3001justin-tse\u3001krahets\u3001night-cruise\u3001nuomi1 \u548c Reanon \u5b8c\u6210\uff08\u6309\u7167\u9996\u5b57\u6bcd\u987a\u5e8f\u6392\u5217\uff09\u3002\u611f\u8c22\u4ed6\u4eec\u4ed8\u51fa\u7684\u65f6\u95f4\u4e0e\u7cbe\u529b\uff0c\u6b63\u662f\u4ed6\u4eec\u786e\u4fdd\u4e86\u5404\u8bed\u8a00\u4ee3\u7801\u7684\u89c4\u8303\u4e0e\u7edf\u4e00\u3002</p> <p>\u5728\u672c\u4e66\u7684\u521b\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u5f97\u5230\u4e86\u8bb8\u591a\u4eba\u7684\u5e2e\u52a9\u3002</p> <ul> <li>\u611f\u8c22\u6211\u5728\u516c\u53f8\u7684\u5bfc\u5e08\u674e\u6c50\u535a\u58eb\uff0c\u5728\u4e00\u6b21\u7545\u8c08\u4e2d\u4f60\u9f13\u52b1\u6211\u201c\u5feb\u884c\u52a8\u8d77\u6765\u201d\uff0c\u575a\u5b9a\u4e86\u6211\u5199\u8fd9\u672c\u4e66\u7684\u51b3\u5fc3\uff1b</li> <li>\u611f\u8c22\u6211\u7684\u5973\u670b\u53cb\u6ce1\u6ce1\u4f5c\u4e3a\u672c\u4e66\u7684\u9996\u4f4d\u8bfb\u8005\uff0c\u4ece\u7b97\u6cd5\u5c0f\u767d\u7684\u89d2\u5ea6\u63d0\u51fa\u8bb8\u591a\u5b9d\u8d35\u5efa\u8bae\uff0c\u4f7f\u5f97\u672c\u4e66\u66f4\u9002\u5408\u65b0\u624b\u9605\u8bfb\uff1b</li> <li>\u611f\u8c22\u817e\u5b9d\u3001\u7426\u5b9d\u3001\u98de\u5b9d\u4e3a\u672c\u4e66\u8d77\u4e86\u4e00\u4e2a\u5bcc\u6709\u521b\u610f\u7684\u540d\u5b57\uff0c\u5524\u8d77\u5927\u5bb6\u5199\u4e0b\u7b2c\u4e00\u884c\u4ee3\u7801\u201cHello World!\u201d\u7684\u7f8e\u597d\u56de\u5fc6\uff1b</li> <li>\u611f\u8c22\u6821\u94e8\u5728\u77e5\u8bc6\u4ea7\u6743\u65b9\u9762\u63d0\u4f9b\u7684\u4e13\u4e1a\u5e2e\u52a9\uff0c\u8fd9\u5bf9\u672c\u5f00\u6e90\u4e66\u7684\u5b8c\u5584\u8d77\u5230\u4e86\u91cd\u8981\u4f5c\u7528\uff1b</li> <li>\u611f\u8c22\u82cf\u6f7c\u4e3a\u672c\u4e66\u8bbe\u8ba1\u4e86\u7cbe\u7f8e\u7684\u5c01\u9762\u548c logo \uff0c\u5e76\u5728\u6211\u7684\u5f3a\u8feb\u75c7\u7684\u9a71\u4f7f\u4e0b\u591a\u6b21\u8010\u5fc3\u4fee\u6539\uff1b</li> <li>\u611f\u8c22 @squidfunk \u63d0\u4f9b\u7684\u6392\u7248\u5efa\u8bae\uff0c\u4ee5\u53ca\u4ed6\u5f00\u53d1\u7684\u5f00\u6e90\u6587\u6863\u4e3b\u9898 Material-for-MkDocs \u3002</li> </ul> <p>\u5728\u5199\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u9605\u8bfb\u4e86\u8bb8\u591a\u5173\u4e8e\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u6559\u6750\u548c\u6587\u7ae0\u3002\u8fd9\u4e9b\u4f5c\u54c1\u4e3a\u672c\u4e66\u63d0\u4f9b\u4e86\u4f18\u79c0\u7684\u8303\u672c\uff0c\u786e\u4fdd\u4e86\u672c\u4e66\u5185\u5bb9\u7684\u51c6\u786e\u6027\u4e0e\u54c1\u8d28\u3002\u5728\u6b64\u611f\u8c22\u6240\u6709\u8001\u5e08\u548c\u524d\u8f88\u7684\u6770\u51fa\u8d21\u732e\uff01</p> <p>\u672c\u4e66\u5021\u5bfc\u624b\u8111\u5e76\u7528\u7684\u5b66\u4e60\u65b9\u5f0f\uff0c\u5728\u8fd9\u4e00\u70b9\u4e0a\u6211\u6df1\u53d7\u300a\u52a8\u624b\u5b66\u6df1\u5ea6\u5b66\u4e60\u300b\u7684\u542f\u53d1\u3002\u5728\u6b64\u5411\u5404\u4f4d\u8bfb\u8005\u5f3a\u70c8\u63a8\u8350\u8fd9\u672c\u4f18\u79c0\u7684\u8457\u4f5c\u3002</p> <p>\u8877\u5fc3\u611f\u8c22\u6211\u7684\u7236\u6bcd\uff0c\u6b63\u662f\u4f60\u4eec\u4e00\u76f4\u4ee5\u6765\u7684\u652f\u6301\u4e0e\u9f13\u52b1\uff0c\u8ba9\u6211\u6709\u673a\u4f1a\u505a\u8fd9\u4ef6\u5bcc\u6709\u8da3\u5473\u7684\u4e8b\u3002</p>"},{"location":"chapter_preface/suggestions/","title":"0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66","text":"<p>Tip</p> <p>\u4e3a\u4e86\u83b7\u5f97\u6700\u4f73\u7684\u9605\u8bfb\u4f53\u9a8c\uff0c\u5efa\u8bae\u4f60\u901a\u8bfb\u672c\u8282\u5185\u5bb9\u3002</p>"},{"location":"chapter_preface/suggestions/#021","title":"0.2.1 \u00a0 \u884c\u6587\u98ce\u683c\u7ea6\u5b9a","text":"<ul> <li>\u6807\u9898\u540e\u6807\u6ce8 <code>*</code> \u7684\u662f\u9009\u8bfb\u7ae0\u8282\uff0c\u5185\u5bb9\u76f8\u5bf9\u56f0\u96be\u3002\u5982\u679c\u4f60\u7684\u65f6\u95f4\u6709\u9650\uff0c\u53ef\u4ee5\u5148\u8df3\u8fc7\u3002</li> <li>\u4e13\u4e1a\u672f\u8bed\u4f1a\u4f7f\u7528\u9ed1\u4f53\uff08\u7eb8\u8d28\u7248\u548c PDF \u7248\uff09\u6216\u6dfb\u52a0\u4e0b\u5212\u7ebf\uff08\u7f51\u9875\u7248\uff09\uff0c\u4f8b\u5982\u6570\u7ec4\uff08array\uff09\u3002\u5efa\u8bae\u8bb0\u4f4f\u5b83\u4eec\uff0c\u4ee5\u4fbf\u9605\u8bfb\u6587\u732e\u3002</li> <li>\u91cd\u70b9\u5185\u5bb9\u548c\u603b\u7ed3\u6027\u8bed\u53e5\u4f1a \u52a0\u7c97\uff0c\u8fd9\u7c7b\u6587\u5b57\u503c\u5f97\u7279\u522b\u5173\u6ce8\u3002</li> <li>\u6709\u7279\u6307\u542b\u4e49\u7684\u8bcd\u53e5\u4f1a\u4f7f\u7528\u201c\u5f15\u53f7\u201d\u6807\u6ce8\uff0c\u4ee5\u907f\u514d\u6b67\u4e49\u3002</li> <li>\u5f53\u6d89\u53ca\u7f16\u7a0b\u8bed\u8a00\u4e4b\u95f4\u4e0d\u4e00\u81f4\u7684\u540d\u8bcd\u65f6\uff0c\u672c\u4e66\u5747\u4ee5 Python \u4e3a\u51c6\uff0c\u4f8b\u5982\u4f7f\u7528 <code>None</code> \u6765\u8868\u793a\u201c\u7a7a\u201d\u3002</li> <li>\u672c\u4e66\u90e8\u5206\u653e\u5f03\u4e86\u7f16\u7a0b\u8bed\u8a00\u7684\u6ce8\u91ca\u89c4\u8303\uff0c\u4ee5\u6362\u53d6\u66f4\u52a0\u7d27\u51d1\u7684\u5185\u5bb9\u6392\u7248\u3002\u6ce8\u91ca\u4e3b\u8981\u5206\u4e3a\u4e09\u79cd\u7c7b\u578b\uff1a\u6807\u9898\u6ce8\u91ca\u3001\u5185\u5bb9\u6ce8\u91ca\u3001\u591a\u884c\u6ce8\u91ca\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>\"\"\"\u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\"\"\"\n\n# \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n\"\"\"\n\u591a\u884c\n\u6ce8\u91ca\n\"\"\"\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>\n</code></pre> <pre><code>// \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n// \u591a\u884c\n// \u6ce8\u91ca\n</code></pre>"},{"location":"chapter_preface/suggestions/#022","title":"0.2.2 \u00a0 \u5728\u52a8\u753b\u56fe\u89e3\u4e2d\u9ad8\u6548\u5b66\u4e60","text":"<p>\u76f8\u8f83\u4e8e\u6587\u5b57\uff0c\u89c6\u9891\u548c\u56fe\u7247\u5177\u6709\u66f4\u9ad8\u7684\u4fe1\u606f\u5bc6\u5ea6\u548c\u7ed3\u6784\u5316\u7a0b\u5ea6\uff0c\u66f4\u6613\u4e8e\u7406\u89e3\u3002\u5728\u672c\u4e66\u4e2d\uff0c\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u5c06\u4e3b\u8981\u901a\u8fc7\u52a8\u753b\u4ee5\u56fe\u89e3\u5f62\u5f0f\u5c55\u793a\uff0c\u800c\u6587\u5b57\u5219\u4f5c\u4e3a\u89e3\u91ca\u4e0e\u8865\u5145\u3002</p> <p>\u5982\u679c\u4f60\u5728\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u53d1\u73b0\u67d0\u6bb5\u5185\u5bb9\u63d0\u4f9b\u4e86\u5982\u56fe 0-2 \u6240\u793a\u7684\u52a8\u753b\u56fe\u89e3\uff0c\u8bf7\u4ee5\u56fe\u4e3a\u4e3b\u3001\u4ee5\u6587\u5b57\u4e3a\u8f85\uff0c\u7efc\u5408\u4e24\u8005\u6765\u7406\u89e3\u5185\u5bb9\u3002</p> <p></p> <p> \u56fe 0-2 \u00a0 \u52a8\u753b\u56fe\u89e3\u793a\u4f8b </p>"},{"location":"chapter_preface/suggestions/#023","title":"0.2.3 \u00a0 \u5728\u4ee3\u7801\u5b9e\u8df5\u4e2d\u52a0\u6df1\u7406\u89e3","text":"<p>\u672c\u4e66\u7684\u914d\u5957\u4ee3\u7801\u6258\u7ba1\u5728 GitHub \u4ed3\u5e93\u3002\u5982\u56fe 0-3 \u6240\u793a\uff0c\u6e90\u4ee3\u7801\u9644\u6709\u6d4b\u8bd5\u6837\u4f8b\uff0c\u53ef\u4e00\u952e\u8fd0\u884c\u3002</p> <p>\u5982\u679c\u65f6\u95f4\u5141\u8bb8\uff0c\u5efa\u8bae\u4f60\u53c2\u7167\u4ee3\u7801\u81ea\u884c\u6572\u4e00\u904d\u3002\u5982\u679c\u5b66\u4e60\u65f6\u95f4\u6709\u9650\uff0c\u8bf7\u81f3\u5c11\u901a\u8bfb\u5e76\u8fd0\u884c\u6240\u6709\u4ee3\u7801\u3002</p> <p>\u4e0e\u9605\u8bfb\u4ee3\u7801\u76f8\u6bd4\uff0c\u7f16\u5199\u4ee3\u7801\u7684\u8fc7\u7a0b\u5f80\u5f80\u80fd\u5e26\u6765\u66f4\u591a\u6536\u83b7\u3002\u52a8\u624b\u5b66\uff0c\u624d\u662f\u771f\u7684\u5b66\u3002</p> <p></p> <p> \u56fe 0-3 \u00a0 \u8fd0\u884c\u4ee3\u7801\u793a\u4f8b </p> <p>\u8fd0\u884c\u4ee3\u7801\u7684\u524d\u7f6e\u5de5\u4f5c\u4e3b\u8981\u5206\u4e3a\u4e09\u6b65\u3002</p> <p>\u7b2c\u4e00\u6b65\uff1a\u5b89\u88c5\u672c\u5730\u7f16\u7a0b\u73af\u5883\u3002\u8bf7\u53c2\u7167\u9644\u5f55\u6240\u793a\u7684\u6559\u7a0b\u8fdb\u884c\u5b89\u88c5\uff0c\u5982\u679c\u5df2\u5b89\u88c5\uff0c\u5219\u53ef\u8df3\u8fc7\u6b64\u6b65\u9aa4\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u514b\u9686\u6216\u4e0b\u8f7d\u4ee3\u7801\u4ed3\u5e93\u3002\u524d\u5f80 GitHub \u4ed3\u5e93\u3002\u5982\u679c\u5df2\u7ecf\u5b89\u88c5 Git \uff0c\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u547d\u4ee4\u514b\u9686\u672c\u4ed3\u5e93\uff1a</p> <pre><code>git clone https://github.com/krahets/hello-algo.git\n</code></pre> <p>\u5f53\u7136\uff0c\u4f60\u4e5f\u53ef\u4ee5\u5728\u56fe 0-4 \u6240\u793a\u7684\u4f4d\u7f6e\uff0c\u70b9\u51fb\u201cDownload ZIP\u201d\u6309\u94ae\u76f4\u63a5\u4e0b\u8f7d\u4ee3\u7801\u538b\u7f29\u5305\uff0c\u7136\u540e\u5728\u672c\u5730\u89e3\u538b\u5373\u53ef\u3002</p> <p></p> <p> \u56fe 0-4 \u00a0 \u514b\u9686\u4ed3\u5e93\u4e0e\u4e0b\u8f7d\u4ee3\u7801 </p> <p>\u7b2c\u4e09\u6b65\uff1a\u8fd0\u884c\u6e90\u4ee3\u7801\u3002\u5982\u56fe 0-5 \u6240\u793a\uff0c\u5bf9\u4e8e\u9876\u90e8\u6807\u6709\u6587\u4ef6\u540d\u79f0\u7684\u4ee3\u7801\u5757\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u4ed3\u5e93\u7684 <code>codes</code> \u6587\u4ef6\u5939\u5185\u627e\u5230\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6\u3002\u6e90\u4ee3\u7801\u6587\u4ef6\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u5c06\u5e2e\u52a9\u4f60\u8282\u7701\u4e0d\u5fc5\u8981\u7684\u8c03\u8bd5\u65f6\u95f4\uff0c\u8ba9\u4f60\u80fd\u591f\u4e13\u6ce8\u4e8e\u5b66\u4e60\u5185\u5bb9\u3002</p> <p></p> <p> \u56fe 0-5 \u00a0 \u4ee3\u7801\u5757\u4e0e\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6 </p> <p>\u9664\u4e86\u672c\u5730\u8fd0\u884c\u4ee3\u7801\uff0c\u7f51\u9875\u7248\u8fd8\u652f\u6301 Python \u4ee3\u7801\u7684\u53ef\u89c6\u5316\u8fd0\u884c\uff08\u57fa\u4e8e pythontutor \u5b9e\u73b0\uff09\u3002\u5982\u56fe 0-6 \u6240\u793a\uff0c\u4f60\u53ef\u4ee5\u70b9\u51fb\u4ee3\u7801\u5757\u4e0b\u65b9\u7684\u201c\u53ef\u89c6\u5316\u8fd0\u884c\u201d\u6765\u5c55\u5f00\u89c6\u56fe\uff0c\u89c2\u5bdf\u7b97\u6cd5\u4ee3\u7801\u7684\u6267\u884c\u8fc7\u7a0b\uff1b\u4e5f\u53ef\u4ee5\u70b9\u51fb\u201c\u5168\u5c4f\u89c2\u770b\u201d\uff0c\u4ee5\u83b7\u5f97\u66f4\u597d\u7684\u9605\u89c8\u4f53\u9a8c\u3002</p> <p></p> <p> \u56fe 0-6 \u00a0 Python \u4ee3\u7801\u7684\u53ef\u89c6\u5316\u8fd0\u884c </p>"},{"location":"chapter_preface/suggestions/#024","title":"0.2.4 \u00a0 \u5728\u63d0\u95ee\u8ba8\u8bba\u4e2d\u5171\u540c\u6210\u957f","text":"<p>\u5728\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u8bf7\u4e0d\u8981\u8f7b\u6613\u8df3\u8fc7\u90a3\u4e9b\u6ca1\u5b66\u660e\u767d\u7684\u77e5\u8bc6\u70b9\u3002\u6b22\u8fce\u5728\u8bc4\u8bba\u533a\u63d0\u51fa\u4f60\u7684\u95ee\u9898\uff0c\u6211\u548c\u5c0f\u4f19\u4f34\u4eec\u5c06\u7aed\u8bda\u4e3a\u4f60\u89e3\u7b54\uff0c\u4e00\u822c\u60c5\u51b5\u4e0b\u53ef\u5728\u4e24\u5929\u5185\u56de\u590d\u3002</p> <p>\u5982\u56fe 0-7 \u6240\u793a\uff0c\u7f51\u9875\u7248\u6bcf\u4e2a\u7ae0\u8282\u7684\u5e95\u90e8\u90fd\u914d\u6709\u8bc4\u8bba\u533a\u3002\u5e0c\u671b\u4f60\u80fd\u591a\u5173\u6ce8\u8bc4\u8bba\u533a\u7684\u5185\u5bb9\u3002\u4e00\u65b9\u9762\uff0c\u4f60\u53ef\u4ee5\u4e86\u89e3\u5927\u5bb6\u9047\u5230\u7684\u95ee\u9898\uff0c\u4ece\u800c\u67e5\u6f0f\u8865\u7f3a\uff0c\u6fc0\u53d1\u66f4\u6df1\u5165\u7684\u601d\u8003\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u671f\u5f85\u4f60\u80fd\u6177\u6168\u5730\u56de\u7b54\u5176\u4ed6\u5c0f\u4f19\u4f34\u7684\u95ee\u9898\uff0c\u5206\u4eab\u4f60\u7684\u89c1\u89e3\uff0c\u5e2e\u52a9\u4ed6\u4eba\u8fdb\u6b65\u3002</p> <p></p> <p> \u56fe 0-7 \u00a0 \u8bc4\u8bba\u533a\u793a\u4f8b </p>"},{"location":"chapter_preface/suggestions/#025","title":"0.2.5 \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf","text":"<p>\u4ece\u603b\u4f53\u4e0a\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8fc7\u7a0b\u5212\u5206\u4e3a\u4e09\u4e2a\u9636\u6bb5\u3002</p> <ol> <li>\u9636\u6bb5\u4e00\uff1a\u7b97\u6cd5\u5165\u95e8\u3002\u6211\u4eec\u9700\u8981\u719f\u6089\u5404\u79cd\u6570\u636e\u7ed3\u6784\u7684\u7279\u70b9\u548c\u7528\u6cd5\uff0c\u5b66\u4e60\u4e0d\u540c\u7b97\u6cd5\u7684\u539f\u7406\u3001\u6d41\u7a0b\u3001\u7528\u9014\u548c\u6548\u7387\u7b49\u65b9\u9762\u7684\u5185\u5bb9\u3002</li> <li>\u9636\u6bb5\u4e8c\uff1a\u5237\u7b97\u6cd5\u9898\u3002\u5efa\u8bae\u4ece\u70ed\u95e8\u9898\u76ee\u5f00\u5237\uff0c\u5148\u79ef\u7d2f\u81f3\u5c11 100 \u9053\u9898\u76ee\uff0c\u719f\u6089\u4e3b\u6d41\u7684\u7b97\u6cd5\u95ee\u9898\u3002\u521d\u6b21\u5237\u9898\u65f6\uff0c\u201c\u77e5\u8bc6\u9057\u5fd8\u201d\u53ef\u80fd\u662f\u4e00\u4e2a\u6311\u6218\uff0c\u4f46\u8bf7\u653e\u5fc3\uff0c\u8fd9\u662f\u5f88\u6b63\u5e38\u7684\u3002\u6211\u4eec\u53ef\u4ee5\u6309\u7167\u201c\u827e\u5bbe\u6d69\u65af\u9057\u5fd8\u66f2\u7ebf\u201d\u6765\u590d\u4e60\u9898\u76ee\uff0c\u901a\u5e38\u5728\u8fdb\u884c 3\uff5e5 \u8f6e\u7684\u91cd\u590d\u540e\uff0c\u5c31\u80fd\u5c06\u5176\u7262\u8bb0\u5728\u5fc3\u3002\u63a8\u8350\u7684\u9898\u5355\u548c\u5237\u9898\u8ba1\u5212\u8bf7\u89c1\u6b64 GitHub \u4ed3\u5e93\u3002</li> <li>\u9636\u6bb5\u4e09\uff1a\u642d\u5efa\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5b66\u4e60\u65b9\u9762\uff0c\u6211\u4eec\u53ef\u4ee5\u9605\u8bfb\u7b97\u6cd5\u4e13\u680f\u6587\u7ae0\u3001\u89e3\u9898\u6846\u67b6\u548c\u7b97\u6cd5\u6559\u6750\uff0c\u4ee5\u4e0d\u65ad\u4e30\u5bcc\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5237\u9898\u65b9\u9762\uff0c\u53ef\u4ee5\u5c1d\u8bd5\u91c7\u7528\u8fdb\u9636\u5237\u9898\u7b56\u7565\uff0c\u5982\u6309\u4e13\u9898\u5206\u7c7b\u3001\u4e00\u9898\u591a\u89e3\u3001\u4e00\u89e3\u591a\u9898\u7b49\uff0c\u76f8\u5173\u7684\u5237\u9898\u5fc3\u5f97\u53ef\u4ee5\u5728\u5404\u4e2a\u793e\u533a\u627e\u5230\u3002</li> </ol> <p>\u5982\u56fe 0-8 \u6240\u793a\uff0c\u672c\u4e66\u5185\u5bb9\u4e3b\u8981\u6db5\u76d6\u201c\u9636\u6bb5\u4e00\u201d\uff0c\u65e8\u5728\u5e2e\u52a9\u4f60\u66f4\u9ad8\u6548\u5730\u5c55\u5f00\u9636\u6bb5\u4e8c\u548c\u9636\u6bb5\u4e09\u7684\u5b66\u4e60\u3002</p> <p></p> <p> \u56fe 0-8 \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf </p>"},{"location":"chapter_preface/summary/","title":"0.3 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u672c\u4e66\u7684\u4e3b\u8981\u53d7\u4f17\u662f\u7b97\u6cd5\u521d\u5b66\u8005\u3002\u5982\u679c\u4f60\u5df2\u6709\u4e00\u5b9a\u57fa\u7840\uff0c\u672c\u4e66\u80fd\u5e2e\u52a9\u4f60\u7cfb\u7edf\u56de\u987e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u4e66\u4e2d\u6e90\u4ee3\u7801\u4e5f\u53ef\u4f5c\u4e3a\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u4f7f\u7528\u3002</li> <li>\u4e66\u4e2d\u5185\u5bb9\u4e3b\u8981\u5305\u62ec\u590d\u6742\u5ea6\u5206\u6790\u3001\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e09\u90e8\u5206\uff0c\u6db5\u76d6\u4e86\u8be5\u9886\u57df\u7684\u5927\u90e8\u5206\u4e3b\u9898\u3002</li> <li>\u5bf9\u4e8e\u7b97\u6cd5\u65b0\u624b\uff0c\u5728\u521d\u5b66\u9636\u6bb5\u9605\u8bfb\u4e00\u672c\u5165\u95e8\u4e66\u81f3\u5173\u91cd\u8981\uff0c\u53ef\u4ee5\u5c11\u8d70\u8bb8\u591a\u5f2f\u8def\u3002</li> <li>\u4e66\u4e2d\u7684\u52a8\u753b\u56fe\u89e3\u901a\u5e38\u7528\u4e8e\u4ecb\u7ecd\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u3002\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u5e94\u7ed9\u4e88\u8fd9\u4e9b\u5185\u5bb9\u66f4\u591a\u5173\u6ce8\u3002</li> <li>\u5b9e\u8df5\u4e43\u5b66\u4e60\u7f16\u7a0b\u4e4b\u6700\u4f73\u9014\u5f84\u3002\u5f3a\u70c8\u5efa\u8bae\u8fd0\u884c\u6e90\u4ee3\u7801\u5e76\u4eb2\u81ea\u6572\u4ee3\u7801\u3002</li> <li>\u672c\u4e66\u7f51\u9875\u7248\u7684\u6bcf\u4e2a\u7ae0\u8282\u90fd\u8bbe\u6709\u8bc4\u8bba\u533a\uff0c\u6b22\u8fce\u968f\u65f6\u5206\u4eab\u4f60\u7684\u7591\u60d1\u4e0e\u89c1\u89e3\u3002</li> </ul>"},{"location":"chapter_reference/","title":"\u53c2\u8003\u6587\u732e","text":"<p>[1] Thomas H. Cormen, et al. Introduction to Algorithms (3<sup>rd</sup> Edition).</p> <p>[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1<sup>st</sup> Edition).</p> <p>[3] Robert Sedgewick, et al. Algorithms (4<sup>th</sup> Edition).</p> <p>[4] \u4e25\u851a\u654f. \u6570\u636e\u7ed3\u6784\uff08C \u8bed\u8a00\u7248\uff09.</p> <p>[5] \u9093\u4fca\u8f89. \u6570\u636e\u7ed3\u6784\uff08C++ \u8bed\u8a00\u7248\uff0c\u7b2c\u4e09\u7248\uff09.</p> <p>[6] \u9a6c\u514b \u827e\u4f26 \u7ef4\u65af\u8457\uff0c\u9648\u8d8a\u8bd1. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790\uff1aJava\u8bed\u8a00\u63cf\u8ff0\uff08\u7b2c\u4e09\u7248\uff09.</p> <p>[7] \u7a0b\u6770. \u5927\u8bdd\u6570\u636e\u7ed3\u6784.</p> <p>[8] \u738b\u4e89. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u7f8e.</p> <p>[9] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6<sup>th</sup> Edition).</p> <p>[10] Aston Zhang, et al. Dive into Deep Learning.</p>"},{"location":"chapter_searching/","title":"\u7b2c 10 \u7ae0 \u00a0 \u641c\u7d22","text":"<p>Abstract</p> <p>\u641c\u7d22\u662f\u4e00\u573a\u672a\u77e5\u7684\u5192\u9669\uff0c\u6211\u4eec\u6216\u8bb8\u9700\u8981\u8d70\u904d\u795e\u79d8\u7a7a\u95f4\u7684\u6bcf\u4e2a\u89d2\u843d\uff0c\u53c8\u6216\u8bb8\u53ef\u4ee5\u5feb\u901f\u9501\u5b9a\u76ee\u6807\u3002</p> <p>\u5728\u8fd9\u573a\u5bfb\u89c5\u4e4b\u65c5\u4e2d\uff0c\u6bcf\u4e00\u6b21\u63a2\u7d22\u90fd\u53ef\u80fd\u5f97\u5230\u4e00\u4e2a\u672a\u66fe\u6599\u60f3\u7684\u7b54\u6848\u3002</p>"},{"location":"chapter_searching/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>10.1 \u00a0 \u4e8c\u5206\u67e5\u627e</li> <li>10.2 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9</li> <li>10.3 \u00a0 \u4e8c\u5206\u67e5\u627e\u8fb9\u754c</li> <li>10.4 \u00a0 \u54c8\u5e0c\u4f18\u5316\u7b56\u7565</li> <li>10.5 \u00a0 \u91cd\u8bc6\u641c\u7d22\u7b97\u6cd5</li> <li>10.6 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_searching/binary_search/","title":"10.1 \u00a0 \u4e8c\u5206\u67e5\u627e","text":"<p>\u4e8c\u5206\u67e5\u627e\uff08binary search\uff09\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u9ad8\u6548\u641c\u7d22\u7b97\u6cd5\u3002\u5b83\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\u641c\u7d22\u8303\u56f4\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u4e3a\u6b62\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 <code>nums</code> \uff0c\u5143\u7d20\u6309\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\u4e14\u4e0d\u91cd\u590d\u3002\u8bf7\u67e5\u627e\u5e76\u8fd4\u56de\u5143\u7d20 <code>target</code> \u5728\u8be5\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u56fe 10-1 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 10-1 \u00a0 \u4e8c\u5206\u67e5\u627e\u793a\u4f8b\u6570\u636e </p> <p>\u5982\u56fe 10-2 \u6240\u793a\uff0c\u6211\u4eec\u5148\u521d\u59cb\u5316\u6307\u9488 \\(i = 0\\) \u548c \\(j = n - 1\\) \uff0c\u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u548c\u5c3e\u5143\u7d20\uff0c\u4ee3\u8868\u641c\u7d22\u533a\u95f4 \\([0, n - 1]\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u4e2d\u62ec\u53f7\u8868\u793a\u95ed\u533a\u95f4\uff0c\u5176\u5305\u542b\u8fb9\u754c\u503c\u672c\u8eab\u3002</p> <p>\u63a5\u4e0b\u6765\uff0c\u5faa\u73af\u6267\u884c\u4ee5\u4e0b\u4e24\u6b65\u3002</p> <ol> <li>\u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m = \\lfloor {(i + j) / 2} \\rfloor\\) \uff0c\u5176\u4e2d \\(\\lfloor \\: \\rfloor\\) \u8868\u793a\u5411\u4e0b\u53d6\u6574\u64cd\u4f5c\u3002</li> <li>\u5224\u65ad <code>nums[m]</code> \u548c <code>target</code> \u7684\u5927\u5c0f\u5173\u7cfb\uff0c\u5206\u4e3a\u4ee5\u4e0b\u4e09\u79cd\u60c5\u51b5\u3002<ol> <li>\u5f53 <code>nums[m] &lt; target</code> \u65f6\uff0c\u8bf4\u660e <code>target</code> \u5728\u533a\u95f4 \\([m + 1, j]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(i = m + 1\\) \u3002</li> <li>\u5f53 <code>nums[m] &gt; target</code> \u65f6\uff0c\u8bf4\u660e <code>target</code> \u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(j = m - 1\\) \u3002</li> <li>\u5f53 <code>nums[m] = target</code> \u65f6\uff0c\u8bf4\u660e\u627e\u5230 <code>target</code> \uff0c\u56e0\u6b64\u8fd4\u56de\u7d22\u5f15 \\(m\\) \u3002</li> </ol> </li> </ol> <p>\u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u76ee\u6807\u5143\u7d20\uff0c\u641c\u7d22\u533a\u95f4\u6700\u7ec8\u4f1a\u7f29\u5c0f\u4e3a\u7a7a\u3002\u6b64\u65f6\u8fd4\u56de \\(-1\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 10-2 \u00a0 \u4e8c\u5206\u67e5\u627e\u6d41\u7a0b </p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e \\(i\\) \u548c \\(j\\) \u90fd\u662f <code>int</code> \u7c7b\u578b\uff0c\u56e0\u6b64 \\(i + j\\) \u53ef\u80fd\u4f1a\u8d85\u51fa <code>int</code> \u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u3002\u4e3a\u4e86\u907f\u514d\u5927\u6570\u8d8a\u754c\uff0c\u6211\u4eec\u901a\u5e38\u91c7\u7528\u516c\u5f0f \\(m = \\lfloor {i + (j - i) / 2} \\rfloor\\) \u6765\u8ba1\u7b97\u4e2d\u70b9\u3002</p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search.py<pre><code>def binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\"\"\"\n    # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    i, j = 0, len(nums) - 1\n    # \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt;= j:\n        # \u7406\u8bba\u4e0a Python \u7684\u6570\u5b57\u53ef\u4ee5\u65e0\u9650\u5927\uff08\u53d6\u51b3\u4e8e\u5185\u5b58\u5927\u5c0f\uff09\uff0c\u65e0\u987b\u8003\u8651\u5927\u6570\u8d8a\u754c\u95ee\u9898\n        m = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        elif nums[m] &gt; target:\n            j = m - 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        else:\n            return m  # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n</code></pre> binary_search.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = nums.size() - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = nums.length - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint BinarySearch(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = nums.Length - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2;   // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        else                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    i, j := 0, len(nums)-1\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    for i &lt;= j {\n        m := i + (j-i)/2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums, target) {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    let i = 0,\n        j = nums.length - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        else return m; // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums: number[], target: number): number {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    let i = 0,\n        j = nums.length - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        } else {\n            // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    return -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n</code></pre> binary_search.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n  int i = 0, j = nums.length - 1;\n  // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n      j = m - 1;\n    } else {\n      // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n      return m;\n    }\n  }\n  // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    let mut i = 0;\n    let mut j = nums.len() as i32 - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        } else {\n            // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int *nums, int len, int target) {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = len - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    var i = 0\n    var j = nums.size - 1\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        else  // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>[class]{}-[func]{binary_search}\n</code></pre> binary_search.zig<pre><code>// \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\nfn binarySearch(comptime T: type, nums: std.ArrayList(T), target: T) T {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    var i: usize = 0;\n    var j: usize = nums.items.len - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        var m = i + (j - i) / 2;                // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums.items[m] &lt; target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        } else if (nums.items[m] &gt; target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        } else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return @intCast(m);\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \uff1a\u5728\u4e8c\u5206\u5faa\u73af\u4e2d\uff0c\u533a\u95f4\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\uff0c\u56e0\u6b64\u5faa\u73af\u6b21\u6570\u4e3a \\(\\log_2 n\\) \u3002</p> <p>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7a7a\u95f4\u3002</p>"},{"location":"chapter_searching/binary_search/#1011","title":"10.1.1 \u00a0 \u533a\u95f4\u8868\u793a\u65b9\u6cd5","text":"<p>\u9664\u4e86\u4e0a\u8ff0\u53cc\u95ed\u533a\u95f4\u5916\uff0c\u5e38\u89c1\u7684\u533a\u95f4\u8868\u793a\u8fd8\u6709\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u533a\u95f4\uff0c\u5b9a\u4e49\u4e3a \\([0, n)\\) \uff0c\u5373\u5de6\u8fb9\u754c\u5305\u542b\u81ea\u8eab\uff0c\u53f3\u8fb9\u754c\u4e0d\u5305\u542b\u81ea\u8eab\u3002\u5728\u8be5\u8868\u793a\u4e0b\uff0c\u533a\u95f4 \\([i, j)\\) \u5728 \\(i = j\\) \u65f6\u4e3a\u7a7a\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u8be5\u8868\u793a\u5b9e\u73b0\u5177\u6709\u76f8\u540c\u529f\u80fd\u7684\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search.py<pre><code>def binary_search_lcro(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09\"\"\"\n    # \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    i, j = 0, len(nums)\n    # \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt; j:\n        m = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n        elif nums[m] &gt; target:\n            j = m  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n        else:\n            return m  # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n</code></pre> binary_search.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(vector&lt;int&gt; &amp;nums, int target) {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = nums.size();\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = nums.length;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint BinarySearchLCRO(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = nums.Length;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2;   // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        else                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunc binarySearchLCRO(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    i, j := 0, len(nums)\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    for i &lt; j {\n        m := i + (j-i)/2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m\n        } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunc binarySearchLCRO(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    var i = nums.startIndex\n    var j = nums.endIndex\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt; j {\n        let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m\n        } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunction binarySearchLCRO(nums, target) {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    let i = 0,\n        j = nums.length;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        else return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunction binarySearchLCRO(nums: number[], target: number): number {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    let i = 0,\n        j = nums.length;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        } else {\n            // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    return -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n</code></pre> binary_search.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(List&lt;int&gt; nums, int target) {\n  // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n  int i = 0, j = nums.length;\n  // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n  while (i &lt; j) {\n    int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n      j = m;\n    } else {\n      // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n      return m;\n    }\n  }\n  // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfn binary_search_lcro(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    let mut i = 0;\n    let mut j = nums.len() as i32;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt; j {\n        let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        } else {\n            // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(int *nums, int len, int target) {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = len;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfun binarySearchLCRO(nums: IntArray, target: Int): Int {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    var i = 0\n    var j = nums.size\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        val m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m\n        else  // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>[class]{}-[func]{binary_search_lcro}\n</code></pre> binary_search.zig<pre><code>// \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09\nfn binarySearchLCRO(comptime T: type, nums: std.ArrayList(T), target: T) T {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    var i: usize = 0;\n    var j: usize = nums.items.len;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        var m = i + (j - i) / 2;                // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums.items[m] &lt; target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        } else if (nums.items[m] &gt; target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        } else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return @intCast(m);\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u5982\u56fe 10-3 \u6240\u793a\uff0c\u5728\u4e24\u79cd\u533a\u95f4\u8868\u793a\u4e0b\uff0c\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u521d\u59cb\u5316\u3001\u5faa\u73af\u6761\u4ef6\u548c\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\u7686\u6709\u6240\u4e0d\u540c\u3002</p> <p>\u7531\u4e8e\u201c\u53cc\u95ed\u533a\u95f4\u201d\u8868\u793a\u4e2d\u7684\u5de6\u53f3\u8fb9\u754c\u90fd\u88ab\u5b9a\u4e49\u4e3a\u95ed\u533a\u95f4\uff0c\u56e0\u6b64\u901a\u8fc7\u6307\u9488 \\(i\\) \u548c\u6307\u9488 \\(j\\) \u7f29\u5c0f\u533a\u95f4\u7684\u64cd\u4f5c\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u8fd9\u6837\u66f4\u4e0d\u5bb9\u6613\u51fa\u9519\uff0c\u56e0\u6b64\u4e00\u822c\u5efa\u8bae\u91c7\u7528\u201c\u53cc\u95ed\u533a\u95f4\u201d\u7684\u5199\u6cd5\u3002</p> <p></p> <p> \u56fe 10-3 \u00a0 \u4e24\u79cd\u533a\u95f4\u5b9a\u4e49 </p>"},{"location":"chapter_searching/binary_search/#1012","title":"10.1.2 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u4e8c\u5206\u67e5\u627e\u5728\u65f6\u95f4\u548c\u7a7a\u95f4\u65b9\u9762\u90fd\u6709\u8f83\u597d\u7684\u6027\u80fd\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\u7684\u65f6\u95f4\u6548\u7387\u9ad8\u3002\u5728\u5927\u6570\u636e\u91cf\u4e0b\uff0c\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5177\u6709\u663e\u8457\u4f18\u52bf\u3002\u4f8b\u5982\uff0c\u5f53\u6570\u636e\u5927\u5c0f \\(n = 2^{20}\\) \u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u9700\u8981 \\(2^{20} = 1048576\\) \u8f6e\u5faa\u73af\uff0c\u800c\u4e8c\u5206\u67e5\u627e\u4ec5\u9700 \\(\\log_2 2^{20} = 20\\) \u8f6e\u5faa\u73af\u3002</li> <li>\u4e8c\u5206\u67e5\u627e\u65e0\u987b\u989d\u5916\u7a7a\u95f4\u3002\u76f8\u8f83\u4e8e\u9700\u8981\u501f\u52a9\u989d\u5916\u7a7a\u95f4\u7684\u641c\u7d22\u7b97\u6cd5\uff08\u4f8b\u5982\u54c8\u5e0c\u67e5\u627e\uff09\uff0c\u4e8c\u5206\u67e5\u627e\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002</li> </ul> <p>\u7136\u800c\uff0c\u4e8c\u5206\u67e5\u627e\u5e76\u975e\u9002\u7528\u4e8e\u6240\u6709\u60c5\u51b5\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6709\u5e8f\u6570\u636e\u3002\u82e5\u8f93\u5165\u6570\u636e\u65e0\u5e8f\uff0c\u4e3a\u4e86\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u800c\u4e13\u95e8\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u4e0d\u507f\u5931\u3002\u56e0\u4e3a\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u6bd4\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u90fd\u66f4\u9ad8\u3002\u5bf9\u4e8e\u9891\u7e41\u63d2\u5165\u5143\u7d20\u7684\u573a\u666f\uff0c\u4e3a\u4fdd\u6301\u6570\u7ec4\u6709\u5e8f\u6027\uff0c\u9700\u8981\u5c06\u5143\u7d20\u63d2\u5165\u5230\u7279\u5b9a\u4f4d\u7f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u4e5f\u662f\u975e\u5e38\u6602\u8d35\u7684\u3002</li> <li>\u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002\u4e8c\u5206\u67e5\u627e\u9700\u8981\u8df3\u8dc3\u5f0f\uff08\u975e\u8fde\u7eed\u5730\uff09\u8bbf\u95ee\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u6267\u884c\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u7684\u6548\u7387\u8f83\u4f4e\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5e94\u7528\u5728\u94fe\u8868\u6216\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u5c0f\u6570\u636e\u91cf\u4e0b\uff0c\u7ebf\u6027\u67e5\u627e\u6027\u80fd\u66f4\u4f73\u3002\u5728\u7ebf\u6027\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u9700 1 \u6b21\u5224\u65ad\u64cd\u4f5c\uff1b\u800c\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u9700\u8981 1 \u6b21\u52a0\u6cd5\u30011 \u6b21\u9664\u6cd5\u30011 ~ 3 \u6b21\u5224\u65ad\u64cd\u4f5c\u30011 \u6b21\u52a0\u6cd5\uff08\u51cf\u6cd5\uff09\uff0c\u5171 4 ~ 6 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u56e0\u6b64\uff0c\u5f53\u6570\u636e\u91cf \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u53cd\u800c\u6bd4\u4e8c\u5206\u67e5\u627e\u66f4\u5feb\u3002</li> </ul>"},{"location":"chapter_searching/binary_search_edge/","title":"10.3 \u00a0 \u4e8c\u5206\u67e5\u627e\u8fb9\u754c","text":""},{"location":"chapter_searching/binary_search_edge/#1031","title":"10.3.1 \u00a0 \u67e5\u627e\u5de6\u8fb9\u754c","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\u3002\u8bf7\u8fd4\u56de\u6570\u7ec4\u4e2d\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 <code>target</code> \u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e2d\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002</p> <p>\u56de\u5fc6\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\u7684\u65b9\u6cd5\uff0c\u641c\u7d22\u5b8c\u6210\u540e \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u4e2a <code>target</code> \uff0c\u56e0\u6b64\u67e5\u627e\u63d2\u5165\u70b9\u672c\u8d28\u4e0a\u662f\u5728\u67e5\u627e\u6700\u5de6\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\u3002</p> <p>\u8003\u8651\u901a\u8fc7\u67e5\u627e\u63d2\u5165\u70b9\u7684\u51fd\u6570\u5b9e\u73b0\u67e5\u627e\u5de6\u8fb9\u754c\u3002\u8bf7\u6ce8\u610f\uff0c\u6570\u7ec4\u4e2d\u53ef\u80fd\u4e0d\u5305\u542b <code>target</code> \uff0c\u8fd9\u79cd\u60c5\u51b5\u53ef\u80fd\u5bfc\u81f4\u4ee5\u4e0b\u4e24\u79cd\u7ed3\u679c\u3002</p> <ul> <li>\u63d2\u5165\u70b9\u7684\u7d22\u5f15 \\(i\\) \u8d8a\u754c\u3002</li> <li>\u5143\u7d20 <code>nums[i]</code> \u4e0e <code>target</code> \u4e0d\u76f8\u7b49\u3002</li> </ul> <p>\u5f53\u9047\u5230\u4ee5\u4e0a\u4e24\u79cd\u60c5\u51b5\u65f6\uff0c\u76f4\u63a5\u8fd4\u56de \\(-1\\) \u5373\u53ef\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_edge.py<pre><code>def binary_search_left_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target\"\"\"\n    # \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    i = binary_search_insertion(nums, target)\n    # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == len(nums) or nums[i] != target:\n        return -1\n    # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n</code></pre> binary_search_edge.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    int i = binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.size() || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(int[] nums, int target) {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    int i = binary_search_insertion.binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.length || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint BinarySearchLeftEdge(int[] nums, int target) {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.Length || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunc binarySearchLeftEdge(nums []int, target int) int {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    i := binarySearchInsertion(nums, target)\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == len(nums) || nums[i] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n}\n</code></pre> binary_search_edge.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunc binarySearchLeftEdge(nums: [Int], target: Int) -&gt; Int {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    let i = binarySearchInsertion(nums: nums, target: target)\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == nums.endIndex || nums[i] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n}\n</code></pre> binary_search_edge.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunction binarySearchLeftEdge(nums, target) {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    const i = binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunction binarySearchLeftEdge(nums: Array&lt;number&gt;, target: number): number {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    const i = binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(List&lt;int&gt; nums, int target) {\n  // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n  int i = binarySearchInsertion(nums, target);\n  // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n  if (i == nums.length || nums[i] != target) {\n    return -1;\n  }\n  // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n  return i;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfn binary_search_left_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    let i = binary_search_insertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == nums.len() as i32 || nums[i as usize] != target {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    i\n}\n</code></pre> binary_search_edge.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(int *nums, int numSize, int target) {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    int i = binarySearchInsertion(nums, numSize, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == numSize || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfun binarySearchLeftEdge(nums: IntArray, target: Int): Int {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    val i = binarySearchInsertion(nums, target)\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.size || nums[i] != target) {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n}\n</code></pre> binary_search_edge.rb<pre><code>[class]{}-[func]{binary_search_left_edge}\n</code></pre> binary_search_edge.zig<pre><code>[class]{}-[func]{binarySearchLeftEdge}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_searching/binary_search_edge/#1032","title":"10.3.2 \u00a0 \u67e5\u627e\u53f3\u8fb9\u754c","text":"<p>\u90a3\u4e48\u5982\u4f55\u67e5\u627e\u6700\u53f3\u4e00\u4e2a <code>target</code> \u5462\uff1f\u6700\u76f4\u63a5\u7684\u65b9\u5f0f\u662f\u4fee\u6539\u4ee3\u7801\uff0c\u66ff\u6362\u5728 <code>nums[m] == target</code> \u60c5\u51b5\u4e0b\u7684\u6307\u9488\u6536\u7f29\u64cd\u4f5c\u3002\u4ee3\u7801\u5728\u6b64\u7701\u7565\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002</p> <p>\u4e0b\u9762\u6211\u4eec\u4ecb\u7ecd\u4e24\u79cd\u66f4\u52a0\u53d6\u5de7\u7684\u65b9\u6cd5\u3002</p>"},{"location":"chapter_searching/binary_search_edge/#1","title":"1. \u00a0 \u590d\u7528\u67e5\u627e\u5de6\u8fb9\u754c","text":"<p>\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u67e5\u627e\u6700\u5de6\u5143\u7d20\u7684\u51fd\u6570\u6765\u67e5\u627e\u6700\u53f3\u5143\u7d20\uff0c\u5177\u4f53\u65b9\u6cd5\u4e3a\uff1a\u5c06\u67e5\u627e\u6700\u53f3\u4e00\u4e2a <code>target</code> \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a <code>target + 1</code>\u3002</p> <p>\u5982\u56fe 10-7 \u6240\u793a\uff0c\u67e5\u627e\u5b8c\u6210\u540e\uff0c\u6307\u9488 \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u4e2a <code>target + 1</code>\uff08\u5982\u679c\u5b58\u5728\uff09\uff0c\u800c \\(j\\) \u6307\u5411\u6700\u53f3\u4e00\u4e2a <code>target</code> \uff0c\u56e0\u6b64\u8fd4\u56de \\(j\\) \u5373\u53ef\u3002</p> <p></p> <p> \u56fe 10-7 \u00a0 \u5c06\u67e5\u627e\u53f3\u8fb9\u754c\u8f6c\u5316\u4e3a\u67e5\u627e\u5de6\u8fb9\u754c </p> <p>\u8bf7\u6ce8\u610f\uff0c\u8fd4\u56de\u7684\u63d2\u5165\u70b9\u662f \\(i\\) \uff0c\u56e0\u6b64\u9700\u8981\u5c06\u5176\u51cf \\(1\\) \uff0c\u4ece\u800c\u83b7\u5f97 \\(j\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_edge.py<pre><code>def binary_search_right_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target\"\"\"\n    # \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    i = binary_search_insertion(nums, target + 1)\n    # j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    j = i - 1\n    # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 or nums[j] != target:\n        return -1\n    # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n</code></pre> binary_search_edge.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    int i = binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(int[] nums, int target) {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    int i = binary_search_insertion.binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint BinarySearchRightEdge(int[] nums, int target) {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunc binarySearchRightEdge(nums []int, target int) int {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    i := binarySearchInsertion(nums, target+1)\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    j := i - 1\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n}\n</code></pre> binary_search_edge.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunc binarySearchRightEdge(nums: [Int], target: Int) -&gt; Int {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    let i = binarySearchInsertion(nums: nums, target: target + 1)\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    let j = i - 1\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n}\n</code></pre> binary_search_edge.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunction binarySearchRightEdge(nums, target) {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    const i = binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    const j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunction binarySearchRightEdge(nums: Array&lt;number&gt;, target: number): number {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    const i = binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    const j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(List&lt;int&gt; nums, int target) {\n  // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n  int i = binarySearchInsertion(nums, target + 1);\n  // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n  int j = i - 1;\n  // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n  if (j == -1 || nums[j] != target) {\n    return -1;\n  }\n  // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n  return j;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfn binary_search_right_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    let i = binary_search_insertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    let j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 || nums[j as usize] != target {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    j\n}\n</code></pre> binary_search_edge.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(int *nums, int numSize, int target) {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    int i = binarySearchInsertion(nums, numSize, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfun binarySearchRightEdge(nums: IntArray, target: Int): Int {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    val i = binarySearchInsertion(nums, target + 1)\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    val j = i - 1\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n}\n</code></pre> binary_search_edge.rb<pre><code>[class]{}-[func]{binary_search_right_edge}\n</code></pre> binary_search_edge.zig<pre><code>[class]{}-[func]{binarySearchRightEdge}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_searching/binary_search_edge/#2","title":"2. \u00a0 \u8f6c\u5316\u4e3a\u67e5\u627e\u5143\u7d20","text":"<p>\u6211\u4eec\u77e5\u9053\uff0c\u5f53\u6570\u7ec4\u4e0d\u5305\u542b <code>target</code> \u65f6\uff0c\u6700\u7ec8 \\(i\\) \u548c \\(j\\) \u4f1a\u5206\u522b\u6307\u5411\u9996\u4e2a\u5927\u4e8e\u3001\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u3002</p> <p>\u56e0\u6b64\uff0c\u5982\u56fe 10-8 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u6784\u9020\u4e00\u4e2a\u6570\u7ec4\u4e2d\u4e0d\u5b58\u5728\u7684\u5143\u7d20\uff0c\u7528\u4e8e\u67e5\u627e\u5de6\u53f3\u8fb9\u754c\u3002</p> <ul> <li>\u67e5\u627e\u6700\u5de6\u4e00\u4e2a <code>target</code> \uff1a\u53ef\u4ee5\u8f6c\u5316\u4e3a\u67e5\u627e <code>target - 0.5</code> \uff0c\u5e76\u8fd4\u56de\u6307\u9488 \\(i\\) \u3002</li> <li>\u67e5\u627e\u6700\u53f3\u4e00\u4e2a <code>target</code> \uff1a\u53ef\u4ee5\u8f6c\u5316\u4e3a\u67e5\u627e <code>target + 0.5</code> \uff0c\u5e76\u8fd4\u56de\u6307\u9488 \\(j\\) \u3002</li> </ul> <p></p> <p> \u56fe 10-8 \u00a0 \u5c06\u67e5\u627e\u8fb9\u754c\u8f6c\u5316\u4e3a\u67e5\u627e\u5143\u7d20 </p> <p>\u4ee3\u7801\u5728\u6b64\u7701\u7565\uff0c\u4ee5\u4e0b\u4e24\u70b9\u503c\u5f97\u6ce8\u610f\u3002</p> <ul> <li>\u7ed9\u5b9a\u6570\u7ec4\u4e0d\u5305\u542b\u5c0f\u6570\uff0c\u8fd9\u610f\u5473\u7740\u6211\u4eec\u65e0\u987b\u5173\u5fc3\u5982\u4f55\u5904\u7406\u76f8\u7b49\u7684\u60c5\u51b5\u3002</li> <li>\u56e0\u4e3a\u8be5\u65b9\u6cd5\u5f15\u5165\u4e86\u5c0f\u6570\uff0c\u6240\u4ee5\u9700\u8981\u5c06\u51fd\u6570\u4e2d\u7684\u53d8\u91cf <code>target</code> \u6539\u4e3a\u6d6e\u70b9\u6570\u7c7b\u578b\uff08Python \u65e0\u987b\u6539\u52a8\uff09\u3002</li> </ul>"},{"location":"chapter_searching/binary_search_insertion/","title":"10.2 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9","text":"<p>\u4e8c\u5206\u67e5\u627e\u4e0d\u4ec5\u53ef\u7528\u4e8e\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff0c\u8fd8\u53ef\u7528\u4e8e\u89e3\u51b3\u8bb8\u591a\u53d8\u79cd\u95ee\u9898\uff0c\u6bd4\u5982\u641c\u7d22\u76ee\u6807\u5143\u7d20\u7684\u63d2\u5165\u4f4d\u7f6e\u3002</p>"},{"location":"chapter_searching/binary_search_insertion/#1021","title":"10.2.1 \u00a0 \u65e0\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u5143\u7d20 <code>target</code> \uff0c\u6570\u7ec4\u4e0d\u5b58\u5728\u91cd\u590d\u5143\u7d20\u3002\u73b0\u5c06 <code>target</code> \u63d2\u5165\u6570\u7ec4 <code>nums</code> \u4e2d\uff0c\u5e76\u4fdd\u6301\u5176\u6709\u5e8f\u6027\u3002\u82e5\u6570\u7ec4\u4e2d\u5df2\u5b58\u5728\u5143\u7d20 <code>target</code> \uff0c\u5219\u63d2\u5165\u5230\u5176\u5de6\u65b9\u3002\u8bf7\u8fd4\u56de\u63d2\u5165\u540e <code>target</code> \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15\u3002\u793a\u4f8b\u5982\u56fe 10-4 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 10-4 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\u793a\u4f8b\u6570\u636e </p> <p>\u5982\u679c\u60f3\u590d\u7528\u4e0a\u4e00\u8282\u7684\u4e8c\u5206\u67e5\u627e\u4ee3\u7801\uff0c\u5219\u9700\u8981\u56de\u7b54\u4ee5\u4e0b\u4e24\u4e2a\u95ee\u9898\u3002</p> <p>\u95ee\u9898\u4e00\uff1a\u5f53\u6570\u7ec4\u4e2d\u5305\u542b <code>target</code> \u65f6\uff0c\u63d2\u5165\u70b9\u7684\u7d22\u5f15\u662f\u5426\u662f\u8be5\u5143\u7d20\u7684\u7d22\u5f15\uff1f</p> <p>\u9898\u76ee\u8981\u6c42\u5c06 <code>target</code> \u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u5de6\u8fb9\uff0c\u8fd9\u610f\u5473\u7740\u65b0\u63d2\u5165\u7684 <code>target</code> \u66ff\u6362\u4e86\u539f\u6765 <code>target</code> \u7684\u4f4d\u7f6e\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5f53\u6570\u7ec4\u5305\u542b <code>target</code> \u65f6\uff0c\u63d2\u5165\u70b9\u7684\u7d22\u5f15\u5c31\u662f\u8be5 <code>target</code> \u7684\u7d22\u5f15\u3002</p> <p>\u95ee\u9898\u4e8c\uff1a\u5f53\u6570\u7ec4\u4e2d\u4e0d\u5b58\u5728 <code>target</code> \u65f6\uff0c\u63d2\u5165\u70b9\u662f\u54ea\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff1f</p> <p>\u8fdb\u4e00\u6b65\u601d\u8003\u4e8c\u5206\u67e5\u627e\u8fc7\u7a0b\uff1a\u5f53 <code>nums[m] &lt; target</code> \u65f6 \\(i\\) \u79fb\u52a8\uff0c\u8fd9\u610f\u5473\u7740\u6307\u9488 \\(i\\) \u5728\u5411\u5927\u4e8e\u7b49\u4e8e <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002\u540c\u7406\uff0c\u6307\u9488 \\(j\\) \u59cb\u7ec8\u5728\u5411\u5c0f\u4e8e\u7b49\u4e8e <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002</p> <p>\u56e0\u6b64\u4e8c\u5206\u7ed3\u675f\u65f6\u4e00\u5b9a\u6709\uff1a\\(i\\) \u6307\u5411\u9996\u4e2a\u5927\u4e8e <code>target</code> \u7684\u5143\u7d20\uff0c\\(j\\) \u6307\u5411\u9996\u4e2a\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u3002\u6613\u5f97\u5f53\u6570\u7ec4\u4e0d\u5305\u542b <code>target</code> \u65f6\uff0c\u63d2\u5165\u7d22\u5f15\u4e3a \\(i\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_insertion.py<pre><code>def binary_search_insertion_simple(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09\"\"\"\n    i, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while i &lt;= j:\n        m = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        elif nums[m] &gt; target:\n            j = m - 1  # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        else:\n            return m  # \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n    # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint BinarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        } else {\n            // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n            return m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target {\n            i = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if nums[m] &gt; target {\n            j = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(\n    nums: Array&lt;number&gt;,\n    target: number\n): number {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n    } else {\n      return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n    }\n  }\n  // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfn binary_search_insertion_simple(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m;\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfun binarySearchInsertionSimple(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>[class]{}-[func]{binary_search_insertion_simple}\n</code></pre> binary_search_insertion.zig<pre><code>[class]{}-[func]{binarySearchInsertionSimple}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_searching/binary_search_insertion/#1022","title":"10.2.2 \u00a0 \u5b58\u5728\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u5728\u4e0a\u4e00\u9898\u7684\u57fa\u7840\u4e0a\uff0c\u89c4\u5b9a\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u5176\u4f59\u4e0d\u53d8\u3002</p> <p>\u5047\u8bbe\u6570\u7ec4\u4e2d\u5b58\u5728\u591a\u4e2a <code>target</code> \uff0c\u5219\u666e\u901a\u4e8c\u5206\u67e5\u627e\u53ea\u80fd\u8fd4\u56de\u5176\u4e2d\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\uff0c\u800c\u65e0\u6cd5\u786e\u5b9a\u8be5\u5143\u7d20\u7684\u5de6\u8fb9\u548c\u53f3\u8fb9\u8fd8\u6709\u591a\u5c11 <code>target</code>\u3002</p> <p>\u9898\u76ee\u8981\u6c42\u5c06\u76ee\u6807\u5143\u7d20\u63d2\u5165\u5230\u6700\u5de6\u8fb9\uff0c\u6240\u4ee5\u6211\u4eec\u9700\u8981\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5de6\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\u3002\u521d\u6b65\u8003\u8651\u901a\u8fc7\u56fe 10-5 \u6240\u793a\u7684\u6b65\u9aa4\u5b9e\u73b0\u3002</p> <ol> <li>\u6267\u884c\u4e8c\u5206\u67e5\u627e\uff0c\u5f97\u5230\u4efb\u610f\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\uff0c\u8bb0\u4e3a \\(k\\) \u3002</li> <li>\u4ece\u7d22\u5f15 \\(k\\) \u5f00\u59cb\uff0c\u5411\u5de6\u8fdb\u884c\u7ebf\u6027\u904d\u5386\uff0c\u5f53\u627e\u5230\u6700\u5de6\u8fb9\u7684 <code>target</code> \u65f6\u8fd4\u56de\u3002</li> </ol> <p></p> <p> \u56fe 10-5 \u00a0 \u7ebf\u6027\u67e5\u627e\u91cd\u590d\u5143\u7d20\u7684\u63d2\u5165\u70b9 </p> <p>\u6b64\u65b9\u6cd5\u867d\u7136\u53ef\u7528\uff0c\u4f46\u5176\u5305\u542b\u7ebf\u6027\u67e5\u627e\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5f53\u6570\u7ec4\u4e2d\u5b58\u5728\u5f88\u591a\u91cd\u590d\u7684 <code>target</code> \u65f6\uff0c\u8be5\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\u3002</p> <p>\u73b0\u8003\u8651\u62d3\u5c55\u4e8c\u5206\u67e5\u627e\u4ee3\u7801\u3002\u5982\u56fe 10-6 \u6240\u793a\uff0c\u6574\u4f53\u6d41\u7a0b\u4fdd\u6301\u4e0d\u53d8\uff0c\u6bcf\u8f6e\u5148\u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m\\) \uff0c\u518d\u5224\u65ad <code>target</code> \u548c <code>nums[m]</code> \u7684\u5927\u5c0f\u5173\u7cfb\uff0c\u5206\u4e3a\u4ee5\u4e0b\u51e0\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u5f53 <code>nums[m] &lt; target</code> \u6216 <code>nums[m] &gt; target</code> \u65f6\uff0c\u8bf4\u660e\u8fd8\u6ca1\u6709\u627e\u5230 <code>target</code> \uff0c\u56e0\u6b64\u91c7\u7528\u666e\u901a\u4e8c\u5206\u67e5\u627e\u7684\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5411 <code>target</code> \u9760\u8fd1\u3002</li> <li>\u5f53 <code>nums[m] == target</code> \u65f6\uff0c\u8bf4\u660e\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u91c7\u7528 \\(j = m - 1\\) \u6765\u7f29\u5c0f\u533a\u95f4\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(j\\) \u5411\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002</li> </ul> <p>\u5faa\u73af\u5b8c\u6210\u540e\uff0c\\(i\\) \u6307\u5411\u6700\u5de6\u8fb9\u7684 <code>target</code> \uff0c\\(j\\) \u6307\u5411\u9996\u4e2a\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\uff0c\u56e0\u6b64\u7d22\u5f15 \\(i\\) \u5c31\u662f\u63d2\u5165\u70b9\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 10-6 \u00a0 \u4e8c\u5206\u67e5\u627e\u91cd\u590d\u5143\u7d20\u7684\u63d2\u5165\u70b9\u7684\u6b65\u9aa4 </p> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u5224\u65ad\u5206\u652f <code>nums[m] &gt; target</code> \u548c <code>nums[m] == target</code> \u7684\u64cd\u4f5c\u76f8\u540c\uff0c\u56e0\u6b64\u4e24\u8005\u53ef\u4ee5\u5408\u5e76\u3002</p> <p>\u5373\u4fbf\u5982\u6b64\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u5c06\u5224\u65ad\u6761\u4ef6\u4fdd\u6301\u5c55\u5f00\uff0c\u56e0\u4e3a\u5176\u903b\u8f91\u66f4\u52a0\u6e05\u6670\u3001\u53ef\u8bfb\u6027\u66f4\u597d\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_insertion.py<pre><code>def binary_search_insertion(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09\"\"\"\n    i, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while i &lt;= j:\n        m = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        elif nums[m] &gt; target:\n            j = m - 1  # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        else:\n            j = m - 1  # \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n    # \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint BinarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        } else {\n            // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target {\n            i = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if nums[m] &gt; target {\n            j = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1 // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums: Array&lt;number&gt;, target: number): number {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n    } else {\n      j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n    }\n  }\n  // \u8fd4\u56de\u63d2\u5165\u70b9 i\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\npub fn binary_search_insertion(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfun binarySearchInsertion(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1 // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>[class]{}-[func]{binary_search_insertion}\n</code></pre> binary_search_insertion.zig<pre><code>[class]{}-[func]{binarySearchInsertion}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>Tip</p> <p>\u672c\u8282\u7684\u4ee3\u7801\u90fd\u662f\u201c\u53cc\u95ed\u533a\u95f4\u201d\u5199\u6cd5\u3002\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u5199\u6cd5\u3002</p> <p>\u603b\u7684\u6765\u770b\uff0c\u4e8c\u5206\u67e5\u627e\u65e0\u975e\u5c31\u662f\u7ed9\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5206\u522b\u8bbe\u5b9a\u641c\u7d22\u76ee\u6807\uff0c\u76ee\u6807\u53ef\u80fd\u662f\u4e00\u4e2a\u5177\u4f53\u7684\u5143\u7d20\uff08\u4f8b\u5982 <code>target</code> \uff09\uff0c\u4e5f\u53ef\u80fd\u662f\u4e00\u4e2a\u5143\u7d20\u8303\u56f4\uff08\u4f8b\u5982\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\uff09\u3002</p> <p>\u5728\u4e0d\u65ad\u7684\u5faa\u73af\u4e8c\u5206\u4e2d\uff0c\u6307\u9488 \\(i\\) \u548c \\(j\\) \u90fd\u9010\u6e10\u903c\u8fd1\u9884\u5148\u8bbe\u5b9a\u7684\u76ee\u6807\u3002\u6700\u7ec8\uff0c\u5b83\u4eec\u6216\u662f\u6210\u529f\u627e\u5230\u7b54\u6848\uff0c\u6216\u662f\u8d8a\u8fc7\u8fb9\u754c\u540e\u505c\u6b62\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/","title":"10.4 \u00a0 \u54c8\u5e0c\u4f18\u5316\u7b56\u7565","text":"<p>\u5728\u7b97\u6cd5\u9898\u4e2d\uff0c\u6211\u4eec\u5e38\u901a\u8fc7\u5c06\u7ebf\u6027\u67e5\u627e\u66ff\u6362\u4e3a\u54c8\u5e0c\u67e5\u627e\u6765\u964d\u4f4e\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u7b97\u6cd5\u9898\u6765\u52a0\u6df1\u7406\u89e3\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6574\u6570\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u76ee\u6807\u5143\u7d20 <code>target</code> \uff0c\u8bf7\u5728\u6570\u7ec4\u4e2d\u641c\u7d22\u201c\u548c\u201d\u4e3a <code>target</code> \u7684\u4e24\u4e2a\u5143\u7d20\uff0c\u5e76\u8fd4\u56de\u5b83\u4eec\u7684\u6570\u7ec4\u7d22\u5f15\u3002\u8fd4\u56de\u4efb\u610f\u4e00\u4e2a\u89e3\u5373\u53ef\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/#1041","title":"10.4.1 \u00a0 \u7ebf\u6027\u67e5\u627e\uff1a\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4","text":"<p>\u8003\u8651\u76f4\u63a5\u904d\u5386\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\u3002\u5982\u56fe 10-9 \u6240\u793a\uff0c\u6211\u4eec\u5f00\u542f\u4e00\u4e2a\u4e24\u5c42\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u5224\u65ad\u4e24\u4e2a\u6574\u6570\u7684\u548c\u662f\u5426\u4e3a <code>target</code> \uff0c\u82e5\u662f\uff0c\u5219\u8fd4\u56de\u5b83\u4eec\u7684\u7d22\u5f15\u3002</p> <p></p> <p> \u56fe 10-9 \u00a0 \u7ebf\u6027\u67e5\u627e\u6c42\u89e3\u4e24\u6570\u4e4b\u548c </p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig two_sum.py<pre><code>def two_sum_brute_force(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"\u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\"\"\"\n    # \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for i in range(len(nums) - 1):\n        for j in range(i + 1, len(nums)):\n            if nums[i] + nums[j] == target:\n                return [i, j]\n    return []\n</code></pre> two_sum.cpp<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nvector&lt;int&gt; twoSumBruteForce(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return {i, j};\n        }\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] twoSumBruteForce(int[] nums, int target) {\n    int size = nums.length;\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return new int[] { i, j };\n        }\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] TwoSumBruteForce(int[] nums, int target) {\n    int size = nums.Length;\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return [i, j];\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums []int, target int) []int {\n    size := len(nums)\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for i := 0; i &lt; size-1; i++ {\n        for j := i + 1; i &lt; size; j++ {\n            if nums[i]+nums[j] == target {\n                return []int{i, j}\n            }\n        }\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums: [Int], target: Int) -&gt; [Int] {\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for i in nums.indices.dropLast() {\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[i] + nums[j] == target {\n                return [i, j]\n            }\n        }\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums, target) {\n    const n = nums.length;\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\n    const n = nums.length;\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* \u65b9\u6cd5\u4e00\uff1a \u66b4\u529b\u679a\u4e3e */\nList&lt;int&gt; twoSumBruteForce(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n  for (var i = 0; i &lt; size - 1; i++) {\n    for (var j = i + 1; j &lt; size; j++) {\n      if (nums[i] + nums[j] == target) return [i, j];\n    }\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\npub fn two_sum_brute_force(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    let size = nums.len();\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for i in 0..size - 1 {\n        for j in i + 1..size {\n            if nums[i] + nums[j] == target {\n                return Some(vec![i as i32, j as i32]);\n            }\n        }\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint *twoSumBruteForce(int *nums, int numsSize, int target, int *returnSize) {\n    for (int i = 0; i &lt; numsSize; ++i) {\n        for (int j = i + 1; j &lt; numsSize; ++j) {\n            if (nums[i] + nums[j] == target) {\n                int *res = malloc(sizeof(int) * 2);\n                res[0] = i, res[1] = j;\n                *returnSize = 2;\n                return res;\n            }\n        }\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfun twoSumBruteForce(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (i in 0..&lt;size - 1) {\n        for (j in i + 1..&lt;size) {\n            if (nums[i] + nums[j] == target) return intArrayOf(i, j)\n        }\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>[class]{}-[func]{two_sum_brute_force}\n</code></pre> two_sum.zig<pre><code>// \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\nfn twoSumBruteForce(nums: []i32, target: i32) ?[2]i32 {\n    var size: usize = nums.len;\n    var i: usize = 0;\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    while (i &lt; size - 1) : (i += 1) {\n        var j = i + 1;\n        while (j &lt; size) : (j += 1) {\n            if (nums[i] + nums[j] == target) {\n                return [_]i32{@intCast(i), @intCast(j)};\n            }\n        }\n    }\n    return null;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u6b64\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u975e\u5e38\u8017\u65f6\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/#1042","title":"10.4.2 \u00a0 \u54c8\u5e0c\u67e5\u627e\uff1a\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4","text":"<p>\u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u952e\u503c\u5bf9\u5206\u522b\u4e3a\u6570\u7ec4\u5143\u7d20\u548c\u5143\u7d20\u7d22\u5f15\u3002\u5faa\u73af\u904d\u5386\u6570\u7ec4\uff0c\u6bcf\u8f6e\u6267\u884c\u56fe 10-10 \u6240\u793a\u7684\u6b65\u9aa4\u3002</p> <ol> <li>\u5224\u65ad\u6570\u5b57 <code>target - nums[i]</code> \u662f\u5426\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u82e5\u662f\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u3002</li> <li>\u5c06\u952e\u503c\u5bf9 <code>nums[i]</code> \u548c\u7d22\u5f15 <code>i</code> \u6dfb\u52a0\u8fdb\u54c8\u5e0c\u8868\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> \u56fe 10-10 \u00a0 \u8f85\u52a9\u54c8\u5e0c\u8868\u6c42\u89e3\u4e24\u6570\u4e4b\u548c </p> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff0c\u4ec5\u9700\u5355\u5c42\u5faa\u73af\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig two_sum.py<pre><code>def two_sum_hash_table(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"\u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\"\"\"\n    # \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    dic = {}\n    # \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for i in range(len(nums)):\n        if target - nums[i] in dic:\n            return [dic[target - nums[i]], i]\n        dic[nums[i]] = i\n    return []\n</code></pre> two_sum.cpp<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nvector&lt;int&gt; twoSumHashTable(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    unordered_map&lt;int, int&gt; dic;\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.find(target - nums[i]) != dic.end()) {\n            return {dic[target - nums[i]], i};\n        }\n        dic.emplace(nums[i], i);\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] twoSumHashTable(int[] nums, int target) {\n    int size = nums.length;\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    Map&lt;Integer, Integer&gt; dic = new HashMap&lt;&gt;();\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.containsKey(target - nums[i])) {\n            return new int[] { dic.get(target - nums[i]), i };\n        }\n        dic.put(nums[i], i);\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] TwoSumHashTable(int[] nums, int target) {\n    int size = nums.Length;\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    Dictionary&lt;int, int&gt; dic = [];\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.ContainsKey(target - nums[i])) {\n            return [dic[target - nums[i]], i];\n        }\n        dic.Add(nums[i], i);\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums []int, target int) []int {\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    hashTable := map[int]int{}\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for idx, val := range nums {\n        if preIdx, ok := hashTable[target-val]; ok {\n            return []int{preIdx, idx}\n        }\n        hashTable[val] = idx\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums: [Int], target: Int) -&gt; [Int] {\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    var dic: [Int: Int] = [:]\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for i in nums.indices {\n        if let j = dic[target - nums[i]] {\n            return [j, i]\n        }\n        dic[nums[i]] = i\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums, target) {\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    let m = {};\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (m[target - nums[i]] !== undefined) {\n            return [m[target - nums[i]], i];\n        } else {\n            m[nums[i]] = i;\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    let m: Map&lt;number, number&gt; = new Map();\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (let i = 0; i &lt; nums.length; i++) {\n        let index = m.get(target - nums[i]);\n        if (index !== undefined) {\n            return [index, i];\n        } else {\n            m.set(nums[i], i);\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a \u8f85\u52a9\u54c8\u5e0c\u8868 */\nList&lt;int&gt; twoSumHashTable(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n  Map&lt;int, int&gt; dic = HashMap();\n  // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n  for (var i = 0; i &lt; size; i++) {\n    if (dic.containsKey(target - nums[i])) {\n      return [dic[target - nums[i]]!, i];\n    }\n    dic.putIfAbsent(nums[i], () =&gt; i);\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\npub fn two_sum_hash_table(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    let mut dic = HashMap::new();\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (i, num) in nums.iter().enumerate() {\n        match dic.get(&amp;(target - num)) {\n            Some(v) =&gt; return Some(vec![*v as i32, i as i32]),\n            None =&gt; dic.insert(num, i as i32),\n        };\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* \u54c8\u5e0c\u8868 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n\n/* \u54c8\u5e0c\u8868\u67e5\u8be2 */\nHashTable *find(HashTable *h, int key) {\n    HashTable *tmp;\n    HASH_FIND_INT(h, &amp;key, tmp);\n    return tmp;\n}\n\n/* \u54c8\u5e0c\u8868\u5143\u7d20\u63d2\u5165 */\nvoid insert(HashTable *h, int key, int val) {\n    HashTable *t = find(h, key);\n    if (t == NULL) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = key, tmp-&gt;val = val;\n        HASH_ADD_INT(h, key, tmp);\n    } else {\n        t-&gt;val = val;\n    }\n}\n\n/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint *twoSumHashTable(int *nums, int numsSize, int target, int *returnSize) {\n    HashTable *hashtable = NULL;\n    for (int i = 0; i &lt; numsSize; i++) {\n        HashTable *t = find(hashtable, target - nums[i]);\n        if (t != NULL) {\n            int *res = malloc(sizeof(int) * 2);\n            res[0] = t-&gt;val, res[1] = i;\n            *returnSize = 2;\n            return res;\n        }\n        insert(hashtable, nums[i], i);\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfun twoSumHashTable(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    val dic = HashMap&lt;Int, Int&gt;()\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (i in 0..&lt;size) {\n        if (dic.containsKey(target - nums[i])) {\n            return intArrayOf(dic[target - nums[i]]!!, i)\n        }\n        dic[nums[i]] = i\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>[class]{}-[func]{two_sum_hash_table}\n</code></pre> two_sum.zig<pre><code>// \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\nfn twoSumHashTable(nums: []i32, target: i32) !?[2]i32 {\n    var size: usize = nums.len;\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    var dic = std.AutoHashMap(i32, i32).init(std.heap.page_allocator);\n    defer dic.deinit();\n    var i: usize = 0;\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    while (i &lt; size) : (i += 1) {\n        if (dic.contains(target - nums[i])) {\n            return [_]i32{dic.get(target - nums[i]).?, @intCast(i)};\n        }\n        try dic.put(nums[i], @intCast(i));\n    }\n    return null;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u6b64\u65b9\u6cd5\u901a\u8fc7\u54c8\u5e0c\u67e5\u627e\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \uff0c\u5927\u5e45\u63d0\u5347\u8fd0\u884c\u6548\u7387\u3002</p> <p>\u7531\u4e8e\u9700\u8981\u7ef4\u62a4\u4e00\u4e2a\u989d\u5916\u7684\u54c8\u5e0c\u8868\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5c3d\u7ba1\u5982\u6b64\uff0c\u8be5\u65b9\u6cd5\u7684\u6574\u4f53\u65f6\u7a7a\u6548\u7387\u66f4\u4e3a\u5747\u8861\uff0c\u56e0\u6b64\u5b83\u662f\u672c\u9898\u7684\u6700\u4f18\u89e3\u6cd5\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/","title":"10.5 \u00a0 \u91cd\u8bc6\u641c\u7d22\u7b97\u6cd5","text":"<p>\u641c\u7d22\u7b97\u6cd5\uff08searching algorithm\uff09\u7528\u4e8e\u5728\u6570\u636e\u7ed3\u6784\uff08\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u6216\u56fe\uff09\u4e2d\u641c\u7d22\u4e00\u4e2a\u6216\u4e00\u7ec4\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5143\u7d20\u3002</p> <p>\u641c\u7d22\u7b97\u6cd5\u53ef\u6839\u636e\u5b9e\u73b0\u601d\u8def\u5206\u4e3a\u4ee5\u4e0b\u4e24\u7c7b\u3002</p> <ul> <li>\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\uff0c\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u548c\u56fe\u7684\u904d\u5386\u7b49\u3002</li> <li>\u5229\u7528\u6570\u636e\u7ec4\u7ec7\u7ed3\u6784\u6216\u6570\u636e\u5305\u542b\u7684\u5148\u9a8c\u4fe1\u606f\uff0c\u5b9e\u73b0\u9ad8\u6548\u5143\u7d20\u67e5\u627e\uff0c\u4f8b\u5982\u4e8c\u5206\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u548c\u4e8c\u53c9\u641c\u7d22\u6811\u67e5\u627e\u7b49\u3002</li> </ul> <p>\u4e0d\u96be\u53d1\u73b0\uff0c\u8fd9\u4e9b\u77e5\u8bc6\u70b9\u90fd\u5df2\u5728\u524d\u9762\u7684\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u56e0\u6b64\u641c\u7d22\u7b97\u6cd5\u5bf9\u4e8e\u6211\u4eec\u6765\u8bf4\u5e76\u4e0d\u964c\u751f\u3002\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u4ece\u66f4\u52a0\u7cfb\u7edf\u7684\u89c6\u89d2\u5207\u5165\uff0c\u91cd\u65b0\u5ba1\u89c6\u641c\u7d22\u7b97\u6cd5\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1051","title":"10.5.1 \u00a0 \u66b4\u529b\u641c\u7d22","text":"<p>\u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u7684\u6bcf\u4e2a\u5143\u7d20\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002</p> <ul> <li>\u201c\u7ebf\u6027\u641c\u7d22\u201d\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\u7b49\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u5b83\u4ece\u6570\u636e\u7ed3\u6784\u7684\u4e00\u7aef\u5f00\u59cb\uff0c\u9010\u4e2a\u8bbf\u95ee\u5143\u7d20\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u5230\u8fbe\u53e6\u4e00\u7aef\u4ecd\u6ca1\u6709\u627e\u5230\u76ee\u6807\u5143\u7d20\u4e3a\u6b62\u3002</li> <li>\u201c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u548c\u201c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u662f\u56fe\u548c\u6811\u7684\u4e24\u79cd\u904d\u5386\u7b56\u7565\u3002\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\u9010\u5c42\u641c\u7d22\uff0c\u7531\u8fd1\u53ca\u8fdc\u5730\u8bbf\u95ee\u5404\u4e2a\u8282\u70b9\u3002\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\uff0c\u6cbf\u7740\u4e00\u6761\u8def\u5f84\u8d70\u5230\u5934\uff0c\u518d\u56de\u6eaf\u5e76\u5c1d\u8bd5\u5176\u4ed6\u8def\u5f84\uff0c\u76f4\u5230\u904d\u5386\u5b8c\u6574\u4e2a\u6570\u636e\u7ed3\u6784\u3002</li> </ul> <p>\u66b4\u529b\u641c\u7d22\u7684\u4f18\u70b9\u662f\u7b80\u5355\u4e14\u901a\u7528\u6027\u597d\uff0c\u65e0\u987b\u5bf9\u6570\u636e\u505a\u9884\u5904\u7406\u548c\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u7136\u800c\uff0c\u6b64\u7c7b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u5143\u7d20\u6570\u91cf\uff0c\u56e0\u6b64\u5728\u6570\u636e\u91cf\u8f83\u5927\u7684\u60c5\u51b5\u4e0b\u6027\u80fd\u8f83\u5dee\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1052","title":"10.5.2 \u00a0 \u81ea\u9002\u5e94\u641c\u7d22","text":"<p>\u81ea\u9002\u5e94\u641c\u7d22\u5229\u7528\u6570\u636e\u7684\u7279\u6709\u5c5e\u6027\uff08\u4f8b\u5982\u6709\u5e8f\u6027\uff09\u6765\u4f18\u5316\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ece\u800c\u66f4\u9ad8\u6548\u5730\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002</p> <ul> <li>\u201c\u4e8c\u5206\u67e5\u627e\u201d\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\u5b9e\u73b0\u9ad8\u6548\u67e5\u627e\uff0c\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002</li> <li>\u201c\u54c8\u5e0c\u67e5\u627e\u201d\u5229\u7528\u54c8\u5e0c\u8868\u5c06\u641c\u7d22\u6570\u636e\u548c\u76ee\u6807\u6570\u636e\u5efa\u7acb\u4e3a\u952e\u503c\u5bf9\u6620\u5c04\uff0c\u4ece\u800c\u5b9e\u73b0\u67e5\u8be2\u64cd\u4f5c\u3002</li> <li>\u201c\u6811\u67e5\u627e\u201d\u5728\u7279\u5b9a\u7684\u6811\u7ed3\u6784\uff08\u4f8b\u5982\u4e8c\u53c9\u641c\u7d22\u6811\uff09\u4e2d\uff0c\u57fa\u4e8e\u6bd4\u8f83\u8282\u70b9\u503c\u6765\u5feb\u901f\u6392\u9664\u8282\u70b9\uff0c\u4ece\u800c\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002</li> </ul> <p>\u6b64\u7c7b\u7b97\u6cd5\u7684\u4f18\u70b9\u662f\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002</p> <p>\u7136\u800c\uff0c\u4f7f\u7528\u8fd9\u4e9b\u7b97\u6cd5\u5f80\u5f80\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u9884\u5904\u7406\u3002\u4f8b\u5982\uff0c\u4e8c\u5206\u67e5\u627e\u9700\u8981\u9884\u5148\u5bf9\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\uff0c\u54c8\u5e0c\u67e5\u627e\u548c\u6811\u67e5\u627e\u90fd\u9700\u8981\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u7ef4\u62a4\u8fd9\u4e9b\u6570\u636e\u7ed3\u6784\u4e5f\u9700\u8981\u989d\u5916\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u5f00\u9500\u3002</p> <p>Tip</p> <p>\u81ea\u9002\u5e94\u641c\u7d22\u7b97\u6cd5\u5e38\u88ab\u79f0\u4e3a\u67e5\u627e\u7b97\u6cd5\uff0c\u4e3b\u8981\u7528\u4e8e\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u68c0\u7d22\u76ee\u6807\u5143\u7d20\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1053","title":"10.5.3 \u00a0 \u641c\u7d22\u65b9\u6cd5\u9009\u53d6","text":"<p>\u7ed9\u5b9a\u5927\u5c0f\u4e3a \\(n\\) \u7684\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u7ebf\u6027\u641c\u7d22\u3001\u4e8c\u5206\u67e5\u627e\u3001\u6811\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u7b49\u591a\u79cd\u65b9\u6cd5\u4ece\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\u3002\u5404\u4e2a\u65b9\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u5982\u56fe 10-11 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 10-11 \u00a0 \u591a\u79cd\u641c\u7d22\u7b56\u7565 </p> <p>\u4e0a\u8ff0\u51e0\u79cd\u65b9\u6cd5\u7684\u64cd\u4f5c\u6548\u7387\u4e0e\u7279\u6027\u5982\u8868 10-1 \u6240\u793a\u3002</p> <p> \u8868 10-1 \u00a0 \u67e5\u627e\u7b97\u6cd5\u6548\u7387\u5bf9\u6bd4 </p> \u7ebf\u6027\u641c\u7d22 \u4e8c\u5206\u67e5\u627e \u6811\u67e5\u627e \u54c8\u5e0c\u67e5\u627e \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u989d\u5916\u7a7a\u95f4 \\(O(1)\\) \\(O(1)\\) \\(O(n)\\) \\(O(n)\\) \u6570\u636e\u9884\u5904\u7406 / \u6392\u5e8f \\(O(n \\log n)\\) \u5efa\u6811 \\(O(n \\log n)\\) \u5efa\u54c8\u5e0c\u8868 \\(O(n)\\) \u6570\u636e\u662f\u5426\u6709\u5e8f \u65e0\u5e8f \u6709\u5e8f \u6709\u5e8f \u65e0\u5e8f <p>\u641c\u7d22\u7b97\u6cd5\u7684\u9009\u62e9\u8fd8\u53d6\u51b3\u4e8e\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u4e0e\u66f4\u65b0\u9891\u7387\u7b49\u3002</p> <p>\u7ebf\u6027\u641c\u7d22</p> <ul> <li>\u901a\u7528\u6027\u8f83\u597d\uff0c\u65e0\u987b\u4efb\u4f55\u6570\u636e\u9884\u5904\u7406\u64cd\u4f5c\u3002\u5047\u5982\u6211\u4eec\u4ec5\u9700\u67e5\u8be2\u4e00\u6b21\u6570\u636e\uff0c\u90a3\u4e48\u5176\u4ed6\u4e09\u79cd\u65b9\u6cd5\u7684\u6570\u636e\u9884\u5904\u7406\u7684\u65f6\u95f4\u6bd4\u7ebf\u6027\u641c\u7d22\u7684\u65f6\u95f4\u8fd8\u8981\u66f4\u957f\u3002</li> <li>\u9002\u7528\u4e8e\u4f53\u91cf\u8f83\u5c0f\u7684\u6570\u636e\uff0c\u6b64\u60c5\u51b5\u4e0b\u65f6\u95f4\u590d\u6742\u5ea6\u5bf9\u6548\u7387\u5f71\u54cd\u8f83\u5c0f\u3002</li> <li>\u9002\u7528\u4e8e\u6570\u636e\u66f4\u65b0\u9891\u7387\u8f83\u9ad8\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u8be5\u65b9\u6cd5\u4e0d\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u4efb\u4f55\u989d\u5916\u7ef4\u62a4\u3002</li> </ul> <p>\u4e8c\u5206\u67e5\u627e</p> <ul> <li>\u9002\u7528\u4e8e\u5927\u6570\u636e\u91cf\u7684\u60c5\u51b5\uff0c\u6548\u7387\u8868\u73b0\u7a33\u5b9a\uff0c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002</li> <li>\u6570\u636e\u91cf\u4e0d\u80fd\u8fc7\u5927\uff0c\u56e0\u4e3a\u5b58\u50a8\u6570\u7ec4\u9700\u8981\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u4e0d\u9002\u7528\u4e8e\u9ad8\u9891\u589e\u5220\u6570\u636e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u7ef4\u62a4\u6709\u5e8f\u6570\u7ec4\u7684\u5f00\u9500\u8f83\u5927\u3002</li> </ul> <p>\u54c8\u5e0c\u67e5\u627e</p> <ul> <li>\u9002\u5408\u5bf9\u67e5\u8be2\u6027\u80fd\u8981\u6c42\u5f88\u9ad8\u7684\u573a\u666f\uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</li> <li>\u4e0d\u9002\u5408\u9700\u8981\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u65e0\u6cd5\u7ef4\u62a4\u6570\u636e\u7684\u6709\u5e8f\u6027\u3002</li> <li>\u5bf9\u54c8\u5e0c\u51fd\u6570\u548c\u54c8\u5e0c\u51b2\u7a81\u5904\u7406\u7b56\u7565\u7684\u4f9d\u8d56\u6027\u8f83\u9ad8\uff0c\u5177\u6709\u8f83\u5927\u7684\u6027\u80fd\u52a3\u5316\u98ce\u9669\u3002</li> <li>\u4e0d\u9002\u5408\u6570\u636e\u91cf\u8fc7\u5927\u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u9700\u8981\u989d\u5916\u7a7a\u95f4\u6765\u6700\u5927\u7a0b\u5ea6\u5730\u51cf\u5c11\u51b2\u7a81\uff0c\u4ece\u800c\u63d0\u4f9b\u826f\u597d\u7684\u67e5\u8be2\u6027\u80fd\u3002</li> </ul> <p>\u6811\u67e5\u627e</p> <ul> <li>\u9002\u7528\u4e8e\u6d77\u91cf\u6570\u636e\uff0c\u56e0\u4e3a\u6811\u8282\u70b9\u5728\u5185\u5b58\u4e2d\u662f\u5206\u6563\u5b58\u50a8\u7684\u3002</li> <li>\u9002\u5408\u9700\u8981\u7ef4\u62a4\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\u3002</li> <li>\u5728\u6301\u7eed\u589e\u5220\u8282\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u4ea7\u751f\u503e\u659c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>\u82e5\u4f7f\u7528 AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u5219\u5404\u9879\u64cd\u4f5c\u53ef\u5728 \\(O(\\log n)\\) \u6548\u7387\u4e0b\u7a33\u5b9a\u8fd0\u884c\uff0c\u4f46\u7ef4\u62a4\u6811\u5e73\u8861\u7684\u64cd\u4f5c\u4f1a\u589e\u52a0\u989d\u5916\u7684\u5f00\u9500\u3002</li> </ul>"},{"location":"chapter_searching/summary/","title":"10.6 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u4e8c\u5206\u67e5\u627e\u4f9d\u8d56\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u901a\u8fc7\u5faa\u73af\u9010\u6b65\u7f29\u51cf\u4e00\u534a\u641c\u7d22\u533a\u95f4\u6765\u8fdb\u884c\u67e5\u627e\u3002\u5b83\u8981\u6c42\u8f93\u5165\u6570\u636e\u6709\u5e8f\uff0c\u4e14\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u6216\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u6570\u636e\u3002\u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\uff0c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u548c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u9002\u7528\u4e8e\u56fe\u548c\u6811\u3002\u6b64\u7c7b\u7b97\u6cd5\u901a\u7528\u6027\u597d\uff0c\u65e0\u987b\u5bf9\u6570\u636e\u8fdb\u884c\u9884\u5904\u7406\uff0c\u4f46\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u8f83\u9ad8\u3002</li> <li>\u54c8\u5e0c\u67e5\u627e\u3001\u6811\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u5c5e\u4e8e\u9ad8\u6548\u641c\u7d22\u65b9\u6cd5\uff0c\u53ef\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002\u6b64\u7c7b\u7b97\u6cd5\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \uff0c\u4f46\u901a\u5e38\u9700\u8981\u501f\u52a9\u989d\u5916\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u5bf9\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u548c\u66f4\u65b0\u9891\u7387\u7b49\u56e0\u7d20\u8fdb\u884c\u5177\u4f53\u5206\u6790\uff0c\u4ece\u800c\u9009\u62e9\u5408\u9002\u7684\u641c\u7d22\u65b9\u6cd5\u3002</li> <li>\u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u5c0f\u578b\u6216\u9891\u7e41\u66f4\u65b0\u7684\u6570\u636e\uff1b\u4e8c\u5206\u67e5\u627e\u9002\u7528\u4e8e\u5927\u578b\u3001\u6392\u5e8f\u7684\u6570\u636e\uff1b\u54c8\u5e0c\u67e5\u627e\u9002\u7528\u4e8e\u5bf9\u67e5\u8be2\u6548\u7387\u8981\u6c42\u8f83\u9ad8\u4e14\u65e0\u987b\u8303\u56f4\u67e5\u8be2\u7684\u6570\u636e\uff1b\u6811\u67e5\u627e\u9002\u7528\u4e8e\u9700\u8981\u7ef4\u62a4\u987a\u5e8f\u548c\u652f\u6301\u8303\u56f4\u67e5\u8be2\u7684\u5927\u578b\u52a8\u6001\u6570\u636e\u3002</li> <li>\u7528\u54c8\u5e0c\u67e5\u627e\u66ff\u6362\u7ebf\u6027\u67e5\u627e\u662f\u4e00\u79cd\u5e38\u7528\u7684\u4f18\u5316\u8fd0\u884c\u65f6\u95f4\u7684\u7b56\u7565\uff0c\u53ef\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002</li> </ul>"},{"location":"chapter_sorting/","title":"\u7b2c 11 \u7ae0 \u00a0 \u6392\u5e8f","text":"<p>Abstract</p> <p>\u6392\u5e8f\u72b9\u5982\u4e00\u628a\u5c06\u6df7\u4e71\u53d8\u4e3a\u79e9\u5e8f\u7684\u9b54\u6cd5\u94a5\u5319\uff0c\u4f7f\u6211\u4eec\u80fd\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u7406\u89e3\u4e0e\u5904\u7406\u6570\u636e\u3002</p> <p>\u65e0\u8bba\u662f\u7b80\u5355\u7684\u5347\u5e8f\uff0c\u8fd8\u662f\u590d\u6742\u7684\u5206\u7c7b\u6392\u5217\uff0c\u6392\u5e8f\u90fd\u5411\u6211\u4eec\u5c55\u793a\u4e86\u6570\u636e\u7684\u548c\u8c10\u7f8e\u611f\u3002</p>"},{"location":"chapter_sorting/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>11.1 \u00a0 \u6392\u5e8f\u7b97\u6cd5</li> <li>11.2 \u00a0 \u9009\u62e9\u6392\u5e8f</li> <li>11.3 \u00a0 \u5192\u6ce1\u6392\u5e8f</li> <li>11.4 \u00a0 \u63d2\u5165\u6392\u5e8f</li> <li>11.5 \u00a0 \u5feb\u901f\u6392\u5e8f</li> <li>11.6 \u00a0 \u5f52\u5e76\u6392\u5e8f</li> <li>11.7 \u00a0 \u5806\u6392\u5e8f</li> <li>11.8 \u00a0 \u6876\u6392\u5e8f</li> <li>11.9 \u00a0 \u8ba1\u6570\u6392\u5e8f</li> <li>11.10 \u00a0 \u57fa\u6570\u6392\u5e8f</li> <li>11.11 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_sorting/bubble_sort/","title":"11.3 \u00a0 \u5192\u6ce1\u6392\u5e8f","text":"<p>\u5192\u6ce1\u6392\u5e8f\uff08bubble sort\uff09\u901a\u8fc7\u8fde\u7eed\u5730\u6bd4\u8f83\u4e0e\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u5b9e\u73b0\u6392\u5e8f\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u5c31\u50cf\u6c14\u6ce1\u4ece\u5e95\u90e8\u5347\u5230\u9876\u90e8\u4e00\u6837\uff0c\u56e0\u6b64\u5f97\u540d\u5192\u6ce1\u6392\u5e8f\u3002</p> <p>\u5982\u56fe 11-4 \u6240\u793a\uff0c\u5192\u6ce1\u8fc7\u7a0b\u53ef\u4ee5\u5229\u7528\u5143\u7d20\u4ea4\u6362\u64cd\u4f5c\u6765\u6a21\u62df\uff1a\u4ece\u6570\u7ec4\u6700\u5de6\u7aef\u5f00\u59cb\u5411\u53f3\u904d\u5386\uff0c\u4f9d\u6b21\u6bd4\u8f83\u76f8\u90bb\u5143\u7d20\u5927\u5c0f\uff0c\u5982\u679c\u201c\u5de6\u5143\u7d20 &gt; \u53f3\u5143\u7d20\u201d\u5c31\u4ea4\u6362\u4e8c\u8005\u3002\u904d\u5386\u5b8c\u6210\u540e\uff0c\u6700\u5927\u7684\u5143\u7d20\u4f1a\u88ab\u79fb\u52a8\u5230\u6570\u7ec4\u7684\u6700\u53f3\u7aef\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-4 \u00a0 \u5229\u7528\u5143\u7d20\u4ea4\u6362\u64cd\u4f5c\u6a21\u62df\u5192\u6ce1 </p>"},{"location":"chapter_sorting/bubble_sort/#1131","title":"11.3.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6b65\u9aa4\u5982\u56fe 11-5 \u6240\u793a\u3002</p> <ol> <li>\u9996\u5148\uff0c\u5bf9 \\(n\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002</li> <li>\u63a5\u4e0b\u6765\uff0c\u5bf9\u5269\u4f59 \\(n - 1\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u7b2c\u4e8c\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002</li> <li>\u4ee5\u6b64\u7c7b\u63a8\uff0c\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u201c\u5192\u6ce1\u201d\u540e\uff0c\u524d \\(n - 1\\) \u5927\u7684\u5143\u7d20\u90fd\u88ab\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002</li> <li>\u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5c0f\u5143\u7d20\uff0c\u65e0\u987b\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002</li> </ol> <p></p> <p> \u56fe 11-5 \u00a0 \u5192\u6ce1\u6392\u5e8f\u6d41\u7a0b </p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig bubble_sort.py<pre><code>def bubble_sort(nums: list[int]):\n    \"\"\"\u5192\u6ce1\u6392\u5e8f\"\"\"\n    n = len(nums)\n    # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in range(n - 1, 0, -1):\n        # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n</code></pre> bubble_sort.cpp<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(vector&lt;int&gt; &amp;nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                // \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\n                swap(nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid BubbleSort(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums []int) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums: inout [Int]) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums.swapAt(j, j + 1)\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums: number[]): void {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(List&lt;int&gt; nums) {\n  // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n      }\n    }\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfn bubble_sort(nums: &amp;mut [i32]) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in (1..nums.len()).rev() {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int nums[], int size) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = size - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfun bubbleSort(nums: IntArray) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j] = nums[j+1].also { nums[j+1] = nums[j] }\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>[class]{}-[func]{bubble_sort}\n</code></pre> bubble_sort.zig<pre><code>// \u5192\u6ce1\u6392\u5e8f\nfn bubbleSort(nums: []i32) void {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    var i: usize = nums.len - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var j: usize = 0;\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/bubble_sort/#1132","title":"11.3.2 \u00a0 \u6548\u7387\u4f18\u5316","text":"<p>\u6211\u4eec\u53d1\u73b0\uff0c\u5982\u679c\u67d0\u8f6e\u201c\u5192\u6ce1\u201d\u4e2d\u6ca1\u6709\u6267\u884c\u4efb\u4f55\u4ea4\u6362\u64cd\u4f5c\uff0c\u8bf4\u660e\u6570\u7ec4\u5df2\u7ecf\u5b8c\u6210\u6392\u5e8f\uff0c\u53ef\u76f4\u63a5\u8fd4\u56de\u7ed3\u679c\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u589e\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d <code>flag</code> \u6765\u76d1\u6d4b\u8fd9\u79cd\u60c5\u51b5\uff0c\u4e00\u65e6\u51fa\u73b0\u5c31\u7acb\u5373\u8fd4\u56de\u3002</p> <p>\u7ecf\u8fc7\u4f18\u5316\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(n^2)\\) \uff1b\u4f46\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u53ef\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig bubble_sort.py<pre><code>def bubble_sort_with_flag(nums: list[int]):\n    \"\"\"\u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\"\"\"\n    n = len(nums)\n    # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in range(n - 1, 0, -1):\n        flag = False  # \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n                flag = True  # \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n        if not flag:\n            break  # \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n</code></pre> bubble_sort.cpp<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(vector&lt;int&gt; &amp;nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        bool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                // \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\n                swap(nums[j], nums[j + 1]);\n                flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag)\n            break; // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nvoid bubbleSortWithFlag(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        boolean flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag)\n            break; // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid BubbleSortWithFlag(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        bool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                flag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag) break;     // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums []int) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        flag := false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n                flag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if flag == false { // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        var flag = false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums.swapAt(j, j + 1)\n                flag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if !flag { // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag) break; // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums: number[]): void {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag) break; // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(List&lt;int&gt; nums) {\n  // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    bool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n    // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n      }\n    }\n    if (!flag) break; // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nfn bubble_sort_with_flag(nums: &amp;mut [i32]) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in (1..nums.len()).rev() {\n        let mut flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if !flag {\n            break; // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n        };\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(int nums[], int size) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = size - 1; i &gt; 0; i--) {\n        bool flag = false;\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n                flag = true;\n            }\n        }\n        if (!flag)\n            break;\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nfun bubbleSortWithFlag(nums: IntArray) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        var flag = false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j] = nums[j + 1].also { nums[j] = nums[j + 1] }\n                flag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag) break // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>[class]{}-[func]{bubble_sort_with_flag}\n</code></pre> bubble_sort.zig<pre><code>// \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\nfn bubbleSortWithFlag(nums: []i32) void {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    var i: usize = nums.len - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var flag = false;   // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        var j: usize = 0;\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true;\n            }\n        }\n        if (!flag) break;   // \u6b64\u8f6e\u201c\u5192\u6ce1\u201d\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/bubble_sort/#1133","title":"11.3.3 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5404\u8f6e\u201c\u5192\u6ce1\u201d\u904d\u5386\u7684\u6570\u7ec4\u957f\u5ea6\u4f9d\u6b21\u4e3a \\(n - 1\\)\u3001\\(n - 2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u603b\u548c\u4e3a \\((n - 1) n / 2\\) \u3002\u5728\u5f15\u5165 <code>flag</code> \u4f18\u5316\u540e\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5728\u201c\u5192\u6ce1\u201d\u4e2d\u9047\u5230\u76f8\u7b49\u5143\u7d20\u4e0d\u4ea4\u6362\u3002</li> </ul>"},{"location":"chapter_sorting/bucket_sort/","title":"11.8 \u00a0 \u6876\u6392\u5e8f","text":"<p>\u524d\u8ff0\u51e0\u79cd\u6392\u5e8f\u7b97\u6cd5\u90fd\u5c5e\u4e8e\u201c\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u901a\u8fc7\u6bd4\u8f83\u5143\u7d20\u95f4\u7684\u5927\u5c0f\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u6b64\u7c7b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u65e0\u6cd5\u8d85\u8d8a \\(O(n \\log n)\\) \u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u63a2\u8ba8\u51e0\u79cd\u201c\u975e\u6bd4\u8f83\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230\u7ebf\u6027\u9636\u3002</p> <p>\u6876\u6392\u5e8f\uff08bucket sort\uff09\u662f\u5206\u6cbb\u7b56\u7565\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u901a\u8fc7\u8bbe\u7f6e\u4e00\u4e9b\u5177\u6709\u5927\u5c0f\u987a\u5e8f\u7684\u6876\uff0c\u6bcf\u4e2a\u6876\u5bf9\u5e94\u4e00\u4e2a\u6570\u636e\u8303\u56f4\uff0c\u5c06\u6570\u636e\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff1b\u7136\u540e\uff0c\u5728\u6bcf\u4e2a\u6876\u5185\u90e8\u5206\u522b\u6267\u884c\u6392\u5e8f\uff1b\u6700\u7ec8\u6309\u7167\u6876\u7684\u987a\u5e8f\u5c06\u6240\u6709\u6570\u636e\u5408\u5e76\u3002</p>"},{"location":"chapter_sorting/bucket_sort/#1181","title":"11.8.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8003\u8651\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u5176\u5143\u7d20\u662f\u8303\u56f4 \\([0, 1)\\) \u5185\u7684\u6d6e\u70b9\u6570\u3002\u6876\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-13 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316 \\(k\\) \u4e2a\u6876\uff0c\u5c06 \\(n\\) \u4e2a\u5143\u7d20\u5206\u914d\u5230 \\(k\\) \u4e2a\u6876\u4e2d\u3002</li> <li>\u5bf9\u6bcf\u4e2a\u6876\u5206\u522b\u6267\u884c\u6392\u5e8f\uff08\u8fd9\u91cc\u91c7\u7528\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff09\u3002</li> <li>\u6309\u7167\u6876\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u5408\u5e76\u7ed3\u679c\u3002</li> </ol> <p></p> <p> \u56fe 11-13 \u00a0 \u6876\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b </p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig bucket_sort.py<pre><code>def bucket_sort(nums: list[float]):\n    \"\"\"\u6876\u6392\u5e8f\"\"\"\n    # \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    k = len(nums) // 2\n    buckets = [[] for _ in range(k)]\n    # 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for num in nums:\n        # \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        i = int(num * k)\n        # \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].append(num)\n    # 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for bucket in buckets:\n        # \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.sort()\n    # 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    i = 0\n    for bucket in buckets:\n        for num in bucket:\n            nums[i] = num\n            i += 1\n</code></pre> bucket_sort.cpp<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(vector&lt;float&gt; &amp;nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    int k = nums.size() / 2;\n    vector&lt;vector&lt;float&gt;&gt; buckets(k);\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (float num : nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        int i = num * k;\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\n        buckets[i].push_back(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        sort(bucket.begin(), bucket.end());\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    int i = 0;\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.java<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(float[] nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    int k = nums.length / 2;\n    List&lt;List&lt;Float&gt;&gt; buckets = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; k; i++) {\n        buckets.add(new ArrayList&lt;&gt;());\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (float num : nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        int i = (int) (num * k);\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets.get(i).add(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (List&lt;Float&gt; bucket : buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        Collections.sort(bucket);\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    int i = 0;\n    for (List&lt;Float&gt; bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.cs<pre><code>/* \u6876\u6392\u5e8f */\nvoid BucketSort(float[] nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    int k = nums.Length / 2;\n    List&lt;List&lt;float&gt;&gt; buckets = [];\n    for (int i = 0; i &lt; k; i++) {\n        buckets.Add([]);\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    foreach (float num in nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        int i = (int)(num * k);\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].Add(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    foreach (List&lt;float&gt; bucket in buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.Sort();\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    int j = 0;\n    foreach (List&lt;float&gt; bucket in buckets) {\n        foreach (float num in bucket) {\n            nums[j++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.go<pre><code>/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums []float64) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    k := len(nums) / 2\n    buckets := make([][]float64, k)\n    for i := 0; i &lt; k; i++ {\n        buckets[i] = make([]float64, 0)\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for _, num := range nums {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        i := int(num * float64(k))\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i] = append(buckets[i], num)\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for i := 0; i &lt; k; i++ {\n        // \u4f7f\u7528\u5185\u7f6e\u5207\u7247\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        sort.Float64s(buckets[i])\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    i := 0\n    for _, bucket := range buckets {\n        for _, num := range bucket {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> bucket_sort.swift<pre><code>/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums: inout [Double]) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    let k = nums.count / 2\n    var buckets = (0 ..&lt; k).map { _ in [Double]() }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for num in nums {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        let i = Int(num * Double(k))\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].append(num)\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for i in buckets.indices {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        buckets[i].sort()\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    var i = nums.startIndex\n    for bucket in buckets {\n        for num in bucket {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> bucket_sort.js<pre><code>/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    const k = nums.length / 2;\n    const buckets = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (const num of nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        const i = Math.floor(num * k);\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].push(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (const bucket of buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.ts<pre><code>/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums: number[]): void {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    const k = nums.length / 2;\n    const buckets: number[][] = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (const num of nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        const i = Math.floor(num * k);\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].push(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (const bucket of buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.dart<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(List&lt;double&gt; nums) {\n  // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n  int k = nums.length ~/ 2;\n  List&lt;List&lt;double&gt;&gt; buckets = List.generate(k, (index) =&gt; []);\n\n  // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n  for (double _num in nums) {\n    // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 _num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n    int i = (_num * k).toInt();\n    // \u5c06 _num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\n    buckets[i].add(_num);\n  }\n  // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n  for (List&lt;double&gt; bucket in buckets) {\n    bucket.sort();\n  }\n  // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n  int i = 0;\n  for (List&lt;double&gt; bucket in buckets) {\n    for (double _num in bucket) {\n      nums[i++] = _num;\n    }\n  }\n}\n</code></pre> bucket_sort.rs<pre><code>/* \u6876\u6392\u5e8f */\nfn bucket_sort(nums: &amp;mut [f64]) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    let k = nums.len() / 2;\n    let mut buckets = vec![vec![]; k];\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for &amp;mut num in &amp;mut *nums {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        let i = (num * k as f64) as usize;\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].push(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for bucket in &amp;mut buckets {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.sort_by(|a, b| a.partial_cmp(b).unwrap());\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    let mut i = 0;\n    for bucket in &amp;mut buckets {\n        for &amp;mut num in bucket {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> bucket_sort.c<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(float nums[], int size) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    int k = size / 2;\n    float **buckets = calloc(k, sizeof(float *));\n    for (int i = 0; i &lt; k; i++) {\n        // \u6bcf\u4e2a\u6876\u6700\u591a\u53ef\u4ee5\u5206\u914d size \u4e2a\u5143\u7d20\n        buckets[i] = calloc(size, sizeof(float));\n    }\n\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (int i = 0; i &lt; size; i++) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        int bucket_idx = nums[i] * k;\n        int j = 0;\n        // \u5982\u679c\u6876\u4e2d\u6709\u6570\u636e\u4e14\u6570\u636e\u5c0f\u4e8e\u5f53\u524d\u503c nums[i], \u8981\u5c06\u5176\u653e\u5230\u5f53\u524d\u6876\u7684\u540e\u9762\uff0c\u76f8\u5f53\u4e8e cpp \u4e2d\u7684 push_back\n        while (buckets[bucket_idx][j] &gt; 0 &amp;&amp; buckets[bucket_idx][j] &lt; nums[i]) {\n            j++;\n        }\n        float temp = nums[i];\n        while (j &lt; size &amp;&amp; buckets[bucket_idx][j] &gt; 0) {\n            swap(&amp;temp, &amp;buckets[bucket_idx][j]);\n            j++;\n        }\n        buckets[bucket_idx][j] = temp;\n    }\n\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (int i = 0; i &lt; k; i++) {\n        qsort(buckets[i], size, sizeof(float), compare_float);\n    }\n\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    for (int i = 0, j = 0; j &lt; k; j++) {\n        for (int l = 0; l &lt; size; l++) {\n            if (buckets[j][l] &gt; 0) {\n                nums[i++] = buckets[j][l];\n            }\n        }\n    }\n\n    // \u91ca\u653e\u4e0a\u8ff0\u5206\u914d\u7684\u5185\u5b58\n    for (int i = 0; i &lt; k; i++) {\n        free(buckets[i]);\n    }\n    free(buckets);\n}\n</code></pre> bucket_sort.kt<pre><code>/* \u6876\u6392\u5e8f */\nfun bucketSort(nums: FloatArray) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    val k = nums.size / 2\n    val buckets = ArrayList&lt;ArrayList&lt;Float&gt;&gt;()\n    for (i in 0..&lt;k) {\n        buckets.add(ArrayList())\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (num in nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        val i = (num * k).toInt()\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].add(num)\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (bucket in buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.sort()\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    var i = 0\n    for (bucket in buckets) {\n        for (num in bucket) {\n            nums[i++] = num\n        }\n    }\n}\n</code></pre> bucket_sort.rb<pre><code>[class]{}-[func]{bucket_sort}\n</code></pre> bucket_sort.zig<pre><code>[class]{}-[func]{bucketSort}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/bucket_sort/#1182","title":"11.8.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<p>\u6876\u6392\u5e8f\u9002\u7528\u4e8e\u5904\u7406\u4f53\u91cf\u5f88\u5927\u7684\u6570\u636e\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u6570\u636e\u5305\u542b 100 \u4e07\u4e2a\u5143\u7d20\uff0c\u7531\u4e8e\u7a7a\u95f4\u9650\u5236\uff0c\u7cfb\u7edf\u5185\u5b58\u65e0\u6cd5\u4e00\u6b21\u6027\u52a0\u8f7d\u6240\u6709\u6570\u636e\u3002\u6b64\u65f6\uff0c\u53ef\u4ee5\u5c06\u6570\u636e\u5206\u6210 1000 \u4e2a\u6876\uff0c\u7136\u540e\u5206\u522b\u5bf9\u6bcf\u4e2a\u6876\u8fdb\u884c\u6392\u5e8f\uff0c\u6700\u540e\u5c06\u7ed3\u679c\u5408\u5e76\u3002</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n + k)\\) \uff1a\u5047\u8bbe\u5143\u7d20\u5728\u5404\u4e2a\u6876\u5185\u5e73\u5747\u5206\u5e03\uff0c\u90a3\u4e48\u6bcf\u4e2a\u6876\u5185\u7684\u5143\u7d20\u6570\u91cf\u4e3a \\(\\frac{n}{k}\\) \u3002\u5047\u8bbe\u6392\u5e8f\u5355\u4e2a\u6876\u4f7f\u7528 \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) \u65f6\u95f4\uff0c\u5219\u6392\u5e8f\u6240\u6709\u6876\u4f7f\u7528 \\(O(n \\log\\frac{n}{k})\\) \u65f6\u95f4\u3002\u5f53\u6876\u6570\u91cf \\(k\\) \u6bd4\u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5219\u8d8b\u5411\u4e8e \\(O(n)\\) \u3002\u5408\u5e76\u7ed3\u679c\u65f6\u9700\u8981\u904d\u5386\u6240\u6709\u6876\u548c\u5143\u7d20\uff0c\u82b1\u8d39 \\(O(n + k)\\) \u65f6\u95f4\u3002</li> <li>\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u6570\u636e\u88ab\u5206\u914d\u5230\u4e00\u4e2a\u6876\u4e2d\uff0c\u4e14\u6392\u5e8f\u8be5\u6876\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n + k)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u9700\u8981\u501f\u52a9 \\(k\\) \u4e2a\u6876\u548c\u603b\u5171 \\(n\\) \u4e2a\u5143\u7d20\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u6876\u6392\u5e8f\u662f\u5426\u7a33\u5b9a\u53d6\u51b3\u4e8e\u6392\u5e8f\u6876\u5185\u5143\u7d20\u7684\u7b97\u6cd5\u662f\u5426\u7a33\u5b9a\u3002</li> </ul>"},{"location":"chapter_sorting/bucket_sort/#1183","title":"11.8.3 \u00a0 \u5982\u4f55\u5b9e\u73b0\u5e73\u5747\u5206\u914d","text":"<p>\u6876\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7406\u8bba\u4e0a\u53ef\u4ee5\u8fbe\u5230 \\(O(n)\\) \uff0c\u5173\u952e\u5728\u4e8e\u5c06\u5143\u7d20\u5747\u5300\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u56e0\u4e3a\u5b9e\u9645\u6570\u636e\u5f80\u5f80\u4e0d\u662f\u5747\u5300\u5206\u5e03\u7684\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u60f3\u8981\u5c06\u6dd8\u5b9d\u4e0a\u7684\u6240\u6709\u5546\u54c1\u6309\u4ef7\u683c\u8303\u56f4\u5e73\u5747\u5206\u914d\u5230 10 \u4e2a\u6876\u4e2d\uff0c\u4f46\u5546\u54c1\u4ef7\u683c\u5206\u5e03\u4e0d\u5747\uff0c\u4f4e\u4e8e 100 \u5143\u7684\u975e\u5e38\u591a\uff0c\u9ad8\u4e8e 1000 \u5143\u7684\u975e\u5e38\u5c11\u3002\u82e5\u5c06\u4ef7\u683c\u533a\u95f4\u5e73\u5747\u5212\u5206\u4e3a 10 \u4e2a\uff0c\u5404\u4e2a\u6876\u4e2d\u7684\u5546\u54c1\u6570\u91cf\u5dee\u8ddd\u4f1a\u975e\u5e38\u5927\u3002</p> <p>\u4e3a\u5b9e\u73b0\u5e73\u5747\u5206\u914d\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u8bbe\u5b9a\u4e00\u6761\u5927\u81f4\u7684\u5206\u754c\u7ebf\uff0c\u5c06\u6570\u636e\u7c97\u7565\u5730\u5206\u5230 3 \u4e2a\u6876\u4e2d\u3002\u5206\u914d\u5b8c\u6bd5\u540e\uff0c\u518d\u5c06\u5546\u54c1\u8f83\u591a\u7684\u6876\u7ee7\u7eed\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u76f4\u81f3\u6240\u6709\u6876\u4e2d\u7684\u5143\u7d20\u6570\u91cf\u5927\u81f4\u76f8\u7b49\u3002</p> <p>\u5982\u56fe 11-14 \u6240\u793a\uff0c\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u521b\u5efa\u4e00\u68f5\u9012\u5f52\u6811\uff0c\u76ee\u6807\u662f\u8ba9\u53f6\u8282\u70b9\u7684\u503c\u5c3d\u53ef\u80fd\u5e73\u5747\u3002\u5f53\u7136\uff0c\u4e0d\u4e00\u5b9a\u8981\u6bcf\u8f6e\u5c06\u6570\u636e\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u5177\u4f53\u5212\u5206\u65b9\u5f0f\u53ef\u6839\u636e\u6570\u636e\u7279\u70b9\u7075\u6d3b\u9009\u62e9\u3002</p> <p></p> <p> \u56fe 11-14 \u00a0 \u9012\u5f52\u5212\u5206\u6876 </p> <p>\u5982\u679c\u6211\u4eec\u63d0\u524d\u77e5\u9053\u5546\u54c1\u4ef7\u683c\u7684\u6982\u7387\u5206\u5e03\uff0c\u5219\u53ef\u4ee5\u6839\u636e\u6570\u636e\u6982\u7387\u5206\u5e03\u8bbe\u7f6e\u6bcf\u4e2a\u6876\u7684\u4ef7\u683c\u5206\u754c\u7ebf\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6570\u636e\u5206\u5e03\u5e76\u4e0d\u4e00\u5b9a\u9700\u8981\u7279\u610f\u7edf\u8ba1\uff0c\u4e5f\u53ef\u4ee5\u6839\u636e\u6570\u636e\u7279\u70b9\u91c7\u7528\u67d0\u79cd\u6982\u7387\u6a21\u578b\u8fdb\u884c\u8fd1\u4f3c\u3002</p> <p>\u5982\u56fe 11-15 \u6240\u793a\uff0c\u6211\u4eec\u5047\u8bbe\u5546\u54c1\u4ef7\u683c\u670d\u4ece\u6b63\u6001\u5206\u5e03\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5408\u7406\u5730\u8bbe\u5b9a\u4ef7\u683c\u533a\u95f4\uff0c\u4ece\u800c\u5c06\u5546\u54c1\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\u3002</p> <p></p> <p> \u56fe 11-15 \u00a0 \u6839\u636e\u6982\u7387\u5206\u5e03\u5212\u5206\u6876 </p>"},{"location":"chapter_sorting/counting_sort/","title":"11.9 \u00a0 \u8ba1\u6570\u6392\u5e8f","text":"<p>\u8ba1\u6570\u6392\u5e8f\uff08counting sort\uff09\u901a\u8fc7\u7edf\u8ba1\u5143\u7d20\u6570\u91cf\u6765\u5b9e\u73b0\u6392\u5e8f\uff0c\u901a\u5e38\u5e94\u7528\u4e8e\u6574\u6570\u6570\u7ec4\u3002</p>"},{"location":"chapter_sorting/counting_sort/#1191","title":"11.9.1 \u00a0 \u7b80\u5355\u5b9e\u73b0","text":"<p>\u5148\u6765\u770b\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\u3002\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d\u7684\u5143\u7d20\u90fd\u662f\u201c\u975e\u8d1f\u6574\u6570\u201d\uff0c\u8ba1\u6570\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-16 \u6240\u793a\u3002</p> <ol> <li>\u904d\u5386\u6570\u7ec4\uff0c\u627e\u51fa\u5176\u4e2d\u7684\u6700\u5927\u6570\u5b57\uff0c\u8bb0\u4e3a \\(m\\) \uff0c\u7136\u540e\u521b\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(m + 1\\) \u7684\u8f85\u52a9\u6570\u7ec4 <code>counter</code> \u3002</li> <li>\u501f\u52a9 <code>counter</code> \u7edf\u8ba1 <code>nums</code> \u4e2d\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\uff0c\u5176\u4e2d <code>counter[num]</code> \u5bf9\u5e94\u6570\u5b57 <code>num</code> \u7684\u51fa\u73b0\u6b21\u6570\u3002\u7edf\u8ba1\u65b9\u6cd5\u5f88\u7b80\u5355\uff0c\u53ea\u9700\u904d\u5386 <code>nums</code>\uff08\u8bbe\u5f53\u524d\u6570\u5b57\u4e3a <code>num</code>\uff09\uff0c\u6bcf\u8f6e\u5c06 <code>counter[num]</code> \u589e\u52a0 \\(1\\) \u5373\u53ef\u3002</li> <li>\u7531\u4e8e <code>counter</code> \u7684\u5404\u4e2a\u7d22\u5f15\u5929\u7136\u6709\u5e8f\uff0c\u56e0\u6b64\u76f8\u5f53\u4e8e\u6240\u6709\u6570\u5b57\u5df2\u7ecf\u6392\u5e8f\u597d\u4e86\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u904d\u5386 <code>counter</code> \uff0c\u6839\u636e\u5404\u6570\u5b57\u51fa\u73b0\u6b21\u6570\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u586b\u5165 <code>nums</code> \u5373\u53ef\u3002</li> </ol> <p></p> <p> \u56fe 11-16 \u00a0 \u8ba1\u6570\u6392\u5e8f\u6d41\u7a0b </p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig counting_sort.py<pre><code>def counting_sort_naive(nums: list[int]):\n    \"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n    # \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\n    # 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    m = 0\n    for num in nums:\n        m = max(m, num)\n    # 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    i = 0\n    for num in range(m + 1):\n        for _ in range(counter[num]):\n            nums[i] = num\n            i += 1\n</code></pre> counting_sort.cpp<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(vector&lt;int&gt; &amp;nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.java<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int[] nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid CountingSortNaive(int[] nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums []int) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    for i, num := 0, 0; num &lt; m+1; num++ {\n        for j := 0; j &lt; counter[num]; j++ {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.swift<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums: inout [Int]) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = nums.max()!\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    var i = 0\n    for num in 0 ..&lt; m + 1 {\n        for _ in 0 ..&lt; counter[num] {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums: number[]): void {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(List&lt;int&gt; nums) {\n  // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n  // counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73b0\u6b21\u6570\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n  int i = 0;\n  for (int _num = 0; _num &lt; m + 1; _num++) {\n    for (int j = 0; j &lt; counter[_num]; j++, i++) {\n      nums[i] = _num;\n    }\n  }\n}\n</code></pre> counting_sort.rs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfn counting_sort_naive(nums: &amp;mut [i32]) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = *nums.into_iter().max().unwrap();\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    let mut counter = vec![0; m as usize + 1];\n    for &amp;num in &amp;*nums {\n        counter[num as usize] += 1;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    let mut i = 0;\n    for num in 0..m + 1 {\n        for _ in 0..counter[num as usize] {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> counting_sort.c<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int nums[], int size) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int *counter = calloc(m + 1, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n    // 4. \u91ca\u653e\u5185\u5b58\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfun countingSortNaive(nums: IntArray) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    var m = 0\n    for (num in nums) {\n        m = max(m.toDouble(), num.toDouble()).toInt()\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    var i = 0\n    for (num in 0..&lt;m + 1) {\n        var j = 0\n        while (j &lt; counter[num]) {\n            nums[i] = num\n            j++\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.rb<pre><code>[class]{}-[func]{counting_sort_naive}\n</code></pre> counting_sort.zig<pre><code>[class]{}-[func]{countingSortNaive}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u8ba1\u6570\u6392\u5e8f\u4e0e\u6876\u6392\u5e8f\u7684\u8054\u7cfb</p> <p>\u4ece\u6876\u6392\u5e8f\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u8ba1\u6570\u6392\u5e8f\u4e2d\u7684\u8ba1\u6570\u6570\u7ec4 <code>counter</code> \u7684\u6bcf\u4e2a\u7d22\u5f15\u89c6\u4e3a\u4e00\u4e2a\u6876\uff0c\u5c06\u7edf\u8ba1\u6570\u91cf\u7684\u8fc7\u7a0b\u770b\u4f5c\u5c06\u5404\u4e2a\u5143\u7d20\u5206\u914d\u5230\u5bf9\u5e94\u7684\u6876\u4e2d\u3002\u672c\u8d28\u4e0a\uff0c\u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u5728\u6574\u578b\u6570\u636e\u4e0b\u7684\u4e00\u4e2a\u7279\u4f8b\u3002</p>"},{"location":"chapter_sorting/counting_sort/#1192","title":"11.9.2 \u00a0 \u5b8c\u6574\u5b9e\u73b0","text":"<p>\u7ec6\u5fc3\u7684\u8bfb\u8005\u53ef\u80fd\u53d1\u73b0\u4e86\uff0c\u5982\u679c\u8f93\u5165\u6570\u636e\u662f\u5bf9\u8c61\uff0c\u4e0a\u8ff0\u6b65\u9aa4 <code>3.</code> \u5c31\u5931\u6548\u4e86\u3002\u5047\u8bbe\u8f93\u5165\u6570\u636e\u662f\u5546\u54c1\u5bf9\u8c61\uff0c\u6211\u4eec\u60f3\u6309\u7167\u5546\u54c1\u4ef7\u683c\uff08\u7c7b\u7684\u6210\u5458\u53d8\u91cf\uff09\u5bf9\u5546\u54c1\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u4e0a\u8ff0\u7b97\u6cd5\u53ea\u80fd\u7ed9\u51fa\u4ef7\u683c\u7684\u6392\u5e8f\u7ed3\u679c\u3002</p> <p>\u90a3\u4e48\u5982\u4f55\u624d\u80fd\u5f97\u5230\u539f\u6570\u636e\u7684\u6392\u5e8f\u7ed3\u679c\u5462\uff1f\u6211\u4eec\u9996\u5148\u8ba1\u7b97 <code>counter</code> \u7684\u201c\u524d\u7f00\u548c\u201d\u3002\u987e\u540d\u601d\u4e49\uff0c\u7d22\u5f15 <code>i</code> \u5904\u7684\u524d\u7f00\u548c <code>prefix[i]</code> \u7b49\u4e8e\u6570\u7ec4\u524d <code>i</code> \u4e2a\u5143\u7d20\u4e4b\u548c\uff1a</p> \\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\] <p>\u524d\u7f00\u548c\u5177\u6709\u660e\u786e\u7684\u610f\u4e49\uff0c<code>prefix[num] - 1</code> \u4ee3\u8868\u5143\u7d20 <code>num</code> \u5728\u7ed3\u679c\u6570\u7ec4 <code>res</code> \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\u3002\u8fd9\u4e2a\u4fe1\u606f\u975e\u5e38\u5173\u952e\uff0c\u56e0\u4e3a\u5b83\u544a\u8bc9\u6211\u4eec\u5404\u4e2a\u5143\u7d20\u5e94\u8be5\u51fa\u73b0\u5728\u7ed3\u679c\u6570\u7ec4\u7684\u54ea\u4e2a\u4f4d\u7f6e\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5012\u5e8f\u904d\u5386\u539f\u6570\u7ec4 <code>nums</code> \u7684\u6bcf\u4e2a\u5143\u7d20 <code>num</code> \uff0c\u5728\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\u6267\u884c\u4ee5\u4e0b\u4e24\u6b65\u3002</p> <ol> <li>\u5c06 <code>num</code> \u586b\u5165\u6570\u7ec4 <code>res</code> \u7684\u7d22\u5f15 <code>prefix[num] - 1</code> \u5904\u3002</li> <li>\u4ee4\u524d\u7f00\u548c <code>prefix[num]</code> \u51cf\u5c0f \\(1\\) \uff0c\u4ece\u800c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e <code>num</code> \u7684\u7d22\u5f15\u3002</li> </ol> <p>\u904d\u5386\u5b8c\u6210\u540e\uff0c\u6570\u7ec4 <code>res</code> \u4e2d\u5c31\u662f\u6392\u5e8f\u597d\u7684\u7ed3\u679c\uff0c\u6700\u540e\u4f7f\u7528 <code>res</code> \u8986\u76d6\u539f\u6570\u7ec4 <code>nums</code> \u5373\u53ef\u3002\u56fe 11-17 \u5c55\u793a\u4e86\u5b8c\u6574\u7684\u8ba1\u6570\u6392\u5e8f\u6d41\u7a0b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-17 \u00a0 \u8ba1\u6570\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u8ba1\u6570\u6392\u5e8f\u7684\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig counting_sort.py<pre><code>def counting_sort(nums: list[int]):\n    \"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n    # \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\n    # 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    m = max(nums)\n    # 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    # \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for i in range(m):\n        counter[i + 1] += counter[i]\n    # 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    # \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    n = len(nums)\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        num = nums[i]\n        res[counter[num] - 1] = num  # \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num] -= 1  # \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    # \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in range(n):\n        nums[i] = res[i]\n</code></pre> counting_sort.cpp<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(vector&lt;int&gt; &amp;nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    int n = nums.size();\n    vector&lt;int&gt; res(n);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--;              // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    nums = res;\n}\n</code></pre> counting_sort.java<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int[] nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    int n = nums.length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid CountingSort(int[] nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    int n = nums.Length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums []int) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for i := 0; i &lt; m; i++ {\n        counter[i+1] += counter[i]\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    n := len(nums)\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        num := nums[i]\n        // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        res[counter[num]-1] = num\n        // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n        counter[num]--\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    copy(nums, res)\n}\n</code></pre> counting_sort.swift<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums: inout [Int]) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = nums.max()!\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for i in 0 ..&lt; m {\n        counter[i + 1] += counter[i]\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let num = nums[i]\n        res[counter[num] - 1] = num // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num] -= 1 // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    const n = nums.length;\n    const res = new Array(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums: number[]): void {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    const n = nums.length;\n    const res: number[] = new Array&lt;number&gt;(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(List&lt;int&gt; nums) {\n  // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n  // counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73b0\u6b21\u6570\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n  // \u5373 counter[_num]-1 \u662f _num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n  for (int i = 0; i &lt; m; i++) {\n    counter[i + 1] += counter[i];\n  }\n  // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n  // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n  int n = nums.length;\n  List&lt;int&gt; res = List.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int _num = nums[i];\n    res[counter[_num] - 1] = _num; // \u5c06 _num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n    counter[_num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e _num \u7684\u7d22\u5f15\n  }\n  // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n  nums.setAll(0, res);\n}\n</code></pre> counting_sort.rs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfn counting_sort(nums: &amp;mut [i32]) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = *nums.into_iter().max().unwrap();\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    let mut counter = vec![0; m as usize + 1];\n    for &amp;num in &amp;*nums {\n        counter[num as usize] += 1;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for i in 0..m as usize {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    let n = nums.len();\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let num = nums[i];\n        res[counter[num as usize] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num as usize] -= 1; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in 0..n {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.c<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int nums[], int size) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int *counter = calloc(m, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    int *res = malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--;              // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    memcpy(nums, res, size * sizeof(int));\n    // 5. \u91ca\u653e\u5185\u5b58\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfun countingSort(nums: IntArray) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    var m = 0\n    for (num in nums) {\n        m = max(m.toDouble(), num.toDouble()).toInt()\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (i in 0..&lt;m) {\n        counter[i + 1] += counter[i]\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    val n = nums.size\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val num = nums[i]\n        res[counter[num] - 1] = num // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]-- // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (i in 0..&lt;n) {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.rb<pre><code>[class]{}-[func]{counting_sort}\n</code></pre> counting_sort.zig<pre><code>[class]{}-[func]{countingSort}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/counting_sort/#1193","title":"11.9.3 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n + m)\\) \uff1a\u6d89\u53ca\u904d\u5386 <code>nums</code> \u548c\u904d\u5386 <code>counter</code> \uff0c\u90fd\u4f7f\u7528\u7ebf\u6027\u65f6\u95f4\u3002\u4e00\u822c\u60c5\u51b5\u4e0b \\(n \\gg m\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u4e8e \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n + m)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u501f\u52a9\u4e86\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) \u548c \\(m\\) \u7684\u6570\u7ec4 <code>res</code> \u548c <code>counter</code> \u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5411 <code>res</code> \u4e2d\u586b\u5145\u5143\u7d20\u7684\u987a\u5e8f\u662f\u201c\u4ece\u53f3\u5411\u5de6\u201d\u7684\uff0c\u56e0\u6b64\u5012\u5e8f\u904d\u5386 <code>nums</code> \u53ef\u4ee5\u907f\u514d\u6539\u53d8\u76f8\u7b49\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b9e\u73b0\u7a33\u5b9a\u6392\u5e8f\u3002\u5b9e\u9645\u4e0a\uff0c\u6b63\u5e8f\u904d\u5386 <code>nums</code> \u4e5f\u53ef\u4ee5\u5f97\u5230\u6b63\u786e\u7684\u6392\u5e8f\u7ed3\u679c\uff0c\u4f46\u7ed3\u679c\u662f\u975e\u7a33\u5b9a\u7684\u3002</li> </ul>"},{"location":"chapter_sorting/counting_sort/#1194","title":"11.9.4 \u00a0 \u5c40\u9650\u6027","text":"<p>\u770b\u5230\u8fd9\u91cc\uff0c\u4f60\u4e5f\u8bb8\u4f1a\u89c9\u5f97\u8ba1\u6570\u6392\u5e8f\u975e\u5e38\u5de7\u5999\uff0c\u4ec5\u901a\u8fc7\u7edf\u8ba1\u6570\u91cf\u5c31\u53ef\u4ee5\u5b9e\u73b0\u9ad8\u6548\u7684\u6392\u5e8f\u3002\u7136\u800c\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\u76f8\u5bf9\u8f83\u4e3a\u4e25\u683c\u3002</p> <p>\u8ba1\u6570\u6392\u5e8f\u53ea\u9002\u7528\u4e8e\u975e\u8d1f\u6574\u6570\u3002\u82e5\u60f3\u5c06\u5176\u7528\u4e8e\u5176\u4ed6\u7c7b\u578b\u7684\u6570\u636e\uff0c\u9700\u8981\u786e\u4fdd\u8fd9\u4e9b\u6570\u636e\u53ef\u4ee5\u8f6c\u6362\u4e3a\u975e\u8d1f\u6574\u6570\uff0c\u5e76\u4e14\u5728\u8f6c\u6362\u8fc7\u7a0b\u4e2d\u4e0d\u80fd\u6539\u53d8\u5404\u4e2a\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u5927\u5c0f\u5173\u7cfb\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u5305\u542b\u8d1f\u6570\u7684\u6574\u6570\u6570\u7ec4\uff0c\u53ef\u4ee5\u5148\u7ed9\u6240\u6709\u6570\u5b57\u52a0\u4e0a\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5168\u90e8\u6570\u5b57\u8f6c\u5316\u4e3a\u6b63\u6570\uff0c\u6392\u5e8f\u5b8c\u6210\u540e\u518d\u8f6c\u6362\u56de\u53bb\u3002</p> <p>\u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u6bd4\u5982\uff0c\u5728\u4e0a\u8ff0\u793a\u4f8b\u4e2d \\(m\\) \u4e0d\u80fd\u592a\u5927\uff0c\u5426\u5219\u4f1a\u5360\u7528\u8fc7\u591a\u7a7a\u95f4\u3002\u800c\u5f53 \\(n \\ll m\\) \u65f6\uff0c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\uff0c\u53ef\u80fd\u6bd4 \\(O(n \\log n)\\) \u7684\u6392\u5e8f\u7b97\u6cd5\u8fd8\u8981\u6162\u3002</p>"},{"location":"chapter_sorting/heap_sort/","title":"11.7 \u00a0 \u5806\u6392\u5e8f","text":"<p>Tip</p> <p>\u9605\u8bfb\u672c\u8282\u524d\uff0c\u8bf7\u786e\u4fdd\u5df2\u5b66\u5b8c\u201c\u5806\u201c\u7ae0\u8282\u3002</p> <p>\u5806\u6392\u5e8f\uff08heap sort\uff09\u662f\u4e00\u79cd\u57fa\u4e8e\u5806\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\u7684\u9ad8\u6548\u6392\u5e8f\u7b97\u6cd5\u3002\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u5df2\u7ecf\u5b66\u8fc7\u7684\u201c\u5efa\u5806\u64cd\u4f5c\u201d\u548c\u201c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u201d\u5b9e\u73b0\u5806\u6392\u5e8f\u3002</p> <ol> <li>\u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5c0f\u9876\u5806\uff0c\u6b64\u65f6\u6700\u5c0f\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002</li> <li>\u4e0d\u65ad\u6267\u884c\u51fa\u5806\u64cd\u4f5c\uff0c\u4f9d\u6b21\u8bb0\u5f55\u51fa\u5806\u5143\u7d20\uff0c\u5373\u53ef\u5f97\u5230\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u7684\u5e8f\u5217\u3002</li> </ol> <p>\u4ee5\u4e0a\u65b9\u6cd5\u867d\u7136\u53ef\u884c\uff0c\u4f46\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u989d\u5916\u6570\u7ec4\u6765\u4fdd\u5b58\u5f39\u51fa\u7684\u5143\u7d20\uff0c\u6bd4\u8f83\u6d6a\u8d39\u7a7a\u95f4\u3002\u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u4e00\u79cd\u66f4\u52a0\u4f18\u96c5\u7684\u5b9e\u73b0\u65b9\u5f0f\u3002</p>"},{"location":"chapter_sorting/heap_sort/#1171","title":"11.7.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5806\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-12 \u6240\u793a\u3002</p> <ol> <li>\u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5927\u9876\u5806\u3002\u5b8c\u6210\u540e\uff0c\u6700\u5927\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002</li> <li>\u5c06\u5806\u9876\u5143\u7d20\uff08\u7b2c\u4e00\u4e2a\u5143\u7d20\uff09\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u6700\u540e\u4e00\u4e2a\u5143\u7d20\uff09\u4ea4\u6362\u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u5806\u7684\u957f\u5ea6\u51cf \\(1\\) \uff0c\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u52a0 \\(1\\) \u3002</li> <li>\u4ece\u5806\u9876\u5143\u7d20\u5f00\u59cb\uff0c\u4ece\u9876\u5230\u5e95\u6267\u884c\u5806\u5316\u64cd\u4f5c\uff08sift down\uff09\u3002\u5b8c\u6210\u5806\u5316\u540e\uff0c\u5806\u7684\u6027\u8d28\u5f97\u5230\u4fee\u590d\u3002</li> <li>\u5faa\u73af\u6267\u884c\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\u3002\u5faa\u73af \\(n - 1\\) \u8f6e\u540e\uff0c\u5373\u53ef\u5b8c\u6210\u6570\u7ec4\u6392\u5e8f\u3002</li> </ol> <p>Tip</p> <p>\u5b9e\u9645\u4e0a\uff0c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u4e2d\u4e5f\u5305\u542b\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\uff0c\u53ea\u662f\u591a\u4e86\u4e00\u4e2a\u5f39\u51fa\u5143\u7d20\u7684\u6b65\u9aa4\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-12 \u00a0 \u5806\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528\u4e86\u4e0e\u201c\u5806\u201d\u7ae0\u8282\u76f8\u540c\u7684\u4ece\u9876\u81f3\u5e95\u5806\u5316 <code>sift_down()</code> \u51fd\u6570\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e\u5806\u7684\u957f\u5ea6\u4f1a\u968f\u7740\u63d0\u53d6\u6700\u5927\u5143\u7d20\u800c\u51cf\u5c0f\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u7ed9 <code>sift_down()</code> \u51fd\u6570\u6dfb\u52a0\u4e00\u4e2a\u957f\u5ea6\u53c2\u6570 \\(n\\) \uff0c\u7528\u4e8e\u6307\u5b9a\u5806\u7684\u5f53\u524d\u6709\u6548\u957f\u5ea6\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig heap_sort.py<pre><code>def sift_down(nums: list[int], n: int, i: int):\n    \"\"\"\u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\n    while True:\n        # \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        l = 2 * i + 1\n        r = 2 * i + 2\n        ma = i\n        if l &lt; n and nums[l] &gt; nums[ma]:\n            ma = l\n        if r &lt; n and nums[r] &gt; nums[ma]:\n            ma = r\n        # \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i:\n            break\n        # \u4ea4\u6362\u4e24\u8282\u70b9\n        nums[i], nums[ma] = nums[ma], nums[i]\n        # \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n\ndef heap_sort(nums: list[int]):\n    \"\"\"\u5806\u6392\u5e8f\"\"\"\n    # \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in range(len(nums) // 2 - 1, -1, -1):\n        sift_down(nums, len(nums), i)\n    # \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for i in range(len(nums) - 1, 0, -1):\n        # \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        nums[0], nums[i] = nums[i], nums[0]\n        # \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        sift_down(nums, i, 0)\n</code></pre> heap_sort.cpp<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(vector&lt;int&gt; &amp;nums, int n, int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(nums[i], nums[ma]);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(vector&lt;int&gt; &amp;nums) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = nums.size() / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, nums.size(), i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (int i = nums.size() - 1; i &gt; 0; --i) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        swap(nums[0], nums[i]);\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.java<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int[] nums, int n, int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int[] nums) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = nums.length / 2 - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.cs<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid SiftDown(int[] nums, int n, int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        (nums[ma], nums[i]) = (nums[i], nums[ma]);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid HeapSort(int[] nums) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = nums.Length / 2 - 1; i &gt;= 0; i--) {\n        SiftDown(nums, nums.Length, i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        (nums[i], nums[0]) = (nums[0], nums[i]);\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        SiftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.go<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums *[]int, n, i int) {\n    for true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        l := 2*i + 1\n        r := 2*i + 2\n        ma := i\n        if l &lt; n &amp;&amp; (*nums)[l] &gt; (*nums)[ma] {\n            ma = l\n        }\n        if r &lt; n &amp;&amp; (*nums)[r] &gt; (*nums)[ma] {\n            ma = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        (*nums)[i], (*nums)[ma] = (*nums)[ma], (*nums)[i]\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums *[]int) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i := len(*nums)/2 - 1; i &gt;= 0; i-- {\n        siftDown(nums, len(*nums), i)\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for i := len(*nums) - 1; i &gt; 0; i-- {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        (*nums)[0], (*nums)[i] = (*nums)[i], (*nums)[0]\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.swift<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums: inout [Int], n: Int, i: Int) {\n    var i = i\n    while true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = 2 * i + 1\n        let r = 2 * i + 2\n        var ma = i\n        if l &lt; n, nums[l] &gt; nums[ma] {\n            ma = l\n        }\n        if r &lt; n, nums[r] &gt; nums[ma] {\n            ma = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        nums.swapAt(i, ma)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums: inout [Int]) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in stride(from: nums.count / 2 - 1, through: 0, by: -1) {\n        siftDown(nums: &amp;nums, n: nums.count, i: i)\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for i in nums.indices.dropFirst().reversed() {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        nums.swapAt(0, i)\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums: &amp;nums, n: i, i: 0)\n    }\n}\n</code></pre> heap_sort.js<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums, n, i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.ts<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums: number[], n: number, i: number): void {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums: number[]): void {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.dart<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(List&lt;int&gt; nums, int n, int i) {\n  while (true) {\n    // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n    int l = 2 * i + 1;\n    int r = 2 * i + 2;\n    int ma = i;\n    if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) ma = l;\n    if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) ma = r;\n    // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n    if (ma == i) break;\n    // \u4ea4\u6362\u4e24\u8282\u70b9\n    int temp = nums[i];\n    nums[i] = nums[ma];\n    nums[ma] = temp;\n    // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n    i = ma;\n  }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(List&lt;int&gt; nums) {\n  // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n  for (int i = nums.length ~/ 2 - 1; i &gt;= 0; i--) {\n    siftDown(nums, nums.length, i);\n  }\n  // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    int tmp = nums[0];\n    nums[0] = nums[i];\n    nums[i] = tmp;\n    // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n    siftDown(nums, i, 0);\n  }\n}\n</code></pre> heap_sort.rs<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfn sift_down(nums: &amp;mut [i32], n: usize, mut i: usize) {\n    loop {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let mut ma = i;\n        if l &lt; n &amp;&amp; nums[l] &gt; nums[ma] {\n            ma = l;\n        }\n        if r &lt; n &amp;&amp; nums[r] &gt; nums[ma] {\n            ma = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        let temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfn heap_sort(nums: &amp;mut [i32]) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in (0..=nums.len() / 2 - 1).rev() {\n        sift_down(nums, nums.len(), i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for i in (1..=nums.len() - 1).rev() {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        let tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        sift_down(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.c<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int nums[], int n, int i) {\n    while (1) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int nums[], int n) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = n / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, n, i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (int i = n - 1; i &gt; 0; --i) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.kt<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfun siftDown(nums: IntArray, n: Int, li: Int) {\n    var i = li\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        val l = 2 * i + 1\n        val r = 2 * i + 2\n        var ma = i\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) ma = l\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) ma = r\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) break\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        nums[i] = nums[ma].also { nums[ma] = nums[i] }\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfun heapSort(nums: IntArray) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (i in nums.size / 2 - 1 downTo 0) {\n        siftDown(nums, nums.size, i)\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (i in nums.size - 1 downTo 1) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        nums[0] = nums[i].also { nums[i] = nums[0] }\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.rb<pre><code>[class]{}-[func]{sift_down}\n\n[class]{}-[func]{heap_sort}\n</code></pre> heap_sort.zig<pre><code>[class]{}-[func]{siftDown}\n\n[class]{}-[func]{heapSort}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/heap_sort/#1172","title":"11.7.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5efa\u5806\u64cd\u4f5c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \uff0c\u5171\u5faa\u73af \\(n - 1\\) \u8f6e\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u51e0\u4e2a\u6307\u9488\u53d8\u91cf\u4f7f\u7528 \\(O(1)\\) \u7a7a\u95f4\u3002\u5143\u7d20\u4ea4\u6362\u548c\u5806\u5316\u64cd\u4f5c\u90fd\u662f\u5728\u539f\u6570\u7ec4\u4e0a\u8fdb\u884c\u7684\u3002</li> <li>\u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u4ea4\u6362\u5806\u9876\u5143\u7d20\u548c\u5806\u5e95\u5143\u7d20\u65f6\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002</li> </ul>"},{"location":"chapter_sorting/insertion_sort/","title":"11.4 \u00a0 \u63d2\u5165\u6392\u5e8f","text":"<p>\u63d2\u5165\u6392\u5e8f\uff08insertion sort\uff09\u662f\u4e00\u79cd\u7b80\u5355\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5b83\u7684\u5de5\u4f5c\u539f\u7406\u4e0e\u624b\u52a8\u6574\u7406\u4e00\u526f\u724c\u7684\u8fc7\u7a0b\u975e\u5e38\u76f8\u4f3c\u3002</p> <p>\u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u5728\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u4e00\u4e2a\u57fa\u51c6\u5143\u7d20\uff0c\u5c06\u8be5\u5143\u7d20\u4e0e\u5176\u5de6\u4fa7\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u5143\u7d20\u9010\u4e00\u6bd4\u8f83\u5927\u5c0f\uff0c\u5e76\u5c06\u8be5\u5143\u7d20\u63d2\u5165\u5230\u6b63\u786e\u7684\u4f4d\u7f6e\u3002</p> <p>\u56fe 11-6 \u5c55\u793a\u4e86\u6570\u7ec4\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\u6d41\u7a0b\u3002\u8bbe\u57fa\u51c6\u5143\u7d20\u4e3a <code>base</code> \uff0c\u6211\u4eec\u9700\u8981\u5c06\u4ece\u76ee\u6807\u7d22\u5f15\u5230 <code>base</code> \u4e4b\u95f4\u7684\u6240\u6709\u5143\u7d20\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\uff0c\u7136\u540e\u5c06 <code>base</code> \u8d4b\u503c\u7ed9\u76ee\u6807\u7d22\u5f15\u3002</p> <p></p> <p> \u56fe 11-6 \u00a0 \u5355\u6b21\u63d2\u5165\u64cd\u4f5c </p>"},{"location":"chapter_sorting/insertion_sort/#1141","title":"11.4.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u63d2\u5165\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-7 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6570\u7ec4\u7684\u7b2c 1 \u4e2a\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002</li> <li>\u9009\u53d6\u6570\u7ec4\u7684\u7b2c 2 \u4e2a\u5143\u7d20\u4f5c\u4e3a <code>base</code> \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u9009\u53d6\u7b2c 3 \u4e2a\u5143\u7d20\u4f5c\u4e3a <code>base</code> \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 3 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5728\u6700\u540e\u4e00\u8f6e\u4e2d\uff0c\u9009\u53d6\u6700\u540e\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a <code>base</code> \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6240\u6709\u5143\u7d20\u5747\u5df2\u6392\u5e8f\u3002</li> </ol> <p></p> <p> \u56fe 11-7 \u00a0 \u63d2\u5165\u6392\u5e8f\u6d41\u7a0b </p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig insertion_sort.py<pre><code>def insertion_sort(nums: list[int]):\n    \"\"\"\u63d2\u5165\u6392\u5e8f\"\"\"\n    # \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for i in range(1, len(nums)):\n        base = nums[i]\n        j = i - 1\n        # \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while j &gt;= 0 and nums[j] &gt; base:\n            nums[j + 1] = nums[j]  # \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j -= 1\n        nums[j + 1] = base  # \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n</code></pre> insertion_sort.cpp<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(vector&lt;int&gt; &amp;nums) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for (int i = 1; i &lt; nums.size(); i++) {\n        int base = nums[i], j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.java<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for (int i = 1; i &lt; nums.length; i++) {\n        int base = nums[i], j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base;        // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.cs<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid InsertionSort(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for (int i = 1; i &lt; nums.Length; i++) {\n        int bas = nums[i], j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; bas) {\n            nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = bas;         // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.go<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums []int) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for i := 1; i &lt; len(nums); i++ {\n        base := nums[i]\n        j := i - 1\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        for j &gt;= 0 &amp;&amp; nums[j] &gt; base {\n            nums[j+1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--\n        }\n        nums[j+1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.swift<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums: inout [Int]) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for i in nums.indices.dropFirst() {\n        let base = nums[i]\n        var j = i - 1\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while j &gt;= 0, nums[j] &gt; base {\n            nums[j + 1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j -= 1\n        }\n        nums[j + 1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.js<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for (let i = 1; i &lt; nums.length; i++) {\n        let base = nums[i],\n            j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.ts<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums: number[]): void {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for (let i = 1; i &lt; nums.length; i++) {\n        const base = nums[i];\n        let j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.dart<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(List&lt;int&gt; nums) {\n  // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n  for (int i = 1; i &lt; nums.length; i++) {\n    int base = nums[i], j = i - 1;\n    // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n    while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n      nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n      j--;\n    }\n    nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n  }\n}\n</code></pre> insertion_sort.rs<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfn insertion_sort(nums: &amp;mut [i32]) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for i in 1..nums.len() {\n        let (base, mut j) = (nums[i], (i - 1) as i32);\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while j &gt;= 0 &amp;&amp; nums[j as usize] &gt; base {\n            nums[(j + 1) as usize] = nums[j as usize]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j -= 1;\n        }\n        nums[(j + 1) as usize] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.c<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int nums[], int size) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for (int i = 1; i &lt; size; i++) {\n        int base = nums[i], j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            nums[j + 1] = nums[j];\n            j--;\n        }\n        // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n        nums[j + 1] = base;\n    }\n}\n</code></pre> insertion_sort.kt<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfun insertionSort(nums: IntArray) {\n    //\u5916\u5faa\u73af: \u5df2\u6392\u5e8f\u5143\u7d20\u4e3a 1, 2, ..., n\n    for (i in nums.indices) {\n        val base = nums[i]\n        var j = i - 1\n        // \u5185\u5faa\u73af: \u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--\n        }\n        nums[j + 1] = base        // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.rb<pre><code>[class]{}-[func]{insertion_sort}\n</code></pre> insertion_sort.zig<pre><code>// \u63d2\u5165\u6392\u5e8f\nfn insertionSort(nums: []i32) void {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    var i: usize = 1;\n    while (i &lt; nums.len) : (i += 1) {\n        var base = nums[i];\n        var j: usize = i;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 1 and nums[j - 1] &gt; base) : (j -= 1) {\n            nums[j] = nums[j - 1];  // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n        }\n        nums[j] = base;             // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/insertion_sort/#1142","title":"11.4.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u6b21\u63d2\u5165\u64cd\u4f5c\u5206\u522b\u9700\u8981\u5faa\u73af \\(n - 1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u6c42\u548c\u5f97\u5230 \\((n - 1) n / 2\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u5728\u9047\u5230\u6709\u5e8f\u6570\u636e\u65f6\uff0c\u63d2\u5165\u64cd\u4f5c\u4f1a\u63d0\u524d\u7ec8\u6b62\u3002\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u63d2\u5165\u6392\u5e8f\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u63d2\u5165\u64cd\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u4f1a\u5c06\u5143\u7d20\u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u4fa7\uff0c\u4e0d\u4f1a\u6539\u53d8\u5b83\u4eec\u7684\u987a\u5e8f\u3002</li> </ul>"},{"location":"chapter_sorting/insertion_sort/#1143","title":"11.4.3 \u00a0 \u63d2\u5165\u6392\u5e8f\u7684\u4f18\u52bf","text":"<p>\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u800c\u6211\u4eec\u5373\u5c06\u5b66\u4e60\u7684\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u5c3d\u7ba1\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u66f4\u9ad8\uff0c\u4f46\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u7684\u60c5\u51b5\u4e0b\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u66f4\u5feb\u3002</p> <p>\u8fd9\u4e2a\u7ed3\u8bba\u4e0e\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u7684\u9002\u7528\u60c5\u51b5\u7684\u7ed3\u8bba\u7c7b\u4f3c\u3002\u5feb\u901f\u6392\u5e8f\u8fd9\u7c7b \\(O(n \\log n)\\) \u7684\u7b97\u6cd5\u5c5e\u4e8e\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5f80\u5f80\u5305\u542b\u66f4\u591a\u5355\u5143\u8ba1\u7b97\u64cd\u4f5c\u3002\u800c\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\\(n^2\\) \u548c \\(n \\log n\\) \u7684\u6570\u503c\u6bd4\u8f83\u63a5\u8fd1\uff0c\u590d\u6742\u5ea6\u4e0d\u5360\u4e3b\u5bfc\u5730\u4f4d\uff0c\u6bcf\u8f6e\u4e2d\u7684\u5355\u5143\u64cd\u4f5c\u6570\u91cf\u8d77\u5230\u51b3\u5b9a\u6027\u4f5c\u7528\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java\uff09\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\u91c7\u7528\u4e86\u63d2\u5165\u6392\u5e8f\uff0c\u5927\u81f4\u601d\u8def\u4e3a\uff1a\u5bf9\u4e8e\u957f\u6570\u7ec4\uff0c\u91c7\u7528\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\uff1b\u5bf9\u4e8e\u77ed\u6570\u7ec4\uff0c\u76f4\u63a5\u4f7f\u7528\u63d2\u5165\u6392\u5e8f\u3002</p> <p>\u867d\u7136\u5192\u6ce1\u6392\u5e8f\u3001\u9009\u62e9\u6392\u5e8f\u548c\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u63d2\u5165\u6392\u5e8f\u7684\u4f7f\u7528\u9891\u7387\u663e\u8457\u9ad8\u4e8e\u5192\u6ce1\u6392\u5e8f\u548c\u9009\u62e9\u6392\u5e8f\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u5192\u6ce1\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u4ea4\u6362\u5b9e\u73b0\uff0c\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u4e34\u65f6\u53d8\u91cf\uff0c\u5171\u6d89\u53ca 3 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u63d2\u5165\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u8d4b\u503c\u5b9e\u73b0\uff0c\u4ec5\u9700 1 \u4e2a\u5355\u5143\u64cd\u4f5c\u3002\u56e0\u6b64\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u8ba1\u7b97\u5f00\u9500\u901a\u5e38\u6bd4\u63d2\u5165\u6392\u5e8f\u66f4\u9ad8\u3002</li> <li>\u9009\u62e9\u6392\u5e8f\u5728\u4efb\u4f55\u60c5\u51b5\u4e0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \u3002\u5982\u679c\u7ed9\u5b9a\u4e00\u7ec4\u90e8\u5206\u6709\u5e8f\u7684\u6570\u636e\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u6bd4\u9009\u62e9\u6392\u5e8f\u6548\u7387\u66f4\u9ad8\u3002</li> <li>\u9009\u62e9\u6392\u5e8f\u4e0d\u7a33\u5b9a\uff0c\u65e0\u6cd5\u5e94\u7528\u4e8e\u591a\u7ea7\u6392\u5e8f\u3002</li> </ul>"},{"location":"chapter_sorting/merge_sort/","title":"11.6 \u00a0 \u5f52\u5e76\u6392\u5e8f","text":"<p>\u5f52\u5e76\u6392\u5e8f\uff08merge sort\uff09\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5305\u542b\u56fe 11-10 \u6240\u793a\u7684\u201c\u5212\u5206\u201d\u548c\u201c\u5408\u5e76\u201d\u9636\u6bb5\u3002</p> <ol> <li>\u5212\u5206\u9636\u6bb5\uff1a\u901a\u8fc7\u9012\u5f52\u4e0d\u65ad\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5206\u5f00\uff0c\u5c06\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u8f6c\u6362\u4e3a\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002</li> <li>\u5408\u5e76\u9636\u6bb5\uff1a\u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u5212\u5206\uff0c\u5f00\u59cb\u5408\u5e76\uff0c\u6301\u7eed\u5730\u5c06\u5de6\u53f3\u4e24\u4e2a\u8f83\u77ed\u7684\u6709\u5e8f\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u8f83\u957f\u7684\u6709\u5e8f\u6570\u7ec4\uff0c\u76f4\u81f3\u7ed3\u675f\u3002</li> </ol> <p></p> <p> \u56fe 11-10 \u00a0 \u5f52\u5e76\u6392\u5e8f\u7684\u5212\u5206\u4e0e\u5408\u5e76\u9636\u6bb5 </p>"},{"location":"chapter_sorting/merge_sort/#1161","title":"11.6.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u5982\u56fe 11-11 \u6240\u793a\uff0c\u201c\u5212\u5206\u9636\u6bb5\u201d\u4ece\u9876\u81f3\u5e95\u9012\u5f52\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5207\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\u3002</p> <ol> <li>\u8ba1\u7b97\u6570\u7ec4\u4e2d\u70b9 <code>mid</code> \uff0c\u9012\u5f52\u5212\u5206\u5de6\u5b50\u6570\u7ec4\uff08\u533a\u95f4 <code>[left, mid]</code> \uff09\u548c\u53f3\u5b50\u6570\u7ec4\uff08\u533a\u95f4 <code>[mid + 1, right]</code> \uff09\u3002</li> <li>\u9012\u5f52\u6267\u884c\u6b65\u9aa4 <code>1.</code> \uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u533a\u95f4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u3002</li> </ol> <p>\u201c\u5408\u5e76\u9636\u6bb5\u201d\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ece\u957f\u5ea6\u4e3a 1 \u7684\u5b50\u6570\u7ec4\u5f00\u59cb\u5408\u5e76\uff0c\u5408\u5e76\u9636\u6bb5\u4e2d\u7684\u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u662f\u6709\u5e8f\u7684\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-11 \u00a0 \u5f52\u5e76\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u5f52\u5e76\u6392\u5e8f\u4e0e\u4e8c\u53c9\u6811\u540e\u5e8f\u904d\u5386\u7684\u9012\u5f52\u987a\u5e8f\u662f\u4e00\u81f4\u7684\u3002</p> <ul> <li>\u540e\u5e8f\u904d\u5386\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6811\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6811\uff0c\u6700\u540e\u5904\u7406\u6839\u8282\u70b9\u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6570\u7ec4\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6570\u7ec4\uff0c\u6700\u540e\u5904\u7406\u5408\u5e76\u3002</li> </ul> <p>\u5f52\u5e76\u6392\u5e8f\u7684\u5b9e\u73b0\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\u3002\u8bf7\u6ce8\u610f\uff0c<code>nums</code> \u7684\u5f85\u5408\u5e76\u533a\u95f4\u4e3a <code>[left, right]</code> \uff0c\u800c <code>tmp</code> \u7684\u5bf9\u5e94\u533a\u95f4\u4e3a <code>[0, right - left]</code> \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig merge_sort.py<pre><code>def merge(nums: list[int], left: int, mid: int, right: int):\n    \"\"\"\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\"\"\"\n    # \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    # \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    tmp = [0] * (right - left + 1)\n    # \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    i, j, k = left, mid + 1, 0\n    # \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid and j &lt;= right:\n        if nums[i] &lt;= nums[j]:\n            tmp[k] = nums[i]\n            i += 1\n        else:\n            tmp[k] = nums[j]\n            j += 1\n        k += 1\n    # \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid:\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    while j &lt;= right:\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    # \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for k in range(0, len(tmp)):\n        nums[left + k] = tmp[k]\n\ndef merge_sort(nums: list[int], left: int, right: int):\n    \"\"\"\u5f52\u5e76\u6392\u5e8f\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6\n    if left &gt;= right:\n        return  # \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    # \u5212\u5206\u9636\u6bb5\n    mid = (left + right) // 2  # \u8ba1\u7b97\u4e2d\u70b9\n    merge_sort(nums, left, mid)  # \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    merge_sort(nums, mid + 1, right)  # \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    # \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right)\n</code></pre> merge_sort.cpp<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    vector&lt;int&gt; tmp(right - left + 1);\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmp.size(); k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right)\n        return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    int mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.java<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(int[] nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    int[] tmp = new int[right - left + 1];\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right)\n        return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    int mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.cs<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid Merge(int[] nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    int[] tmp = new int[right - left + 1];\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmp.Length; ++k) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid MergeSort(int[] nums, int left, int right) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right) return;       // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    int mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\n    MergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    MergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    Merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.go<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunc merge(nums []int, left, mid, right int) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    tmp := make([]int, right-left+1)\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    i, j, k := left, mid+1, 0\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    for i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i++\n        } else {\n            tmp[k] = nums[j]\n            j++\n        }\n        k++\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    for i &lt;= mid {\n        tmp[k] = nums[i]\n        i++\n        k++\n    }\n    for j &lt;= right {\n        tmp[k] = nums[j]\n        j++\n        k++\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for k := 0; k &lt; len(tmp); k++ {\n        nums[left+k] = tmp[k]\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums []int, left, right int) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if left &gt;= right {\n        return\n    }\n    // \u5212\u5206\u9636\u6bb5\n    mid := (left + right) / 2\n    mergeSort(nums, left, mid)\n    mergeSort(nums, mid+1, right)\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.swift<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    var tmp = Array(repeating: 0, count: right - left + 1)\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    var i = left, j = mid + 1, k = 0\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid, j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i += 1\n        } else {\n            tmp[k] = nums[j]\n            j += 1\n        }\n        k += 1\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid {\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for k in tmp.indices {\n        nums[left + k] = tmp[k]\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if left &gt;= right { // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n        return\n    }\n    // \u5212\u5206\u9636\u6bb5\n    let mid = (left + right) / 2 // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums: &amp;nums, left: left, right: mid) // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums: &amp;nums, left: mid + 1, right: right) // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums: &amp;nums, left: left, mid: mid, right: right)\n}\n</code></pre> merge_sort.js<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunction merge(nums, left, mid, right) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    const tmp = new Array(right - left + 1);\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums, left, right) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    let mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.ts<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    const tmp = new Array(right - left + 1);\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    let mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.dart<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(List&lt;int&gt; nums, int left, int mid, int right) {\n  // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n  // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n  List&lt;int&gt; tmp = List.filled(right - left + 1, 0);\n  // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n  int i = left, j = mid + 1, k = 0;\n  // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n  while (i &lt;= mid &amp;&amp; j &lt;= right) {\n    if (nums[i] &lt;= nums[j])\n      tmp[k++] = nums[i++];\n    else\n      tmp[k++] = nums[j++];\n  }\n  // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n  while (i &lt;= mid) {\n    tmp[k++] = nums[i++];\n  }\n  while (j &lt;= right) {\n    tmp[k++] = nums[j++];\n  }\n  // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n  for (k = 0; k &lt; tmp.length; k++) {\n    nums[left + k] = tmp[k];\n  }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(List&lt;int&gt; nums, int left, int right) {\n  // \u7ec8\u6b62\u6761\u4ef6\n  if (left &gt;= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n  // \u5212\u5206\u9636\u6bb5\n  int mid = (left + right) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\n  mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n  mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n  // \u5408\u5e76\u9636\u6bb5\n  merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.rs<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfn merge(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    let tmp_size = right - left + 1;\n    let mut tmp = vec![0; tmp_size];\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    let (mut i, mut j, mut k) = (left, mid + 1, 0);\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i];\n            i += 1;\n        } else {\n            tmp[k] = nums[j];\n            j += 1;\n        }\n        k += 1;\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid {\n        tmp[k] = nums[i];\n        k += 1;\n        i += 1;\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j];\n        k += 1;\n        j += 1;\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for k in 0..tmp_size {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfn merge_sort(nums: &amp;mut [i32], left: usize, right: usize) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if left &gt;= right {\n        return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    }\n\n    // \u5212\u5206\u9636\u6bb5\n    let mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n    merge_sort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    merge_sort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.c<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(int *nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    int tmpSize = right - left + 1;\n    int *tmp = (int *)malloc(tmpSize * sizeof(int));\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmpSize; ++k) {\n        nums[left + k] = tmp[k];\n    }\n    // \u91ca\u653e\u5185\u5b58\n    free(tmp);\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int *nums, int left, int right) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right)\n        return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    int mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.kt<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfun merge(nums: IntArray, left: Int, mid: Int, right: Int) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4\u4e3a [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    val tmp = IntArray(right - left + 1)\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    var i = left\n    var j = mid + 1\n    var k = 0\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u8fdb\u884c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) tmp[k++] = nums[i++]\n        else tmp[k++] = nums[j++]\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++]\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++]\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (l in tmp.indices) {\n        nums[left + l] = tmp[l]\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfun mergeSort(nums: IntArray, left: Int, right: Int) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right) return  // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    val mid = (left + right) / 2 // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid) // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right) // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.rb<pre><code>[class]{}-[func]{merge}\n\n[class]{}-[func]{merge_sort}\n</code></pre> merge_sort.zig<pre><code>// \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfn merge(nums: []i32, left: usize, mid: usize, right: usize) !void {\n    // \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\n    var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n    defer mem_arena.deinit();\n    const mem_allocator = mem_arena.allocator();\n    var tmp = try mem_allocator.alloc(i32, right + 1 - left);\n    std.mem.copy(i32, tmp, nums[left..right+1]);\n    // \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \n    var leftStart = left - left;\n    var leftEnd = mid - left;\n    // \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \n    var rightStart = mid + 1 - left;\n    var rightEnd = right - left;\n    // i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\n    var i = leftStart;\n    var j = rightStart;\n    // \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\n    var k = left;\n    while (k &lt;= right) : (k += 1) {\n        // \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n        if (i &gt; leftEnd) {\n            nums[k] = tmp[j];\n            j += 1;\n        // \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 &lt;= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n        } else if  (j &gt; rightEnd or tmp[i] &lt;= tmp[j]) {\n            nums[k] = tmp[i];\n            i += 1;\n        // \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 &gt; \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n        } else {\n            nums[k] = tmp[j];\n            j += 1;\n        }\n    }\n}\n\n// \u5f52\u5e76\u6392\u5e8f\nfn mergeSort(nums: []i32, left: usize, right: usize) !void {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right) return;              // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    var mid = (left + right) / 2;           // \u8ba1\u7b97\u4e2d\u70b9\n    try mergeSort(nums, left, mid);         // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    try mergeSort(nums, mid + 1, right);    // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    try merge(nums, left, mid, right);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/merge_sort/#1162","title":"11.6.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5212\u5206\u4ea7\u751f\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u5c42\u5408\u5e76\u7684\u603b\u64cd\u4f5c\u6570\u91cf\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u9012\u5f52\u6df1\u5ea6\u4e3a \\(\\log n\\) \uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002\u5408\u5e76\u64cd\u4f5c\u9700\u8981\u501f\u52a9\u8f85\u52a9\u6570\u7ec4\u5b9e\u73b0\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u5408\u5e76\u8fc7\u7a0b\u4e2d\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u6b21\u5e8f\u4fdd\u6301\u4e0d\u53d8\u3002</li> </ul>"},{"location":"chapter_sorting/merge_sort/#1163","title":"11.6.3 \u00a0 \u94fe\u8868\u6392\u5e8f","text":"<p>\u5bf9\u4e8e\u94fe\u8868\uff0c\u5f52\u5e76\u6392\u5e8f\u76f8\u8f83\u4e8e\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\u5177\u6709\u663e\u8457\u4f18\u52bf\uff0c\u53ef\u4ee5\u5c06\u94fe\u8868\u6392\u5e8f\u4efb\u52a1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(1)\\) \u3002</p> <ul> <li>\u5212\u5206\u9636\u6bb5\uff1a\u53ef\u4ee5\u4f7f\u7528\u201c\u8fed\u4ee3\u201d\u66ff\u4ee3\u201c\u9012\u5f52\u201d\u6765\u5b9e\u73b0\u94fe\u8868\u5212\u5206\u5de5\u4f5c\uff0c\u4ece\u800c\u7701\u53bb\u9012\u5f52\u4f7f\u7528\u7684\u6808\u5e27\u7a7a\u95f4\u3002</li> <li>\u5408\u5e76\u9636\u6bb5\uff1a\u5728\u94fe\u8868\u4e2d\uff0c\u8282\u70b9\u589e\u5220\u64cd\u4f5c\u4ec5\u9700\u6539\u53d8\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\u5b9e\u73b0\uff0c\u56e0\u6b64\u5408\u5e76\u9636\u6bb5\uff08\u5c06\u4e24\u4e2a\u77ed\u6709\u5e8f\u94fe\u8868\u5408\u5e76\u4e3a\u4e00\u4e2a\u957f\u6709\u5e8f\u94fe\u8868\uff09\u65e0\u987b\u521b\u5efa\u989d\u5916\u94fe\u8868\u3002</li> </ul> <p>\u5177\u4f53\u5b9e\u73b0\u7ec6\u8282\u6bd4\u8f83\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u76f8\u5173\u8d44\u6599\u8fdb\u884c\u5b66\u4e60\u3002</p>"},{"location":"chapter_sorting/quick_sort/","title":"11.5 \u00a0 \u5feb\u901f\u6392\u5e8f","text":"<p>\u5feb\u901f\u6392\u5e8f\uff08quick sort\uff09\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u8fd0\u884c\u9ad8\u6548\uff0c\u5e94\u7528\u5e7f\u6cdb\u3002</p> <p>\u5feb\u901f\u6392\u5e8f\u7684\u6838\u5fc3\u64cd\u4f5c\u662f\u201c\u54e8\u5175\u5212\u5206\u201d\uff0c\u5176\u76ee\u6807\u662f\uff1a\u9009\u62e9\u6570\u7ec4\u4e2d\u7684\u67d0\u4e2a\u5143\u7d20\u4f5c\u4e3a\u201c\u57fa\u51c6\u6570\u201d\uff0c\u5c06\u6240\u6709\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u5de6\u4fa7\uff0c\u800c\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u53f3\u4fa7\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u54e8\u5175\u5212\u5206\u7684\u6d41\u7a0b\u5982\u56fe 11-8 \u6240\u793a\u3002</p> <ol> <li>\u9009\u53d6\u6570\u7ec4\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u521d\u59cb\u5316\u4e24\u4e2a\u6307\u9488 <code>i</code> \u548c <code>j</code> \u5206\u522b\u6307\u5411\u6570\u7ec4\u7684\u4e24\u7aef\u3002</li> <li>\u8bbe\u7f6e\u4e00\u4e2a\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u4f7f\u7528 <code>i</code>\uff08<code>j</code>\uff09\u5206\u522b\u5bfb\u627e\u7b2c\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u5927\uff08\u5c0f\uff09\u7684\u5143\u7d20\uff0c\u7136\u540e\u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\u3002</li> <li>\u5faa\u73af\u6267\u884c\u6b65\u9aa4 <code>2.</code> \uff0c\u76f4\u5230 <code>i</code> \u548c <code>j</code> \u76f8\u9047\u65f6\u505c\u6b62\uff0c\u6700\u540e\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-8 \u00a0 \u54e8\u5175\u5212\u5206\u6b65\u9aa4 </p> <p>\u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u539f\u6570\u7ec4\u88ab\u5212\u5206\u6210\u4e09\u90e8\u5206\uff1a\u5de6\u5b50\u6570\u7ec4\u3001\u57fa\u51c6\u6570\u3001\u53f3\u5b50\u6570\u7ec4\uff0c\u4e14\u6ee1\u8db3\u201c\u5de6\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20 \\(\\leq\\) \u57fa\u51c6\u6570 \\(\\leq\\) \u53f3\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20\u201d\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u63a5\u4e0b\u6765\u53ea\u9700\u5bf9\u8fd9\u4e24\u4e2a\u5b50\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\u3002</p> <p>\u5feb\u901f\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565</p> <p>\u54e8\u5175\u5212\u5206\u7684\u5b9e\u8d28\u662f\u5c06\u4e00\u4e2a\u8f83\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u7b80\u5316\u4e3a\u4e24\u4e2a\u8f83\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"\u54e8\u5175\u5212\u5206\"\"\"\n    # \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        # \u5143\u7d20\u4ea4\u6362\n        nums[i], nums[j] = nums[j], nums[i]\n    # \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n</code></pre> quick_sort.cpp<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(vector&lt;int&gt; &amp;nums, int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.java<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int[] nums, int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint partition(int[] nums, int left, int right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid Swap(int[] nums, int i, int j) {\n    (nums[j], nums[i]) = (nums[i], nums[j]);\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint Partition(int[] nums, int left, int right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        Swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    Swap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.go<pre><code>/* \u54e8\u5175\u5212\u5206 */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        // \u5143\u7d20\u4ea4\u6362\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    nums[i], nums[left] = nums[left], nums[i]\n    return i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.swift<pre><code>/* \u54e8\u5175\u5212\u5206 */\nfunc partition(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    var i = left\n    var j = right\n    while i &lt; j {\n        while i &lt; j, nums[j] &gt;= nums[left] {\n            j -= 1 // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while i &lt; j, nums[i] &lt;= nums[left] {\n            i += 1 // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        nums.swapAt(i, j) // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    nums.swapAt(i, left) // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.js<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nswap(nums, i, j) {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums, left, right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        // \u5143\u7d20\u4ea4\u6362\n        this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nswap(nums: number[], i: number, j: number): void {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums: number[], left: number, right: number): number {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        // \u5143\u7d20\u4ea4\u6362\n        this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid _swap(List&lt;int&gt; nums, int i, int j) {\n  int tmp = nums[i];\n  nums[i] = nums[j];\n  nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n    _swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n  }\n  _swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n  return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rs<pre><code>/* \u54e8\u5175\u5212\u5206 */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        nums.swap(i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    nums.swap(i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.c<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int nums[], int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n            j--;\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n            i++;\n        }\n        // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n        swap(nums, i, j);\n    }\n    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    swap(nums, i, left);\n    // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n    return i;\n}\n</code></pre> quick_sort.kt<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nfun swap(nums: IntArray, i: Int, j: Int) {\n    nums[i] = nums[j].also { nums[j] = nums[i] }\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nfun partition(nums: IntArray, left: Int, right: Int): Int {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--           // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++           // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j)  // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left)   // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i              // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rb<pre><code>[class]{QuickSort}-[func]{partition}\n</code></pre> quick_sort.zig<pre><code>// \u5143\u7d20\u4ea4\u6362\nfn swap(nums: []i32, i: usize, j: usize) void {\n    var tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n// \u54e8\u5175\u5212\u5206\nfn partition(nums: []i32, left: usize, right: usize) usize {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    var i = left;\n    var j = right;\n    while (i &lt; j) {\n        while (i &lt; j and nums[j] &gt;= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j and nums[i] &lt;= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/quick_sort/#1151","title":"11.5.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u5feb\u901f\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-9 \u6240\u793a\u3002</p> <ol> <li>\u9996\u5148\uff0c\u5bf9\u539f\u6570\u7ec4\u6267\u884c\u4e00\u6b21\u201c\u54e8\u5175\u5212\u5206\u201d\uff0c\u5f97\u5230\u672a\u6392\u5e8f\u7684\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u3002</li> <li>\u7136\u540e\uff0c\u5bf9\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5206\u522b\u9012\u5f52\u6267\u884c\u201c\u54e8\u5175\u5212\u5206\u201d\u3002</li> <li>\u6301\u7eed\u9012\u5f52\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\uff0c\u4ece\u800c\u5b8c\u6210\u6574\u4e2a\u6570\u7ec4\u7684\u6392\u5e8f\u3002</li> </ol> <p></p> <p> \u56fe 11-9 \u00a0 \u5feb\u901f\u6392\u5e8f\u6d41\u7a0b </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"\u5feb\u901f\u6392\u5e8f\"\"\"\n    # \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if left &gt;= right:\n        return\n    # \u54e8\u5175\u5212\u5206\n    pivot = self.partition(nums, left, right)\n    # \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    self.quick_sort(nums, left, pivot - 1)\n    self.quick_sort(nums, pivot + 1, right)\n</code></pre> quick_sort.cpp<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right)\n        return;\n    // \u54e8\u5175\u5212\u5206\n    int pivot = partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.java<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right)\n        return;\n    // \u54e8\u5175\u5212\u5206\n    int pivot = partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid QuickSort(int[] nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right)\n        return;\n    // \u54e8\u5175\u5212\u5206\n    int pivot = Partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    QuickSort(nums, left, pivot - 1);\n    QuickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.go<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if left &gt;= right {\n        return\n    }\n    // \u54e8\u5175\u5212\u5206\n    pivot := q.partition(nums, left, right)\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    q.quickSort(nums, left, pivot-1)\n    q.quickSort(nums, pivot+1, right)\n}\n</code></pre> quick_sort.swift<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if left &gt;= right {\n        return\n    }\n    // \u54e8\u5175\u5212\u5206\n    let pivot = partition(nums: &amp;nums, left: left, right: right)\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums: &amp;nums, left: left, right: pivot - 1)\n    quickSort(nums: &amp;nums, left: pivot + 1, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums, left, right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) return;\n    // \u54e8\u5175\u5212\u5206\n    const pivot = this.partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums: number[], left: number, right: number): void {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) {\n        return;\n    }\n    // \u54e8\u5175\u5212\u5206\n    const pivot = this.partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n  if (left &gt;= right) return;\n  // \u54e8\u5175\u5212\u5206\n  int pivot = _partition(nums, left, right);\n  // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n  quickSort(nums, left, pivot - 1);\n  quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.rs<pre><code>/* \u5feb\u901f\u6392\u5e8f */\npub fn quick_sort(left: i32, right: i32, nums: &amp;mut [i32]) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if left &gt;= right {\n        return;\n    }\n    // \u54e8\u5175\u5212\u5206\n    let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    Self::quick_sort(left, pivot - 1, nums);\n    Self::quick_sort(pivot + 1, right, nums);\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n            j--;\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n            i++;\n        }\n        // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n        swap(nums, i, j);\n    }\n    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    swap(nums, i, left);\n    // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n    return i;\n}\n\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u5feb\u901f\u6392\u5e8f\nvoid quickSort(int nums[], int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) {\n        return;\n    }\n    // \u54e8\u5175\u5212\u5206\n    int pivot = partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.kt<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfun quickSort(nums: IntArray, left: Int, right: Int) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) return\n    // \u54e8\u5175\u5212\u5206\n    val pivot = partition(nums, left, right)\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1)\n    quickSort(nums, pivot + 1, right)\n}\n</code></pre> quick_sort.rb<pre><code>[class]{QuickSort}-[func]{quick_sort}\n</code></pre> quick_sort.zig<pre><code>// \u5feb\u901f\u6392\u5e8f\nfn quickSort(nums: []i32, left: usize, right: usize) void {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) return;\n    // \u54e8\u5175\u5212\u5206\n    var pivot = partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/quick_sort/#1152","title":"11.5.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u5e73\u5747\u60c5\u51b5\u4e0b\uff0c\u54e8\u5175\u5212\u5206\u7684\u9012\u5f52\u5c42\u6570\u4e3a \\(\\log n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u603b\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n \\log n)\\) \u65f6\u95f4\u3002\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u90fd\u5c06\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u5212\u5206\u4e3a\u957f\u5ea6\u4e3a \\(0\\) \u548c \\(n - 1\\) \u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u6b64\u65f6\u9012\u5f52\u5c42\u6570\u8fbe\u5230 \\(n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u5728\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u5012\u5e8f\u7684\u60c5\u51b5\u4e0b\uff0c\u8fbe\u5230\u6700\u5dee\u9012\u5f52\u6df1\u5ea6 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002\u6392\u5e8f\u64cd\u4f5c\u662f\u5728\u539f\u6570\u7ec4\u4e0a\u8fdb\u884c\u7684\uff0c\u672a\u501f\u52a9\u989d\u5916\u6570\u7ec4\u3002</li> <li>\u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u54e8\u5175\u5212\u5206\u7684\u6700\u540e\u4e00\u6b65\uff0c\u57fa\u51c6\u6570\u53ef\u80fd\u4f1a\u88ab\u4ea4\u6362\u81f3\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u4fa7\u3002</li> </ul>"},{"location":"chapter_sorting/quick_sort/#1153","title":"11.5.3 \u00a0 \u5feb\u901f\u6392\u5e8f\u4e3a\u4ec0\u4e48\u5feb","text":"<p>\u4ece\u540d\u79f0\u4e0a\u5c31\u80fd\u770b\u51fa\uff0c\u5feb\u901f\u6392\u5e8f\u5728\u6548\u7387\u65b9\u9762\u5e94\u8be5\u5177\u6709\u4e00\u5b9a\u7684\u4f18\u52bf\u3002\u5c3d\u7ba1\u5feb\u901f\u6392\u5e8f\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e0e\u201c\u5f52\u5e76\u6392\u5e8f\u201d\u548c\u201c\u5806\u6392\u5e8f\u201d\u76f8\u540c\uff0c\u4f46\u901a\u5e38\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u51fa\u73b0\u6700\u5dee\u60c5\u51b5\u7684\u6982\u7387\u5f88\u4f4e\uff1a\u867d\u7136\u5feb\u901f\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u6ca1\u6709\u5f52\u5e76\u6392\u5e8f\u7a33\u5b9a\uff0c\u4f46\u5728\u7edd\u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u80fd\u5728 \\(O(n \\log n)\\) \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u8fd0\u884c\u3002</li> <li>\u7f13\u5b58\u4f7f\u7528\u6548\u7387\u9ad8\uff1a\u5728\u6267\u884c\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u65f6\uff0c\u7cfb\u7edf\u53ef\u5c06\u6574\u4e2a\u5b50\u6570\u7ec4\u52a0\u8f7d\u5230\u7f13\u5b58\uff0c\u56e0\u6b64\u8bbf\u95ee\u5143\u7d20\u7684\u6548\u7387\u8f83\u9ad8\u3002\u800c\u50cf\u201c\u5806\u6392\u5e8f\u201d\u8fd9\u7c7b\u7b97\u6cd5\u9700\u8981\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u5143\u7d20\uff0c\u4ece\u800c\u7f3a\u4e4f\u8fd9\u4e00\u7279\u6027\u3002</li> <li>\u590d\u6742\u5ea6\u7684\u5e38\u6570\u7cfb\u6570\u5c0f\uff1a\u5728\u4e0a\u8ff0\u4e09\u79cd\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u6bd4\u8f83\u3001\u8d4b\u503c\u3001\u4ea4\u6362\u7b49\u64cd\u4f5c\u7684\u603b\u6570\u91cf\u6700\u5c11\u3002\u8fd9\u4e0e\u201c\u63d2\u5165\u6392\u5e8f\u201d\u6bd4\u201c\u5192\u6ce1\u6392\u5e8f\u201d\u66f4\u5feb\u7684\u539f\u56e0\u7c7b\u4f3c\u3002</li> </ul>"},{"location":"chapter_sorting/quick_sort/#1154","title":"11.5.4 \u00a0 \u57fa\u51c6\u6570\u4f18\u5316","text":"<p>\u5feb\u901f\u6392\u5e8f\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\u7684\u65f6\u95f4\u6548\u7387\u53ef\u80fd\u964d\u4f4e\u3002\u4e3e\u4e00\u4e2a\u6781\u7aef\u4f8b\u5b50\uff0c\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u662f\u5b8c\u5168\u5012\u5e8f\u7684\uff0c\u7531\u4e8e\u6211\u4eec\u9009\u62e9\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u5728\u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u57fa\u51c6\u6570\u88ab\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u53f3\u7aef\uff0c\u5bfc\u81f4\u5de6\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n - 1\\)\u3001\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(0\\) \u3002\u5982\u6b64\u9012\u5f52\u4e0b\u53bb\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u540e\u90fd\u6709\u4e00\u4e2a\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(0\\) \uff0c\u5206\u6cbb\u7b56\u7565\u5931\u6548\uff0c\u5feb\u901f\u6392\u5e8f\u9000\u5316\u4e3a\u201c\u5192\u6ce1\u6392\u5e8f\u201d\u7684\u8fd1\u4f3c\u5f62\u5f0f\u3002</p> <p>\u4e3a\u4e86\u5c3d\u91cf\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u53d1\u751f\uff0c\u6211\u4eec\u53ef\u4ee5\u4f18\u5316\u54e8\u5175\u5212\u5206\u4e2d\u7684\u57fa\u51c6\u6570\u7684\u9009\u53d6\u7b56\u7565\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u53ef\u4ee5\u968f\u673a\u9009\u53d6\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\u3002\u7136\u800c\uff0c\u5982\u679c\u8fd0\u6c14\u4e0d\u4f73\uff0c\u6bcf\u6b21\u90fd\u9009\u5230\u4e0d\u7406\u60f3\u7684\u57fa\u51c6\u6570\uff0c\u6548\u7387\u4ecd\u7136\u4e0d\u5c3d\u5982\u4eba\u610f\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u751f\u6210\u7684\u662f\u201c\u4f2a\u968f\u673a\u6570\u201d\u3002\u5982\u679c\u6211\u4eec\u9488\u5bf9\u4f2a\u968f\u673a\u6570\u5e8f\u5217\u6784\u5efa\u4e00\u4e2a\u7279\u5b9a\u7684\u6d4b\u8bd5\u6837\u4f8b\uff0c\u90a3\u4e48\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u4ecd\u7136\u53ef\u80fd\u52a3\u5316\u3002</p> <p>\u4e3a\u4e86\u8fdb\u4e00\u6b65\u6539\u8fdb\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6570\u7ec4\u4e2d\u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\uff08\u901a\u5e38\u4e3a\u6570\u7ec4\u7684\u9996\u3001\u5c3e\u3001\u4e2d\u70b9\u5143\u7d20\uff09\uff0c\u5e76\u5c06\u8fd9\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\u4f5c\u4e3a\u57fa\u51c6\u6570\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u57fa\u51c6\u6570\u201c\u65e2\u4e0d\u592a\u5c0f\u4e5f\u4e0d\u592a\u5927\u201d\u7684\u6982\u7387\u5c06\u5927\u5e45\u63d0\u5347\u3002\u5f53\u7136\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u9009\u53d6\u66f4\u591a\u5019\u9009\u5143\u7d20\uff0c\u4ee5\u8fdb\u4e00\u6b65\u63d0\u9ad8\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002\u91c7\u7528\u8fd9\u79cd\u65b9\u6cd5\u540e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u7684\u6982\u7387\u5927\u5927\u964d\u4f4e\u3002</p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def median_three(self, nums: list[int], left: int, mid: int, right: int) -&gt; int:\n    \"\"\"\u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\"\"\"\n    l, m, r = nums[left], nums[mid], nums[right]\n    if (l &lt;= m &lt;= r) or (r &lt;= m &lt;= l):\n        return mid  # m \u5728 l \u548c r \u4e4b\u95f4\n    if (m &lt;= l &lt;= r) or (r &lt;= l &lt;= m):\n        return left  # l \u5728 m \u548c r \u4e4b\u95f4\n    return right\n\ndef partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"\u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\"\"\"\n    # \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    med = self.median_three(nums, left, (left + right) // 2, right)\n    # \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    nums[left], nums[med] = nums[med], nums[left]\n    # \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        # \u5143\u7d20\u4ea4\u6362\n        nums[i], nums[j] = nums[j], nums[i]\n    # \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n</code></pre> quick_sort.cpp<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u95f4\n    return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.java<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u95f4\n    return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right) {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.cs<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint MedianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u95f4\n    return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint Partition(int[] nums, int left, int right) {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    int med = MedianThree(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    Swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        Swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    Swap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.go<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n    l, m, r := nums[left], nums[mid], nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m \u5728 l \u548c r \u4e4b\u95f4\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l \u5728 m \u548c r \u4e4b\u95f4\n    }\n    return right\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09*/\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    med := q.medianThree(nums, left, (left+right)/2, right)\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    nums[left], nums[med] = nums[med], nums[left]\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- //\u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ //\u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        //\u5143\u7d20\u4ea4\u6362\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    //\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    nums[i], nums[left] = nums[left], nums[i]\n    return i //\u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.swift<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -&gt; Int {\n    let l = nums[left]\n    let m = nums[mid]\n    let r = nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m \u5728 l \u548c r \u4e4b\u95f4\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l \u5728 m \u548c r \u4e4b\u95f4\n    }\n    return right\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    let med = medianThree(nums: nums, left: left, mid: (left + right) / 2, right: right)\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    nums.swapAt(left, med)\n    return partition(nums: &amp;nums, left: left, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(nums, left, mid, right) {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l \u5728 m \u548c r \u4e4b\u95f4\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums, left, right) {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    this.swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.ts<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(\n    nums: number[],\n    left: number,\n    mid: number,\n    right: number\n): number {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l \u5728 m \u548c r \u4e4b\u95f4\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums: number[], left: number, right: number): number {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    this.swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.dart<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint _medianThree(List&lt;int&gt; nums, int left, int mid, int right) {\n  int l = nums[left], m = nums[mid], r = nums[right];\n  if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n    return mid; // m \u5728 l \u548c r \u4e4b\u95f4\n  if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n    return left; // l \u5728 m \u548c r \u4e4b\u95f4\n  return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n  int med = _medianThree(nums, left, (left + right) ~/ 2, right);\n  // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n  _swap(nums, left, med);\n  // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n    _swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n  }\n  _swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n  return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rs<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfn median_three(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) -&gt; usize {\n    let (l, m, r) = (nums[left], nums[mid], nums[right]);\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid; // m \u5728 l \u548c r \u4e4b\u95f4\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left; // l \u5728 m \u548c r \u4e4b\u95f4\n    }\n    right\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    let med = Self::median_three(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    nums.swap(left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        nums.swap(i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    nums.swap(i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u4e2d\u4f4d\u57fa\u51c6\u6570\u4f18\u5316\uff09 */\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint medianThree(int nums[], int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u95f4\n    return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */ \nint partitionMedian(int nums[], int left, int right) {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.kt<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfun medianThree(nums: IntArray, left: Int, mid: Int, right: Int): Int {\n    val l = nums[left]\n    val m = nums[mid]\n    val r = nums[right]\n    if ((m in l..r) || (m in r..l))\n        return mid  // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((l in m..r) || (l in r..m))\n        return left // l \u5728 m \u548c r \u4e4b\u95f4\n    return right\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nfun partition(nums: IntArray, left: Int, right: Int): Int {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--           // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++           // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j)  // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left)   // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i              // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rb<pre><code>[class]{QuickSortMedian}-[func]{median_three}\n\n[class]{QuickSortMedian}-[func]{partition}\n</code></pre> quick_sort.zig<pre><code>// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nfn medianThree(nums: []i32, left: usize, mid: usize, right: usize) usize {\n    var l = nums[left];\n    var m = nums[mid];\n    var r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m \u5728 l \u548c r \u4e4b\u95f4\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l \u5728 m \u548c r \u4e4b\u95f4\n    return right;\n}\n\n// \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\nfn partition(nums: []i32, left: usize, right: usize) usize {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    var med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    var i = left;\n    var j = right;\n    while (i &lt; j) {\n        while (i &lt; j and nums[j] &gt;= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j and nums[i] &lt;= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/quick_sort/#1155","title":"11.5.5 \u00a0 \u5c3e\u9012\u5f52\u4f18\u5316","text":"<p>\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u53ef\u80fd\u5360\u7528\u7a7a\u95f4\u8f83\u591a\u3002\u4ee5\u5b8c\u5168\u6709\u5e8f\u7684\u8f93\u5165\u6570\u7ec4\u4e3a\u4f8b\uff0c\u8bbe\u9012\u5f52\u4e2d\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(m\\) \uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u90fd\u5c06\u4ea7\u751f\u957f\u5ea6\u4e3a \\(0\\) \u7684\u5de6\u5b50\u6570\u7ec4\u548c\u957f\u5ea6\u4e3a \\(m - 1\\) \u7684\u53f3\u5b50\u6570\u7ec4\uff0c\u8fd9\u610f\u5473\u7740\u6bcf\u4e00\u5c42\u9012\u5f52\u8c03\u7528\u51cf\u5c11\u7684\u95ee\u9898\u89c4\u6a21\u975e\u5e38\u5c0f\uff08\u53ea\u51cf\u5c11\u4e00\u4e2a\u5143\u7d20\uff09\uff0c\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\u4f1a\u8fbe\u5230 \\(n - 1\\) \uff0c\u6b64\u65f6\u9700\u8981\u5360\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002</p> <p>\u4e3a\u4e86\u9632\u6b62\u6808\u5e27\u7a7a\u95f4\u7684\u7d2f\u79ef\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6bcf\u8f6e\u54e8\u5175\u6392\u5e8f\u5b8c\u6210\u540e\uff0c\u6bd4\u8f83\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\uff0c\u4ec5\u5bf9\u8f83\u77ed\u7684\u5b50\u6570\u7ec4\u8fdb\u884c\u9012\u5f52\u3002\u7531\u4e8e\u8f83\u77ed\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(n / 2\\) \uff0c\u56e0\u6b64\u8fd9\u79cd\u65b9\u6cd5\u80fd\u786e\u4fdd\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff0c\u4ece\u800c\u5c06\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"\u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\"\"\"\n    # \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while left &lt; right:\n        # \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        pivot = self.partition(nums, left, right)\n        # \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if pivot - left &lt; right - pivot:\n            self.quick_sort(nums, left, pivot - 1)  # \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1  # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        else:\n            self.quick_sort(nums, pivot + 1, right)  # \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1  # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n</code></pre> quick_sort.cpp<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        int pivot = partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1;                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1;                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.java<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        int pivot = partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid QuickSort(int[] nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        int pivot = Partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            QuickSort(nums, left, pivot - 1);  // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1;  // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            QuickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.go<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09*/\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    for left &lt; right {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        pivot := q.partition(nums, left, right)\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if pivot-left &lt; right-pivot {\n            q.quickSort(nums, left, pivot-1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            q.quickSort(nums, pivot+1, right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.swift<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\n    var left = left\n    var right = right\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while left &lt; right {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        let pivot = partition(nums: &amp;nums, left: left, right: right)\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left) &lt; (right - pivot) {\n            quickSortTailCall(nums: &amp;nums, left: left, right: pivot - 1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            quickSortTailCall(nums: &amp;nums, left: pivot + 1, right: right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.js<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums, left, right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        let pivot = this.partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            this.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums: number[], left: number, right: number): void {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        let pivot = this.partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            this.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n  while (left &lt; right) {\n    // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n    int pivot = _partition(nums, left, right);\n    // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n    if (pivot - left &lt; right - pivot) {\n      quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n      left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n    } else {\n      quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n      right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n    }\n  }\n}\n</code></pre> quick_sort.rs<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\npub fn quick_sort(mut left: i32, mut right: i32, nums: &amp;mut [i32]) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while left &lt; right {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if pivot - left &lt; right - pivot {\n            Self::quick_sort(left, pivot - 1, nums); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            Self::quick_sort(pivot + 1, right, nums); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\n// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nvoid quickSortTailCall(int nums[], int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        int pivot = partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSortTailCall(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1;                         // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            quickSortTailCall(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1;                         // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.kt<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfun quickSort(nums: IntArray, left: Int, right: Int) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) return\n    // \u54e8\u5175\u5212\u5206\n    val pivot = partition(nums, left, right)\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1)\n    quickSort(nums, pivot + 1, right)\n}\n</code></pre> quick_sort.rb<pre><code>[class]{QuickSortTailCall}-[func]{quick_sort}\n</code></pre> quick_sort.zig<pre><code>// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nfn quickSort(nums: []i32, left_: usize, right_: usize) void {\n    var left = left_;\n    var right = right_;\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        var pivot = partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1);   // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1;                   // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right);  // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1;                  // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/radix_sort/","title":"11.10 \u00a0 \u57fa\u6570\u6392\u5e8f","text":"<p>\u4e0a\u4e00\u8282\u4ecb\u7ecd\u4e86\u8ba1\u6570\u6392\u5e8f\uff0c\u5b83\u9002\u7528\u4e8e\u6570\u636e\u91cf \\(n\\) \u8f83\u5927\u4f46\u6570\u636e\u8303\u56f4 \\(m\\) \u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u5bf9 \\(n = 10^6\\) \u4e2a\u5b66\u53f7\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u5b66\u53f7\u662f\u4e00\u4e2a \\(8\\) \u4f4d\u6570\u5b57\uff0c\u8fd9\u610f\u5473\u7740\u6570\u636e\u8303\u56f4 \\(m = 10^8\\) \u975e\u5e38\u5927\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u9700\u8981\u5206\u914d\u5927\u91cf\u5185\u5b58\u7a7a\u95f4\uff0c\u800c\u57fa\u6570\u6392\u5e8f\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u3002</p> <p>\u57fa\u6570\u6392\u5e8f\uff08radix sort\uff09\u7684\u6838\u5fc3\u601d\u60f3\u4e0e\u8ba1\u6570\u6392\u5e8f\u4e00\u81f4\uff0c\u4e5f\u901a\u8fc7\u7edf\u8ba1\u4e2a\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u57fa\u6570\u6392\u5e8f\u5229\u7528\u6570\u5b57\u5404\u4f4d\u4e4b\u95f4\u7684\u9012\u8fdb\u5173\u7cfb\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e00\u4f4d\u8fdb\u884c\u6392\u5e8f\uff0c\u4ece\u800c\u5f97\u5230\u6700\u7ec8\u7684\u6392\u5e8f\u7ed3\u679c\u3002</p>"},{"location":"chapter_sorting/radix_sort/#11101","title":"11.10.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u4ee5\u5b66\u53f7\u6570\u636e\u4e3a\u4f8b\uff0c\u5047\u8bbe\u6570\u5b57\u7684\u6700\u4f4e\u4f4d\u662f\u7b2c \\(1\\) \u4f4d\uff0c\u6700\u9ad8\u4f4d\u662f\u7b2c \\(8\\) \u4f4d\uff0c\u57fa\u6570\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-18 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316\u4f4d\u6570 \\(k = 1\\) \u3002</li> <li>\u5bf9\u5b66\u53f7\u7684\u7b2c \\(k\\) \u4f4d\u6267\u884c\u201c\u8ba1\u6570\u6392\u5e8f\u201d\u3002\u5b8c\u6210\u540e\uff0c\u6570\u636e\u4f1a\u6839\u636e\u7b2c \\(k\\) \u4f4d\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u3002</li> <li>\u5c06 \\(k\\) \u589e\u52a0 \\(1\\) \uff0c\u7136\u540e\u8fd4\u56de\u6b65\u9aa4 <code>2.</code> \u7ee7\u7eed\u8fed\u4ee3\uff0c\u76f4\u5230\u6240\u6709\u4f4d\u90fd\u6392\u5e8f\u5b8c\u6210\u540e\u7ed3\u675f\u3002</li> </ol> <p></p> <p> \u56fe 11-18 \u00a0 \u57fa\u6570\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b </p> <p>\u4e0b\u9762\u5256\u6790\u4ee3\u7801\u5b9e\u73b0\u3002\u5bf9\u4e8e\u4e00\u4e2a \\(d\\) \u8fdb\u5236\u7684\u6570\u5b57 \\(x\\) \uff0c\u8981\u83b7\u53d6\u5176\u7b2c \\(k\\) \u4f4d \\(x_k\\) \uff0c\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u8ba1\u7b97\u516c\u5f0f\uff1a</p> \\[ x_k = \\lfloor\\frac{x}{d^{k-1}}\\rfloor \\bmod d \\] <p>\u5176\u4e2d \\(\\lfloor a \\rfloor\\) \u8868\u793a\u5bf9\u6d6e\u70b9\u6570 \\(a\\) \u5411\u4e0b\u53d6\u6574\uff0c\u800c \\(\\bmod \\: d\\) \u8868\u793a\u5bf9 \\(d\\) \u53d6\u6a21\uff08\u53d6\u4f59\uff09\u3002\u5bf9\u4e8e\u5b66\u53f7\u6570\u636e\uff0c\\(d = 10\\) \u4e14 \\(k \\in [1, 8]\\) \u3002</p> <p>\u6b64\u5916\uff0c\u6211\u4eec\u9700\u8981\u5c0f\u5e45\u6539\u52a8\u8ba1\u6570\u6392\u5e8f\u4ee3\u7801\uff0c\u4f7f\u4e4b\u53ef\u4ee5\u6839\u636e\u6570\u5b57\u7684\u7b2c \\(k\\) \u4f4d\u8fdb\u884c\u6392\u5e8f\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig radix_sort.py<pre><code>def digit(num: int, exp: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\"\"\"\n    # \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num // exp) % 10\n\ndef counting_sort_digit(nums: list[int], exp: int):\n    \"\"\"\u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\"\"\"\n    # \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    counter = [0] * 10\n    n = len(nums)\n    # \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for i in range(n):\n        d = digit(nums[i], exp)  # \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d] += 1  # \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    # \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for i in range(1, 10):\n        counter[i] += counter[i - 1]\n    # \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        d = digit(nums[i], exp)\n        j = counter[d] - 1  # \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]  # \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1  # \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    # \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in range(n):\n        nums[i] = res[i]\n\ndef radix_sort(nums: list[int]):\n    \"\"\"\u57fa\u6570\u6392\u5e8f\"\"\"\n    # \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    m = max(nums)\n    # \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    exp = 1\n    while exp &lt;= m:\n        # \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        # k = 1 -&gt; exp = 1\n        # k = 2 -&gt; exp = 10\n        # \u5373 exp = 10^(k-1)\n        counting_sort_digit(nums, exp)\n        exp *= 10\n</code></pre> radix_sort.cpp<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(vector&lt;int&gt; &amp;nums, int exp) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    vector&lt;int&gt; counter(10, 0);\n    int n = nums.size();\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    vector&lt;int&gt; res(n, 0);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(vector&lt;int&gt; &amp;nums) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    int m = *max_element(nums.begin(), nums.end());\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (int exp = 1; exp &lt;= m; exp *= 10)\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.java<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int[] nums, int exp) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    int[] counter = new int[10];\n    int n = nums.length;\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int[] nums) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    int m = Integer.MIN_VALUE;\n    for (int num : nums)\n        if (num &gt; m)\n            m = num;\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (int exp = 1; exp &lt;= m; exp *= 10)\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.cs<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint Digit(int num, int exp) {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid CountingSortDigit(int[] nums, int exp) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    int[] counter = new int[10];\n    int n = nums.Length;\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (int i = 0; i &lt; n; i++) {\n        int d = Digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = Digit(nums[i], exp);\n        int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid RadixSort(int[] nums) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    int m = int.MinValue;\n    foreach (int num in nums) {\n        if (num &gt; m) m = num;\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (int exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        CountingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.go<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num, exp int) int {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums []int, exp int) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    counter := make([]int, 10)\n    n := len(nums)\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for i := 0; i &lt; n; i++ {\n        d := digit(nums[i], exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++             // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for i := 1; i &lt; 10; i++ {\n        counter[i] += counter[i-1]\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        d := digit(nums[i], exp)\n        j := counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]    // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--        // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i := 0; i &lt; n; i++ {\n        nums[i] = res[i]\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums []int) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    max := math.MinInt\n    for _, num := range nums {\n        if num &gt; max {\n            max = num\n        }\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for exp := 1; max &gt;= exp; exp *= 10 {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp)\n    }\n}\n</code></pre> radix_sort.swift<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num: Int, exp: Int) -&gt; Int {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    (num / exp) % 10\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums: inout [Int], exp: Int) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    var counter = Array(repeating: 0, count: 10)\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for i in nums.indices {\n        let d = digit(num: nums[i], exp: exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d] += 1 // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for i in 1 ..&lt; 10 {\n        counter[i] += counter[i - 1]\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let d = digit(num: nums[i], exp: exp)\n        let j = counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i] // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1 // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums: inout [Int]) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    var m = Int.min\n    for num in nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for exp in sequence(first: 1, next: { m &gt;= ($0 * 10) ? $0 * 10 : nil }) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums: &amp;nums, exp: exp)\n    }\n}\n</code></pre> radix_sort.js<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num, exp) {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return Math.floor(num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums, exp) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    let m = Number.MIN_VALUE;\n    for (const num of nums) {\n        if (num &gt; m) {\n            m = num;\n        }\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.ts<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num: number, exp: number): number {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return Math.floor(num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums: number[], exp: number): void {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums: number[]): void {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    let m = Number.MIN_VALUE;\n    for (const num of nums) {\n        if (num &gt; m) {\n            m = num;\n        }\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.dart<pre><code>/* \u83b7\u53d6\u5143\u7d20 _num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int _num, int exp) {\n  // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n  return (_num ~/ exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(List&lt;int&gt; nums, int exp) {\n  // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n  List&lt;int&gt; counter = List&lt;int&gt;.filled(10, 0);\n  int n = nums.length;\n  // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n  for (int i = 0; i &lt; n; i++) {\n    int d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n    counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n  }\n  // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n  for (int i = 1; i &lt; 10; i++) {\n    counter[i] += counter[i - 1];\n  }\n  // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n  List&lt;int&gt; res = List&lt;int&gt;.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int d = digit(nums[i], exp);\n    int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n    res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n    counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n  }\n  // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n  for (int i = 0; i &lt; n; i++) nums[i] = res[i];\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(List&lt;int&gt; nums) {\n  // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n  // dart \u4e2d int \u7684\u957f\u5ea6\u662f 64 \u4f4d\u7684\n  int m = -1 &lt;&lt; 63;\n  for (int _num in nums) if (_num &gt; m) m = _num;\n  // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n  for (int exp = 1; exp &lt;= m; exp *= 10)\n    // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n    // k = 1 -&gt; exp = 1\n    // k = 2 -&gt; exp = 10\n    // \u5373 exp = 10^(k-1)\n    countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.rs<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfn digit(num: i32, exp: i32) -&gt; usize {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return ((num / exp) % 10) as usize;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfn counting_sort_digit(nums: &amp;mut [i32], exp: i32) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    let mut counter = [0; 10];\n    let n = nums.len();\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for i in 0..n {\n        let d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d] += 1; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for i in 1..10 {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let d = digit(nums[i], exp);\n        let j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in 0..n {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfn radix_sort(nums: &amp;mut [i32]) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    let m = *nums.into_iter().max().unwrap();\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    let mut exp = 1;\n    while exp &lt;= m {\n        counting_sort_digit(nums, exp);\n        exp *= 10;\n    }\n}\n</code></pre> radix_sort.c<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int nums[], int size, int exp) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    int *counter = (int *)malloc((sizeof(int) * 10));\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (int i = 0; i &lt; size; i++) {\n        // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        int d = digit(nums[i], exp);\n        // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n        counter[d]++;\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    int *res = (int *)malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; size; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int nums[], int size) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    int max = INT32_MIN;\n    for (size_t i = 0; i &lt; size - 1; i++) {\n        if (nums[i] &gt; max) {\n            max = nums[i];\n        }\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (int exp = 1; max &gt;= exp; exp *= 10)\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, size, exp);\n}\n</code></pre> radix_sort.kt<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfun digit(num: Int, exp: Int): Int {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfun countingSortDigit(nums: IntArray, exp: Int) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    val counter = IntArray(10)\n    val n = nums.size\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (i in 0..&lt;n) {\n        val d = digit(nums[i], exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++ // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (i in 1..9) {\n        counter[i] += counter[i - 1]\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val d = digit(nums[i], exp)\n        val j = counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i] // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]-- // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (i in 0..&lt;n) nums[i] = res[i]\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfun radixSort(nums: IntArray) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    var m = Int.MIN_VALUE\n    for (num in nums) if (num &gt; m) m = num\n    var exp = 1\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    while (exp &lt;= m) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp)\n        exp *= 10\n    }\n}\n</code></pre> radix_sort.rb<pre><code>[class]{}-[func]{digit}\n\n[class]{}-[func]{counting_sort_digit}\n\n[class]{}-[func]{radix_sort}\n</code></pre> radix_sort.zig<pre><code>// \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\nfn digit(num: i32, exp: i32) i32 {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return @mod(@divFloor(num, exp), 10);\n}\n\n// \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\nfn countingSortDigit(nums: []i32, exp: i32) !void {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n    // defer mem_arena.deinit();\n    const mem_allocator = mem_arena.allocator();\n    var counter = try mem_allocator.alloc(usize, 10);\n    @memset(counter, 0);\n    var n = nums.len;\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (nums) |num| {\n        var d: u32 = @bitCast(digit(num, exp)); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d] += 1; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    var i: usize = 1;\n    while (i &lt; 10) : (i += 1) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    var res = try mem_allocator.alloc(i32, n);\n    i = n - 1;\n    while (i &gt;= 0) : (i -= 1) {\n        var d: u32 = @bitCast(digit(nums[i], exp));\n        var j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1;        // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n        if (i == 0) break;\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    i = 0;\n    while (i &lt; n) : (i += 1) {\n        nums[i] = res[i];\n    }\n}\n\n// \u57fa\u6570\u6392\u5e8f\nfn radixSort(nums: []i32) !void {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    var m: i32 = std.math.minInt(i32);\n    for (nums) |num| {\n        if (num &gt; m) m = num;\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    var exp: i32 = 1;\n    while (exp &lt;= m) : (exp *= 10) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        try countingSortDigit(nums, exp);    \n    }\n} \n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u4e3a\u4ec0\u4e48\u4ece\u6700\u4f4e\u4f4d\u5f00\u59cb\u6392\u5e8f\uff1f</p> <p>\u5728\u8fde\u7eed\u7684\u6392\u5e8f\u8f6e\u6b21\u4e2d\uff0c\u540e\u4e00\u8f6e\u6392\u5e8f\u4f1a\u8986\u76d6\u524d\u4e00\u8f6e\u6392\u5e8f\u7684\u7ed3\u679c\u3002\u4e3e\u4f8b\u6765\u8bf4\uff0c\u5982\u679c\u7b2c\u4e00\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a &lt; b\\) \uff0c\u800c\u7b2c\u4e8c\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a &gt; b\\) \uff0c\u90a3\u4e48\u7b2c\u4e8c\u8f6e\u7684\u7ed3\u679c\u5c06\u53d6\u4ee3\u7b2c\u4e00\u8f6e\u7684\u7ed3\u679c\u3002\u7531\u4e8e\u6570\u5b57\u7684\u9ad8\u4f4d\u4f18\u5148\u7ea7\u9ad8\u4e8e\u4f4e\u4f4d\uff0c\u56e0\u6b64\u5e94\u8be5\u5148\u6392\u5e8f\u4f4e\u4f4d\u518d\u6392\u5e8f\u9ad8\u4f4d\u3002</p>"},{"location":"chapter_sorting/radix_sort/#11102","title":"11.10.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<p>\u76f8\u8f83\u4e8e\u8ba1\u6570\u6392\u5e8f\uff0c\u57fa\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u503c\u8303\u56f4\u8f83\u5927\u7684\u60c5\u51b5\uff0c\u4f46\u524d\u63d0\u662f\u6570\u636e\u5fc5\u987b\u53ef\u4ee5\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u683c\u5f0f\uff0c\u4e14\u4f4d\u6570\u4e0d\u80fd\u8fc7\u5927\u3002\u4f8b\u5982\uff0c\u6d6e\u70b9\u6570\u4e0d\u9002\u5408\u4f7f\u7528\u57fa\u6570\u6392\u5e8f\uff0c\u56e0\u4e3a\u5176\u4f4d\u6570 \\(k\\) \u8fc7\u5927\uff0c\u53ef\u80fd\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(nk) \\gg O(n^2)\\) \u3002</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nk)\\)\uff1a\u8bbe\u6570\u636e\u91cf\u4e3a \\(n\\)\u3001\u6570\u636e\u4e3a \\(d\\) \u8fdb\u5236\u3001\u6700\u5927\u4f4d\u6570\u4e3a \\(k\\) \uff0c\u5219\u5bf9\u67d0\u4e00\u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(n + d)\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6240\u6709 \\(k\\) \u4f4d\u4f7f\u7528 \\(O((n + d)k)\\) \u65f6\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\\(d\\) \u548c \\(k\\) \u90fd\u76f8\u5bf9\u8f83\u5c0f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n + d)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u4e0e\u8ba1\u6570\u6392\u5e8f\u76f8\u540c\uff0c\u57fa\u6570\u6392\u5e8f\u9700\u8981\u501f\u52a9\u957f\u5ea6\u4e3a \\(n\\) \u548c \\(d\\) \u7684\u6570\u7ec4 <code>res</code> \u548c <code>counter</code> \u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u5f53\u8ba1\u6570\u6392\u5e8f\u7a33\u5b9a\u65f6\uff0c\u57fa\u6570\u6392\u5e8f\u4e5f\u7a33\u5b9a\uff1b\u5f53\u8ba1\u6570\u6392\u5e8f\u4e0d\u7a33\u5b9a\u65f6\uff0c\u57fa\u6570\u6392\u5e8f\u65e0\u6cd5\u4fdd\u8bc1\u5f97\u5230\u6b63\u786e\u7684\u6392\u5e8f\u7ed3\u679c\u3002</li> </ul>"},{"location":"chapter_sorting/selection_sort/","title":"11.2 \u00a0 \u9009\u62e9\u6392\u5e8f","text":"<p>\u9009\u62e9\u6392\u5e8f\uff08selection sort\uff09\u7684\u5de5\u4f5c\u539f\u7406\u975e\u5e38\u7b80\u5355\uff1a\u5f00\u542f\u4e00\u4e2a\u5faa\u73af\uff0c\u6bcf\u8f6e\u4ece\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u6700\u5c0f\u7684\u5143\u7d20\uff0c\u5c06\u5176\u653e\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u672b\u5c3e\u3002</p> <p>\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u9009\u62e9\u6392\u5e8f\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u56fe 11-2 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u672a\u6392\u5e8f\uff0c\u5373\u672a\u6392\u5e8f\uff08\u7d22\u5f15\uff09\u533a\u95f4\u4e3a \\([0, n-1]\\) \u3002</li> <li>\u9009\u53d6\u533a\u95f4 \\([0, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(0\\) \u5904\u7684\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 1 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u9009\u53d6\u533a\u95f4 \\([1, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(1\\) \u5904\u7684\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ee5\u6b64\u7c7b\u63a8\u3002\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u9009\u62e9\u4e0e\u4ea4\u6362\u540e\uff0c\u6570\u7ec4\u524d \\(n - 1\\) \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5927\u5143\u7d20\uff0c\u65e0\u987b\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-2 \u00a0 \u9009\u62e9\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u7528 \\(k\\) \u6765\u8bb0\u5f55\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig selection_sort.py<pre><code>def selection_sort(nums: list[int]):\n    \"\"\"\u9009\u62e9\u6392\u5e8f\"\"\"\n    n = len(nums)\n    # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for i in range(n - 1):\n        # \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        k = i\n        for j in range(i + 1, n):\n            if nums[j] &lt; nums[k]:\n                k = j  # \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        # \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        nums[i], nums[k] = nums[k], nums[i]\n</code></pre> selection_sort.cpp<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(vector&lt;int&gt; &amp;nums) {\n    int n = nums.size();\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        swap(nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.java<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int[] nums) {\n    int n = nums.length;\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.cs<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid SelectionSort(int[] nums) {\n    int n = nums.Length;\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        (nums[k], nums[i]) = (nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.go<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums []int) {\n    n := len(nums)\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for i := 0; i &lt; n-1; i++ {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        k := i\n        for j := i + 1; j &lt; n; j++ {\n            if nums[j] &lt; nums[k] {\n                // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n                k = j\n            }\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        nums[i], nums[k] = nums[k], nums[i]\n\n    }\n}\n</code></pre> selection_sort.swift<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums: inout [Int]) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for i in nums.indices.dropLast() {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        var k = i\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[j] &lt; nums[k] {\n                k = j // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        nums.swapAt(i, k)\n    }\n}\n</code></pre> selection_sort.js<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums) {\n    let n = nums.length;\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (let i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.ts<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums: number[]): void {\n    let n = nums.length;\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (let i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.dart<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(List&lt;int&gt; nums) {\n  int n = nums.length;\n  // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n  for (int i = 0; i &lt; n - 1; i++) {\n    // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n    int k = i;\n    for (int j = i + 1; j &lt; n; j++) {\n      if (nums[j] &lt; nums[k]) k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n    }\n    // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n    int temp = nums[i];\n    nums[i] = nums[k];\n    nums[k] = temp;\n  }\n}\n</code></pre> selection_sort.rs<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfn selection_sort(nums: &amp;mut [i32]) {\n    if nums.is_empty() {\n        return;\n    }\n    let n = nums.len();\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for i in 0..n - 1 {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        let mut k = i;\n        for j in i + 1..n {\n            if nums[j] &lt; nums[k] {\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        nums.swap(i, k);\n    }\n}\n</code></pre> selection_sort.c<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int nums[], int n) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.kt<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfun selectionSort(nums: IntArray) {\n    val n = nums.size\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (i in 0..&lt;n - 1) {\n        var k = i\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        for (j in i + 1..&lt;n) {\n            if (nums[j] &lt; nums[k]) k = j // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        nums[i] = nums[k].also { nums[k] = nums[i] }\n    }\n}\n</code></pre> selection_sort.rb<pre><code>[class]{}-[func]{selection_sort}\n</code></pre> selection_sort.zig<pre><code>[class]{}-[func]{selectionSort}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_sorting/selection_sort/#1121","title":"11.2.1 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5916\u5faa\u73af\u5171 \\(n - 1\\) \u8f6e\uff0c\u7b2c\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u6700\u540e\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(2\\) \uff0c\u5373\u5404\u8f6e\u5916\u5faa\u73af\u5206\u522b\u5305\u542b \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(3\\)\u3001\\(2\\) \u8f6e\u5185\u5faa\u73af\uff0c\u6c42\u548c\u4e3a \\(\\frac{(n - 1)(n + 2)}{2}\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5982\u56fe 11-3 \u6240\u793a\uff0c\u5143\u7d20 <code>nums[i]</code> \u6709\u53ef\u80fd\u88ab\u4ea4\u6362\u81f3\u4e0e\u5176\u76f8\u7b49\u7684\u5143\u7d20\u7684\u53f3\u8fb9\uff0c\u5bfc\u81f4\u4e24\u8005\u7684\u76f8\u5bf9\u987a\u5e8f\u53d1\u751f\u6539\u53d8\u3002</li> </ul> <p> \u56fe 11-3 \u00a0 \u9009\u62e9\u6392\u5e8f\u975e\u7a33\u5b9a\u793a\u4f8b </p>"},{"location":"chapter_sorting/sorting_algorithm/","title":"11.1 \u00a0 \u6392\u5e8f\u7b97\u6cd5","text":"<p>\u6392\u5e8f\u7b97\u6cd5\uff08sorting algorithm\uff09\u7528\u4e8e\u5bf9\u4e00\u7ec4\u6570\u636e\u6309\u7167\u7279\u5b9a\u987a\u5e8f\u8fdb\u884c\u6392\u5217\u3002\u6392\u5e8f\u7b97\u6cd5\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u56e0\u4e3a\u6709\u5e8f\u6570\u636e\u901a\u5e38\u80fd\u591f\u88ab\u66f4\u9ad8\u6548\u5730\u67e5\u627e\u3001\u5206\u6790\u548c\u5904\u7406\u3002</p> <p>\u5982\u56fe 11-1 \u6240\u793a\uff0c\u6392\u5e8f\u7b97\u6cd5\u4e2d\u7684\u6570\u636e\u7c7b\u578b\u53ef\u4ee5\u662f\u6574\u6570\u3001\u6d6e\u70b9\u6570\u3001\u5b57\u7b26\u6216\u5b57\u7b26\u4e32\u7b49\u3002\u6392\u5e8f\u7684\u5224\u65ad\u89c4\u5219\u53ef\u6839\u636e\u9700\u6c42\u8bbe\u5b9a\uff0c\u5982\u6570\u5b57\u5927\u5c0f\u3001\u5b57\u7b26 ASCII \u7801\u987a\u5e8f\u6216\u81ea\u5b9a\u4e49\u89c4\u5219\u3002</p> <p></p> <p> \u56fe 11-1 \u00a0 \u6570\u636e\u7c7b\u578b\u548c\u5224\u65ad\u89c4\u5219\u793a\u4f8b </p>"},{"location":"chapter_sorting/sorting_algorithm/#1111","title":"11.1.1 \u00a0 \u8bc4\u4ef7\u7ef4\u5ea6","text":"<p>\u8fd0\u884c\u6548\u7387\uff1a\u6211\u4eec\u671f\u671b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c3d\u91cf\u4f4e\uff0c\u4e14\u603b\u4f53\u64cd\u4f5c\u6570\u91cf\u8f83\u5c11\uff08\u65f6\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\u53d8\u5c0f\uff09\u3002\u5bf9\u4e8e\u5927\u6570\u636e\u91cf\u7684\u60c5\u51b5\uff0c\u8fd0\u884c\u6548\u7387\u663e\u5f97\u5c24\u4e3a\u91cd\u8981\u3002</p> <p>\u5c31\u5730\u6027\uff1a\u987e\u540d\u601d\u4e49\uff0c\u539f\u5730\u6392\u5e8f\u901a\u8fc7\u5728\u539f\u6570\u7ec4\u4e0a\u76f4\u63a5\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\uff0c\u65e0\u987b\u501f\u52a9\u989d\u5916\u7684\u8f85\u52a9\u6570\u7ec4\uff0c\u4ece\u800c\u8282\u7701\u5185\u5b58\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u539f\u5730\u6392\u5e8f\u7684\u6570\u636e\u642c\u8fd0\u64cd\u4f5c\u8f83\u5c11\uff0c\u8fd0\u884c\u901f\u5ea6\u4e5f\u66f4\u5feb\u3002</p> <p>\u7a33\u5b9a\u6027\uff1a\u7a33\u5b9a\u6392\u5e8f\u5728\u5b8c\u6210\u6392\u5e8f\u540e\uff0c\u76f8\u7b49\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u76f8\u5bf9\u987a\u5e8f\u4e0d\u53d1\u751f\u6539\u53d8\u3002</p> <p>\u7a33\u5b9a\u6392\u5e8f\u662f\u591a\u7ea7\u6392\u5e8f\u573a\u666f\u7684\u5fc5\u8981\u6761\u4ef6\u3002\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u5b58\u50a8\u5b66\u751f\u4fe1\u606f\u7684\u8868\u683c\uff0c\u7b2c 1 \u5217\u548c\u7b2c 2 \u5217\u5206\u522b\u662f\u59d3\u540d\u548c\u5e74\u9f84\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u975e\u7a33\u5b9a\u6392\u5e8f\u53ef\u80fd\u5bfc\u81f4\u8f93\u5165\u6570\u636e\u7684\u6709\u5e8f\u6027\u4e27\u5931\uff1a</p> <pre><code># \u8f93\u5165\u6570\u636e\u662f\u6309\u7167\u59d3\u540d\u6392\u5e8f\u597d\u7684\n# (name, age)\n  ('A', 19)\n  ('B', 18)\n  ('C', 21)\n  ('D', 19)\n  ('E', 23)\n\n# \u5047\u8bbe\u4f7f\u7528\u975e\u7a33\u5b9a\u6392\u5e8f\u7b97\u6cd5\u6309\u5e74\u9f84\u6392\u5e8f\u5217\u8868\uff0c\n# \u7ed3\u679c\u4e2d ('D', 19) \u548c ('A', 19) \u7684\u76f8\u5bf9\u4f4d\u7f6e\u6539\u53d8\uff0c\n# \u8f93\u5165\u6570\u636e\u6309\u59d3\u540d\u6392\u5e8f\u7684\u6027\u8d28\u4e22\u5931\n  ('B', 18)\n  ('D', 19)\n  ('A', 19)\n  ('C', 21)\n  ('E', 23)\n</code></pre> <p>\u81ea\u9002\u5e94\u6027\uff1a\u81ea\u9002\u5e94\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u53d7\u8f93\u5165\u6570\u636e\u7684\u5f71\u54cd\uff0c\u5373\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5e76\u4e0d\u5b8c\u5168\u76f8\u7b49\u3002</p> <p>\u81ea\u9002\u5e94\u6027\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u6765\u8bc4\u4f30\u3002\u5982\u679c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u5dee\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u8bf4\u660e\u6392\u5e8f\u7b97\u6cd5\u5728\u67d0\u4e9b\u6570\u636e\u4e0b\u6027\u80fd\u53ef\u80fd\u52a3\u5316\uff0c\u56e0\u6b64\u88ab\u89c6\u4e3a\u8d1f\u9762\u5c5e\u6027\uff1b\u800c\u5982\u679c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u5219\u88ab\u89c6\u4e3a\u6b63\u9762\u5c5e\u6027\u3002</p> <p>\u662f\u5426\u57fa\u4e8e\u6bd4\u8f83\uff1a\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u4f9d\u8d56\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff08\\(&lt;\\)\u3001\\(=\\)\u3001\\(&gt;\\)\uff09\u6765\u5224\u65ad\u5143\u7d20\u7684\u76f8\u5bf9\u987a\u5e8f\uff0c\u4ece\u800c\u6392\u5e8f\u6574\u4e2a\u6570\u7ec4\uff0c\u7406\u8bba\u6700\u4f18\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u800c\u975e\u6bd4\u8f83\u6392\u5e8f\u4e0d\u4f7f\u7528\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(n)\\) \uff0c\u4f46\u5176\u901a\u7528\u6027\u76f8\u5bf9\u8f83\u5dee\u3002</p>"},{"location":"chapter_sorting/sorting_algorithm/#1112","title":"11.1.2 \u00a0 \u7406\u60f3\u6392\u5e8f\u7b97\u6cd5","text":"<p>\u8fd0\u884c\u5feb\u3001\u539f\u5730\u3001\u7a33\u5b9a\u3001\u6b63\u5411\u81ea\u9002\u5e94\u3001\u901a\u7528\u6027\u597d\u3002\u663e\u7136\uff0c\u8fc4\u4eca\u4e3a\u6b62\u5c1a\u672a\u53d1\u73b0\u517c\u5177\u4ee5\u4e0a\u6240\u6709\u7279\u6027\u7684\u6392\u5e8f\u7b97\u6cd5\u3002\u56e0\u6b64\uff0c\u5728\u9009\u62e9\u6392\u5e8f\u7b97\u6cd5\u65f6\uff0c\u9700\u8981\u6839\u636e\u5177\u4f53\u7684\u6570\u636e\u7279\u70b9\u548c\u95ee\u9898\u9700\u6c42\u6765\u51b3\u5b9a\u3002</p> <p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u5171\u540c\u5b66\u4e60\u5404\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5e76\u57fa\u4e8e\u4e0a\u8ff0\u8bc4\u4ef7\u7ef4\u5ea6\u5bf9\u5404\u4e2a\u6392\u5e8f\u7b97\u6cd5\u7684\u4f18\u7f3a\u70b9\u8fdb\u884c\u5206\u6790\u3002</p>"},{"location":"chapter_sorting/summary/","title":"11.11 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_sorting/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u5192\u6ce1\u6392\u5e8f\u901a\u8fc7\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u901a\u8fc7\u6dfb\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d\u6765\u5b9e\u73b0\u63d0\u524d\u8fd4\u56de\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(n)\\) \u3002</li> <li>\u63d2\u5165\u6392\u5e8f\u6bcf\u8f6e\u5c06\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u5143\u7d20\u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u6b63\u786e\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b8c\u6210\u6392\u5e8f\u3002\u867d\u7136\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u7531\u4e8e\u5355\u5143\u64cd\u4f5c\u76f8\u5bf9\u8f83\u5c11\uff0c\u56e0\u6b64\u5728\u5c0f\u6570\u636e\u91cf\u7684\u6392\u5e8f\u4efb\u52a1\u4e2d\u975e\u5e38\u53d7\u6b22\u8fce\u3002</li> <li>\u5feb\u901f\u6392\u5e8f\u57fa\u4e8e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u54e8\u5175\u5212\u5206\u4e2d\uff0c\u6709\u53ef\u80fd\u6bcf\u6b21\u90fd\u9009\u53d6\u5230\u6700\u5dee\u7684\u57fa\u51c6\u6570\uff0c\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u3002\u5f15\u5165\u4e2d\u4f4d\u6570\u57fa\u51c6\u6570\u6216\u968f\u673a\u57fa\u51c6\u6570\u53ef\u4ee5\u964d\u4f4e\u8fd9\u79cd\u52a3\u5316\u7684\u6982\u7387\u3002\u5c3e\u9012\u5f52\u65b9\u6cd5\u53ef\u4ee5\u6709\u6548\u5730\u51cf\u5c11\u9012\u5f52\u6df1\u5ea6\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(\\log n)\\) \u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\u5305\u62ec\u5212\u5206\u548c\u5408\u5e76\u4e24\u4e2a\u9636\u6bb5\uff0c\u5178\u578b\u5730\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\u3002\u5728\u5f52\u5e76\u6392\u5e8f\u4e2d\uff0c\u6392\u5e8f\u6570\u7ec4\u9700\u8981\u521b\u5efa\u8f85\u52a9\u6570\u7ec4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b\u7136\u800c\u6392\u5e8f\u94fe\u8868\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \u3002</li> <li>\u6876\u6392\u5e8f\u5305\u542b\u4e09\u4e2a\u6b65\u9aa4\uff1a\u6570\u636e\u5206\u6876\u3001\u6876\u5185\u6392\u5e8f\u548c\u5408\u5e76\u7ed3\u679c\u3002\u5b83\u540c\u6837\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u9002\u7528\u4e8e\u6570\u636e\u4f53\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u3002\u6876\u6392\u5e8f\u7684\u5173\u952e\u5728\u4e8e\u5bf9\u6570\u636e\u8fdb\u884c\u5e73\u5747\u5206\u914d\u3002</li> <li>\u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5b83\u901a\u8fc7\u7edf\u8ba1\u6570\u636e\u51fa\u73b0\u7684\u6b21\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u6709\u9650\u7684\u60c5\u51b5\uff0c\u5e76\u4e14\u8981\u6c42\u6570\u636e\u80fd\u591f\u8f6c\u6362\u4e3a\u6b63\u6574\u6570\u3002</li> <li>\u57fa\u6570\u6392\u5e8f\u901a\u8fc7\u9010\u4f4d\u6392\u5e8f\u6765\u5b9e\u73b0\u6570\u636e\u6392\u5e8f\uff0c\u8981\u6c42\u6570\u636e\u80fd\u591f\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u6570\u5b57\u3002</li> <li>\u603b\u7684\u6765\u8bf4\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5177\u6709\u9ad8\u6548\u7387\u3001\u7a33\u5b9a\u3001\u539f\u5730\u4ee5\u53ca\u6b63\u5411\u81ea\u9002\u5e94\u6027\u7b49\u4f18\u70b9\u3002\u7136\u800c\uff0c\u6b63\u5982\u5176\u4ed6\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e00\u6837\uff0c\u6ca1\u6709\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\u80fd\u591f\u540c\u65f6\u6ee1\u8db3\u6240\u6709\u8fd9\u4e9b\u6761\u4ef6\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u6839\u636e\u6570\u636e\u7684\u7279\u6027\u6765\u9009\u62e9\u5408\u9002\u7684\u6392\u5e8f\u7b97\u6cd5\u3002</li> <li>\u56fe 11-19 \u5bf9\u6bd4\u4e86\u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u6548\u7387\u3001\u7a33\u5b9a\u6027\u3001\u5c31\u5730\u6027\u548c\u81ea\u9002\u5e94\u6027\u7b49\u3002</li> </ul> <p> \u56fe 11-19 \u00a0 \u6392\u5e8f\u7b97\u6cd5\u5bf9\u6bd4 </p>"},{"location":"chapter_sorting/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u6392\u5e8f\u7b97\u6cd5\u7a33\u5b9a\u6027\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u662f\u5fc5\u9700\u7684\uff1f</p> <p>\u5728\u73b0\u5b9e\u4e2d\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u57fa\u4e8e\u5bf9\u8c61\u7684\u67d0\u4e2a\u5c5e\u6027\u8fdb\u884c\u6392\u5e8f\u3002\u4f8b\u5982\uff0c\u5b66\u751f\u6709\u59d3\u540d\u548c\u8eab\u9ad8\u4e24\u4e2a\u5c5e\u6027\uff0c\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u4e00\u4e2a\u591a\u7ea7\u6392\u5e8f\uff1a\u5148\u6309\u7167\u59d3\u540d\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u5230 <code>(A, 180) (B, 185) (C, 170) (D, 170)</code> \uff1b\u518d\u5bf9\u8eab\u9ad8\u8fdb\u884c\u6392\u5e8f\u3002\u7531\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e0d\u7a33\u5b9a\uff0c\u56e0\u6b64\u53ef\u80fd\u5f97\u5230 <code>(D, 170) (C, 170) (A, 180) (B, 185)</code> \u3002</p> <p>\u53ef\u4ee5\u53d1\u73b0\uff0c\u5b66\u751f D \u548c C \u7684\u4f4d\u7f6e\u53d1\u751f\u4e86\u4ea4\u6362\uff0c\u59d3\u540d\u7684\u6709\u5e8f\u6027\u88ab\u7834\u574f\u4e86\uff0c\u800c\u8fd9\u662f\u6211\u4eec\u4e0d\u5e0c\u671b\u770b\u5230\u7684\u3002</p> <p>Q\uff1a\u54e8\u5175\u5212\u5206\u4e2d\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u4e0e\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u7684\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\u5417\uff1f</p> <p>\u4e0d\u884c\uff0c\u5f53\u6211\u4eec\u4ee5\u6700\u5de6\u7aef\u5143\u7d20\u4e3a\u57fa\u51c6\u6570\u65f6\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u518d\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002\u8fd9\u4e2a\u7ed3\u8bba\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u6211\u4eec\u6765\u5256\u6790\u4e00\u4e0b\u539f\u56e0\u3002</p> <p>\u54e8\u5175\u5212\u5206 <code>partition()</code> \u7684\u6700\u540e\u4e00\u6b65\u662f\u4ea4\u6362 <code>nums[left]</code> \u548c <code>nums[i]</code> \u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u57fa\u51c6\u6570\u5de6\u8fb9\u7684\u5143\u7d20\u90fd <code>&lt;=</code> \u57fa\u51c6\u6570\uff0c\u8fd9\u5c31\u8981\u6c42\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u524d <code>nums[left] &gt;= nums[i]</code> \u5fc5\u987b\u6210\u7acb\u3002\u5047\u8bbe\u6211\u4eec\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\uff0c\u90a3\u4e48\u5982\u679c\u627e\u4e0d\u5230\u6bd4\u57fa\u51c6\u6570\u66f4\u5927\u7684\u5143\u7d20\uff0c\u5219\u4f1a\u5728 <code>i == j</code> \u65f6\u8df3\u51fa\u5faa\u73af\uff0c\u6b64\u65f6\u53ef\u80fd <code>nums[j] == nums[i] &gt; nums[left]</code>\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u6b64\u65f6\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u64cd\u4f5c\u4f1a\u628a\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u66f4\u5927\u7684\u5143\u7d20\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\uff0c\u5bfc\u81f4\u54e8\u5175\u5212\u5206\u5931\u8d25\u3002</p> <p>\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u6570\u7ec4 <code>[0, 0, 0, 0, 1]</code> \uff0c\u5982\u679c\u5148\u201c\u4ece\u5de6\u5411\u53f3\u67e5\u627e\u201d\uff0c\u54e8\u5175\u5212\u5206\u540e\u6570\u7ec4\u4e3a <code>[1, 0, 0, 0, 0]</code> \uff0c\u8fd9\u4e2a\u7ed3\u679c\u662f\u4e0d\u6b63\u786e\u7684\u3002</p> <p>\u518d\u6df1\u5165\u601d\u8003\u4e00\u4e0b\uff0c\u5982\u679c\u6211\u4eec\u9009\u62e9 <code>nums[right]</code> \u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u6b63\u597d\u53cd\u8fc7\u6765\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002</p> <p>Q\uff1a\u5173\u4e8e\u5c3e\u9012\u5f52\u4f18\u5316\uff0c\u4e3a\u4ec0\u4e48\u9009\u77ed\u7684\u6570\u7ec4\u80fd\u4fdd\u8bc1\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff1f</p> <p>\u9012\u5f52\u6df1\u5ea6\u5c31\u662f\u5f53\u524d\u672a\u8fd4\u56de\u7684\u9012\u5f52\u65b9\u6cd5\u7684\u6570\u91cf\u3002\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u6211\u4eec\u5c06\u539f\u6570\u7ec4\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\u3002\u5728\u5c3e\u9012\u5f52\u4f18\u5316\u540e\uff0c\u5411\u4e0b\u9012\u5f52\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u6700\u5927\u4e3a\u539f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a\u3002\u5047\u8bbe\u6700\u5dee\u60c5\u51b5\uff0c\u4e00\u76f4\u4e3a\u4e00\u534a\u957f\u5ea6\uff0c\u90a3\u4e48\u6700\u7ec8\u7684\u9012\u5f52\u6df1\u5ea6\u5c31\u662f \\(\\log n\\) \u3002</p> <p>\u56de\u987e\u539f\u59cb\u7684\u5feb\u901f\u6392\u5e8f\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u4f1a\u8fde\u7eed\u5730\u9012\u5f52\u957f\u5ea6\u8f83\u5927\u7684\u6570\u7ec4\uff0c\u6700\u5dee\u60c5\u51b5\u4e0b\u4e3a \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \u3002\u5c3e\u9012\u5f52\u4f18\u5316\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u51fa\u73b0\u3002</p> <p>Q\uff1a\u5f53\u6570\u7ec4\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u76f8\u7b49\u65f6\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n^2)\\) \u5417\uff1f\u8be5\u5982\u4f55\u5904\u7406\u8fd9\u79cd\u9000\u5316\u60c5\u51b5\uff1f</p> <p>\u662f\u7684\u3002\u5bf9\u4e8e\u8fd9\u79cd\u60c5\u51b5\uff0c\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u54e8\u5175\u5212\u5206\u5c06\u6570\u7ec4\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\uff1a\u5c0f\u4e8e\u3001\u7b49\u4e8e\u3001\u5927\u4e8e\u57fa\u51c6\u6570\u3002\u4ec5\u5411\u4e0b\u9012\u5f52\u5c0f\u4e8e\u548c\u5927\u4e8e\u7684\u4e24\u90e8\u5206\u3002\u5728\u8be5\u65b9\u6cd5\u4e0b\uff0c\u8f93\u5165\u5143\u7d20\u5168\u90e8\u76f8\u7b49\u7684\u6570\u7ec4\uff0c\u4ec5\u4e00\u8f6e\u54e8\u5175\u5212\u5206\u5373\u53ef\u5b8c\u6210\u6392\u5e8f\u3002</p> <p>Q\uff1a\u6876\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u4ec0\u4e48\u662f \\(O(n^2)\\) \uff1f</p> <p>\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u88ab\u5206\u81f3\u540c\u4e00\u4e2a\u6876\u4e2d\u3002\u5982\u679c\u6211\u4eec\u91c7\u7528\u4e00\u4e2a \\(O(n^2)\\) \u7b97\u6cd5\u6765\u6392\u5e8f\u8fd9\u4e9b\u5143\u7d20\uff0c\u5219\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_stack_and_queue/","title":"\u7b2c 5 \u7ae0 \u00a0 \u6808\u4e0e\u961f\u5217","text":"<p>Abstract</p> <p>\u6808\u5982\u540c\u53e0\u732b\u732b\uff0c\u800c\u961f\u5217\u5c31\u50cf\u732b\u732b\u6392\u961f\u3002</p> <p>\u4e24\u8005\u5206\u522b\u4ee3\u8868\u5148\u5165\u540e\u51fa\u548c\u5148\u5165\u5148\u51fa\u7684\u903b\u8f91\u5173\u7cfb\u3002</p>"},{"location":"chapter_stack_and_queue/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>5.1 \u00a0 \u6808</li> <li>5.2 \u00a0 \u961f\u5217</li> <li>5.3 \u00a0 \u53cc\u5411\u961f\u5217</li> <li>5.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_stack_and_queue/deque/","title":"5.3 \u00a0 \u53cc\u5411\u961f\u5217","text":"<p>\u5728\u961f\u5217\u4e2d\uff0c\u6211\u4eec\u4ec5\u80fd\u5220\u9664\u5934\u90e8\u5143\u7d20\u6216\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u3002\u5982\u56fe 5-7 \u6240\u793a\uff0c\u53cc\u5411\u961f\u5217\uff08double-ended queue\uff09\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u7075\u6d3b\u6027\uff0c\u5141\u8bb8\u5728\u5934\u90e8\u548c\u5c3e\u90e8\u6267\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u6216\u5220\u9664\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 5-7 \u00a0 \u53cc\u5411\u961f\u5217\u7684\u64cd\u4f5c </p>"},{"location":"chapter_stack_and_queue/deque/#531","title":"5.3.1 \u00a0 \u53cc\u5411\u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"<p>\u53cc\u5411\u961f\u5217\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-3 \u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u79f0\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002</p> <p> \u8868 5-3 \u00a0 \u53cc\u5411\u961f\u5217\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 <code>push_first()</code> \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u9996 \\(O(1)\\) <code>push_last()</code> \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) <code>pop_first()</code> \u5220\u9664\u961f\u9996\u5143\u7d20 \\(O(1)\\) <code>pop_last()</code> \u5220\u9664\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) <code>peek_first()</code> \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) <code>peek_last()</code> \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) <p>\u540c\u6837\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u5df2\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig deque.py<pre><code>from collections import deque\n\n# \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217\ndeque: deque[int] = deque()\n\n# \u5143\u7d20\u5165\u961f\ndeque.append(2)      # \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.appendleft(3)  # \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.appendleft(1)\n\n# \u8bbf\u95ee\u5143\u7d20\nfront: int = deque[0]  # \u961f\u9996\u5143\u7d20\nrear: int = deque[-1]  # \u961f\u5c3e\u5143\u7d20\n\n# \u5143\u7d20\u51fa\u961f\npop_front: int = deque.popleft()  # \u961f\u9996\u5143\u7d20\u51fa\u961f\npop_rear: int = deque.pop()       # \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n# \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(deque)\n\n# \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(deque) == 0\n</code></pre> deque.cpp<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\ndeque&lt;int&gt; deque;\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint front = deque.front(); // \u961f\u9996\u5143\u7d20\nint back = deque.back();   // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\ndeque.pop_front();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.pop_back();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = deque.empty();\n</code></pre> deque.java<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nDeque&lt;Integer&gt; deque = new LinkedList&lt;&gt;();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.offerLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.offerFirst(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.peekFirst();  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.peekLast();    // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.pollFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.pollLast();    // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = deque.isEmpty();\n</code></pre> deque.cs<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 C# \u4e2d\uff0c\u5c06\u94fe\u8868 LinkedList \u770b\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nLinkedList&lt;int&gt; deque = new();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.AddLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.AddFirst(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.First.Value;  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.Last.Value;    // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\ndeque.RemoveFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.RemoveLast();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.Count;\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.Count == 0;\n</code></pre> deque_test.go<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u4f7f\u7528\ndeque := list.New()\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.PushBack(2)      // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.PushFront(1)\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nfront := deque.Front() // \u961f\u9996\u5143\u7d20\nrear := deque.Back()   // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\ndeque.Remove(front)    // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.Remove(rear)     // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize := deque.Len()\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := deque.Len() == 0\n</code></pre> deque.swift<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nvar deque: [Int] = []\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.append(2) // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.insert(1, at: 0)\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nlet peekFirst = deque.first! // \u961f\u9996\u5143\u7d20\nlet peekLast = deque.last! // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u4f7f\u7528 Array \u6a21\u62df\u65f6 popFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet popFirst = deque.removeFirst() // \u961f\u9996\u5143\u7d20\u51fa\u961f\nlet popLast = deque.removeLast() // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.count\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = deque.isEmpty\n</code></pre> deque.js<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque = [];\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst = deque[0];\nconst peekLast = deque[deque.length - 1];\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront = deque.shift();\nconst popBack = deque.pop();\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = deque.length;\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty = size === 0;\n</code></pre> deque.ts<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque: number[] = [];\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst: number = deque[0];\nconst peekLast: number = deque[deque.length - 1];\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront: number = deque.shift() as number;\nconst popBack: number = deque.pop() as number;\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size: number = deque.length;\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty: boolean = size === 0;\n</code></pre> deque.dart<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0cQueue \u88ab\u5b9a\u4e49\u4e3a\u53cc\u5411\u961f\u5217\nQueue&lt;int&gt; deque = Queue&lt;int&gt;();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.addLast(2);  // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.addFirst(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.first; // \u961f\u9996\u5143\u7d20\nint peekLast = deque.last;   // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.removeFirst(); // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.removeLast();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.length;\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.isEmpty;\n</code></pre> deque.rs<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2);  // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nif let Some(front) = deque.front() { // \u961f\u9996\u5143\u7d20\n}\nif let Some(rear) = deque.back() {   // \u961f\u5c3e\u5143\u7d20\n}\n\n/* \u5143\u7d20\u51fa\u961f */\nif let Some(pop_front) = deque.pop_front() { // \u961f\u9996\u5143\u7d20\u51fa\u961f\n}\nif let Some(pop_rear) = deque.pop_back() {   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.len();\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = deque.is_empty();\n</code></pre> deque.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u53cc\u5411\u961f\u5217\n</code></pre> deque.kt<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nval deque = LinkedList&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.offerLast(2)  // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.offerLast(5)\ndeque.offerLast(4)\ndeque.offerFirst(3) // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.offerFirst(1)\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nval peekFirst = deque.peekFirst() // \u961f\u9996\u5143\u7d20\nval peekLast = deque.peekLast()   // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\nval popFirst = deque.pollFirst() // \u961f\u9996\u5143\u7d20\u51fa\u961f\nval popLast = deque.pollLast()   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nval size = deque.size\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nval isEmpty = deque.isEmpty()\n</code></pre> deque.rb<pre><code>\n</code></pre> deque.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/deque/#532","title":"5.3.2 \u00a0 \u53cc\u5411\u961f\u5217\u5b9e\u73b0 *","text":"<p>\u53cc\u5411\u961f\u5217\u7684\u5b9e\u73b0\u4e0e\u961f\u5217\u7c7b\u4f3c\uff0c\u53ef\u4ee5\u9009\u62e9\u94fe\u8868\u6216\u6570\u7ec4\u4f5c\u4e3a\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002</p>"},{"location":"chapter_stack_and_queue/deque/#1","title":"1. \u00a0 \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u7684\u5b9e\u73b0","text":"<p>\u56de\u987e\u4e0a\u4e00\u8282\u5185\u5bb9\uff0c\u6211\u4eec\u4f7f\u7528\u666e\u901a\u5355\u5411\u94fe\u8868\u6765\u5b9e\u73b0\u961f\u5217\uff0c\u56e0\u4e3a\u5b83\u53ef\u4ee5\u65b9\u4fbf\u5730\u5220\u9664\u5934\u8282\u70b9\uff08\u5bf9\u5e94\u51fa\u961f\u64cd\u4f5c\uff09\u548c\u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0\u65b0\u8282\u70b9\uff08\u5bf9\u5e94\u5165\u961f\u64cd\u4f5c\uff09\u3002</p> <p>\u5bf9\u4e8e\u53cc\u5411\u961f\u5217\u800c\u8a00\uff0c\u5934\u90e8\u548c\u5c3e\u90e8\u90fd\u53ef\u4ee5\u6267\u884c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u53cc\u5411\u961f\u5217\u9700\u8981\u5b9e\u73b0\u53e6\u4e00\u4e2a\u5bf9\u79f0\u65b9\u5411\u7684\u64cd\u4f5c\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u91c7\u7528\u201c\u53cc\u5411\u94fe\u8868\u201d\u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u5982\u56fe 5-8 \u6240\u793a\uff0c\u6211\u4eec\u5c06\u53cc\u5411\u94fe\u8868\u7684\u5934\u8282\u70b9\u548c\u5c3e\u8282\u70b9\u89c6\u4e3a\u53cc\u5411\u961f\u5217\u7684\u961f\u9996\u548c\u961f\u5c3e\uff0c\u540c\u65f6\u5b9e\u73b0\u5728\u4e24\u7aef\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u7684\u529f\u80fd\u3002</p> LinkedListDequepush_last()push_first()pop_last()pop_first() <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 5-8 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_deque.py<pre><code>class ListNode:\n    \"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\"\"\"\n\n    def __init__(self, val: int):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self.val: int = val\n        self.next: ListNode | None = None  # \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n        self.prev: ListNode | None = None  # \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\nclass LinkedListDeque:\n    \"\"\"\u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._front: ListNode | None = None  # \u5934\u8282\u70b9 front\n        self._rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\n        self._size: int = 0  # \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self.size() == 0\n\n    def push(self, num: int, is_front: bool):\n        \"\"\"\u5165\u961f\u64cd\u4f5c\"\"\"\n        node = ListNode(num)\n        # \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if self.is_empty():\n            self._front = self._rear = node\n        # \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        elif is_front:\n            # \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            self._front.prev = node\n            node.next = self._front\n            self._front = node  # \u66f4\u65b0\u5934\u8282\u70b9\n        # \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else:\n            # \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            self._rear.next = node\n            node.prev = self._rear\n            self._rear = node  # \u66f4\u65b0\u5c3e\u8282\u70b9\n        self._size += 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n\n    def push_first(self, num: int):\n        \"\"\"\u961f\u9996\u5165\u961f\"\"\"\n        self.push(num, True)\n\n    def push_last(self, num: int):\n        \"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\n        self.push(num, False)\n\n    def pop(self, is_front: bool) -&gt; int:\n        \"\"\"\u51fa\u961f\u64cd\u4f5c\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        # \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if is_front:\n            val: int = self._front.val  # \u6682\u5b58\u5934\u8282\u70b9\u503c\n            # \u5220\u9664\u5934\u8282\u70b9\n            fnext: ListNode | None = self._front.next\n            if fnext != None:\n                fnext.prev = None\n                self._front.next = None\n            self._front = fnext  # \u66f4\u65b0\u5934\u8282\u70b9\n        # \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else:\n            val: int = self._rear.val  # \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            # \u5220\u9664\u5c3e\u8282\u70b9\n            rprev: ListNode | None = self._rear.prev\n            if rprev != None:\n                rprev.next = None\n                self._rear.prev = None\n            self._rear = rprev  # \u66f4\u65b0\u5c3e\u8282\u70b9\n        self._size -= 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val\n\n    def pop_first(self) -&gt; int:\n        \"\"\"\u961f\u9996\u51fa\u961f\"\"\"\n        return self.pop(True)\n\n    def pop_last(self) -&gt; int:\n        \"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\n        return self.pop(False)\n\n    def peek_first(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        return self._front.val\n\n    def peek_last(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        return self._rear.val\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n        node = self._front\n        res = [0] * self.size()\n        for i in range(self.size()):\n            res[i] = node.val\n            node = node.next\n        return res\n</code></pre> linkedlist_deque.cpp<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nstruct DoublyListNode {\n    int val;              // \u8282\u70b9\u503c\n    DoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n    DoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\u6307\u9488\n    DoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n    }\n};\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n  private:\n    DoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    int queSize = 0;              // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    LinkedListDeque() : front(nullptr), rear(nullptr) {\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~LinkedListDeque() {\n        // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n        DoublyListNode *pre, *cur = front;\n        while (cur != nullptr) {\n            pre = cur;\n            cur = cur-&gt;next;\n            delete pre;\n        }\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    void push(int num, bool isFront) {\n        DoublyListNode *node = new DoublyListNode(num);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty())\n            front = rear = node;\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front-&gt;prev = node;\n            node-&gt;next = front;\n            front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear-&gt;next = node;\n            node-&gt;prev = rear;\n            rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    int pop(bool isFront) {\n        if (isEmpty())\n            throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n        int val;\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            val = front-&gt;val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            DoublyListNode *fNext = front-&gt;next;\n            if (fNext != nullptr) {\n                fNext-&gt;prev = nullptr;\n                front-&gt;next = nullptr;\n            }\n            delete front;\n            front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        } else {\n            val = rear-&gt;val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            DoublyListNode *rPrev = rear-&gt;prev;\n            if (rPrev != nullptr) {\n                rPrev-&gt;next = nullptr;\n                rear-&gt;prev = nullptr;\n            }\n            delete rear;\n            rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    int popFirst() {\n        return pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    int popLast() {\n        return pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        return front-&gt;val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        return rear-&gt;val;\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    vector&lt;int&gt; toVector() {\n        DoublyListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_deque.java<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    int val; // \u8282\u70b9\u503c\n    ListNode next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    ListNode prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n    ListNode(int val) {\n        this.val = val;\n        prev = next = null;\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private int queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    public LinkedListDeque() {\n        front = rear = null;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    private void push(int num, boolean isFront) {\n        ListNode node = new ListNode(num);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty())\n            front = rear = node;\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front.prev = node;\n            node.next = front;\n            front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear.next = node;\n            node.prev = rear;\n            rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    private int pop(boolean isFront) {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        int val;\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            val = front.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            ListNode fNext = front.next;\n            if (fNext != null) {\n                fNext.prev = null;\n                front.next = null;\n            }\n            front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        } else {\n            val = rear.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            ListNode rPrev = rear.prev;\n            if (rPrev != null) {\n                rPrev.next = null;\n                rear.prev = null;\n            }\n            rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int popFirst() {\n        return pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int popLast() {\n        return pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return rear.val;\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.cs<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode(int val) {\n    public int val = val;       // \u8282\u70b9\u503c\n    public ListNode? next = null; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    public ListNode? prev = null; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    ListNode? front, rear; // \u5934\u8282\u70b9 front, \u5c3e\u8282\u70b9 rear\n    int queSize = 0;      // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    public LinkedListDeque() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    void Push(int num, bool isFront) {\n        ListNode node = new(num);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (IsEmpty()) {\n            front = node;\n            rear = node;\n        }\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front!.prev = node;\n            node.next = front;\n            front = node; // \u66f4\u65b0\u5934\u8282\u70b9                           \n        }\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear!.next = node;\n            node.prev = rear;\n            rear = node;  // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n\n        queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void PushFirst(int num) {\n        Push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void PushLast(int num) {\n        Push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    int? Pop(bool isFront) {\n        if (IsEmpty())\n            throw new Exception();\n        int? val;\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            val = front?.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            ListNode? fNext = front?.next;\n            if (fNext != null) {\n                fNext.prev = null;\n                front!.next = null;\n            }\n            front = fNext;   // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else {\n            val = rear?.val;  // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            ListNode? rPrev = rear?.prev;\n            if (rPrev != null) {\n                rPrev.next = null;\n                rear!.prev = null;\n            }\n            rear = rPrev;    // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n\n        queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int? PopFirst() {\n        return Pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int? PopLast() {\n        return Pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int? PeekFirst() {\n        if (IsEmpty())\n            throw new Exception();\n        return front?.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int? PeekLast() {\n        if (IsEmpty())\n            throw new Exception();\n        return rear?.val;\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int?[] ToArray() {\n        ListNode? node = front;\n        int?[] res = new int?[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node?.val;\n            node = node?.next;\n        }\n\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.go<pre><code>/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype linkedListDeque struct {\n    // \u4f7f\u7528\u5185\u7f6e\u5305 list\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u53cc\u7aef\u961f\u5217 */\nfunc newLinkedListDeque() *linkedListDeque {\n    return &amp;linkedListDeque{\n        data: list.New(),\n    }\n}\n\n/* \u961f\u9996\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushFirst(value any) {\n    s.data.PushFront(value)\n}\n\n/* \u961f\u5c3e\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushLast(value any) {\n    s.data.PushBack(value)\n}\n\n/* \u961f\u9996\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u961f\u5c3e\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListDeque) peekFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (s *linkedListDeque) peekLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListDeque) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListDeque) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListDeque) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_deque.swift<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    var val: Int // \u8282\u70b9\u503c\n    var next: ListNode? // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    weak var prev: ListNode? // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n    init(val: Int) {\n        self.val = val\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private var front: ListNode? // \u5934\u8282\u70b9 front\n    private var rear: ListNode? // \u5c3e\u8282\u70b9 rear\n    private var _size: Int // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    init() {\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    private func push(num: Int, isFront: Bool) {\n        let node = ListNode(val: num)\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if isEmpty() {\n            front = node\n            rear = node\n        }\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if isFront {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front?.prev = node\n            node.next = front\n            front = node // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear?.next = node\n            node.prev = rear\n            rear = node // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        _size += 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    func pushFirst(num: Int) {\n        push(num: num, isFront: true)\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    func pushLast(num: Int) {\n        push(num: num, isFront: false)\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    private func pop(isFront: Bool) -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        let val: Int\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if isFront {\n            val = front!.val // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            let fNext = front?.next\n            if fNext != nil {\n                fNext?.prev = nil\n                front?.next = nil\n            }\n            front = fNext // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else {\n            val = rear!.val // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            let rPrev = rear?.prev\n            if rPrev != nil {\n                rPrev?.next = nil\n                rear?.prev = nil\n            }\n            rear = rPrev // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        _size -= 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    func popFirst() -&gt; Int {\n        pop(isFront: true)\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    func popLast() -&gt; Int {\n        pop(isFront: false)\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return front!.val\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return rear!.val\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.js<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    val; // \u8282\u70b9\u503c\n\n    constructor(val) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    #front; // \u5934\u8282\u70b9 front\n    #rear; // \u5c3e\u8282\u70b9 rear\n    #queSize; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n        this.#queSize = 0;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\n    pushLast(val) {\n        const node = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            this.#rear.next = node;\n            node.prev = this.#rear;\n            this.#rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        this.#queSize++;\n    }\n\n    /* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\n    pushFirst(val) {\n        const node = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            this.#front.prev = node;\n            node.next = this.#front;\n            this.#front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        this.#queSize++;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\n    popLast() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5c3e\u8282\u70b9\n        let temp = this.#rear.prev;\n        if (temp !== null) {\n            temp.next = null;\n            this.#rear.prev = null;\n        }\n        this.#rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        this.#queSize--;\n        return value;\n    }\n\n    /* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\n    popFirst() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5934\u8282\u70b9\n        let temp = this.#front.next;\n        if (temp !== null) {\n            temp.prev = null;\n            this.#front.next = null;\n        }\n        this.#front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\n        this.#queSize--;\n        return value;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast() {\n        return this.#queSize === 0 ? null : this.#rear.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst() {\n        return this.#queSize === 0 ? null : this.#front.val;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u6253\u5370\u53cc\u5411\u961f\u5217 */\n    print() {\n        const arr = [];\n        let temp = this.#front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.ts<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    prev: ListNode; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    next: ListNode; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    val: number; // \u8282\u70b9\u503c\n\n    constructor(val: number) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private front: ListNode; // \u5934\u8282\u70b9 front\n    private rear: ListNode; // \u5c3e\u8282\u70b9 rear\n    private queSize: number; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n        this.queSize = 0;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\n    pushLast(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            this.rear.next = node;\n            node.prev = this.rear;\n            this.rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        this.queSize++;\n    }\n\n    /* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\n    pushFirst(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            this.front.prev = node;\n            node.next = this.front;\n            this.front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        this.queSize++;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\n    popLast(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5c3e\u8282\u70b9\n        let temp: ListNode = this.rear.prev;\n        if (temp !== null) {\n            temp.next = null;\n            this.rear.prev = null;\n        }\n        this.rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        this.queSize--;\n        return value;\n    }\n\n    /* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\n    popFirst(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5934\u8282\u70b9\n        let temp: ListNode = this.front.next;\n        if (temp !== null) {\n            temp.prev = null;\n            this.front.next = null;\n        }\n        this.front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\n        this.queSize--;\n        return value;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast(): number {\n        return this.queSize === 0 ? null : this.rear.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst(): number {\n        return this.queSize === 0 ? null : this.front.val;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u6253\u5370\u53cc\u5411\u961f\u5217 */\n    print(): void {\n        const arr: number[] = [];\n        let temp: ListNode = this.front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.dart<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n  int val; // \u8282\u70b9\u503c\n  ListNode? next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n  ListNode? prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n  ListNode(this.val, {this.next, this.prev});\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u5bf9\u5217 */\nclass LinkedListDeque {\n  late ListNode? _front; // \u5934\u8282\u70b9 _front\n  late ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\n  int _queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n  LinkedListDeque() {\n    this._front = null;\n    this._rear = null;\n  }\n\n  /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u957f\u5ea6 */\n  int size() {\n    return this._queSize;\n  }\n\n  /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return size() == 0;\n  }\n\n  /* \u5165\u961f\u64cd\u4f5c */\n  void push(int _num, bool isFront) {\n    final ListNode node = ListNode(_num);\n    if (isEmpty()) {\n      // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 _front \u548c _rear \u90fd\u6307\u5411 node\n      _front = _rear = node;\n    } else if (isFront) {\n      // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n      // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n      _front!.prev = node;\n      node.next = _front;\n      _front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n    } else {\n      // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n      // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n      _rear!.next = node;\n      node.prev = _rear;\n      _rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n    }\n    _queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n  }\n\n  /* \u961f\u9996\u5165\u961f */\n  void pushFirst(int _num) {\n    push(_num, true);\n  }\n\n  /* \u961f\u5c3e\u5165\u961f */\n  void pushLast(int _num) {\n    push(_num, false);\n  }\n\n  /* \u51fa\u961f\u64cd\u4f5c */\n  int? pop(bool isFront) {\n    // \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\n    if (isEmpty()) {\n      return null;\n    }\n    final int val;\n    if (isFront) {\n      // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n      val = _front!.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n      // \u5220\u9664\u5934\u8282\u70b9\n      ListNode? fNext = _front!.next;\n      if (fNext != null) {\n        fNext.prev = null;\n        _front!.next = null;\n      }\n      _front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n    } else {\n      // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n      val = _rear!.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n      // \u5220\u9664\u5c3e\u8282\u70b9\n      ListNode? rPrev = _rear!.prev;\n      if (rPrev != null) {\n        rPrev.next = null;\n        _rear!.prev = null;\n      }\n      _rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n    }\n    _queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    return val;\n  }\n\n  /* \u961f\u9996\u51fa\u961f */\n  int? popFirst() {\n    return pop(true);\n  }\n\n  /* \u961f\u5c3e\u51fa\u961f */\n  int? popLast() {\n    return pop(false);\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int? peekFirst() {\n    return _front?.val;\n  }\n\n  /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n  int? peekLast() {\n    return _rear?.val;\n  }\n\n  /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; _queSize; i++) {\n      res.add(node!.val);\n      node = node.next;\n    }\n    return res;\n  }\n}\n</code></pre> linkedlist_deque.rs<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\npub struct ListNode&lt;T&gt; {\n    pub val: T,                                 // \u8282\u70b9\u503c\n    pub next: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n    pub prev: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u524d\u9a71\u8282\u70b9\u6307\u9488\n}\n\nimpl&lt;T&gt; ListNode&lt;T&gt; {\n    pub fn new(val: T) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n        Rc::new(RefCell::new(ListNode {\n            val,\n            next: None,\n            prev: None,\n        }))\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListDeque&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5934\u8282\u70b9 front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // \u5c3e\u8282\u70b9 rear\n    que_size: usize,                         // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListDeque&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    pub fn push(&amp;mut self, num: T, is_front: bool) {\n        let node = ListNode::new(num);\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        if is_front {\n            match self.front.take() {\n                // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n                None =&gt; {\n                    self.rear = Some(node.clone());\n                    self.front = Some(node);\n                }\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n                Some(old_front) =&gt; {\n                    old_front.borrow_mut().prev = Some(node.clone());\n                    node.borrow_mut().next = Some(old_front);\n                    self.front = Some(node); // \u66f4\u65b0\u5934\u8282\u70b9\n                }\n            }\n        }\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else {\n            match self.rear.take() {\n                // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n                None =&gt; {\n                    self.front = Some(node.clone());\n                    self.rear = Some(node);\n                }\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n                Some(old_rear) =&gt; {\n                    old_rear.borrow_mut().next = Some(node.clone());\n                    node.borrow_mut().prev = Some(old_rear);\n                    self.rear = Some(node); // \u66f4\u65b0\u5c3e\u8282\u70b9\n                }\n            }\n        }\n        self.que_size += 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pub fn push_first(&amp;mut self, num: T) {\n        self.push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pub fn push_last(&amp;mut self, num: T) {\n        self.push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    pub fn pop(&amp;mut self, is_front: bool) -&gt; Option&lt;T&gt; {\n        // \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de None\n        if self.is_empty() {\n            return None;\n        };\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if is_front {\n            self.front.take().map(|old_front| {\n                match old_front.borrow_mut().next.take() {\n                    Some(new_front) =&gt; {\n                        new_front.borrow_mut().prev.take();\n                        self.front = Some(new_front); // \u66f4\u65b0\u5934\u8282\u70b9\n                    }\n                    None =&gt; {\n                        self.rear.take();\n                    }\n                }\n                self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n                Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n            })\n        }\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else {\n            self.rear.take().map(|old_rear| {\n                match old_rear.borrow_mut().prev.take() {\n                    Some(new_rear) =&gt; {\n                        new_rear.borrow_mut().next.take();\n                        self.rear = Some(new_rear); // \u66f4\u65b0\u5c3e\u8282\u70b9\n                    }\n                    None =&gt; {\n                        self.front.take();\n                    }\n                }\n                self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n                Rc::try_unwrap(old_rear).ok().unwrap().into_inner().val\n            })\n        }\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    pub fn pop_first(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    pub fn pop_last(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    pub fn peek_first(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    pub fn peek_last(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.rear.as_ref()\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.insert(0, node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_deque.c<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\ntypedef struct DoublyListNode {\n    int val;                     // \u8282\u70b9\u503c\n    struct DoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\n    struct DoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\n} DoublyListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nDoublyListNode *newDoublyListNode(int num) {\n    DoublyListNode *new = (DoublyListNode *)malloc(sizeof(DoublyListNode));\n    new-&gt;val = num;\n    new-&gt;next = NULL;\n    new-&gt;prev = NULL;\n    return new;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delDoublyListNode(DoublyListNode *node) {\n    free(node);\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\n    DoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    int queSize;                  // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n} LinkedListDeque;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListDeque *newLinkedListDeque() {\n    LinkedListDeque *deque = (LinkedListDeque *)malloc(sizeof(LinkedListDeque));\n    deque-&gt;front = NULL;\n    deque-&gt;rear = NULL;\n    deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListdeque(LinkedListDeque *deque) {\n    // \u91ca\u653e\u6240\u6709\u8282\u70b9\n    for (int i = 0; i &lt; deque-&gt;queSize &amp;&amp; deque-&gt;front != NULL; i++) {\n        DoublyListNode *tmp = deque-&gt;front;\n        deque-&gt;front = deque-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // \u91ca\u653e deque \u7ed3\u6784\u4f53\n    free(deque);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListDeque *deque) {\n    return (size(deque) == 0);\n}\n\n/* \u5165\u961f */\nvoid push(LinkedListDeque *deque, int num, bool isFront) {\n    DoublyListNode *node = newDoublyListNode(num);\n    // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411node\n    if (empty(deque)) {\n        deque-&gt;front = deque-&gt;rear = node;\n    }\n    // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n    else if (isFront) {\n        // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n        deque-&gt;front-&gt;prev = node;\n        node-&gt;next = deque-&gt;front;\n        deque-&gt;front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n    }\n    // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n    else {\n        // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n        deque-&gt;rear-&gt;next = node;\n        node-&gt;prev = deque-&gt;rear;\n        deque-&gt;rear = node;\n    }\n    deque-&gt;queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(LinkedListDeque *deque, int num) {\n    push(deque, num, true);\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(LinkedListDeque *deque, int num) {\n    push(deque, num, false);\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;front);\n    return deque-&gt;front-&gt;val;\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;rear);\n    return deque-&gt;rear-&gt;val;\n}\n\n/* \u51fa\u961f */\nint pop(LinkedListDeque *deque, bool isFront) {\n    if (empty(deque))\n        return -1;\n    int val;\n    // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n    if (isFront) {\n        val = peekFirst(deque); // \u6682\u5b58\u5934\u8282\u70b9\u503c\n        DoublyListNode *fNext = deque-&gt;front-&gt;next;\n        if (fNext) {\n            fNext-&gt;prev = NULL;\n            deque-&gt;front-&gt;next = NULL;\n            delDoublyListNode(deque-&gt;front);\n        }\n        deque-&gt;front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n    }\n    // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n    else {\n        val = peekLast(deque); // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n        DoublyListNode *rPrev = deque-&gt;rear-&gt;prev;\n        if (rPrev) {\n            rPrev-&gt;next = NULL;\n            deque-&gt;rear-&gt;prev = NULL;\n            delDoublyListNode(deque-&gt;rear);\n        }\n        deque-&gt;rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n    }\n    deque-&gt;queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    return val;\n}\n\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(LinkedListDeque *deque) {\n    return pop(deque, true);\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(LinkedListDeque *deque) {\n    return pop(deque, false);\n}\n\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListDeque(LinkedListDeque *deque) {\n    int *arr = malloc(sizeof(int) * deque-&gt;queSize);\n    // \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\n    int i;\n    DoublyListNode *node;\n    for (i = 0, node = deque-&gt;front; i &lt; deque-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, deque-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_deque.kt<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode(var value: Int) {\n    // \u8282\u70b9\u503c\n    var next: ListNode? = null // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    var prev: ListNode? = null // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private var front: ListNode? = null // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private var rear: ListNode? = null\n    private var queSize = 0 // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    fun push(num: Int, isFront: Boolean) {\n        val node = ListNode(num)\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty()) {\n            rear = node\n            front = rear\n            // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        } else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front?.prev = node\n            node.next = front\n            front = node // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear?.next = node\n            node.prev = rear\n            rear = node // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize++ // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    fun pushFirst(num: Int) {\n        push(num, true)\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    fun pushLast(num: Int) {\n        push(num, false)\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    fun pop(isFront: Boolean): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n\n        val value: Int\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            value = front!!.value // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            val fNext = front!!.next\n            if (fNext != null) {\n                fNext.prev = null\n                front!!.next = null\n            }\n            front = fNext // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        } else {\n            value = rear!!.value // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            val rPrev = rear!!.prev\n            if (rPrev != null) {\n                rPrev.next = null\n                rear!!.prev = null\n            }\n            rear = rPrev // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize-- // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return value\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    fun popFirst(): Int {\n        return pop(true)\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    fun popLast(): Int {\n        return pop(false)\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peekFirst(): Int {\n        if (isEmpty()) {\n            throw IndexOutOfBoundsException()\n\n        }\n        return front!!.value\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return rear!!.value\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!.value\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.rb<pre><code>[class]{ListNode}-[func]{}\n\n[class]{LinkedListDeque}-[func]{}\n</code></pre> linkedlist_deque.zig<pre><code>// \u53cc\u5411\u94fe\u8868\u8282\u70b9\nfn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = undefined,     // \u8282\u70b9\u503c\n        next: ?*Self = null,    // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n        prev: ?*Self = null,    // \u524d\u9a71\u8282\u70b9\u6307\u9488\n\n        // Initialize a list node with specific value\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n            self.prev = null;\n        }\n    };\n}\n\n// \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\nfn LinkedListDeque(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        front: ?*ListNode(T) = null,                    // \u5934\u8282\u70b9 front\n        rear: ?*ListNode(T) = null,                     // \u5c3e\u8282\u70b9 rear\n        que_size: usize = 0,                             // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.front = null;\n            self.rear = null;\n            self.que_size = 0;\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.que_size;\n        }\n\n        // \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u5165\u961f\u64cd\u4f5c\n        pub fn push(self: *Self, num: T, is_front: bool) !void {\n            var node = try self.mem_allocator.create(ListNode(T));\n            node.init(num);\n            // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n            if (self.isEmpty()) {\n                self.front = node;\n                self.rear = node;\n            // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n            } else if (is_front) {\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n                self.front.?.prev = node;\n                node.next = self.front;\n                self.front = node;  // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n            } else {\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n                self.rear.?.next = node;\n                node.prev = self.rear;\n                self.rear = node;   // \u66f4\u65b0\u5c3e\u8282\u70b9\n            }\n            self.que_size += 1;      // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        } \n\n        // \u961f\u9996\u5165\u961f\n        pub fn pushFirst(self: *Self, num: T) !void {\n            try self.push(num, true);\n        } \n\n        // \u961f\u5c3e\u5165\u961f\n        pub fn pushLast(self: *Self, num: T) !void {\n            try self.push(num, false);\n        } \n\n        // \u51fa\u961f\u64cd\u4f5c\n        pub fn pop(self: *Self, is_front: bool) T {\n            if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n            var val: T = undefined;\n            // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n            if (is_front) {\n                val = self.front.?.val;     // \u6682\u5b58\u5934\u8282\u70b9\u503c\n                // \u5220\u9664\u5934\u8282\u70b9\n                var fNext = self.front.?.next;\n                if (fNext != null) {\n                    fNext.?.prev = null;\n                    self.front.?.next = null;\n                }\n                self.front = fNext;         // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n            } else {\n                val = self.rear.?.val;      // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n                // \u5220\u9664\u5c3e\u8282\u70b9\n                var rPrev = self.rear.?.prev;\n                if (rPrev != null) {\n                    rPrev.?.next = null;\n                    self.rear.?.prev = null;\n                }\n                self.rear = rPrev;          // \u66f4\u65b0\u5c3e\u8282\u70b9\n            }\n            self.que_size -= 1;              // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n            return val;\n        } \n\n        // \u961f\u9996\u51fa\u961f\n        pub fn popFirst(self: *Self) T {\n            return self.pop(true);\n        } \n\n        // \u961f\u5c3e\u51fa\u961f\n        pub fn popLast(self: *Self) T {\n            return self.pop(false);\n        } \n\n        // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n        pub fn peekFirst(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n            return self.front.?.val;\n        }  \n\n        // \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\n        pub fn peekLast(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n            return self.rear.?.val;\n        }\n\n        // \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.front;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[i] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"<p>\u5982\u56fe 5-9 \u6240\u793a\uff0c\u4e0e\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7c7b\u4f3c\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4f7f\u7528\u73af\u5f62\u6570\u7ec4\u6765\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u3002</p> ArrayDequepush_last()push_first()pop_last()pop_first() <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 5-9 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u5728\u961f\u5217\u7684\u5b9e\u73b0\u57fa\u7840\u4e0a\uff0c\u4ec5\u9700\u589e\u52a0\u201c\u961f\u9996\u5165\u961f\u201d\u548c\u201c\u961f\u5c3e\u51fa\u961f\u201d\u7684\u65b9\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_deque.py<pre><code>class ArrayDeque:\n    \"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\n\n    def __init__(self, capacity: int):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._nums: list[int] = [0] * capacity\n        self._front: int = 0\n        self._size: int = 0\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self._size == 0\n\n    def index(self, i: int) -&gt; int:\n        \"\"\"\u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15\"\"\"\n        # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        # \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        # \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + self.capacity()) % self.capacity()\n\n    def push_first(self, num: int):\n        \"\"\"\u961f\u9996\u5165\u961f\"\"\"\n        if self._size == self.capacity():\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        # \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        self._front = self.index(self._front - 1)\n        # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        self._nums[self._front] = num\n        self._size += 1\n\n    def push_last(self, num: int):\n        \"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\n        if self._size == self.capacity():\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        # \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        rear = self.index(self._front + self._size)\n        # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop_first(self) -&gt; int:\n        \"\"\"\u961f\u9996\u51fa\u961f\"\"\"\n        num = self.peek_first()\n        # \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        self._front = self.index(self._front + 1)\n        self._size -= 1\n        return num\n\n    def pop_last(self) -&gt; int:\n        \"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\n        num = self.peek_last()\n        self._size -= 1\n        return num\n\n    def peek_first(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        return self._nums[self._front]\n\n    def peek_last(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        # \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        last = self.index(self._front + self._size - 1)\n        return self._nums[last]\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n        # \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        res = []\n        for i in range(self._size):\n            res.append(self._nums[self.index(self._front + i)])\n        return res\n</code></pre> array_deque.cpp<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n  private:\n    vector&lt;int&gt; nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;        // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;      // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    ArrayDeque(int capacity) {\n        nums.resize(capacity);\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    int capacity() {\n        return nums.size();\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    int index(int i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    void pushFirst(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    void pushLast(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        int rear = index(front + queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    int popFirst() {\n        int num = peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        return nums[front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    vector&lt;int&gt; toVector() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        vector&lt;int&gt; res(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n};\n</code></pre> array_deque.java<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    private int[] nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public ArrayDeque(int capacity) {\n        this.nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    private int index(int i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void pushFirst(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void pushLast(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        int rear = index(front + queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int popFirst() {\n        int num = peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int[] toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.cs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public ArrayDeque(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    int Index(int i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + Capacity()) % Capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void PushFirst(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = Index(front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void PushLast(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        int rear = Index(front + queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int PopFirst() {\n        int num = PeekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = Index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int PopLast() {\n        int num = PeekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int PeekFirst() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        return nums[front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int PeekLast() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = Index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int[] ToArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[Index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.go<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype arrayDeque struct {\n    nums        []int // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    queSize     int   // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n    queCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\n    return &amp;arrayDeque{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayDeque) size() int {\n    return q.queSize\n}\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayDeque) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nfunc (q *arrayDeque) index(i int) int {\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n    return (i + q.queCapacity) % q.queCapacity\n}\n\n/* \u961f\u9996\u5165\u961f */\nfunc (q *arrayDeque) pushFirst(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n        return\n    }\n    // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n    q.front = q.index(q.front - 1)\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n    q.nums[q.front] = num\n    q.queSize++\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nfunc (q *arrayDeque) pushLast(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n        return\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    rear := q.index(q.front + q.queSize)\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* \u961f\u9996\u51fa\u961f */\nfunc (q *arrayDeque) popFirst() any {\n    num := q.peekFirst()\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    q.front = q.index(q.front + 1)\n    q.queSize--\n    return num\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nfunc (q *arrayDeque) popLast() any {\n    num := q.peekLast()\n    q.queSize--\n    return num\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayDeque) peekFirst() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (q *arrayDeque) peekLast() any {\n    if q.isEmpty() {\n        return nil\n    }\n    // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    last := q.index(q.front + q.queSize - 1)\n    return q.nums[last]\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayDeque) toSlice() []int {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    res := make([]int, q.queSize)\n    for i, j := 0, q.front; i &lt; q.queSize; i++ {\n        res[i] = q.nums[q.index(j)]\n        j++\n    }\n    return res\n}\n</code></pre> array_deque.swift<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    private var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var _size: Int // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init(capacity: Int) {\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    private func index(i: Int) -&gt; Int {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        (i + capacity()) % capacity()\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    func pushFirst(num: Int) {\n        if size() == capacity() {\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(i: front - 1)\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num\n        _size += 1\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    func pushLast(num: Int) {\n        if size() == capacity() {\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        let rear = index(i: front + size())\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    func popFirst() -&gt; Int {\n        let num = peekFirst()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(i: front + 1)\n        _size -= 1\n        return num\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    func popLast() -&gt; Int {\n        let num = peekLast()\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return nums[front]\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        let last = index(i: front + size() - 1)\n        return nums[last]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    func toArray() -&gt; [Int] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        (front ..&lt; front + size()).map { nums[index(i: $0)] }\n    }\n}\n</code></pre> array_deque.js<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    #nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    #front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    #queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n        this.#front = 0;\n        this.#queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    capacity() {\n        return this.#nums.length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    index(i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pushFirst(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        this.#front = this.index(this.#front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        this.#nums[this.#front] = num;\n        this.#queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pushLast(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        const rear = this.index(this.#front + this.#queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    popFirst() {\n        const num = this.peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        this.#front = this.index(this.#front + 1);\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    popLast() {\n        const num = this.peekLast();\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.#nums[this.#front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        const last = this.index(this.#front + this.#queSize - 1);\n        return this.#nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const res = [];\n        for (let i = 0, j = this.#front; i &lt; this.#queSize; i++, j++) {\n            res[i] = this.#nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.ts<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    private nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private queSize: number; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = 0;\n        this.queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    capacity(): number {\n        return this.nums.length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    index(i: number): number {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pushFirst(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        this.front = this.index(this.front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        this.nums[this.front] = num;\n        this.queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pushLast(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        const rear: number = this.index(this.front + this.queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    popFirst(): number {\n        const num: number = this.peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        this.front = this.index(this.front + 1);\n        this.queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    popLast(): number {\n        const num: number = this.peekLast();\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.nums[this.front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        const last = this.index(this.front + this.queSize - 1);\n        return this.nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    toArray(): number[] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const res: number[] = [];\n        for (let i = 0, j = this.front; i &lt; this.queSize; i++, j++) {\n            res[i] = this.nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.dart<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n  late List&lt;int&gt; _nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n  late int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n  late int _queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  ArrayDeque(int capacity) {\n    this._nums = List.filled(capacity, 0);\n    this._front = this._queSize = 0;\n  }\n\n  /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n  int capacity() {\n    return _nums.length;\n  }\n\n  /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n  int index(int i) {\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n    return (i + capacity()) % capacity();\n  }\n\n  /* \u961f\u9996\u5165\u961f */\n  void pushFirst(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n    }\n    // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 _front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n    _front = index(_front - 1);\n    // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u9996\n    _nums[_front] = _num;\n    _queSize++;\n  }\n\n  /* \u961f\u5c3e\u5165\u961f */\n  void pushLast(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    int rear = index(_front + _queSize);\n    // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* \u961f\u9996\u51fa\u961f */\n  int popFirst() {\n    int _num = peekFirst();\n    // \u961f\u9996\u6307\u9488\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n    _front = index(_front + 1);\n    _queSize--;\n    return _num;\n  }\n\n  /* \u961f\u5c3e\u51fa\u961f */\n  int popLast() {\n    int _num = peekLast();\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int peekFirst() {\n    if (isEmpty()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n    }\n    return _nums[_front];\n  }\n\n  /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n  int peekLast() {\n    if (isEmpty()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n    }\n    // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    int last = index(_front + _queSize - 1);\n    return _nums[last];\n  }\n\n  /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n  List&lt;int&gt; toArray() {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[index(j)];\n    }\n    return res;\n  }\n}\n</code></pre> array_deque.rs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nstruct ArrayDeque {\n    nums: Vec&lt;i32&gt;,  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front: usize,    // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    que_size: usize, // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n}\n\nimpl ArrayDeque {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(capacity: usize) -&gt; Self {\n        Self {\n            nums: vec![0; capacity],\n            front: 0,\n            que_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    pub fn capacity(&amp;self) -&gt; usize {\n        self.nums.len()\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        self.que_size\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    fn index(&amp;self, i: i32) -&gt; usize {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return ((i + self.capacity() as i32) % self.capacity() as i32) as usize;\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pub fn push_first(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        self.front = self.index(self.front as i32 - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        self.nums[self.front] = num;\n        self.que_size += 1;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pub fn push_last(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        let rear = self.index(self.front as i32 + self.que_size as i32);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self.nums[rear] = num;\n        self.que_size += 1;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    fn pop_first(&amp;mut self) -&gt; i32 {\n        let num = self.peek_first();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        self.front = self.index(self.front as i32 + 1);\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    fn pop_last(&amp;mut self) -&gt; i32 {\n        let num = self.peek_last();\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fn peek_first(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        };\n        self.nums[self.front]\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fn peek_last(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        };\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        let last = self.index(self.front as i32 + self.que_size as i32 - 1);\n        self.nums[last]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    fn to_array(&amp;self) -&gt; Vec&lt;i32&gt; {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        let mut res = vec![0; self.que_size];\n        let mut j = self.front;\n        for i in 0..self.que_size {\n            res[i] = self.nums[self.index(j as i32)];\n            j += 1;\n        }\n        res\n    }\n}\n</code></pre> array_deque.c<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\n    int *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n    int queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayDeque;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayDeque *newArrayDeque(int capacity) {\n    ArrayDeque *deque = (ArrayDeque *)malloc(sizeof(ArrayDeque));\n    // \u521d\u59cb\u5316\u6570\u7ec4\n    deque-&gt;queCapacity = capacity;\n    deque-&gt;nums = (int *)malloc(sizeof(int) * deque-&gt;queCapacity);\n    deque-&gt;front = deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayDeque(ArrayDeque *deque) {\n    free(deque-&gt;nums);\n    free(deque);\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayDeque *deque) {\n    return deque-&gt;queCapacity;\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayDeque *deque) {\n    return deque-&gt;queSize == 0;\n}\n\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint dequeIndex(ArrayDeque *deque, int i) {\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n    return ((i + capacity(deque)) % capacity(deque));\n}\n\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\n        return;\n    }\n    // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u56de\u5230\u5c3e\u90e8\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front - 1);\n    // \u5c06 num \u6dfb\u52a0\u5230\u961f\u9996\n    deque-&gt;nums[deque-&gt;front] = num;\n    deque-&gt;queSize++;\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\n        return;\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    int rear = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize);\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    deque-&gt;nums[rear] = num;\n    deque-&gt;queSize++;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(ArrayDeque *deque) {\n    // \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\n    assert(empty(deque) == 0);\n    return deque-&gt;nums[deque-&gt;front];\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(ArrayDeque *deque) {\n    // \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\n    assert(empty(deque) == 0);\n    int last = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize - 1);\n    return deque-&gt;nums[last];\n}\n\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(ArrayDeque *deque) {\n    int num = peekFirst(deque);\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front + 1);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(ArrayDeque *deque) {\n    int num = peekLast(deque);\n    deque-&gt;queSize--;\n    return num;\n}\n</code></pre> array_deque.kt<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque(capacity: Int) {\n    private var nums = IntArray(capacity) // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var queSize = 0 // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    private fun index(i: Int): Int {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity()\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    fun pushFirst(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1)\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num\n        queSize++\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    fun pushLast(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        val rear = index(front + queSize)\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        queSize++\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    fun popFirst(): Int {\n        val num = peekFirst()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(front + 1)\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fun popLast(): Int {\n        val num = peekLast()\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        val last = index(front + queSize - 1)\n        return nums[last]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    fun toArray(): IntArray {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[index(j)]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_deque.rb<pre><code>[class]{ArrayDeque}-[func]{}\n</code></pre> array_deque.zig<pre><code>[class]{ArrayDeque}-[func]{}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#533","title":"5.3.3 \u00a0 \u53cc\u5411\u961f\u5217\u5e94\u7528","text":"<p>\u53cc\u5411\u961f\u5217\u517c\u5177\u6808\u4e0e\u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u5b83\u53ef\u4ee5\u5b9e\u73b0\u8fd9\u4e24\u8005\u7684\u6240\u6709\u5e94\u7528\u573a\u666f\uff0c\u540c\u65f6\u63d0\u4f9b\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\u3002</p> <p>\u6211\u4eec\u77e5\u9053\uff0c\u8f6f\u4ef6\u7684\u201c\u64a4\u9500\u201d\u529f\u80fd\u901a\u5e38\u4f7f\u7528\u6808\u6765\u5b9e\u73b0\uff1a\u7cfb\u7edf\u5c06\u6bcf\u6b21\u66f4\u6539\u64cd\u4f5c <code>push</code> \u5230\u6808\u4e2d\uff0c\u7136\u540e\u901a\u8fc7 <code>pop</code> \u5b9e\u73b0\u64a4\u9500\u3002\u7136\u800c\uff0c\u8003\u8651\u5230\u7cfb\u7edf\u8d44\u6e90\u7684\u9650\u5236\uff0c\u8f6f\u4ef6\u901a\u5e38\u4f1a\u9650\u5236\u64a4\u9500\u7684\u6b65\u6570\uff08\u4f8b\u5982\u4ec5\u5141\u8bb8\u4fdd\u5b58 \\(50\\) \u6b65\uff09\u3002\u5f53\u6808\u7684\u957f\u5ea6\u8d85\u8fc7 \\(50\\) \u65f6\uff0c\u8f6f\u4ef6\u9700\u8981\u5728\u6808\u5e95\uff08\u961f\u9996\uff09\u6267\u884c\u5220\u9664\u64cd\u4f5c\u3002\u4f46\u6808\u65e0\u6cd5\u5b9e\u73b0\u8be5\u529f\u80fd\uff0c\u6b64\u65f6\u5c31\u9700\u8981\u4f7f\u7528\u53cc\u5411\u961f\u5217\u6765\u66ff\u4ee3\u6808\u3002\u8bf7\u6ce8\u610f\uff0c\u201c\u64a4\u9500\u201d\u7684\u6838\u5fc3\u903b\u8f91\u4ecd\u7136\u9075\u5faa\u6808\u7684\u5148\u5165\u540e\u51fa\u539f\u5219\uff0c\u53ea\u662f\u53cc\u5411\u961f\u5217\u80fd\u591f\u66f4\u52a0\u7075\u6d3b\u5730\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u903b\u8f91\u3002</p>"},{"location":"chapter_stack_and_queue/queue/","title":"5.2 \u00a0 \u961f\u5217","text":"<p>\u961f\u5217\uff08queue\uff09\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u89c4\u5219\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u987e\u540d\u601d\u4e49\uff0c\u961f\u5217\u6a21\u62df\u4e86\u6392\u961f\u73b0\u8c61\uff0c\u5373\u65b0\u6765\u7684\u4eba\u4e0d\u65ad\u52a0\u5165\u961f\u5217\u5c3e\u90e8\uff0c\u800c\u4f4d\u4e8e\u961f\u5217\u5934\u90e8\u7684\u4eba\u9010\u4e2a\u79bb\u5f00\u3002</p> <p>\u5982\u56fe 5-4 \u6240\u793a\uff0c\u6211\u4eec\u5c06\u961f\u5217\u5934\u90e8\u79f0\u4e3a\u201c\u961f\u9996\u201d\uff0c\u5c3e\u90e8\u79f0\u4e3a\u201c\u961f\u5c3e\u201d\uff0c\u5c06\u628a\u5143\u7d20\u52a0\u5165\u961f\u5c3e\u7684\u64cd\u4f5c\u79f0\u4e3a\u201c\u5165\u961f\u201d\uff0c\u5220\u9664\u961f\u9996\u5143\u7d20\u7684\u64cd\u4f5c\u79f0\u4e3a\u201c\u51fa\u961f\u201d\u3002</p> <p></p> <p> \u56fe 5-4 \u00a0 \u961f\u5217\u7684\u5148\u5165\u5148\u51fa\u89c4\u5219 </p>"},{"location":"chapter_stack_and_queue/queue/#521","title":"5.2.1 \u00a0 \u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"<p>\u961f\u5217\u7684\u5e38\u89c1\u64cd\u4f5c\u5982\u8868 5-2 \u6240\u793a\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u65b9\u6cd5\u540d\u79f0\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002\u6211\u4eec\u5728\u6b64\u91c7\u7528\u4e0e\u6808\u76f8\u540c\u7684\u65b9\u6cd5\u547d\u540d\u3002</p> <p> \u8868 5-2 \u00a0 \u961f\u5217\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 <code>push()</code> \u5143\u7d20\u5165\u961f\uff0c\u5373\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) <code>pop()</code> \u961f\u9996\u5143\u7d20\u51fa\u961f \\(O(1)\\) <code>peek()</code> \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) <p>\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u73b0\u6210\u7684\u961f\u5217\u7c7b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig queue.py<pre><code>from collections import deque\n\n# \u521d\u59cb\u5316\u961f\u5217\n# \u5728 Python \u4e2d\uff0c\u6211\u4eec\u4e00\u822c\u5c06\u53cc\u5411\u961f\u5217\u7c7b deque \u5f53\u4f5c\u961f\u5217\u4f7f\u7528\n# \u867d\u7136 queue.Queue() \u662f\u7eaf\u6b63\u7684\u961f\u5217\u7c7b\uff0c\u4f46\u4e0d\u592a\u597d\u7528\uff0c\u56e0\u6b64\u4e0d\u63a8\u8350\nque: deque[int] = deque()\n\n# \u5143\u7d20\u5165\u961f\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n\n# \u8bbf\u95ee\u961f\u9996\u5143\u7d20\nfront: int = que[0]\n\n# \u5143\u7d20\u51fa\u961f\npop: int = que.popleft()\n\n# \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(que)\n\n# \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(que) == 0\n</code></pre> queue.cpp<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\nqueue&lt;int&gt; queue;\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint front = queue.front();\n\n/* \u5143\u7d20\u51fa\u961f */\nqueue.pop();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = queue.empty();\n</code></pre> queue.java<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\nQueue&lt;Integer&gt; queue = new LinkedList&lt;&gt;();\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.peek();\n\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.poll();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = queue.isEmpty();\n</code></pre> queue.cs<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\nQueue&lt;int&gt; queue = new();\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.Peek();\n\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.Dequeue();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.Count;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.Count == 0;\n</code></pre> queue_test.go<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u961f\u5217\u6765\u4f7f\u7528\nqueue := list.New()\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek := queue.Front()\n\n/* \u5143\u7d20\u51fa\u961f */\npop := queue.Front()\nqueue.Remove(pop)\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nsize := queue.Len()\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := queue.Len() == 0\n</code></pre> queue.swift<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nvar queue: [Int] = []\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nlet peek = queue.first!\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u7531\u4e8e\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 removeFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet pool = queue.removeFirst()\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = queue.count\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = queue.isEmpty\n</code></pre> queue.js<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nconst queue = [];\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n</code></pre> queue.ts<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528 \nconst queue: number[] = [];\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n</code></pre> queue.dart<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0c\u961f\u5217\u7c7b Qeque \u662f\u53cc\u5411\u961f\u5217\uff0c\u4e5f\u53ef\u4f5c\u4e3a\u961f\u5217\u4f7f\u7528\nQueue&lt;int&gt; queue = Queue();\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.first;\n\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.removeFirst();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.length;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.isEmpty;\n</code></pre> queue.rs<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Rust \u4e2d\u4f7f\u7528\u53cc\u5411\u961f\u5217\u4f5c\u4e3a\u666e\u901a\u961f\u5217\u6765\u4f7f\u7528\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(1);\ndeque.push_back(3);\ndeque.push_back(2);\ndeque.push_back(5);\ndeque.push_back(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nif let Some(front) = deque.front() {\n}\n\n/* \u5143\u7d20\u51fa\u961f */\nif let Some(pop) = deque.pop_front() {\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.len();\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = deque.is_empty();\n</code></pre> queue.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u961f\u5217\n</code></pre> queue.kt<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\nval queue = LinkedList&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.offer(1)\nqueue.offer(3)\nqueue.offer(2)\nqueue.offer(5)\nqueue.offer(4)\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nval peek = queue.peek()\n\n/* \u5143\u7d20\u51fa\u961f */\nval pop = queue.poll()\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nval size = queue.size\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nval isEmpty = queue.isEmpty()\n</code></pre> queue.rb<pre><code>\n</code></pre> queue.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/queue/#522","title":"5.2.2 \u00a0 \u961f\u5217\u5b9e\u73b0","text":"<p>\u4e3a\u4e86\u5b9e\u73b0\u961f\u5217\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u4ee5\u5728\u4e00\u7aef\u6dfb\u52a0\u5143\u7d20\uff0c\u5e76\u5728\u53e6\u4e00\u7aef\u5220\u9664\u5143\u7d20\uff0c\u94fe\u8868\u548c\u6570\u7ec4\u90fd\u7b26\u5408\u8981\u6c42\u3002</p>"},{"location":"chapter_stack_and_queue/queue/#1","title":"1. \u00a0 \u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"<p>\u5982\u56fe 5-5 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u201c\u5934\u8282\u70b9\u201d\u548c\u201c\u5c3e\u8282\u70b9\u201d\u5206\u522b\u89c6\u4e3a\u201c\u961f\u9996\u201d\u548c\u201c\u961f\u5c3e\u201d\uff0c\u89c4\u5b9a\u961f\u5c3e\u4ec5\u53ef\u6dfb\u52a0\u8282\u70b9\uff0c\u961f\u9996\u4ec5\u53ef\u5220\u9664\u8282\u70b9\u3002</p> LinkedListQueuepush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-5 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u4ee5\u4e0b\u662f\u7528\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_queue.py<pre><code>class LinkedListQueue:\n    \"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._front: ListNode | None = None  # \u5934\u8282\u70b9 front\n        self._rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return not self._front\n\n    def push(self, num: int):\n        \"\"\"\u5165\u961f\"\"\"\n        # \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        node = ListNode(num)\n        # \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if self._front is None:\n            self._front = node\n            self._rear = node\n        # \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        else:\n            self._rear.next = node\n            self._rear = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u961f\"\"\"\n        num = self.peek()\n        # \u5220\u9664\u5934\u8282\u70b9\n        self._front = self._front.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\n        return self._front.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        queue = []\n        temp = self._front\n        while temp:\n            queue.append(temp.val)\n            temp = temp.next\n        return queue\n</code></pre> linkedlist_queue.cpp<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n  private:\n    ListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    int queSize;\n\n  public:\n    LinkedListQueue() {\n        front = nullptr;\n        rear = nullptr;\n        queSize = 0;\n    }\n\n    ~LinkedListQueue() {\n        // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n        freeMemoryLinkedList(front);\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u961f */\n    void push(int num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        ListNode *node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == nullptr) {\n            front = node;\n            rear = node;\n        }\n        // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        else {\n            rear-&gt;next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    int pop() {\n        int num = peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        ListNode *tmp = front;\n        front = front-&gt;next;\n        // \u91ca\u653e\u5185\u5b58\n        delete tmp;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peek() {\n        if (size() == 0)\n            throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n        return front-&gt;val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_queue.java<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    private ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f */\n    public void push(int num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        ListNode node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == null) {\n            front = node;\n            rear = node;\n        // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int pop() {\n        int num = peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front.next;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.cs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    ListNode? front, rear;  // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear \n    int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u961f */\n    public void Push(int num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        ListNode node = new(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == null) {\n            front = node;\n            rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else if (rear != null) {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int Pop() {\n        int num = Peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front?.next;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return front!.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] ToArray() {\n        if (front == null)\n            return [];\n\n        ListNode? node = front;\n        int[] res = new int[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.go<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntype linkedListQueue struct {\n    // \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u961f\u5217\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newLinkedListQueue() *linkedListQueue {\n    return &amp;linkedListQueue{\n        data: list.New(),\n    }\n}\n\n/* \u5165\u961f */\nfunc (s *linkedListQueue) push(value any) {\n    s.data.PushBack(value)\n}\n\n/* \u51fa\u961f */\nfunc (s *linkedListQueue) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListQueue) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListQueue) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListQueue) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListQueue) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_queue.swift<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    private var front: ListNode? // \u5934\u8282\u70b9\n    private var rear: ListNode? // \u5c3e\u8282\u70b9\n    private var _size: Int\n\n    init() {\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u961f */\n    func push(num: Int) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        let node = ListNode(x: num)\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if front == nil {\n            front = node\n            rear = node\n        }\n        // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        else {\n            rear?.next = node\n            rear = node\n        }\n        _size += 1\n    }\n\n    /* \u51fa\u961f */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front?.next\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return front!.val\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.js<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    #front; // \u5934\u8282\u70b9 #front\n    #rear; // \u5c3e\u8282\u70b9 #rear\n    #queSize = 0;\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        const node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (!this.#front) {\n            this.#front = node;\n            this.#rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            this.#rear.next = node;\n            this.#rear = node;\n        }\n        this.#queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop() {\n        const num = this.peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        this.#front = this.#front.next;\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek() {\n        if (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.#front.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray() {\n        let node = this.#front;\n        const res = new Array(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.ts<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    private front: ListNode | null; // \u5934\u8282\u70b9 front\n    private rear: ListNode | null; // \u5c3e\u8282\u70b9 rear\n    private queSize: number = 0;\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num: number): void {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        const node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (!this.front) {\n            this.front = node;\n            this.rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            this.rear!.next = node;\n            this.rear = node;\n        }\n        this.queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop(): number {\n        const num = this.peek();\n        if (!this.front) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        // \u5220\u9664\u5934\u8282\u70b9\n        this.front = this.front.next;\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek(): number {\n        if (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.front!.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray(): number[] {\n        let node = this.front;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.dart<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n  ListNode? _front; // \u5934\u8282\u70b9 _front\n  ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\n  int _queSize = 0; // \u961f\u5217\u957f\u5ea6\n\n  LinkedListQueue() {\n    _front = null;\n    _rear = null;\n  }\n\n  /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u5165\u961f */\n  void push(int _num) {\n    // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 _num\n    final node = ListNode(_num);\n    // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n    if (_front == null) {\n      _front = node;\n      _rear = node;\n    } else {\n      // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n      _rear!.next = node;\n      _rear = node;\n    }\n    _queSize++;\n  }\n\n  /* \u51fa\u961f */\n  int pop() {\n    final int _num = peek();\n    // \u5220\u9664\u5934\u8282\u70b9\n    _front = _front!.next;\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int peek() {\n    if (_queSize == 0) {\n      throw Exception('\u961f\u5217\u4e3a\u7a7a');\n    }\n    return _front!.val;\n  }\n\n  /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; queue = [];\n    while (node != null) {\n      queue.add(node.val);\n      node = node.next;\n    }\n    return queue;\n  }\n}\n</code></pre> linkedlist_queue.rs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListQueue&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5934\u8282\u70b9 front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // \u5c3e\u8282\u70b9 rear\n    que_size: usize,                         // \u961f\u5217\u7684\u957f\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListQueue&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u961f */\n    pub fn push(&amp;mut self, num: T) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        let new_rear = ListNode::new(num);\n        match self.rear.take() {\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n            Some(old_rear) =&gt; {\n                old_rear.borrow_mut().next = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n            // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n            None =&gt; {\n                self.front = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n        }\n        self.que_size += 1;\n    }\n\n    /* \u51fa\u961f */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.front.take().map(|old_front| {\n            match old_front.borrow_mut().next.take() {\n                Some(new_front) =&gt; {\n                    self.front = Some(new_front);\n                }\n                None =&gt; {\n                    self.rear.take();\n                }\n            }\n            self.que_size -= 1;\n            Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n        })\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.insert(0, node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_queue.c<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\n    ListNode *front, *rear;\n    int queSize;\n} LinkedListQueue;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListQueue *newLinkedListQueue() {\n    LinkedListQueue *queue = (LinkedListQueue *)malloc(sizeof(LinkedListQueue));\n    queue-&gt;front = NULL;\n    queue-&gt;rear = NULL;\n    queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListQueue(LinkedListQueue *queue) {\n    // \u91ca\u653e\u6240\u6709\u8282\u70b9\n    while (queue-&gt;front != NULL) {\n        ListNode *tmp = queue-&gt;front;\n        queue-&gt;front = queue-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // \u91ca\u653e queue \u7ed3\u6784\u4f53\n    free(queue);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListQueue *queue) {\n    return (size(queue) == 0);\n}\n\n/* \u5165\u961f */\nvoid push(LinkedListQueue *queue, int num) {\n    // \u5c3e\u8282\u70b9\u5904\u6dfb\u52a0 node\n    ListNode *node = newListNode(num);\n    // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n    if (queue-&gt;front == NULL) {\n        queue-&gt;front = node;\n        queue-&gt;rear = node;\n    }\n    // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n    else {\n        queue-&gt;rear-&gt;next = node;\n        queue-&gt;rear = node;\n    }\n    queue-&gt;queSize++;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(LinkedListQueue *queue) {\n    assert(size(queue) &amp;&amp; queue-&gt;front);\n    return queue-&gt;front-&gt;val;\n}\n\n/* \u51fa\u961f */\nint pop(LinkedListQueue *queue) {\n    int num = peek(queue);\n    ListNode *tmp = queue-&gt;front;\n    queue-&gt;front = queue-&gt;front-&gt;next;\n    free(tmp);\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListQueue(LinkedListQueue *queue) {\n    int *arr = malloc(sizeof(int) * queue-&gt;queSize);\n    // \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\n    int i;\n    ListNode *node;\n    for (i = 0, node = queue-&gt;front; i &lt; queue-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, queue-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_queue.kt<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue(\n    // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private var front: ListNode? = null,\n    private var rear: ListNode? = null,\n    private var queSize: Int = 0\n) {\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u961f */\n    fun push(num: Int) {\n        // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        val node = ListNode(num)\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == null) {\n            front = node\n            rear = node\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            rear?.next = node\n            rear = node\n        }\n        queSize++\n    }\n\n    /* \u51fa\u961f */\n    fun pop(): Int {\n        val num = peek()\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front?.next\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!.value\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!.value\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.rb<pre><code>[class]{LinkedListQueue}-[func]{}\n</code></pre> linkedlist_queue.zig<pre><code>// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\nfn LinkedListQueue(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        front: ?*inc.ListNode(T) = null,                // \u5934\u8282\u70b9 front\n        rear: ?*inc.ListNode(T) = null,                 // \u5c3e\u8282\u70b9 rear\n        que_size: usize = 0,                            // \u961f\u5217\u7684\u957f\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.front = null;\n            self.rear = null;\n            self.que_size = 0;\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.que_size;\n        }\n\n        // \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.size() == 0) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\n            return self.front.?.val;\n        }  \n\n        // \u5165\u961f\n        pub fn push(self: *Self, num: T) !void {\n            // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n            var node = try self.mem_allocator.create(inc.ListNode(T));\n            node.init(num);\n            // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n            if (self.front == null) {\n                self.front = node;\n                self.rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n            } else {\n                self.rear.?.next = node;\n                self.rear = node;\n            }\n            self.que_size += 1;\n        } \n\n        // \u51fa\u961f\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            // \u5220\u9664\u5934\u8282\u70b9\n            self.front = self.front.?.next;\n            self.que_size -= 1;\n            return num;\n        } \n\n        // \u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.front;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[i] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/queue/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"<p>\u5728\u6570\u7ec4\u4e2d\u5220\u9664\u9996\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4f1a\u5bfc\u81f4\u51fa\u961f\u64cd\u4f5c\u6548\u7387\u8f83\u4f4e\u3002\u7136\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u5de7\u5999\u65b9\u6cd5\u6765\u907f\u514d\u8fd9\u4e2a\u95ee\u9898\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u53d8\u91cf <code>front</code> \u6307\u5411\u961f\u9996\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u5e76\u7ef4\u62a4\u4e00\u4e2a\u53d8\u91cf <code>size</code> \u7528\u4e8e\u8bb0\u5f55\u961f\u5217\u957f\u5ea6\u3002\u5b9a\u4e49 <code>rear = front + size</code> \uff0c\u8fd9\u4e2a\u516c\u5f0f\u8ba1\u7b97\u51fa\u7684 <code>rear</code> \u6307\u5411\u961f\u5c3e\u5143\u7d20\u4e4b\u540e\u7684\u4e0b\u4e00\u4e2a\u4f4d\u7f6e\u3002</p> <p>\u57fa\u4e8e\u6b64\u8bbe\u8ba1\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u5143\u7d20\u7684\u6709\u6548\u533a\u95f4\u4e3a <code>[front, rear - 1]</code>\uff0c\u5404\u79cd\u64cd\u4f5c\u7684\u5b9e\u73b0\u65b9\u6cd5\u5982\u56fe 5-6 \u6240\u793a\u3002</p> <ul> <li>\u5165\u961f\u64cd\u4f5c\uff1a\u5c06\u8f93\u5165\u5143\u7d20\u8d4b\u503c\u7ed9 <code>rear</code> \u7d22\u5f15\u5904\uff0c\u5e76\u5c06 <code>size</code> \u589e\u52a0 1 \u3002</li> <li>\u51fa\u961f\u64cd\u4f5c\uff1a\u53ea\u9700\u5c06 <code>front</code> \u589e\u52a0 1 \uff0c\u5e76\u5c06 <code>size</code> \u51cf\u5c11 1 \u3002</li> </ul> <p>\u53ef\u4ee5\u770b\u5230\uff0c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u90fd\u53ea\u9700\u8fdb\u884c\u4e00\u6b21\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002</p> ArrayQueuepush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-6 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\u4e00\u4e2a\u95ee\u9898\uff1a\u5728\u4e0d\u65ad\u8fdb\u884c\u5165\u961f\u548c\u51fa\u961f\u7684\u8fc7\u7a0b\u4e2d\uff0c<code>front</code> \u548c <code>rear</code> \u90fd\u5728\u5411\u53f3\u79fb\u52a8\uff0c\u5f53\u5b83\u4eec\u5230\u8fbe\u6570\u7ec4\u5c3e\u90e8\u65f6\u5c31\u65e0\u6cd5\u7ee7\u7eed\u79fb\u52a8\u4e86\u3002\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u89c6\u4e3a\u9996\u5c3e\u76f8\u63a5\u7684\u201c\u73af\u5f62\u6570\u7ec4\u201d\u3002</p> <p>\u5bf9\u4e8e\u73af\u5f62\u6570\u7ec4\uff0c\u6211\u4eec\u9700\u8981\u8ba9 <code>front</code> \u6216 <code>rear</code> \u5728\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u76f4\u63a5\u56de\u5230\u6570\u7ec4\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002\u8fd9\u79cd\u5468\u671f\u6027\u89c4\u5f8b\u53ef\u4ee5\u901a\u8fc7\u201c\u53d6\u4f59\u64cd\u4f5c\u201d\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_queue.py<pre><code>class ArrayQueue:\n    \"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\n\n    def __init__(self, size: int):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._nums: list[int] = [0] * size  # \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n        self._front: int = 0  # \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n        self._size: int = 0  # \u961f\u5217\u957f\u5ea6\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"\u5165\u961f\"\"\"\n        if self._size == self.capacity():\n            raise IndexError(\"\u961f\u5217\u5df2\u6ee1\")\n        # \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        rear: int = (self._front + self._size) % self.capacity()\n        # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u961f\"\"\"\n        num: int = self.peek()\n        # \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        self._front = (self._front + 1) % self.capacity()\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\n        return self._nums[self._front]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        res = [0] * self.size()\n        j: int = self._front\n        for i in range(self.size()):\n            res[i] = self._nums[(j % self.capacity())]\n            j += 1\n        return res\n</code></pre> array_queue.cpp<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n  private:\n    int *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;     // \u961f\u5217\u957f\u5ea6\n    int queCapacity; // \u961f\u5217\u5bb9\u91cf\n\n  public:\n    ArrayQueue(int capacity) {\n        // \u521d\u59cb\u5316\u6570\u7ec4\n        nums = new int[capacity];\n        queCapacity = capacity;\n        front = queSize = 0;\n    }\n\n    ~ArrayQueue() {\n        delete[] nums;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    int capacity() {\n        return queCapacity;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f */\n    void push(int num) {\n        if (queSize == queCapacity) {\n            cout &lt;&lt; \"\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        int rear = (front + queSize) % queCapacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    int pop() {\n        int num = peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % queCapacity;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peek() {\n        if (isEmpty())\n            throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n        return nums[front];\n    }\n\n    /* \u5c06\u6570\u7ec4\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        vector&lt;int&gt; arr(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            arr[i] = nums[j % queCapacity];\n        }\n        return arr;\n    }\n};\n</code></pre> array_queue.java<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    private int[] nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private int queSize; // \u961f\u5217\u957f\u5ea6\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u961f */\n    public void push(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        int rear = (front + queSize) % capacity();\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int pop() {\n        int num = peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % capacity();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    public int[] toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.cs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize; // \u961f\u5217\u957f\u5ea6\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u961f */\n    public void Push(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        int rear = (front + queSize) % Capacity();\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int Pop() {\n        int num = Peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % Capacity();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return nums[front];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    public int[] ToArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % this.Capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.go<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntype arrayQueue struct {\n    nums        []int // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    queSize     int   // \u961f\u5217\u957f\u5ea6\n    queCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\n    return &amp;arrayQueue{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayQueue) size() int {\n    return q.queSize\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayQueue) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* \u5165\u961f */\nfunc (q *arrayQueue) push(num int) {\n    // \u5f53 rear == queCapacity \u8868\u793a\u961f\u5217\u5df2\u6ee1\n    if q.queSize == q.queCapacity {\n        return\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n    rear := (q.front + q.queSize) % q.queCapacity\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* \u51fa\u961f */\nfunc (q *arrayQueue) pop() any {\n    num := q.peek()\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n    q.front = (q.front + 1) % q.queCapacity\n    q.queSize--\n    return num\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayQueue) peek() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayQueue) toSlice() []int {\n    rear := (q.front + q.queSize)\n    if rear &gt;= q.queCapacity {\n        rear %= q.queCapacity\n        return append(q.nums[q.front:], q.nums[:rear]...)\n    }\n    return q.nums[q.front:rear]\n}\n</code></pre> array_queue.swift<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    private var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var _size: Int // \u961f\u5217\u957f\u5ea6\n\n    init(capacity: Int) {\n        // \u521d\u59cb\u5316\u6570\u7ec4\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u961f */\n    func push(num: Int) {\n        if size() == capacity() {\n            print(\"\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        let rear = (front + size()) % capacity()\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* \u51fa\u961f */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % capacity()\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return nums[front]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    func toArray() -&gt; [Int] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        (front ..&lt; front + size()).map { nums[$0 % capacity()] }\n    }\n}\n</code></pre> array_queue.js<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    #nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    #front = 0; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    #queSize = 0; // \u961f\u5217\u957f\u5ea6\n\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    get capacity() {\n        return this.#nums.length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num) {\n        if (this.size === this.capacity) {\n            console.log('\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        const rear = (this.#front + this.size) % this.capacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop() {\n        const num = this.peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        this.#front = (this.#front + 1) % this.capacity;\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek() {\n        if (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.#nums[this.#front];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.#front; i &lt; this.size; i++, j++) {\n            arr[i] = this.#nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.ts<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    private nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private queSize: number; // \u961f\u5217\u957f\u5ea6\n\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = this.queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    get capacity(): number {\n        return this.nums.length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num: number): void {\n        if (this.size === this.capacity) {\n            console.log('\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        const rear = (this.front + this.queSize) % this.capacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop(): number {\n        const num = this.peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        this.front = (this.front + 1) % this.capacity;\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek(): number {\n        if (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.nums[this.front];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray(): number[] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.front; i &lt; this.size; i++, j++) {\n            arr[i] = this.nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.dart<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n  late List&lt;int&gt; _nums; // \u7528\u4e8e\u50a8\u5b58\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n  late int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n  late int _queSize; // \u961f\u5217\u957f\u5ea6\n\n  ArrayQueue(int capacity) {\n    _nums = List.filled(capacity, 0);\n    _front = _queSize = 0;\n  }\n\n  /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n  int capaCity() {\n    return _nums.length;\n  }\n\n  /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u5165\u961f */\n  void push(int _num) {\n    if (_queSize == capaCity()) {\n      throw Exception(\"\u961f\u5217\u5df2\u6ee1\");\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n    int rear = (_front + _queSize) % capaCity();\n    // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* \u51fa\u961f */\n  int pop() {\n    int _num = peek();\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n    _front = (_front + 1) % capaCity();\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"\u961f\u5217\u4e3a\u7a7a\");\n    }\n    return _nums[_front];\n  }\n\n  /* \u8fd4\u56de Array */\n  List&lt;int&gt; toArray() {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    final List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[j % capaCity()];\n    }\n    return res;\n  }\n}\n</code></pre> array_queue.rs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nstruct ArrayQueue {\n    nums: Vec&lt;i32&gt;,    // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front: i32,        // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    que_size: i32,     // \u961f\u5217\u957f\u5ea6\n    que_capacity: i32, // \u961f\u5217\u5bb9\u91cf\n}\n\nimpl ArrayQueue {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(capacity: i32) -&gt; ArrayQueue {\n        ArrayQueue {\n            nums: vec![0; capacity as usize],\n            front: 0,\n            que_size: 0,\n            que_capacity: capacity,\n        }\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    fn capacity(&amp;self) -&gt; i32 {\n        self.que_capacity\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    fn size(&amp;self) -&gt; i32 {\n        self.que_size\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* \u5165\u961f */\n    fn push(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        let rear = (self.front + self.que_size) % self.que_capacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self.nums[rear as usize] = num;\n        self.que_size += 1;\n    }\n\n    /* \u51fa\u961f */\n    fn pop(&amp;mut self) -&gt; i32 {\n        let num = self.peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        self.front = (self.front + 1) % self.que_capacity;\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fn peek(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"index out of bounds\");\n        }\n        self.nums[self.front as usize]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    fn to_vector(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let cap = self.que_capacity;\n        let mut j = self.front;\n        let mut arr = vec![0; self.que_size as usize];\n        for i in 0..self.que_size {\n            arr[i as usize] = self.nums[(j % cap) as usize];\n            j += 1;\n        }\n        arr\n    }\n}\n</code></pre> array_queue.c<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\n    int *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n    int queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayQueue;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayQueue *newArrayQueue(int capacity) {\n    ArrayQueue *queue = (ArrayQueue *)malloc(sizeof(ArrayQueue));\n    // \u521d\u59cb\u5316\u6570\u7ec4\n    queue-&gt;queCapacity = capacity;\n    queue-&gt;nums = (int *)malloc(sizeof(int) * queue-&gt;queCapacity);\n    queue-&gt;front = queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayQueue(ArrayQueue *queue) {\n    free(queue-&gt;nums);\n    free(queue);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayQueue *queue) {\n    return queue-&gt;queCapacity;\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayQueue *queue) {\n    return queue-&gt;queSize == 0;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(ArrayQueue *queue) {\n    assert(size(queue) != 0);\n    return queue-&gt;nums[queue-&gt;front];\n}\n\n/* \u5165\u961f */\nvoid push(ArrayQueue *queue, int num) {\n    if (size(queue) == capacity(queue)) {\n        printf(\"\u961f\u5217\u5df2\u6ee1\\r\\n\");\n        return;\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n    int rear = (queue-&gt;front + queue-&gt;queSize) % queue-&gt;queCapacity;\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    queue-&gt;nums[rear] = num;\n    queue-&gt;queSize++;\n}\n\n/* \u51fa\u961f */\nint pop(ArrayQueue *queue) {\n    int num = peek(queue);\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n    queue-&gt;front = (queue-&gt;front + 1) % queue-&gt;queCapacity;\n    queue-&gt;queSize--;\n    return num;\n}\n</code></pre> array_queue.kt<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue(capacity: Int) {\n    private val nums = IntArray(capacity) // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var queSize = 0 // \u961f\u5217\u957f\u5ea6\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* \u5165\u961f */\n    fun push(num: Int) {\n        if (queSize == capacity()) {\n            println(\"\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        val rear = (front + queSize) % capacity()\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        queSize++\n    }\n\n    /* \u51fa\u961f */\n    fun pop(): Int {\n        val num = peek()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % capacity()\n        queSize--\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    fun toArray(): IntArray {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[j % capacity()]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_queue.rb<pre><code>[class]{ArrayQueue}-[func]{}\n</code></pre> array_queue.zig<pre><code>// \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\nfn ArrayQueue(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        nums: []T = undefined,                          // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4     \n        cap: usize = 0,                                 // \u961f\u5217\u5bb9\u91cf\n        front: usize = 0,                               // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n        queSize: usize = 0,                             // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6570\u7ec4\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator, cap: usize) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.cap = cap;\n            self.nums = try self.mem_allocator.alloc(T, self.cap);\n            @memset(self.nums, @as(T, 0));\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\n        pub fn capacity(self: *Self) usize {\n            return self.cap;\n        }\n\n        // \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.queSize;\n        }\n\n        // \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.queSize == 0;\n        }\n\n        // \u5165\u961f\n        pub fn push(self: *Self, num: T) !void {\n            if (self.size() == self.capacity()) {\n                std.debug.print(\"\u961f\u5217\u5df2\u6ee1\\n\", .{});\n                return;\n            }\n            // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n            // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n            var rear = (self.front + self.queSize) % self.capacity();\n            // \u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n            self.nums[rear] = num;\n            self.queSize += 1;\n        } \n\n        // \u51fa\u961f\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\uff0c\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n            self.front = (self.front + 1) % self.capacity();\n            self.queSize -= 1;\n            return num;\n        } \n\n        // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\n            return self.nums[self.front];\n        } \n\n        // \u8fd4\u56de\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            var j: usize = self.front;\n            while (i &lt; self.size()) : ({ i += 1; j += 1; }) {\n                res[i] = self.nums[j % self.capacity()];\n            }\n            return res;\n        }\n    };\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u4ee5\u4e0a\u5b9e\u73b0\u7684\u961f\u5217\u4ecd\u7136\u5177\u6709\u5c40\u9650\u6027\uff1a\u5176\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u7136\u800c\uff0c\u8fd9\u4e2a\u95ee\u9898\u4e0d\u96be\u89e3\u51b3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u66ff\u6362\u4e3a\u52a8\u6001\u6570\u7ec4\uff0c\u4ece\u800c\u5f15\u5165\u6269\u5bb9\u673a\u5236\u3002\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u5c1d\u8bd5\u81ea\u884c\u5b9e\u73b0\u3002</p> <p>\u4e24\u79cd\u5b9e\u73b0\u7684\u5bf9\u6bd4\u7ed3\u8bba\u4e0e\u6808\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002</p>"},{"location":"chapter_stack_and_queue/queue/#523","title":"5.2.3 \u00a0 \u961f\u5217\u5178\u578b\u5e94\u7528","text":"<ul> <li>\u6dd8\u5b9d\u8ba2\u5355\u3002\u8d2d\u7269\u8005\u4e0b\u5355\u540e\uff0c\u8ba2\u5355\u5c06\u52a0\u5165\u961f\u5217\u4e2d\uff0c\u7cfb\u7edf\u968f\u540e\u4f1a\u6839\u636e\u987a\u5e8f\u5904\u7406\u961f\u5217\u4e2d\u7684\u8ba2\u5355\u3002\u5728\u53cc\u5341\u4e00\u671f\u95f4\uff0c\u77ed\u65f6\u95f4\u5185\u4f1a\u4ea7\u751f\u6d77\u91cf\u8ba2\u5355\uff0c\u9ad8\u5e76\u53d1\u6210\u4e3a\u5de5\u7a0b\u5e08\u4eec\u9700\u8981\u91cd\u70b9\u653b\u514b\u7684\u95ee\u9898\u3002</li> <li>\u5404\u7c7b\u5f85\u529e\u4e8b\u9879\u3002\u4efb\u4f55\u9700\u8981\u5b9e\u73b0\u201c\u5148\u6765\u540e\u5230\u201d\u529f\u80fd\u7684\u573a\u666f\uff0c\u4f8b\u5982\u6253\u5370\u673a\u7684\u4efb\u52a1\u961f\u5217\u3001\u9910\u5385\u7684\u51fa\u9910\u961f\u5217\u7b49\uff0c\u961f\u5217\u5728\u8fd9\u4e9b\u573a\u666f\u4e2d\u53ef\u4ee5\u6709\u6548\u5730\u7ef4\u62a4\u5904\u7406\u987a\u5e8f\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/stack/","title":"5.1 \u00a0 \u6808","text":"<p>\u6808\uff08stack\uff09\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u903b\u8f91\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u5c06\u6808\u7c7b\u6bd4\u4e3a\u684c\u9762\u4e0a\u7684\u4e00\u645e\u76d8\u5b50\uff0c\u5982\u679c\u60f3\u53d6\u51fa\u5e95\u90e8\u7684\u76d8\u5b50\uff0c\u5219\u9700\u8981\u5148\u5c06\u4e0a\u9762\u7684\u76d8\u5b50\u4f9d\u6b21\u79fb\u8d70\u3002\u6211\u4eec\u5c06\u76d8\u5b50\u66ff\u6362\u4e3a\u5404\u79cd\u7c7b\u578b\u7684\u5143\u7d20\uff08\u5982\u6574\u6570\u3001\u5b57\u7b26\u3001\u5bf9\u8c61\u7b49\uff09\uff0c\u5c31\u5f97\u5230\u4e86\u6808\u8fd9\u79cd\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u5982\u56fe 5-1 \u6240\u793a\uff0c\u6211\u4eec\u628a\u5806\u53e0\u5143\u7d20\u7684\u9876\u90e8\u79f0\u4e3a\u201c\u6808\u9876\u201d\uff0c\u5e95\u90e8\u79f0\u4e3a\u201c\u6808\u5e95\u201d\u3002\u5c06\u628a\u5143\u7d20\u6dfb\u52a0\u5230\u6808\u9876\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u5165\u6808\u201d\uff0c\u5220\u9664\u6808\u9876\u5143\u7d20\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u51fa\u6808\u201d\u3002</p> <p></p> <p> \u56fe 5-1 \u00a0 \u6808\u7684\u5148\u5165\u540e\u51fa\u89c4\u5219 </p>"},{"location":"chapter_stack_and_queue/stack/#511","title":"5.1.1 \u00a0 \u6808\u7684\u5e38\u7528\u64cd\u4f5c","text":"<p>\u6808\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-1 \u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002\u5728\u6b64\uff0c\u6211\u4eec\u4ee5\u5e38\u89c1\u7684 <code>push()</code>\u3001<code>pop()</code>\u3001<code>peek()</code> \u547d\u540d\u4e3a\u4f8b\u3002</p> <p> \u8868 5-1 \u00a0 \u6808\u7684\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5 \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 <code>push()</code> \u5143\u7d20\u5165\u6808\uff08\u6dfb\u52a0\u81f3\u6808\u9876\uff09 \\(O(1)\\) <code>pop()</code> \u6808\u9876\u5143\u7d20\u51fa\u6808 \\(O(1)\\) <code>peek()</code> \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \\(O(1)\\) <p>\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u6808\u7c7b\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u8bed\u8a00\u53ef\u80fd\u6ca1\u6709\u4e13\u95e8\u63d0\u4f9b\u6808\u7c7b\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u5c06\u8be5\u8bed\u8a00\u7684\u201c\u6570\u7ec4\u201d\u6216\u201c\u94fe\u8868\u201d\u5f53\u4f5c\u6808\u6765\u4f7f\u7528\uff0c\u5e76\u5728\u7a0b\u5e8f\u903b\u8f91\u4e0a\u5ffd\u7565\u4e0e\u6808\u65e0\u5173\u7684\u64cd\u4f5c\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig stack.py<pre><code># \u521d\u59cb\u5316\u6808\n# Python \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a list \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nstack: list[int] = []\n\n# \u5143\u7d20\u5165\u6808\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n# \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npeek: int = stack[-1]\n\n# \u5143\u7d20\u51fa\u6808\npop: int = stack.pop()\n\n# \u83b7\u53d6\u6808\u7684\u957f\u5ea6\nsize: int = len(stack)\n\n# \u5224\u65ad\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(stack) == 0\n</code></pre> stack.cpp<pre><code>/* \u521d\u59cb\u5316\u6808 */\nstack&lt;int&gt; stack;\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top = stack.top();\n\n/* \u5143\u7d20\u51fa\u6808 */\nstack.pop(); // \u65e0\u8fd4\u56de\u503c\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool empty = stack.empty();\n</code></pre> stack.java<pre><code>/* \u521d\u59cb\u5316\u6808 */\nStack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.peek();\n\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = stack.isEmpty();\n</code></pre> stack.cs<pre><code>/* \u521d\u59cb\u5316\u6808 */\nStack&lt;int&gt; stack = new();\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.Peek();\n\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.Pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.Count;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.Count == 0;\n</code></pre> stack_test.go<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// \u5728 Go \u4e2d\uff0c\u63a8\u8350\u5c06 Slice \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack []int\n\n/* \u5143\u7d20\u5165\u6808 */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek := stack[len(stack)-1]\n\n/* \u5143\u7d20\u51fa\u6808 */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nsize := len(stack)\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nisEmpty := len(stack) == 0\n</code></pre> stack.swift<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack: [Int] = []\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet peek = stack.last!\n\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.removeLast()\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.count\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = stack.isEmpty\n</code></pre> stack.js<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack = [];\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length-1];\n\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n</code></pre> stack.ts<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack: number[] = [];\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length - 1];\n\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n</code></pre> stack.dart<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// Dart \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nList&lt;int&gt; stack = [];\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.last;\n\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.removeLast();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.length;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.isEmpty;\n</code></pre> stack.rs<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// \u628a Vec \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nlet mut stack: Vec&lt;i32&gt; = Vec::new();\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet top = stack.last().unwrap();\n\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.pop().unwrap();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.len();\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = stack.is_empty();\n</code></pre> stack.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u6808\n</code></pre> stack.kt<pre><code>/* \u521d\u59cb\u5316\u6808 */\nval stack = Stack&lt;Int&gt;()\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1)\nstack.push(3)\nstack.push(2)\nstack.push(5)\nstack.push(4)\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nval peek = stack.peek()\n\n/* \u5143\u7d20\u51fa\u6808 */\nval pop = stack.pop()\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nval size = stack.size\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nval isEmpty = stack.isEmpty()\n</code></pre> stack.rb<pre><code>\n</code></pre> stack.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#512","title":"5.1.2 \u00a0 \u6808\u7684\u5b9e\u73b0","text":"<p>\u4e3a\u4e86\u6df1\u5165\u4e86\u89e3\u6808\u7684\u8fd0\u884c\u673a\u5236\uff0c\u6211\u4eec\u6765\u5c1d\u8bd5\u81ea\u5df1\u5b9e\u73b0\u4e00\u4e2a\u6808\u7c7b\u3002</p> <p>\u6808\u9075\u5faa\u5148\u5165\u540e\u51fa\u7684\u539f\u5219\uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u80fd\u5728\u6808\u9876\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\u3002\u7136\u800c\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u90fd\u53ef\u4ee5\u5728\u4efb\u610f\u4f4d\u7f6e\u6dfb\u52a0\u548c\u5220\u9664\u5143\u7d20\uff0c\u56e0\u6b64\u6808\u53ef\u4ee5\u89c6\u4e3a\u4e00\u79cd\u53d7\u9650\u5236\u7684\u6570\u7ec4\u6216\u94fe\u8868\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u201c\u5c4f\u853d\u201d\u6570\u7ec4\u6216\u94fe\u8868\u7684\u90e8\u5206\u65e0\u5173\u64cd\u4f5c\uff0c\u4f7f\u5176\u5bf9\u5916\u8868\u73b0\u7684\u903b\u8f91\u7b26\u5408\u6808\u7684\u7279\u6027\u3002</p>"},{"location":"chapter_stack_and_queue/stack/#1","title":"1. \u00a0 \u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"<p>\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u5934\u8282\u70b9\u89c6\u4e3a\u6808\u9876\uff0c\u5c3e\u8282\u70b9\u89c6\u4e3a\u6808\u5e95\u3002</p> <p>\u5982\u56fe 5-2 \u6240\u793a\uff0c\u5bf9\u4e8e\u5165\u6808\u64cd\u4f5c\uff0c\u6211\u4eec\u53ea\u9700\u5c06\u5143\u7d20\u63d2\u5165\u94fe\u8868\u5934\u90e8\uff0c\u8fd9\u79cd\u8282\u70b9\u63d2\u5165\u65b9\u6cd5\u88ab\u79f0\u4e3a\u201c\u5934\u63d2\u6cd5\u201d\u3002\u800c\u5bf9\u4e8e\u51fa\u6808\u64cd\u4f5c\uff0c\u53ea\u9700\u5c06\u5934\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u5373\u53ef\u3002</p> LinkedListStackpush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-2 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u5165\u6808\u51fa\u6808\u64cd\u4f5c </p> <p>\u4ee5\u4e0b\u662f\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u793a\u4f8b\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig linkedlist_stack.py<pre><code>class LinkedListStack:\n    \"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._peek: ListNode | None = None\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return not self._peek\n\n    def push(self, val: int):\n        \"\"\"\u5165\u6808\"\"\"\n        node = ListNode(val)\n        node.next = self._peek\n        self._peek = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u6808\"\"\"\n        num = self.peek()\n        self._peek = self._peek.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u6808\u4e3a\u7a7a\")\n        return self._peek.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        arr = []\n        node = self._peek\n        while node:\n            arr.append(node.val)\n            node = node.next\n        arr.reverse()\n        return arr\n</code></pre> linkedlist_stack.cpp<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n  private:\n    ListNode *stackTop; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    int stkSize;        // \u6808\u7684\u957f\u5ea6\n\n  public:\n    LinkedListStack() {\n        stackTop = nullptr;\n        stkSize = 0;\n    }\n\n    ~LinkedListStack() {\n        // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n        freeMemoryLinkedList(stackTop);\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    int size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    void push(int num) {\n        ListNode *node = new ListNode(num);\n        node-&gt;next = stackTop;\n        stackTop = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    int pop() {\n        int num = top();\n        ListNode *tmp = stackTop;\n        stackTop = stackTop-&gt;next;\n        // \u91ca\u653e\u5185\u5b58\n        delete tmp;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"\u6808\u4e3a\u7a7a\");\n        return stackTop-&gt;val;\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = stackTop;\n        vector&lt;int&gt; res(size());\n        for (int i = res.size() - 1; i &gt;= 0; i--) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_stack.java<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    private ListNode stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private int stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void push(int num) {\n        ListNode node = new ListNode(num);\n        node.next = stackPeek;\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    public int pop() {\n        int num = peek();\n        stackPeek = stackPeek.next;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stackPeek.val;\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] toArray() {\n        ListNode node = stackPeek;\n        int[] res = new int[size()];\n        for (int i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.cs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    ListNode? stackPeek;  // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    int stkSize = 0;   // \u6808\u7684\u957f\u5ea6\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int Size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void Push(int num) {\n        ListNode node = new(num) {\n            next = stackPeek\n        };\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    public int Pop() {\n        int num = Peek();\n        stackPeek = stackPeek!.next;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stackPeek!.val;\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] ToArray() {\n        if (stackPeek == null)\n            return [];\n\n        ListNode? node = stackPeek;\n        int[] res = new int[Size()];\n        for (int i = res.Length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.go<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntype linkedListStack struct {\n    // \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u6808\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u6808 */\nfunc newLinkedListStack() *linkedListStack {\n    return &amp;linkedListStack{\n        data: list.New(),\n    }\n}\n\n/* \u5165\u6808 */\nfunc (s *linkedListStack) push(value int) {\n    s.data.PushBack(value)\n}\n\n/* \u51fa\u6808 */\nfunc (s *linkedListStack) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc (s *linkedListStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc (s *linkedListStack) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListStack) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListStack) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_stack.swift<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    private var _peek: ListNode? // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private var _size: Int // \u6808\u7684\u957f\u5ea6\n\n    init() {\n        _size = 0\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u6808 */\n    func push(num: Int) {\n        let node = ListNode(x: num)\n        node.next = _peek\n        _peek = node\n        _size += 1\n    }\n\n    /* \u51fa\u6808 */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        _peek = _peek?.next\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u6808\u4e3a\u7a7a\")\n        }\n        return _peek!.val\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        var node = _peek\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices.reversed() {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.js<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    #stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    #stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n    constructor() {\n        this.#stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num) {\n        const node = new ListNode(num);\n        node.next = this.#stackPeek;\n        this.#stackPeek = node;\n        this.#stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    pop() {\n        const num = this.peek();\n        this.#stackPeek = this.#stackPeek.next;\n        this.#stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    peek() {\n        if (!this.#stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.#stackPeek.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray() {\n        let node = this.#stackPeek;\n        const res = new Array(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.ts<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    private stackPeek: ListNode | null; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private stkSize: number = 0; // \u6808\u7684\u957f\u5ea6\n\n    constructor() {\n        this.stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num: number): void {\n        const node = new ListNode(num);\n        node.next = this.stackPeek;\n        this.stackPeek = node;\n        this.stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    pop(): number {\n        const num = this.peek();\n        if (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n        this.stackPeek = this.stackPeek.next;\n        this.stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    peek(): number {\n        if (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.stackPeek.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray(): number[] {\n        let node = this.stackPeek;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.dart<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u7c7b\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n  ListNode? _stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n  int _stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n  LinkedListStack() {\n    _stackPeek = null;\n  }\n\n  /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n  int size() {\n    return _stkSize;\n  }\n\n  /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _stkSize == 0;\n  }\n\n  /* \u5165\u6808 */\n  void push(int _num) {\n    final ListNode node = ListNode(_num);\n    node.next = _stackPeek;\n    _stackPeek = node;\n    _stkSize++;\n  }\n\n  /* \u51fa\u6808 */\n  int pop() {\n    final int _num = peek();\n    _stackPeek = _stackPeek!.next;\n    _stkSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n  int peek() {\n    if (_stackPeek == null) {\n      throw Exception(\"\u6808\u4e3a\u7a7a\");\n    }\n    return _stackPeek!.val;\n  }\n\n  /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a List \u5e76\u8fd4\u56de */\n  List&lt;int&gt; toList() {\n    ListNode? node = _stackPeek;\n    List&lt;int&gt; list = [];\n    while (node != null) {\n      list.add(node.val);\n      node = node.next;\n    }\n    list = list.reversed.toList();\n    return list;\n  }\n}\n</code></pre> linkedlist_stack.rs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\n#[allow(dead_code)]\npub struct LinkedListStack&lt;T&gt; {\n    stack_peek: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    stk_size: usize,                              // \u6808\u7684\u957f\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListStack&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            stack_peek: None,\n            stk_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.stk_size;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    pub fn push(&amp;mut self, num: T) {\n        let node = ListNode::new(num);\n        node.borrow_mut().next = self.stack_peek.take();\n        self.stack_peek = Some(node);\n        self.stk_size += 1;\n    }\n\n    /* \u51fa\u6808 */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack_peek.take().map(|old_head| {\n            match old_head.borrow_mut().next.take() {\n                Some(new_head) =&gt; {\n                    self.stack_peek = Some(new_head);\n                }\n                None =&gt; {\n                    self.stack_peek = None;\n                }\n            }\n            self.stk_size -= 1;\n            Rc::try_unwrap(old_head).ok().unwrap().into_inner().val\n        })\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.stack_peek.as_ref()\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.push(node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_stack.c<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\n    ListNode *top; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    int size;      // \u6808\u7684\u957f\u5ea6\n} LinkedListStack;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListStack *newLinkedListStack() {\n    LinkedListStack *s = malloc(sizeof(LinkedListStack));\n    s-&gt;top = NULL;\n    s-&gt;size = 0;\n    return s;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListStack(LinkedListStack *s) {\n    while (s-&gt;top) {\n        ListNode *n = s-&gt;top-&gt;next;\n        free(s-&gt;top);\n        s-&gt;top = n;\n    }\n    free(s);\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(LinkedListStack *s) {\n    return s-&gt;size;\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(LinkedListStack *s) {\n    return size(s) == 0;\n}\n\n/* \u5165\u6808 */\nvoid push(LinkedListStack *s, int num) {\n    ListNode *node = (ListNode *)malloc(sizeof(ListNode));\n    node-&gt;next = s-&gt;top; // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6307\u9488\u57df\n    node-&gt;val = num;     // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6570\u636e\u57df\n    s-&gt;top = node;       // \u66f4\u65b0\u6808\u9876\n    s-&gt;size++;           // \u66f4\u65b0\u6808\u5927\u5c0f\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(LinkedListStack *s) {\n    if (s-&gt;size == 0) {\n        printf(\"\u6808\u4e3a\u7a7a\\n\");\n        return INT_MAX;\n    }\n    return s-&gt;top-&gt;val;\n}\n\n/* \u51fa\u6808 */\nint pop(LinkedListStack *s) {\n    int val = peek(s);\n    ListNode *tmp = s-&gt;top;\n    s-&gt;top = s-&gt;top-&gt;next;\n    // \u91ca\u653e\u5185\u5b58\n    free(tmp);\n    s-&gt;size--;\n    return val;\n}\n</code></pre> linkedlist_stack.kt<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack(\n    private var stackPeek: ListNode? = null, // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private var stkSize: Int = 0 // \u6808\u7684\u957f\u5ea6\n) {\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return stkSize\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u6808 */\n    fun push(num: Int) {\n        val node = ListNode(num)\n        node.next = stackPeek\n        stackPeek = node\n        stkSize++\n    }\n\n    /* \u51fa\u6808 */\n    fun pop(): Int? {\n        val num = peek()\n        stackPeek = stackPeek?.next\n        stkSize--;\n        return num\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    fun peek(): Int? {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stackPeek?.value\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    fun toArray(): IntArray {\n        var node = stackPeek\n        val res = IntArray(size())\n        for (i in res.size - 1 downTo 0) {\n            res[i] = node?.value!!\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.rb<pre><code>[class]{LinkedListStack}-[func]{}\n</code></pre> linkedlist_stack.zig<pre><code>// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\nfn LinkedListStack(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        stack_top: ?*inc.ListNode(T) = null,             // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n        stk_size: usize = 0,                             // \u6808\u7684\u957f\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,    // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.stack_top = null;\n            self.stk_size = 0;\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u6808\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.stk_size;\n        }\n\n        // \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8bbf\u95ee\u6808\u9876\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.size() == 0) @panic(\"\u6808\u4e3a\u7a7a\");\n            return self.stack_top.?.val;\n        }  \n\n        // \u5165\u6808\n        pub fn push(self: *Self, num: T) !void {\n            var node = try self.mem_allocator.create(inc.ListNode(T));\n            node.init(num);\n            node.next = self.stack_top;\n            self.stack_top = node;\n            self.stk_size += 1;\n        } \n\n        // \u51fa\u6808\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            self.stack_top = self.stack_top.?.next;\n            self.stk_size -= 1;\n            return num;\n        } \n\n        // \u5c06\u6808\u8f6c\u6362\u4e3a\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.stack_top;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[res.len - i - 1] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"<p>\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u7684\u5c3e\u90e8\u4f5c\u4e3a\u6808\u9876\u3002\u5982\u56fe 5-3 \u6240\u793a\uff0c\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u5206\u522b\u5bf9\u5e94\u5728\u6570\u7ec4\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u4e0e\u5220\u9664\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(1)\\) \u3002</p> ArrayStackpush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-3 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u6808\u7684\u5165\u6808\u51fa\u6808\u64cd\u4f5c </p> <p>\u7531\u4e8e\u5165\u6808\u7684\u5143\u7d20\u53ef\u80fd\u4f1a\u6e90\u6e90\u4e0d\u65ad\u5730\u589e\u52a0\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u52a8\u6001\u6570\u7ec4\uff0c\u8fd9\u6837\u5c31\u65e0\u987b\u81ea\u884c\u5904\u7406\u6570\u7ec4\u6269\u5bb9\u95ee\u9898\u3002\u4ee5\u4e0b\u4e3a\u793a\u4f8b\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_stack.py<pre><code>class ArrayStack:\n    \"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._stack: list[int] = []\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\n        return len(self._stack)\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self._stack == []\n\n    def push(self, item: int):\n        \"\"\"\u5165\u6808\"\"\"\n        self._stack.append(item)\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u6808\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u6808\u4e3a\u7a7a\")\n        return self._stack.pop()\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u6808\u4e3a\u7a7a\")\n        return self._stack[-1]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        return self._stack\n</code></pre> array_stack.cpp<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n  private:\n    vector&lt;int&gt; stack;\n\n  public:\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    int size() {\n        return stack.size();\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return stack.size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    void push(int num) {\n        stack.push_back(num);\n    }\n\n    /* \u51fa\u6808 */\n    int pop() {\n        int num = top();\n        stack.pop_back();\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"\u6808\u4e3a\u7a7a\");\n        return stack.back();\n    }\n\n    /* \u8fd4\u56de Vector */\n    vector&lt;int&gt; toVector() {\n        return stack;\n    }\n};\n</code></pre> array_stack.java<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    private ArrayList&lt;Integer&gt; stack;\n\n    public ArrayStack() {\n        // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n        stack = new ArrayList&lt;&gt;();\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int size() {\n        return stack.size();\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void push(int num) {\n        stack.add(num);\n    }\n\n    /* \u51fa\u6808 */\n    public int pop() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.remove(size() - 1);\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.get(size() - 1);\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public Object[] toArray() {\n        return stack.toArray();\n    }\n}\n</code></pre> array_stack.cs<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    List&lt;int&gt; stack;\n    public ArrayStack() {\n        // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n        stack = [];\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int Size() {\n        return stack.Count;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void Push(int num) {\n        stack.Add(num);\n    }\n\n    /* \u51fa\u6808 */\n    public int Pop() {\n        if (IsEmpty())\n            throw new Exception();\n        var val = Peek();\n        stack.RemoveAt(Size() - 1);\n        return val;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stack[Size() - 1];\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] ToArray() {\n        return [.. stack];\n    }\n}\n</code></pre> array_stack.go<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntype arrayStack struct {\n    data []int // \u6570\u636e\n}\n\n/* \u521d\u59cb\u5316\u6808 */\nfunc newArrayStack() *arrayStack {\n    return &amp;arrayStack{\n        // \u8bbe\u7f6e\u6808\u7684\u957f\u5ea6\u4e3a 0\uff0c\u5bb9\u91cf\u4e3a 16\n        data: make([]int, 0, 16),\n    }\n}\n\n/* \u6808\u7684\u957f\u5ea6 */\nfunc (s *arrayStack) size() int {\n    return len(s.data)\n}\n\n/* \u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *arrayStack) isEmpty() bool {\n    return s.size() == 0\n}\n\n/* \u5165\u6808 */\nfunc (s *arrayStack) push(v int) {\n    // \u5207\u7247\u4f1a\u81ea\u52a8\u6269\u5bb9\n    s.data = append(s.data, v)\n}\n\n/* \u51fa\u6808 */\nfunc (s *arrayStack) pop() any {\n    val := s.peek()\n    s.data = s.data[:len(s.data)-1]\n    return val\n}\n\n/* \u83b7\u53d6\u6808\u9876\u5143\u7d20 */\nfunc (s *arrayStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    val := s.data[len(s.data)-1]\n    return val\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (s *arrayStack) toSlice() []int {\n    return s.data\n}\n</code></pre> array_stack.swift<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    private var stack: [Int]\n\n    init() {\n        // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n        stack = []\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        stack.count\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        stack.isEmpty\n    }\n\n    /* \u5165\u6808 */\n    func push(num: Int) {\n        stack.append(num)\n    }\n\n    /* \u51fa\u6808 */\n    @discardableResult\n    func pop() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u6808\u4e3a\u7a7a\")\n        }\n        return stack.removeLast()\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u6808\u4e3a\u7a7a\")\n        }\n        return stack.last!\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        stack\n    }\n}\n</code></pre> array_stack.js<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    #stack;\n    constructor() {\n        this.#stack = [];\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#stack.length;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#stack.length === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num) {\n        this.#stack.push(num);\n    }\n\n    /* \u51fa\u6808 */\n    pop() {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.#stack.pop();\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    top() {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.#stack[this.#stack.length - 1];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        return this.#stack;\n    }\n}\n</code></pre> array_stack.ts<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    private stack: number[];\n    constructor() {\n        this.stack = [];\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.stack.length;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.stack.length === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num: number): void {\n        this.stack.push(num);\n    }\n\n    /* \u51fa\u6808 */\n    pop(): number | undefined {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.stack.pop();\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    top(): number | undefined {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.stack[this.stack.length - 1];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        return this.stack;\n    }\n}\n</code></pre> array_stack.dart<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n  late List&lt;int&gt; _stack;\n  ArrayStack() {\n    _stack = [];\n  }\n\n  /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n  int size() {\n    return _stack.length;\n  }\n\n  /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _stack.isEmpty;\n  }\n\n  /* \u5165\u6808 */\n  void push(int _num) {\n    _stack.add(_num);\n  }\n\n  /* \u51fa\u6808 */\n  int pop() {\n    if (isEmpty()) {\n      throw Exception(\"\u6808\u4e3a\u7a7a\");\n    }\n    return _stack.removeLast();\n  }\n\n  /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"\u6808\u4e3a\u7a7a\");\n    }\n    return _stack.last;\n  }\n\n  /* \u5c06\u6808\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n  List&lt;int&gt; toArray() =&gt; _stack;\n}\n</code></pre> array_stack.rs<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nstruct ArrayStack&lt;T&gt; {\n    stack: Vec&lt;T&gt;,\n}\n\nimpl&lt;T&gt; ArrayStack&lt;T&gt; {\n    /* \u521d\u59cb\u5316\u6808 */\n    fn new() -&gt; ArrayStack&lt;T&gt; {\n        ArrayStack::&lt;T&gt; {\n            stack: Vec::&lt;T&gt;::new(),\n        }\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    fn size(&amp;self) -&gt; usize {\n        self.stack.len()\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.size() == 0\n    }\n\n    /* \u5165\u6808 */\n    fn push(&amp;mut self, num: T) {\n        self.stack.push(num);\n    }\n\n    /* \u51fa\u6808 */\n    fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack.pop()\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    fn peek(&amp;self) -&gt; Option&lt;&amp;T&gt; {\n        if self.is_empty() {\n            panic!(\"\u6808\u4e3a\u7a7a\")\n        };\n        self.stack.last()\n    }\n\n    /* \u8fd4\u56de &amp;Vec */\n    fn to_array(&amp;self) -&gt; &amp;Vec&lt;T&gt; {\n        &amp;self.stack\n    }\n}\n</code></pre> array_stack.c<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\n    int *data;\n    int size;\n} ArrayStack;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayStack *newArrayStack() {\n    ArrayStack *stack = malloc(sizeof(ArrayStack));\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5927\u5bb9\u91cf\uff0c\u907f\u514d\u6269\u5bb9\n    stack-&gt;data = malloc(sizeof(int) * MAX_SIZE);\n    stack-&gt;size = 0;\n    return stack;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayStack(ArrayStack *stack) {\n    free(stack-&gt;data);\n    free(stack);\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(ArrayStack *stack) {\n    return stack-&gt;size;\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(ArrayStack *stack) {\n    return stack-&gt;size == 0;\n}\n\n/* \u5165\u6808 */\nvoid push(ArrayStack *stack, int num) {\n    if (stack-&gt;size == MAX_SIZE) {\n        printf(\"\u6808\u5df2\u6ee1\\n\");\n        return;\n    }\n    stack-&gt;data[stack-&gt;size] = num;\n    stack-&gt;size++;\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(ArrayStack *stack) {\n    if (stack-&gt;size == 0) {\n        printf(\"\u6808\u4e3a\u7a7a\\n\");\n        return INT_MAX;\n    }\n    return stack-&gt;data[stack-&gt;size - 1];\n}\n\n/* \u51fa\u6808 */\nint pop(ArrayStack *stack) {\n    int val = peek(stack);\n    stack-&gt;size--;\n    return val;\n}\n</code></pre> array_stack.kt<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n    private val stack = ArrayList&lt;Int&gt;()\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    fun size(): Int {\n        return stack.size\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* \u5165\u6808 */\n    fun push(num: Int) {\n        stack.add(num)\n    }\n\n    /* \u51fa\u6808 */\n    fun pop(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack.removeAt(size() - 1)\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack[size() - 1]\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    fun toArray(): Array&lt;Any&gt; {\n        return stack.toArray()\n    }\n}\n</code></pre> array_stack.rb<pre><code>[class]{ArrayStack}-[func]{}\n</code></pre> array_stack.zig<pre><code>// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\nfn ArrayStack(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        stack: ?std.ArrayList(T) = null,     \n\n        // \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) void {\n            if (self.stack == null) {\n                self.stack = std.ArrayList(T).init(allocator);\n            }\n        }\n\n        // \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.stack == null) return;\n            self.stack.?.deinit();\n        }\n\n        // \u83b7\u53d6\u6808\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.stack.?.items.len;\n        }\n\n        // \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8bbf\u95ee\u6808\u9876\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u6808\u4e3a\u7a7a\");\n            return self.stack.?.items[self.size() - 1];\n        }  \n\n        // \u5165\u6808\n        pub fn push(self: *Self, num: T) !void {\n            try self.stack.?.append(num);\n        } \n\n        // \u51fa\u6808\n        pub fn pop(self: *Self) T {\n            var num = self.stack.?.pop();\n            return num;\n        } \n\n        // \u8fd4\u56de ArrayList\n        pub fn toList(self: *Self) std.ArrayList(T) {\n            return self.stack.?;\n        }\n    };\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_stack_and_queue/stack/#513","title":"5.1.3 \u00a0 \u4e24\u79cd\u5b9e\u73b0\u5bf9\u6bd4","text":"<p>\u652f\u6301\u64cd\u4f5c</p> <p>\u4e24\u79cd\u5b9e\u73b0\u90fd\u652f\u6301\u6808\u5b9a\u4e49\u4e2d\u7684\u5404\u9879\u64cd\u4f5c\u3002\u6570\u7ec4\u5b9e\u73b0\u989d\u5916\u652f\u6301\u968f\u673a\u8bbf\u95ee\uff0c\u4f46\u8fd9\u5df2\u8d85\u51fa\u4e86\u6808\u7684\u5b9a\u4e49\u8303\u7574\uff0c\u56e0\u6b64\u4e00\u822c\u4e0d\u4f1a\u7528\u5230\u3002</p> <p>\u65f6\u95f4\u6548\u7387</p> <p>\u5728\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0\u4e2d\uff0c\u5165\u6808\u548c\u51fa\u6808\u64cd\u4f5c\u90fd\u5728\u9884\u5148\u5206\u914d\u597d\u7684\u8fde\u7eed\u5185\u5b58\u4e2d\u8fdb\u884c\uff0c\u5177\u6709\u5f88\u597d\u7684\u7f13\u5b58\u672c\u5730\u6027\uff0c\u56e0\u6b64\u6548\u7387\u8f83\u9ad8\u3002\u7136\u800c\uff0c\u5982\u679c\u5165\u6808\u65f6\u8d85\u51fa\u6570\u7ec4\u5bb9\u91cf\uff0c\u4f1a\u89e6\u53d1\u6269\u5bb9\u673a\u5236\uff0c\u5bfc\u81f4\u8be5\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53d8\u4e3a \\(O(n)\\) \u3002</p> <p>\u5728\u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0\u4e2d\uff0c\u94fe\u8868\u7684\u6269\u5bb9\u975e\u5e38\u7075\u6d3b\uff0c\u4e0d\u5b58\u5728\u4e0a\u8ff0\u6570\u7ec4\u6269\u5bb9\u65f6\u6548\u7387\u964d\u4f4e\u7684\u95ee\u9898\u3002\u4f46\u662f\uff0c\u5165\u6808\u64cd\u4f5c\u9700\u8981\u521d\u59cb\u5316\u8282\u70b9\u5bf9\u8c61\u5e76\u4fee\u6539\u6307\u9488\uff0c\u56e0\u6b64\u6548\u7387\u76f8\u5bf9\u8f83\u4f4e\u3002\u4e0d\u8fc7\uff0c\u5982\u679c\u5165\u6808\u5143\u7d20\u672c\u8eab\u5c31\u662f\u8282\u70b9\u5bf9\u8c61\uff0c\u90a3\u4e48\u53ef\u4ee5\u7701\u53bb\u521d\u59cb\u5316\u6b65\u9aa4\uff0c\u4ece\u800c\u63d0\u9ad8\u6548\u7387\u3002</p> <p>\u7efc\u4e0a\u6240\u8ff0\uff0c\u5f53\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u7684\u5143\u7d20\u662f\u57fa\u672c\u6570\u636e\u7c7b\u578b\u65f6\uff0c\u4f8b\u5982 <code>int</code> \u6216 <code>double</code> \uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\u3002</p> <ul> <li>\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u5728\u89e6\u53d1\u6269\u5bb9\u65f6\u6548\u7387\u4f1a\u964d\u4f4e\uff0c\u4f46\u7531\u4e8e\u6269\u5bb9\u662f\u4f4e\u9891\u64cd\u4f5c\uff0c\u56e0\u6b64\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002</li> <li>\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u53ef\u4ee5\u63d0\u4f9b\u66f4\u52a0\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002</li> </ul> <p>\u7a7a\u95f4\u6548\u7387</p> <p>\u5728\u521d\u59cb\u5316\u5217\u8868\u65f6\uff0c\u7cfb\u7edf\u4f1a\u4e3a\u5217\u8868\u5206\u914d\u201c\u521d\u59cb\u5bb9\u91cf\u201d\uff0c\u8be5\u5bb9\u91cf\u53ef\u80fd\u8d85\u51fa\u5b9e\u9645\u9700\u6c42\uff1b\u5e76\u4e14\uff0c\u6269\u5bb9\u673a\u5236\u901a\u5e38\u662f\u6309\u7167\u7279\u5b9a\u500d\u7387\uff08\u4f8b\u5982 2 \u500d\uff09\u8fdb\u884c\u6269\u5bb9\u7684\uff0c\u6269\u5bb9\u540e\u7684\u5bb9\u91cf\u4e5f\u53ef\u80fd\u8d85\u51fa\u5b9e\u9645\u9700\u6c42\u3002\u56e0\u6b64\uff0c\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u53ef\u80fd\u9020\u6210\u4e00\u5b9a\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002</p> <p>\u7136\u800c\uff0c\u7531\u4e8e\u94fe\u8868\u8282\u70b9\u9700\u8981\u989d\u5916\u5b58\u50a8\u6307\u9488\uff0c\u56e0\u6b64\u94fe\u8868\u8282\u70b9\u5360\u7528\u7684\u7a7a\u95f4\u76f8\u5bf9\u8f83\u5927\u3002</p> <p>\u7efc\u4e0a\uff0c\u6211\u4eec\u4e0d\u80fd\u7b80\u5355\u5730\u786e\u5b9a\u54ea\u79cd\u5b9e\u73b0\u66f4\u52a0\u8282\u7701\u5185\u5b58\uff0c\u9700\u8981\u9488\u5bf9\u5177\u4f53\u60c5\u51b5\u8fdb\u884c\u5206\u6790\u3002</p>"},{"location":"chapter_stack_and_queue/stack/#514","title":"5.1.4 \u00a0 \u6808\u7684\u5178\u578b\u5e94\u7528","text":"<ul> <li>\u6d4f\u89c8\u5668\u4e2d\u7684\u540e\u9000\u4e0e\u524d\u8fdb\u3001\u8f6f\u4ef6\u4e2d\u7684\u64a4\u9500\u4e0e\u53cd\u64a4\u9500\u3002\u6bcf\u5f53\u6211\u4eec\u6253\u5f00\u65b0\u7684\u7f51\u9875\uff0c\u6d4f\u89c8\u5668\u5c31\u4f1a\u5bf9\u4e0a\u4e00\u4e2a\u7f51\u9875\u6267\u884c\u5165\u6808\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u540e\u9000\u64cd\u4f5c\u56de\u5230\u4e0a\u4e00\u4e2a\u7f51\u9875\u3002\u540e\u9000\u64cd\u4f5c\u5b9e\u9645\u4e0a\u662f\u5728\u6267\u884c\u51fa\u6808\u3002\u5982\u679c\u8981\u540c\u65f6\u652f\u6301\u540e\u9000\u548c\u524d\u8fdb\uff0c\u90a3\u4e48\u9700\u8981\u4e24\u4e2a\u6808\u6765\u914d\u5408\u5b9e\u73b0\u3002</li> <li>\u7a0b\u5e8f\u5185\u5b58\u7ba1\u7406\u3002\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\uff0c\u7cfb\u7edf\u90fd\u4f1a\u5728\u6808\u9876\u6dfb\u52a0\u4e00\u4e2a\u6808\u5e27\uff0c\u7528\u4e8e\u8bb0\u5f55\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u4fe1\u606f\u3002\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u5411\u4e0b\u9012\u63a8\u9636\u6bb5\u4f1a\u4e0d\u65ad\u6267\u884c\u5165\u6808\u64cd\u4f5c\uff0c\u800c\u5411\u4e0a\u56de\u6eaf\u9636\u6bb5\u5219\u4f1a\u4e0d\u65ad\u6267\u884c\u51fa\u6808\u64cd\u4f5c\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/summary/","title":"5.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_stack_and_queue/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u6808\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002</li> <li>\u5728\u65f6\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u5177\u6709\u8f83\u9ad8\u7684\u5e73\u5747\u6548\u7387\uff0c\u4f46\u5728\u6269\u5bb9\u8fc7\u7a0b\u4e2d\uff0c\u5355\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u52a3\u5316\u81f3 \\(O(n)\\) \u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u6808\u7684\u94fe\u8868\u5b9e\u73b0\u5177\u6709\u66f4\u4e3a\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002</li> <li>\u5728\u7a7a\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u53ef\u80fd\u5bfc\u81f4\u4e00\u5b9a\u7a0b\u5ea6\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002\u4f46\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u94fe\u8868\u8282\u70b9\u6240\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u6bd4\u6570\u7ec4\u5143\u7d20\u66f4\u5927\u3002</li> <li>\u961f\u5217\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u540c\u6837\u53ef\u4ee5\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u7684\u5bf9\u6bd4\u4e0a\uff0c\u961f\u5217\u7684\u7ed3\u8bba\u4e0e\u524d\u8ff0\u6808\u7684\u7ed3\u8bba\u76f8\u4f3c\u3002</li> <li>\u53cc\u5411\u961f\u5217\u662f\u4e00\u79cd\u5177\u6709\u66f4\u9ad8\u81ea\u7531\u5ea6\u7684\u961f\u5217\uff0c\u5b83\u5141\u8bb8\u5728\u4e24\u7aef\u8fdb\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u662f\u5426\u662f\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\uff1f</p> <p>\u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u529f\u80fd\u672c\u8d28\u4e0a\u662f\u201c\u6808\u201d\u7684\u4f53\u73b0\u3002\u5f53\u7528\u6237\u8bbf\u95ee\u4e00\u4e2a\u65b0\u9875\u9762\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u88ab\u6dfb\u52a0\u5230\u6808\u9876\uff1b\u5f53\u7528\u6237\u70b9\u51fb\u540e\u9000\u6309\u94ae\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u4ece\u6808\u9876\u5f39\u51fa\u3002\u4f7f\u7528\u53cc\u5411\u961f\u5217\u53ef\u4ee5\u65b9\u4fbf\u5730\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u64cd\u4f5c\uff0c\u8fd9\u4e2a\u5728\u201c\u53cc\u5411\u961f\u5217\u201d\u7ae0\u8282\u6709\u63d0\u5230\u3002</p> <p>Q\uff1a\u5728\u51fa\u6808\u540e\uff0c\u662f\u5426\u9700\u8981\u91ca\u653e\u51fa\u6808\u8282\u70b9\u7684\u5185\u5b58\uff1f</p> <p>\u5982\u679c\u540e\u7eed\u4ecd\u9700\u8981\u4f7f\u7528\u5f39\u51fa\u8282\u70b9\uff0c\u5219\u4e0d\u9700\u8981\u91ca\u653e\u5185\u5b58\u3002\u82e5\u4e4b\u540e\u4e0d\u9700\u8981\u7528\u5230\uff0c<code>Java</code> \u548c <code>Python</code> \u7b49\u8bed\u8a00\u62e5\u6709\u81ea\u52a8\u5783\u573e\u56de\u6536\u673a\u5236\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\uff1b\u5728 <code>C</code> \u548c <code>C++</code> \u4e2d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\u3002</p> <p>Q\uff1a\u53cc\u5411\u961f\u5217\u50cf\u662f\u4e24\u4e2a\u6808\u62fc\u63a5\u5728\u4e86\u4e00\u8d77\uff0c\u5b83\u7684\u7528\u9014\u662f\u4ec0\u4e48\uff1f</p> <p>\u53cc\u5411\u961f\u5217\u5c31\u50cf\u662f\u6808\u548c\u961f\u5217\u7684\u7ec4\u5408\u6216\u4e24\u4e2a\u6808\u62fc\u5728\u4e86\u4e00\u8d77\u3002\u5b83\u8868\u73b0\u7684\u662f\u6808 + \u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u53ef\u4ee5\u5b9e\u73b0\u6808\u4e0e\u961f\u5217\u7684\u6240\u6709\u5e94\u7528\uff0c\u5e76\u4e14\u66f4\u52a0\u7075\u6d3b\u3002</p> <p>Q\uff1a\u64a4\u9500\uff08undo\uff09\u548c\u53cd\u64a4\u9500\uff08redo\uff09\u5177\u4f53\u662f\u5982\u4f55\u5b9e\u73b0\u7684\uff1f</p> <p>\u4f7f\u7528\u4e24\u4e2a\u6808\uff0c\u6808 <code>A</code> \u7528\u4e8e\u64a4\u9500\uff0c\u6808 <code>B</code> \u7528\u4e8e\u53cd\u64a4\u9500\u3002</p> <ol> <li>\u6bcf\u5f53\u7528\u6237\u6267\u884c\u4e00\u4e2a\u64cd\u4f5c\uff0c\u5c06\u8fd9\u4e2a\u64cd\u4f5c\u538b\u5165\u6808 <code>A</code> \uff0c\u5e76\u6e05\u7a7a\u6808 <code>B</code> \u3002</li> <li>\u5f53\u7528\u6237\u6267\u884c\u201c\u64a4\u9500\u201d\u65f6\uff0c\u4ece\u6808 <code>A</code> \u4e2d\u5f39\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u5e76\u5c06\u5176\u538b\u5165\u6808 <code>B</code> \u3002</li> <li>\u5f53\u7528\u6237\u6267\u884c\u201c\u53cd\u64a4\u9500\u201d\u65f6\uff0c\u4ece\u6808 <code>B</code> \u4e2d\u5f39\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u5e76\u5c06\u5176\u538b\u5165\u6808 <code>A</code> \u3002</li> </ol>"},{"location":"chapter_tree/","title":"\u7b2c 7 \u7ae0 \u00a0 \u6811","text":"<p>Abstract</p> <p>\u53c2\u5929\u5927\u6811\u5145\u6ee1\u751f\u547d\u529b\uff0c\u6839\u6df1\u53f6\u8302\uff0c\u5206\u679d\u6276\u758f\u3002</p> <p>\u5b83\u4e3a\u6211\u4eec\u5c55\u73b0\u4e86\u6570\u636e\u5206\u6cbb\u7684\u751f\u52a8\u5f62\u6001\u3002</p>"},{"location":"chapter_tree/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>7.1 \u00a0 \u4e8c\u53c9\u6811</li> <li>7.2 \u00a0 \u4e8c\u53c9\u6811\u904d\u5386</li> <li>7.3 \u00a0 \u4e8c\u53c9\u6811\u6570\u7ec4\u8868\u793a</li> <li>7.4 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811</li> <li>7.5 \u00a0 AVL \u6811 *</li> <li>7.6 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_tree/array_representation_of_tree/","title":"7.3 \u00a0 \u4e8c\u53c9\u6811\u6570\u7ec4\u8868\u793a","text":"<p>\u5728\u94fe\u8868\u8868\u793a\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u5b58\u50a8\u5355\u5143\u4e3a\u8282\u70b9 <code>TreeNode</code> \uff0c\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u6307\u9488\u76f8\u8fde\u63a5\u3002\u4e0a\u4e00\u8282\u4ecb\u7ecd\u4e86\u94fe\u8868\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7684\u5404\u9879\u57fa\u672c\u64cd\u4f5c\u3002</p> <p>\u90a3\u4e48\uff0c\u6211\u4eec\u80fd\u5426\u7528\u6570\u7ec4\u6765\u8868\u793a\u4e8c\u53c9\u6811\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\u3002</p>"},{"location":"chapter_tree/array_representation_of_tree/#731","title":"7.3.1 \u00a0 \u8868\u793a\u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"<p>\u5148\u5206\u6790\u4e00\u4e2a\u7b80\u5355\u6848\u4f8b\u3002\u7ed9\u5b9a\u4e00\u68f5\u5b8c\u7f8e\u4e8c\u53c9\u6811\uff0c\u6211\u4eec\u5c06\u6240\u6709\u8282\u70b9\u6309\u7167\u5c42\u5e8f\u904d\u5386\u7684\u987a\u5e8f\u5b58\u50a8\u5728\u4e00\u4e2a\u6570\u7ec4\u4e2d\uff0c\u5219\u6bcf\u4e2a\u8282\u70b9\u90fd\u5bf9\u5e94\u552f\u4e00\u7684\u6570\u7ec4\u7d22\u5f15\u3002</p> <p>\u6839\u636e\u5c42\u5e8f\u904d\u5386\u7684\u7279\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa\u7236\u8282\u70b9\u7d22\u5f15\u4e0e\u5b50\u8282\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u201c\u6620\u5c04\u516c\u5f0f\u201d\uff1a\u82e5\u67d0\u8282\u70b9\u7684\u7d22\u5f15\u4e3a \\(i\\) \uff0c\u5219\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \u3002\u56fe 7-12 \u5c55\u793a\u4e86\u5404\u4e2a\u8282\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u6620\u5c04\u5173\u7cfb\u3002</p> <p></p> <p> \u56fe 7-12 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a </p> <p>\u6620\u5c04\u516c\u5f0f\u7684\u89d2\u8272\u76f8\u5f53\u4e8e\u94fe\u8868\u4e2d\u7684\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\u3002\u7ed9\u5b9a\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\u8282\u70b9\uff0c\u6211\u4eec\u90fd\u53ef\u4ee5\u901a\u8fc7\u6620\u5c04\u516c\u5f0f\u6765\u8bbf\u95ee\u5b83\u7684\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u3002</p>"},{"location":"chapter_tree/array_representation_of_tree/#732","title":"7.3.2 \u00a0 \u8868\u793a\u4efb\u610f\u4e8c\u53c9\u6811","text":"<p>\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u4e00\u4e2a\u7279\u4f8b\uff0c\u5728\u4e8c\u53c9\u6811\u7684\u4e2d\u95f4\u5c42\u901a\u5e38\u5b58\u5728\u8bb8\u591a <code>None</code> \u3002\u7531\u4e8e\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5e76\u4e0d\u5305\u542b\u8fd9\u4e9b <code>None</code> \uff0c\u56e0\u6b64\u6211\u4eec\u65e0\u6cd5\u4ec5\u51ed\u8be5\u5e8f\u5217\u6765\u63a8\u6d4b <code>None</code> \u7684\u6570\u91cf\u548c\u5206\u5e03\u4f4d\u7f6e\u3002\u8fd9\u610f\u5473\u7740\u5b58\u5728\u591a\u79cd\u4e8c\u53c9\u6811\u7ed3\u6784\u90fd\u7b26\u5408\u8be5\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002</p> <p>\u5982\u56fe 7-13 \u6240\u793a\uff0c\u7ed9\u5b9a\u4e00\u68f5\u975e\u5b8c\u7f8e\u4e8c\u53c9\u6811\uff0c\u4e0a\u8ff0\u6570\u7ec4\u8868\u793a\u65b9\u6cd5\u5df2\u7ecf\u5931\u6548\u3002</p> <p></p> <p> \u56fe 7-13 \u00a0 \u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5bf9\u5e94\u591a\u79cd\u4e8c\u53c9\u6811\u53ef\u80fd\u6027 </p> <p>\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u4e2d\u663e\u5f0f\u5730\u5199\u51fa\u6240\u6709 <code>None</code> \u3002\u5982\u56fe 7-14 \u6240\u793a\uff0c\u8fd9\u6837\u5904\u7406\u540e\uff0c\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5c31\u53ef\u4ee5\u552f\u4e00\u8868\u793a\u4e8c\u53c9\u6811\u4e86\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code># \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\n# \u4f7f\u7528 None \u6765\u8868\u793a\u7a7a\u4f4d\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c INT_MAX \u6807\u8bb0\u7a7a\u4f4d\nvector&lt;int&gt; tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u7684\u5305\u88c5\u7c7b Integer \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nint?[] tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 any \u7c7b\u578b\u7684\u5207\u7247, \u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 Int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nList&lt;int?&gt; tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 None \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c\u6807\u8bb0\u7a7a\u4f4d\uff0c\u56e0\u6b64\u8981\u6c42\u8282\u70b9\u503c\u4e0d\u80fd\u4e3a INT_MAX\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nval tree = mutableListOf( 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 )\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p></p> <p> \u56fe 7-14 \u00a0 \u4efb\u610f\u7c7b\u578b\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a </p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u4f7f\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u56de\u987e\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u5b9a\u4e49\uff0c<code>None</code> \u53ea\u51fa\u73b0\u5728\u6700\u5e95\u5c42\u4e14\u9760\u53f3\u7684\u4f4d\u7f6e\uff0c\u56e0\u6b64\u6240\u6709 <code>None</code> \u4e00\u5b9a\u51fa\u73b0\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u672b\u5c3e\u3002</p> <p>\u8fd9\u610f\u5473\u7740\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u5b8c\u5168\u4e8c\u53c9\u6811\u65f6\uff0c\u53ef\u4ee5\u7701\u7565\u5b58\u50a8\u6240\u6709 <code>None</code> \uff0c\u975e\u5e38\u65b9\u4fbf\u3002\u56fe 7-15 \u7ed9\u51fa\u4e86\u4e00\u4e2a\u4f8b\u5b50\u3002</p> <p></p> <p> \u56fe 7-15 \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a </p> <p>\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u68f5\u57fa\u4e8e\u6570\u7ec4\u8868\u793a\u7684\u4e8c\u53c9\u6811\uff0c\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u64cd\u4f5c\u3002</p> <ul> <li>\u7ed9\u5b9a\u67d0\u8282\u70b9\uff0c\u83b7\u53d6\u5b83\u7684\u503c\u3001\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u3001\u7236\u8282\u70b9\u3002</li> <li>\u83b7\u53d6\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u3001\u540e\u5e8f\u904d\u5386\u3001\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig array_binary_tree.py<pre><code>class ArrayBinaryTree:\n    \"\"\"\u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b\"\"\"\n\n    def __init__(self, arr: list[int | None]):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._tree = list(arr)\n\n    def size(self):\n        \"\"\"\u5217\u8868\u5bb9\u91cf\"\"\"\n        return len(self._tree)\n\n    def val(self, i: int) -&gt; int:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c\"\"\"\n        # \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 or i &gt;= self.size():\n            return None\n        return self._tree[i]\n\n    def left(self, i: int) -&gt; int | None:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n        return 2 * i + 1\n\n    def right(self, i: int) -&gt; int | None:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n        return 2 * i + 2\n\n    def parent(self, i: int) -&gt; int | None:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n        return (i - 1) // 2\n\n    def level_order(self) -&gt; list[int]:\n        \"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for i in range(self.size()):\n            if self.val(i) is not None:\n                self.res.append(self.val(i))\n        return self.res\n\n    def dfs(self, i: int, order: str):\n        \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\n        if self.val(i) is None:\n            return\n        # \u524d\u5e8f\u904d\u5386\n        if order == \"pre\":\n            self.res.append(self.val(i))\n        self.dfs(self.left(i), order)\n        # \u4e2d\u5e8f\u904d\u5386\n        if order == \"in\":\n            self.res.append(self.val(i))\n        self.dfs(self.right(i), order)\n        # \u540e\u5e8f\u904d\u5386\n        if order == \"post\":\n            self.res.append(self.val(i))\n\n    def pre_order(self) -&gt; list[int]:\n        \"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        self.dfs(0, order=\"pre\")\n        return self.res\n\n    def in_order(self) -&gt; list[int]:\n        \"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        self.dfs(0, order=\"in\")\n        return self.res\n\n    def post_order(self) -&gt; list[int]:\n        \"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        self.dfs(0, order=\"post\")\n        return self.res\n</code></pre> array_binary_tree.cpp<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    ArrayBinaryTree(vector&lt;int&gt; arr) {\n        tree = arr;\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    int size() {\n        return tree.size();\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    int val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size())\n            return INT_MAX;\n        return tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    int left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    int right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    int parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; levelOrder() {\n        vector&lt;int&gt; res;\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != INT_MAX)\n                res.push_back(val(i));\n        }\n        return res;\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; preOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; inOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; postOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"post\", res);\n        return res;\n    }\n\n  private:\n    vector&lt;int&gt; tree;\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    void dfs(int i, string order, vector&lt;int&gt; &amp;res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (val(i) == INT_MAX)\n            return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order == \"pre\")\n            res.push_back(val(i));\n        dfs(left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order == \"in\")\n            res.push_back(val(i));\n        dfs(right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order == \"post\")\n            res.push_back(val(i));\n    }\n};\n</code></pre> array_binary_tree.java<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    private List&lt;Integer&gt; tree;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public ArrayBinaryTree(List&lt;Integer&gt; arr) {\n        tree = new ArrayList&lt;&gt;(arr);\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    public int size() {\n        return tree.size();\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    public Integer val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size())\n            return null;\n        return tree.get(i);\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public Integer left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public Integer right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    public Integer parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; levelOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != null)\n                res.add(val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    private void dfs(Integer i, String order, List&lt;Integer&gt; res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (val(i) == null)\n            return;\n        // \u524d\u5e8f\u904d\u5386\n        if (\"pre\".equals(order))\n            res.add(val(i));\n        dfs(left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (\"in\".equals(order))\n            res.add(val(i));\n        dfs(right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (\"post\".equals(order))\n            res.add(val(i));\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; preOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; inOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; postOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.cs<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree(List&lt;int?&gt; arr) {\n    List&lt;int?&gt; tree = new(arr);\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    public int Size() {\n        return tree.Count;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    public int? Val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= Size())\n            return null;\n        return tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public int Left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public int Right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    public int Parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; LevelOrder() {\n        List&lt;int&gt; res = [];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (int i = 0; i &lt; Size(); i++) {\n            if (Val(i).HasValue)\n                res.Add(Val(i)!.Value);\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    void DFS(int i, string order, List&lt;int&gt; res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (!Val(i).HasValue)\n            return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order == \"pre\")\n            res.Add(Val(i)!.Value);\n        DFS(Left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order == \"in\")\n            res.Add(Val(i)!.Value);\n        DFS(Right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order == \"post\")\n            res.Add(Val(i)!.Value);\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; PreOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; InOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"in\", res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; PostOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.go<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\ntype arrayBinaryTree struct {\n    tree []any\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newArrayBinaryTree(arr []any) *arrayBinaryTree {\n    return &amp;arrayBinaryTree{\n        tree: arr,\n    }\n}\n\n/* \u5217\u8868\u5bb9\u91cf */\nfunc (abt *arrayBinaryTree) size() int {\n    return len(abt.tree)\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nfunc (abt *arrayBinaryTree) val(i int) any {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if i &lt; 0 || i &gt;= abt.size() {\n        return nil\n    }\n    return abt.tree[i]\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) left(i int) int {\n    return 2*i + 1\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) right(i int) int {\n    return 2*i + 2\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) parent(i int) int {\n    return (i - 1) / 2\n}\n\n/* \u5c42\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) levelOrder() []any {\n    var res []any\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n    for i := 0; i &lt; abt.size(); i++ {\n        if abt.val(i) != nil {\n            res = append(res, abt.val(i))\n        }\n    }\n    return res\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nfunc (abt *arrayBinaryTree) dfs(i int, order string, res *[]any) {\n    // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n    if abt.val(i) == nil {\n        return\n    }\n    // \u524d\u5e8f\u904d\u5386\n    if order == \"pre\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.left(i), order, res)\n    // \u4e2d\u5e8f\u904d\u5386\n    if order == \"in\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.right(i), order, res)\n    // \u540e\u5e8f\u904d\u5386\n    if order == \"post\" {\n        *res = append(*res, abt.val(i))\n    }\n}\n\n/* \u524d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) preOrder() []any {\n    var res []any\n    abt.dfs(0, \"pre\", &amp;res)\n    return res\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) inOrder() []any {\n    var res []any\n    abt.dfs(0, \"in\", &amp;res)\n    return res\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) postOrder() []any {\n    var res []any\n    abt.dfs(0, \"post\", &amp;res)\n    return res\n}\n</code></pre> array_binary_tree.swift<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    private var tree: [Int?]\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init(arr: [Int?]) {\n        tree = arr\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    func size() -&gt; Int {\n        tree.count\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    func val(i: Int) -&gt; Int? {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 || i &gt;= size() {\n            return nil\n        }\n        return tree[i]\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    func left(i: Int) -&gt; Int {\n        2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    func right(i: Int) -&gt; Int {\n        2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    func parent(i: Int) -&gt; Int {\n        (i - 1) / 2\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    func levelOrder() -&gt; [Int] {\n        var res: [Int] = []\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for i in 0 ..&lt; size() {\n            if let val = val(i: i) {\n                res.append(val)\n            }\n        }\n        return res\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    private func dfs(i: Int, order: String, res: inout [Int]) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        guard let val = val(i: i) else {\n            return\n        }\n        // \u524d\u5e8f\u904d\u5386\n        if order == \"pre\" {\n            res.append(val)\n        }\n        dfs(i: left(i: i), order: order, res: &amp;res)\n        // \u4e2d\u5e8f\u904d\u5386\n        if order == \"in\" {\n            res.append(val)\n        }\n        dfs(i: right(i: i), order: order, res: &amp;res)\n        // \u540e\u5e8f\u904d\u5386\n        if order == \"post\" {\n            res.append(val)\n        }\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    func preOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"pre\", res: &amp;res)\n        return res\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    func inOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"in\", res: &amp;res)\n        return res\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    func postOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"post\", res: &amp;res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.js<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    #tree;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(arr) {\n        this.#tree = arr;\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    size() {\n        return this.#tree.length;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    val(i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    left(i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    right(i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    parent(i) {\n        return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    levelOrder() {\n        let res = [];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    #dfs(i, order, res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (this.val(i) === null) return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    preOrder() {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    inOrder() {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    postOrder() {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.ts<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    #tree: (number | null)[];\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(arr: (number | null)[]) {\n        this.#tree = arr;\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    size(): number {\n        return this.#tree.length;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    val(i: number): number | null {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    left(i: number): number {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    right(i: number): number {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    parent(i: number): number {\n        return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    levelOrder(): number[] {\n        let res = [];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    #dfs(i: number, order: Order, res: (number | null)[]): void {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (this.val(i) === null) return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    preOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    inOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    postOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.dart<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n  late List&lt;int?&gt; _tree;\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  ArrayBinaryTree(this._tree);\n\n  /* \u5217\u8868\u5bb9\u91cf */\n  int size() {\n    return _tree.length;\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n  int? val(int i) {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if (i &lt; 0 || i &gt;= size()) {\n      return null;\n    }\n    return _tree[i];\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n  int? left(int i) {\n    return 2 * i + 1;\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n  int? right(int i) {\n    return 2 * i + 2;\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n  int? parent(int i) {\n    return (i - 1) ~/ 2;\n  }\n\n  /* \u5c42\u5e8f\u904d\u5386 */\n  List&lt;int&gt; levelOrder() {\n    List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; size(); i++) {\n      if (val(i) != null) {\n        res.add(val(i)!);\n      }\n    }\n    return res;\n  }\n\n  /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n  void dfs(int i, String order, List&lt;int?&gt; res) {\n    // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n    if (val(i) == null) {\n      return;\n    }\n    // \u524d\u5e8f\u904d\u5386\n    if (order == 'pre') {\n      res.add(val(i));\n    }\n    dfs(left(i)!, order, res);\n    // \u4e2d\u5e8f\u904d\u5386\n    if (order == 'in') {\n      res.add(val(i));\n    }\n    dfs(right(i)!, order, res);\n    // \u540e\u5e8f\u904d\u5386\n    if (order == 'post') {\n      res.add(val(i));\n    }\n  }\n\n  /* \u524d\u5e8f\u904d\u5386 */\n  List&lt;int?&gt; preOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'pre', res);\n    return res;\n  }\n\n  /* \u4e2d\u5e8f\u904d\u5386 */\n  List&lt;int?&gt; inOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'in', res);\n    return res;\n  }\n\n  /* \u540e\u5e8f\u904d\u5386 */\n  List&lt;int?&gt; postOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'post', res);\n    return res;\n  }\n}\n</code></pre> array_binary_tree.rs<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nstruct ArrayBinaryTree {\n    tree: Vec&lt;Option&lt;i32&gt;&gt;,\n}\n\nimpl ArrayBinaryTree {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(arr: Vec&lt;Option&lt;i32&gt;&gt;) -&gt; Self {\n        Self { tree: arr }\n    }\n\n    /* \u5217\u8868\u5bb9\u91cf */\n    fn size(&amp;self) -&gt; i32 {\n        self.tree.len() as i32\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    fn val(&amp;self, i: i32) -&gt; Option&lt;i32&gt; {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 || i &gt;= self.size() {\n            None\n        } else {\n            self.tree[i as usize]\n        }\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fn left(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fn right(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    fn parent(&amp;self, i: i32) -&gt; i32 {\n        (i - 1) / 2\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    fn level_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for i in 0..self.size() {\n            if let Some(val) = self.val(i) {\n                res.push(val)\n            }\n        }\n        res\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    fn dfs(&amp;self, i: i32, order: &amp;str, res: &amp;mut Vec&lt;i32&gt;) {\n        if self.val(i).is_none() {\n            return;\n        }\n        let val = self.val(i).unwrap();\n        // \u524d\u5e8f\u904d\u5386\n        if order == \"pre\" {\n            res.push(val);\n        }\n        self.dfs(self.left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if order == \"in\" {\n            res.push(val);\n        }\n        self.dfs(self.right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if order == \"post\" {\n            res.push(val);\n        }\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    fn pre_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"pre\", &amp;mut res);\n        res\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    fn in_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"in\", &amp;mut res);\n        res\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    fn post_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"post\", &amp;mut res);\n        res\n    }\n}\n</code></pre> array_binary_tree.c<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7ed3\u6784\u4f53 */\ntypedef struct {\n    int *tree;\n    int size;\n} ArrayBinaryTree;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayBinaryTree *newArrayBinaryTree(int *arr, int arrSize) {\n    ArrayBinaryTree *abt = (ArrayBinaryTree *)malloc(sizeof(ArrayBinaryTree));\n    abt-&gt;tree = malloc(sizeof(int) * arrSize);\n    memcpy(abt-&gt;tree, arr, sizeof(int) * arrSize);\n    abt-&gt;size = arrSize;\n    return abt;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayBinaryTree(ArrayBinaryTree *abt) {\n    free(abt-&gt;tree);\n    free(abt);\n}\n\n/* \u5217\u8868\u5bb9\u91cf */\nint size(ArrayBinaryTree *abt) {\n    return abt-&gt;size;\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nint val(ArrayBinaryTree *abt, int i) {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if (i &lt; 0 || i &gt;= size(abt))\n        return INT_MAX;\n    return abt-&gt;tree[i];\n}\n\n/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; size(abt); i++) {\n        if (val(abt, i) != INT_MAX)\n            res[index++] = val(abt, i);\n    }\n    *returnSize = index;\n    return res;\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nvoid dfs(ArrayBinaryTree *abt, int i, char *order, int *res, int *index) {\n    // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n    if (val(abt, i) == INT_MAX)\n        return;\n    // \u524d\u5e8f\u904d\u5386\n    if (strcmp(order, \"pre\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, left(i), order, res, index);\n    // \u4e2d\u5e8f\u904d\u5386\n    if (strcmp(order, \"in\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, right(i), order, res, index);\n    // \u540e\u5e8f\u904d\u5386\n    if (strcmp(order, \"post\") == 0)\n        res[(*index)++] = val(abt, i);\n}\n\n/* \u524d\u5e8f\u904d\u5386 */\nint *preOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"pre\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nint *inOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"in\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nint *postOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"post\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n</code></pre> array_binary_tree.kt<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree(val tree: List&lt;Int?&gt;) {\n    /* \u5217\u8868\u5bb9\u91cf */\n    fun size(): Int {\n        return tree.size\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    fun value(i: Int): Int? {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size()) return null\n        return tree[i]\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    fun parent(i: Int): Int {\n        return (i - 1) / 2\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    fun levelOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (i in 0..&lt;size()) {\n            if (value(i) != null) res.add(value(i))\n        }\n        return res\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    fun dfs(i: Int, order: String, res: MutableList&lt;Int?&gt;) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (value(i) == null) return\n        // \u524d\u5e8f\u904d\u5386\n        if (\"pre\" == order) res.add(value(i))\n        dfs(left(i), order, res)\n        // \u4e2d\u5e8f\u904d\u5386\n        if (\"in\" == order) res.add(value(i))\n        dfs(right(i), order, res)\n        // \u540e\u5e8f\u904d\u5386\n        if (\"post\" == order) res.add(value(i))\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    fun preOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        dfs(0, \"pre\", res)\n        return res\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    fun inOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        dfs(0, \"in\", res)\n        return res\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    fun postOrder(): List&lt;Int?&gt; {\n        val res = ArrayList&lt;Int?&gt;()\n        dfs(0, \"post\", res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.rb<pre><code>[class]{ArrayBinaryTree}-[func]{}\n</code></pre> array_binary_tree.zig<pre><code>[class]{ArrayBinaryTree}-[func]{}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_tree/array_representation_of_tree/#733","title":"7.3.3 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\u4e3b\u8981\u6709\u4ee5\u4e0b\u4f18\u70b9\u3002</p> <ul> <li>\u6570\u7ec4\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u5bf9\u7f13\u5b58\u53cb\u597d\uff0c\u8bbf\u95ee\u4e0e\u904d\u5386\u901f\u5ea6\u8f83\u5feb\u3002</li> <li>\u4e0d\u9700\u8981\u5b58\u50a8\u6307\u9488\uff0c\u6bd4\u8f83\u8282\u7701\u7a7a\u95f4\u3002</li> <li>\u5141\u8bb8\u968f\u673a\u8bbf\u95ee\u8282\u70b9\u3002</li> </ul> <p>\u7136\u800c\uff0c\u6570\u7ec4\u8868\u793a\u4e5f\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\u3002</p> <ul> <li>\u6570\u7ec4\u5b58\u50a8\u9700\u8981\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5b58\u50a8\u6570\u636e\u91cf\u8fc7\u5927\u7684\u6811\u3002</li> <li>\u589e\u5220\u8282\u70b9\u9700\u8981\u901a\u8fc7\u6570\u7ec4\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u5b9e\u73b0\uff0c\u6548\u7387\u8f83\u4f4e\u3002</li> <li>\u5f53\u4e8c\u53c9\u6811\u4e2d\u5b58\u5728\u5927\u91cf <code>None</code> \u65f6\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u7684\u8282\u70b9\u6570\u636e\u6bd4\u91cd\u8f83\u4f4e\uff0c\u7a7a\u95f4\u5229\u7528\u7387\u8f83\u4f4e\u3002</li> </ul>"},{"location":"chapter_tree/avl_tree/","title":"7.5 \u00a0 AVL \u6811 *","text":"<p>\u5728\u201c\u4e8c\u53c9\u641c\u7d22\u6811\u201d\u7ae0\u8282\u4e2d\u6211\u4eec\u63d0\u5230\uff0c\u5728\u591a\u6b21\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u9000\u5316\u4e3a\u94fe\u8868\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c06\u4ece \\(O(\\log n)\\) \u52a3\u5316\u4e3a \\(O(n)\\) \u3002</p> <p>\u5982\u56fe 7-24 \u6240\u793a\uff0c\u7ecf\u8fc7\u4e24\u6b21\u5220\u9664\u8282\u70b9\u64cd\u4f5c\uff0c\u8fd9\u68f5\u4e8c\u53c9\u641c\u7d22\u6811\u4fbf\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\u3002</p> <p></p> <p> \u56fe 7-24 \u00a0 AVL \u6811\u5728\u5220\u9664\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316 </p> <p>\u518d\u4f8b\u5982\uff0c\u5728\u56fe 7-25 \u6240\u793a\u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e24\u4e2a\u8282\u70b9\u540e\uff0c\u6811\u5c06\u4e25\u91cd\u5411\u5de6\u503e\u659c\uff0c\u67e5\u627e\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u968f\u4e4b\u52a3\u5316\u3002</p> <p></p> <p> \u56fe 7-25 \u00a0 AVL \u6811\u5728\u63d2\u5165\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316 </p> <p>1962 \u5e74 G. M. Adelson-Velsky \u548c E. M. Landis \u5728\u8bba\u6587\u201cAn algorithm for the organization of information\u201d\u4e2d\u63d0\u51fa\u4e86AVL\uff08\u6811\uff09\u3002\u8bba\u6587\u4e2d\u8be6\u7ec6\u63cf\u8ff0\u4e86\u4e00\u7cfb\u5217\u64cd\u4f5c\uff0c\u786e\u4fdd\u5728\u6301\u7eed\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4e0d\u4f1a\u9000\u5316\uff0c\u4ece\u800c\u4f7f\u5f97\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4fdd\u6301\u5728 \\(O(\\log n)\\) \u7ea7\u522b\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5728\u9700\u8981\u9891\u7e41\u8fdb\u884c\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u7684\u573a\u666f\u4e2d\uff0cAVL \u6811\u80fd\u59cb\u7ec8\u4fdd\u6301\u9ad8\u6548\u7684\u6570\u636e\u64cd\u4f5c\u6027\u80fd\uff0c\u5177\u6709\u5f88\u597d\u7684\u5e94\u7528\u4ef7\u503c\u3002</p>"},{"location":"chapter_tree/avl_tree/#751-avl","title":"7.5.1 \u00a0 AVL \u6811\u5e38\u89c1\u672f\u8bed","text":"<p>AVL \u6811\u65e2\u662f\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u4e5f\u662f\u5e73\u8861\u4e8c\u53c9\u6811\uff0c\u540c\u65f6\u6ee1\u8db3\u8fd9\u4e24\u7c7b\u4e8c\u53c9\u6811\u7684\u6240\u6709\u6027\u8d28\uff0c\u56e0\u6b64\u662f\u4e00\u79cd\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\uff08balanced binary search tree\uff09\u3002</p>"},{"location":"chapter_tree/avl_tree/#1","title":"1. \u00a0 \u8282\u70b9\u9ad8\u5ea6","text":"<p>\u7531\u4e8e AVL \u6811\u7684\u76f8\u5173\u64cd\u4f5c\u9700\u8981\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u4e3a\u8282\u70b9\u7c7b\u6dfb\u52a0 <code>height</code> \u53d8\u91cf\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class TreeNode:\n    \"\"\"AVL \u6811\u8282\u70b9\u7c7b\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                 # \u8282\u70b9\u503c\n        self.height: int = 0                # \u8282\u70b9\u9ad8\u5ea6\n        self.left: TreeNode | None = None   # \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n        self.right: TreeNode | None = None  # \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nstruct TreeNode {\n    int val{};          // \u8282\u70b9\u503c\n    int height = 0;     // \u8282\u70b9\u9ad8\u5ea6\n    TreeNode *left{};   // \u5de6\u5b50\u8282\u70b9\n    TreeNode *right{};  // \u53f3\u5b50\u8282\u70b9\n    TreeNode() = default;\n    explicit TreeNode(int x) : val(x){}\n};\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    public int val;        // \u8282\u70b9\u503c\n    public int height;     // \u8282\u70b9\u9ad8\u5ea6\n    public TreeNode left;  // \u5de6\u5b50\u8282\u70b9\n    public TreeNode right; // \u53f3\u5b50\u8282\u70b9\n    public TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode(int? x) {\n    public int? val = x;    // \u8282\u70b9\u503c\n    public int height;      // \u8282\u70b9\u9ad8\u5ea6\n    public TreeNode? left;  // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    public TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\n    Val    int       // \u8282\u70b9\u503c\n    Height int       // \u8282\u70b9\u9ad8\u5ea6\n    Left   *TreeNode // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    Right  *TreeNode // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    var val: Int // \u8282\u70b9\u503c\n    var height: Int // \u8282\u70b9\u9ad8\u5ea6\n    var left: TreeNode? // \u5de6\u5b50\u8282\u70b9\n    var right: TreeNode? // \u53f3\u5b50\u8282\u70b9\n\n    init(x: Int) {\n        val = x\n        height = 0\n    }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    val; // \u8282\u70b9\u503c\n    height; //\u8282\u70b9\u9ad8\u5ea6\n    left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n    right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n    constructor(val, left, right, height) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    val: number;            // \u8282\u70b9\u503c\n    height: number;         // \u8282\u70b9\u9ad8\u5ea6\n    left: TreeNode | null;  // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n    right: TreeNode | null; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n    constructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height; \n        this.left = left === undefined ? null : left; \n        this.right = right === undefined ? null : right; \n    }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n  int val;         // \u8282\u70b9\u503c\n  int height;      // \u8282\u70b9\u9ad8\u5ea6\n  TreeNode? left;  // \u5de6\u5b50\u8282\u70b9\n  TreeNode? right; // \u53f3\u5b50\u8282\u70b9\n  TreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\n    val: i32,                               // \u8282\u70b9\u503c\n    height: i32,                            // \u8282\u70b9\u9ad8\u5ea6\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // \u5de6\u5b50\u8282\u70b9\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // \u53f3\u5b50\u8282\u70b9\n}\n\nimpl TreeNode {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            height: 0,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nTreeNode struct TreeNode {\n    int val;\n    int height;\n    struct TreeNode *left;\n    struct TreeNode *right;\n} TreeNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode(val _val: Int) {  // \u8282\u70b9\u503c\n    val height: Int = 0          // \u8282\u70b9\u9ad8\u5ea6\n    val left: TreeNode? = null   // \u5de6\u5b50\u8282\u70b9\n    val right: TreeNode? = null  // \u53f3\u5b50\u8282\u70b9\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>\u201c\u8282\u70b9\u9ad8\u5ea6\u201d\u662f\u6307\u4ece\u8be5\u8282\u70b9\u5230\u5b83\u7684\u6700\u8fdc\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u5373\u6240\u7ecf\u8fc7\u7684\u201c\u8fb9\u201d\u7684\u6570\u91cf\u3002\u9700\u8981\u7279\u522b\u6ce8\u610f\u7684\u662f\uff0c\u53f6\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a \\(0\\) \uff0c\u800c\u7a7a\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a \\(-1\\) \u3002\u6211\u4eec\u5c06\u521b\u5efa\u4e24\u4e2a\u5de5\u5177\u51fd\u6570\uff0c\u5206\u522b\u7528\u4e8e\u83b7\u53d6\u548c\u66f4\u65b0\u8282\u70b9\u7684\u9ad8\u5ea6\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def height(self, node: TreeNode | None) -&gt; int:\n    \"\"\"\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\"\"\"\n    # \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    if node is not None:\n        return node.height\n    return -1\n\ndef update_height(self, node: TreeNode | None):\n    \"\"\"\u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\"\"\"\n    # \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height = max([self.height(node.left), self.height(node.right)]) + 1\n</code></pre> avl_tree.cpp<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node == nullptr ? -1 : node-&gt;height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node-&gt;height = max(height(node-&gt;left), height(node-&gt;right)) + 1;\n}\n</code></pre> avl_tree.java<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint Height(TreeNode? node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid UpdateHeight(TreeNode node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height = Math.Max(Height(node.left), Height(node.right)) + 1;\n}\n</code></pre> avl_tree.go<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    if node != nil {\n        return node.Height\n    }\n    return -1\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\n    lh := t.height(node.Left)\n    rh := t.height(node.Right)\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    if lh &gt; rh {\n        node.Height = lh + 1\n    } else {\n        node.Height = rh + 1\n    }\n}\n</code></pre> avl_tree.swift<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc height(node: TreeNode?) -&gt; Int {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    node?.height ?? -1\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc updateHeight(node: TreeNode?) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n</code></pre> avl_tree.js<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\n#updateHeight(node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node: TreeNode): number {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nupdateHeight(node: TreeNode): void {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode? node) {\n  // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n  return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode? node) {\n  // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n  node!.height = max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfn height(node: OptionTreeNodeRc) -&gt; i32 {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    match node {\n        Some(node) =&gt; node.borrow().height,\n        None =&gt; -1,\n    }\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfn update_height(node: OptionTreeNodeRc) {\n    if let Some(node) = node {\n        let left = node.borrow().left.clone();\n        let right = node.borrow().right.clone();\n        // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n        node.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    if (node != NULL) {\n        return node-&gt;height;\n    }\n    return -1;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n    int lh = height(node-&gt;left);\n    int rh = height(node-&gt;right);\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    if (lh &gt; rh) {\n        node-&gt;height = lh + 1;\n    } else {\n        node-&gt;height = rh + 1;\n    }\n}\n</code></pre> avl_tree.kt<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfun height(node: TreeNode?): Int {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node?.height ?: -1\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfun updateHeight(node: TreeNode?) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node?.height = (max(height(node?.left).toDouble(), height(node?.right).toDouble()) + 1).toInt()\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{height}\n\n[class]{AVLTree}-[func]{update_height}\n</code></pre> avl_tree.zig<pre><code>// \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\nfn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n    _ = self;\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return if (node == null) -1 else node.?.height;\n}\n\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nfn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.?.height = @max(self.height(node.?.left), self.height(node.?.right)) + 1;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2","title":"2. \u00a0 \u8282\u70b9\u5e73\u8861\u56e0\u5b50","text":"<p>\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\uff08balance factor\uff09\u5b9a\u4e49\u4e3a\u8282\u70b9\u5de6\u5b50\u6811\u7684\u9ad8\u5ea6\u51cf\u53bb\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\uff0c\u540c\u65f6\u89c4\u5b9a\u7a7a\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4e3a \\(0\\) \u3002\u6211\u4eec\u540c\u6837\u5c06\u83b7\u53d6\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u7684\u529f\u80fd\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def balance_factor(self, node: TreeNode | None) -&gt; int:\n    \"\"\"\u83b7\u53d6\u5e73\u8861\u56e0\u5b50\"\"\"\n    # \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if node is None:\n        return 0\n    # \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return self.height(node.left) - self.height(node.right)\n</code></pre> avl_tree.cpp<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == nullptr)\n        return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.java<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null)\n        return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.cs<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint BalanceFactor(TreeNode? node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return Height(node.left) - Height(node.right);\n}\n</code></pre> avl_tree.go<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if node == nil {\n        return 0\n    }\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return t.height(node.Left) - t.height(node.Right)\n}\n</code></pre> avl_tree.swift<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc balanceFactor(node: TreeNode?) -&gt; Int {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    guard let node = node else { return 0 }\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node: node.left) - height(node: node.right)\n}\n</code></pre> avl_tree.js<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node === null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.ts<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node: TreeNode): number {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node === null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.dart<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node) {\n  // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n  if (node == null) return 0;\n  // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n  return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.rs<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfn balance_factor(node: OptionTreeNodeRc) -&gt; i32 {\n    match node {\n        // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n        None =&gt; 0,\n        // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n        Some(node) =&gt; {\n            Self::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())\n        }\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == NULL) {\n        return 0;\n    }\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.kt<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfun balanceFactor(node: TreeNode?): Int {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null) return 0\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node.left) - height(node.right)\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{balance_factor}\n</code></pre> avl_tree.zig<pre><code>// \u83b7\u53d6\u5e73\u8861\u56e0\u5b50\nfn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return self.height(node.?.left) - self.height(node.?.right);\n}\n</code></pre> <p>Note</p> <p>\u8bbe\u5e73\u8861\u56e0\u5b50\u4e3a \\(f\\) \uff0c\u5219\u4e00\u68f5 AVL \u6811\u7684\u4efb\u610f\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7686\u6ee1\u8db3 \\(-1 \\le f \\le 1\\) \u3002</p>"},{"location":"chapter_tree/avl_tree/#752-avl","title":"7.5.2 \u00a0 AVL \u6811\u65cb\u8f6c","text":"<p>AVL \u6811\u7684\u7279\u70b9\u5728\u4e8e\u201c\u65cb\u8f6c\u201d\u64cd\u4f5c\uff0c\u5b83\u80fd\u591f\u5728\u4e0d\u5f71\u54cd\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u524d\u63d0\u4e0b\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u65cb\u8f6c\u64cd\u4f5c\u65e2\u80fd\u4fdd\u6301\u201c\u4e8c\u53c9\u641c\u7d22\u6811\u201d\u7684\u6027\u8d28\uff0c\u4e5f\u80fd\u4f7f\u6811\u91cd\u65b0\u53d8\u4e3a\u201c\u5e73\u8861\u4e8c\u53c9\u6811\u201d\u3002</p> <p>\u6211\u4eec\u5c06\u5e73\u8861\u56e0\u5b50\u7edd\u5bf9\u503c \\(&gt; 1\\) \u7684\u8282\u70b9\u79f0\u4e3a\u201c\u5931\u8861\u8282\u70b9\u201d\u3002\u6839\u636e\u8282\u70b9\u5931\u8861\u60c5\u51b5\u7684\u4e0d\u540c\uff0c\u65cb\u8f6c\u64cd\u4f5c\u5206\u4e3a\u56db\u79cd\uff1a\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3002\u4e0b\u9762\u8be6\u7ec6\u4ecb\u7ecd\u8fd9\u4e9b\u65cb\u8f6c\u64cd\u4f5c\u3002</p>"},{"location":"chapter_tree/avl_tree/#1_1","title":"1. \u00a0 \u53f3\u65cb","text":"<p>\u5982\u56fe 7-26 \u6240\u793a\uff0c\u8282\u70b9\u4e0b\u65b9\u4e3a\u5e73\u8861\u56e0\u5b50\u3002\u4ece\u5e95\u81f3\u9876\u770b\uff0c\u4e8c\u53c9\u6811\u4e2d\u9996\u4e2a\u5931\u8861\u8282\u70b9\u662f\u201c\u8282\u70b9 3\u201d\u3002\u6211\u4eec\u5173\u6ce8\u4ee5\u8be5\u5931\u8861\u8282\u70b9\u4e3a\u6839\u8282\u70b9\u7684\u5b50\u6811\uff0c\u5c06\u8be5\u8282\u70b9\u8bb0\u4e3a <code>node</code> \uff0c\u5176\u5de6\u5b50\u8282\u70b9\u8bb0\u4e3a <code>child</code> \uff0c\u6267\u884c\u201c\u53f3\u65cb\u201d\u64cd\u4f5c\u3002\u5b8c\u6210\u53f3\u65cb\u540e\uff0c\u5b50\u6811\u6062\u590d\u5e73\u8861\uff0c\u5e76\u4e14\u4ecd\u7136\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-26 \u00a0 \u53f3\u65cb\u64cd\u4f5c\u6b65\u9aa4 </p> <p>\u5982\u56fe 7-27 \u6240\u793a\uff0c\u5f53\u8282\u70b9 <code>child</code> \u6709\u53f3\u5b50\u8282\u70b9\uff08\u8bb0\u4e3a <code>grand_child</code> \uff09\u65f6\uff0c\u9700\u8981\u5728\u53f3\u65cb\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 <code>grand_child</code> \u4f5c\u4e3a <code>node</code> \u7684\u5de6\u5b50\u8282\u70b9\u3002</p> <p></p> <p> \u56fe 7-27 \u00a0 \u6709 grand_child \u7684\u53f3\u65cb\u64cd\u4f5c </p> <p>\u201c\u5411\u53f3\u65cb\u8f6c\u201d\u662f\u4e00\u79cd\u5f62\u8c61\u5316\u7684\u8bf4\u6cd5\uff0c\u5b9e\u9645\u4e0a\u9700\u8981\u901a\u8fc7\u4fee\u6539\u8282\u70b9\u6307\u9488\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def right_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u53f3\u65cb\u64cd\u4f5c\"\"\"\n    child = node.left\n    grand_child = child.right\n    # \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node\n    node.left = grand_child\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    self.update_height(child)\n    # \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;left;\n    TreeNode *grandChild = child-&gt;right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode rightRotate(TreeNode node) {\n    TreeNode child = node.left;\n    TreeNode grandChild = child.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? RightRotate(TreeNode? node) {\n    TreeNode? child = node?.left;\n    TreeNode? grandChild = child?.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\n    child := node.Left\n    grandChild := child.Right\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.Right = node\n    node.Left = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc rightRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.left\n    let grandChild = child?.right\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child?.right = node\n    node?.left = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\n#rightRotate(node) {\n    const child = node.left;\n    const grandChild = child.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nrightRotate(node: TreeNode): TreeNode {\n    const child = node.left;\n    const grandChild = child.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node) {\n  TreeNode? child = node!.left;\n  TreeNode? grandChild = child!.right;\n  // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n  child.right = node;\n  node.left = grandChild;\n  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  updateHeight(node);\n  updateHeight(child);\n  // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfn right_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().left.clone().unwrap();\n            let grand_child = child.borrow().right.clone();\n            // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n            child.borrow_mut().right = Some(node.clone());\n            node.borrow_mut().left = grand_child;\n            // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;left;\n    grandChild = child-&gt;right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfun rightRotate(node: TreeNode?): TreeNode {\n    val child = node!!.left\n    val grandChild = child!!.right\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node\n    node.left = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node)\n    updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{right_rotate}\n</code></pre> avl_tree.zig<pre><code>// \u53f3\u65cb\u64cd\u4f5c\nfn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    var child = node.?.left;\n    var grandChild = child.?.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.?.right = node;\n    node.?.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.updateHeight(node);\n    self.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2_1","title":"2. \u00a0 \u5de6\u65cb","text":"<p>\u76f8\u5e94\u5730\uff0c\u5982\u679c\u8003\u8651\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u201c\u955c\u50cf\u201d\uff0c\u5219\u9700\u8981\u6267\u884c\u56fe 7-28 \u6240\u793a\u7684\u201c\u5de6\u65cb\u201d\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 7-28 \u00a0 \u5de6\u65cb\u64cd\u4f5c </p> <p>\u540c\u7406\uff0c\u5982\u56fe 7-29 \u6240\u793a\uff0c\u5f53\u8282\u70b9 <code>child</code> \u6709\u5de6\u5b50\u8282\u70b9\uff08\u8bb0\u4e3a <code>grand_child</code> \uff09\u65f6\uff0c\u9700\u8981\u5728\u5de6\u65cb\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 <code>grand_child</code> \u4f5c\u4e3a <code>node</code> \u7684\u53f3\u5b50\u8282\u70b9\u3002</p> <p></p> <p> \u56fe 7-29 \u00a0 \u6709 grand_child \u7684\u5de6\u65cb\u64cd\u4f5c </p> <p>\u53ef\u4ee5\u89c2\u5bdf\u5230\uff0c\u53f3\u65cb\u548c\u5de6\u65cb\u64cd\u4f5c\u5728\u903b\u8f91\u4e0a\u662f\u955c\u50cf\u5bf9\u79f0\u7684\uff0c\u5b83\u4eec\u5206\u522b\u89e3\u51b3\u7684\u4e24\u79cd\u5931\u8861\u60c5\u51b5\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u57fa\u4e8e\u5bf9\u79f0\u6027\uff0c\u6211\u4eec\u53ea\u9700\u5c06\u53f3\u65cb\u7684\u5b9e\u73b0\u4ee3\u7801\u4e2d\u7684\u6240\u6709\u7684 <code>left</code> \u66ff\u6362\u4e3a <code>right</code> \uff0c\u5c06\u6240\u6709\u7684 <code>right</code> \u66ff\u6362\u4e3a <code>left</code> \uff0c\u5373\u53ef\u5f97\u5230\u5de6\u65cb\u7684\u5b9e\u73b0\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def left_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u5de6\u65cb\u64cd\u4f5c\"\"\"\n    child = node.right\n    grand_child = child.left\n    # \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node\n    node.right = grand_child\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    self.update_height(child)\n    # \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;right;\n    TreeNode *grandChild = child-&gt;left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode leftRotate(TreeNode node) {\n    TreeNode child = node.right;\n    TreeNode grandChild = child.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? LeftRotate(TreeNode? node) {\n    TreeNode? child = node?.right;\n    TreeNode? grandChild = child?.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\n    child := node.Right\n    grandChild := child.Left\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.Left = node\n    node.Right = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc leftRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.right\n    let grandChild = child?.left\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child?.left = node\n    node?.right = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\n#leftRotate(node) {\n    const child = node.right;\n    const grandChild = child.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nleftRotate(node: TreeNode): TreeNode {\n    const child = node.right;\n    const grandChild = child.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node) {\n  TreeNode? child = node!.right;\n  TreeNode? grandChild = child!.left;\n  // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n  child.left = node;\n  node.right = grandChild;\n  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  updateHeight(node);\n  updateHeight(child);\n  // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfn left_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().right.clone().unwrap();\n            let grand_child = child.borrow().left.clone();\n            // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n            child.borrow_mut().left = Some(node.clone());\n            node.borrow_mut().right = grand_child;\n            // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;right;\n    grandChild = child-&gt;left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfun leftRotate(node: TreeNode?): TreeNode {\n    val child = node!!.right\n    val grandChild = child!!.left\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node\n    node.right = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node)\n    updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{left_rotate}\n</code></pre> avl_tree.zig<pre><code>// \u5de6\u65cb\u64cd\u4f5c\nfn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    var child = node.?.right;\n    var grandChild = child.?.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.?.left = node;\n    node.?.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.updateHeight(node);\n    self.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3","title":"3. \u00a0 \u5148\u5de6\u65cb\u540e\u53f3\u65cb","text":"<p>\u5bf9\u4e8e\u56fe 7-30 \u4e2d\u7684\u5931\u8861\u8282\u70b9 3 \uff0c\u4ec5\u4f7f\u7528\u5de6\u65cb\u6216\u53f3\u65cb\u90fd\u65e0\u6cd5\u4f7f\u5b50\u6811\u6062\u590d\u5e73\u8861\u3002\u6b64\u65f6\u9700\u8981\u5148\u5bf9 <code>child</code> \u6267\u884c\u201c\u5de6\u65cb\u201d\uff0c\u518d\u5bf9 <code>node</code> \u6267\u884c\u201c\u53f3\u65cb\u201d\u3002</p> <p></p> <p> \u56fe 7-30 \u00a0 \u5148\u5de6\u65cb\u540e\u53f3\u65cb </p>"},{"location":"chapter_tree/avl_tree/#4","title":"4. \u00a0 \u5148\u53f3\u65cb\u540e\u5de6\u65cb","text":"<p>\u5982\u56fe 7-31 \u6240\u793a\uff0c\u5bf9\u4e8e\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u955c\u50cf\u60c5\u51b5\uff0c\u9700\u8981\u5148\u5bf9 <code>child</code> \u6267\u884c\u201c\u53f3\u65cb\u201d\uff0c\u518d\u5bf9 <code>node</code> \u6267\u884c\u201c\u5de6\u65cb\u201d\u3002</p> <p></p> <p> \u56fe 7-31 \u00a0 \u5148\u53f3\u65cb\u540e\u5de6\u65cb </p>"},{"location":"chapter_tree/avl_tree/#5","title":"5. \u00a0 \u65cb\u8f6c\u7684\u9009\u62e9","text":"<p>\u56fe 7-32 \u5c55\u793a\u7684\u56db\u79cd\u5931\u8861\u60c5\u51b5\u4e0e\u4e0a\u8ff0\u6848\u4f8b\u9010\u4e2a\u5bf9\u5e94\uff0c\u5206\u522b\u9700\u8981\u91c7\u7528\u53f3\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5de6\u65cb\u7684\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 7-32 \u00a0 AVL \u6811\u7684\u56db\u79cd\u65cb\u8f6c\u60c5\u51b5 </p> <p>\u5982\u4e0b\u8868\u6240\u793a\uff0c\u6211\u4eec\u901a\u8fc7\u5224\u65ad\u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4ee5\u53ca\u8f83\u9ad8\u4e00\u4fa7\u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7684\u6b63\u8d1f\u53f7\uff0c\u6765\u786e\u5b9a\u5931\u8861\u8282\u70b9\u5c5e\u4e8e\u56fe 7-32 \u4e2d\u7684\u54ea\u79cd\u60c5\u51b5\u3002</p> <p> \u8868 7-3 \u00a0 \u56db\u79cd\u65cb\u8f6c\u60c5\u51b5\u7684\u9009\u62e9\u6761\u4ef6 </p> \u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5e94\u91c7\u7528\u7684\u65cb\u8f6c\u65b9\u6cd5 \\(&gt; 1\\) \uff08\u5de6\u504f\u6811\uff09 \\(\\geq 0\\) \u53f3\u65cb \\(&gt; 1\\) \uff08\u5de6\u504f\u6811\uff09 \\(&lt;0\\) \u5148\u5de6\u65cb\u540e\u53f3\u65cb \\(&lt; -1\\) \uff08\u53f3\u504f\u6811\uff09 \\(\\leq 0\\) \u5de6\u65cb \\(&lt; -1\\) \uff08\u53f3\u504f\u6811\uff09 \\(&gt;0\\) \u5148\u53f3\u65cb\u540e\u5de6\u65cb <p>\u4e3a\u4e86\u4fbf\u4e8e\u4f7f\u7528\uff0c\u6211\u4eec\u5c06\u65cb\u8f6c\u64cd\u4f5c\u5c01\u88c5\u6210\u4e00\u4e2a\u51fd\u6570\u3002\u6709\u4e86\u8fd9\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u5c31\u80fd\u5bf9\u5404\u79cd\u5931\u8861\u60c5\u51b5\u8fdb\u884c\u65cb\u8f6c\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\"\"\"\n    # \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    balance_factor = self.balance_factor(node)\n    # \u5de6\u504f\u6811\n    if balance_factor &gt; 1:\n        if self.balance_factor(node.left) &gt;= 0:\n            # \u53f3\u65cb\n            return self.right_rotate(node)\n        else:\n            # \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = self.left_rotate(node.left)\n            return self.right_rotate(node)\n    # \u53f3\u504f\u6811\n    elif balance_factor &lt; -1:\n        if self.balance_factor(node.right) &lt;= 0:\n            # \u5de6\u65cb\n            return self.left_rotate(node)\n        else:\n            # \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = self.right_rotate(node.right)\n            return self.left_rotate(node)\n    # \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n</code></pre> avl_tree.cpp<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int _balanceFactor = balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (_balanceFactor &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (_balanceFactor &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode rotate(TreeNode node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int balanceFactor = balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = leftRotate(node.left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = rightRotate(node.right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? Rotate(TreeNode? node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int balanceFactorInt = BalanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactorInt &gt; 1) {\n        if (BalanceFactor(node?.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return RightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node!.left = LeftRotate(node!.left);\n            return RightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactorInt &lt; -1) {\n        if (BalanceFactor(node?.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return LeftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node!.right = RightRotate(node!.right);\n            return LeftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    // Go \u63a8\u8350\u77ed\u53d8\u91cf\uff0c\u8fd9\u91cc bf \u6307\u4ee3 t.balanceFactor\n    bf := t.balanceFactor(node)\n    // \u5de6\u504f\u6811\n    if bf &gt; 1 {\n        if t.balanceFactor(node.Left) &gt;= 0 {\n            // \u53f3\u65cb\n            return t.rightRotate(node)\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.Left = t.leftRotate(node.Left)\n            return t.rightRotate(node)\n        }\n    }\n    // \u53f3\u504f\u6811\n    if bf &lt; -1 {\n        if t.balanceFactor(node.Right) &lt;= 0 {\n            // \u5de6\u65cb\n            return t.leftRotate(node)\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.Right = t.rightRotate(node.Right)\n            return t.leftRotate(node)\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc rotate(node: TreeNode?) -&gt; TreeNode? {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    let balanceFactor = balanceFactor(node: node)\n    // \u5de6\u504f\u6811\n    if balanceFactor &gt; 1 {\n        if self.balanceFactor(node: node?.left) &gt;= 0 {\n            // \u53f3\u65cb\n            return rightRotate(node: node)\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node?.left = leftRotate(node: node?.left)\n            return rightRotate(node: node)\n        }\n    }\n    // \u53f3\u504f\u6811\n    if balanceFactor &lt; -1 {\n        if self.balanceFactor(node: node?.right) &lt;= 0 {\n            // \u5de6\u65cb\n            return leftRotate(node: node)\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node?.right = rightRotate(node: node?.right)\n            return leftRotate(node: node)\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n#rotate(node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    const balanceFactor = this.balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return this.#rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = this.#leftRotate(node.left);\n            return this.#rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return this.#leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = this.#rightRotate(node.right);\n            return this.#leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nrotate(node: TreeNode): TreeNode {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    const balanceFactor = this.balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return this.rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = this.leftRotate(node.left);\n            return this.rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return this.leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = this.rightRotate(node.right);\n            return this.leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node) {\n  // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n  int factor = balanceFactor(node);\n  // \u5de6\u504f\u6811\n  if (factor &gt; 1) {\n    if (balanceFactor(node!.left) &gt;= 0) {\n      // \u53f3\u65cb\n      return rightRotate(node);\n    } else {\n      // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n      node.left = leftRotate(node.left);\n      return rightRotate(node);\n    }\n  }\n  // \u53f3\u504f\u6811\n  if (factor &lt; -1) {\n    if (balanceFactor(node!.right) &lt;= 0) {\n      // \u5de6\u65cb\n      return leftRotate(node);\n    } else {\n      // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n      node.right = rightRotate(node.right);\n      return leftRotate(node);\n    }\n  }\n  // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfn rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    let balance_factor = Self::balance_factor(node.clone());\n    // \u5de6\u504f\u6811\n    if balance_factor &gt; 1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().left.clone()) &gt;= 0 {\n            // \u53f3\u65cb\n            Self::right_rotate(Some(node))\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            let left = node.borrow().left.clone();\n            node.borrow_mut().left = Self::left_rotate(left);\n            Self::right_rotate(Some(node))\n        }\n    }\n    // \u53f3\u504f\u6811\n    else if balance_factor &lt; -1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().right.clone()) &lt;= 0 {\n            // \u5de6\u65cb\n            Self::left_rotate(Some(node))\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            let right = node.borrow().right.clone();\n            node.borrow_mut().right = Self::right_rotate(right);\n            Self::left_rotate(Some(node))\n        }\n    } else {\n        // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n        node\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int bf = balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (bf &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (bf &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfun rotate(node: TreeNode): TreeNode {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    val balanceFactor = balanceFactor(node)\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node)\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = leftRotate(node.left)\n            return rightRotate(node)\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node)\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = rightRotate(node.right)\n            return leftRotate(node)\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{rotate}\n</code></pre> avl_tree.zig<pre><code>// \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nfn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    var balance_factor = self.balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balance_factor &gt; 1) {\n        if (self.balanceFactor(node.?.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return self.rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.?.left = self.leftRotate(node.?.left);\n            return self.rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balance_factor &lt; -1) {\n        if (self.balanceFactor(node.?.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return self.leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.?.right = self.rightRotate(node.?.right);\n            return self.leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#753-avl","title":"7.5.3 \u00a0 AVL \u6811\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_tree/avl_tree/#1_2","title":"1. \u00a0 \u63d2\u5165\u8282\u70b9","text":"<p>AVL \u6811\u7684\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u4e0e\u4e8c\u53c9\u641c\u7d22\u6811\u5728\u4e3b\u4f53\u4e0a\u7c7b\u4f3c\u3002\u552f\u4e00\u7684\u533a\u522b\u5728\u4e8e\uff0c\u5728 AVL \u6811\u4e2d\u63d2\u5165\u8282\u70b9\u540e\uff0c\u4ece\u8be5\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u53ef\u80fd\u4f1a\u51fa\u73b0\u4e00\u7cfb\u5217\u5931\u8861\u8282\u70b9\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4ece\u8fd9\u4e2a\u8282\u70b9\u5f00\u59cb\uff0c\u81ea\u5e95\u5411\u4e0a\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def insert(self, val):\n    \"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n    self._root = self.insert_helper(self._root, val)\n\ndef insert_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode:\n    \"\"\"\u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\n    if node is None:\n        return TreeNode(val)\n    # 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9\n    if val &lt; node.val:\n        node.left = self.insert_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.insert_helper(node.right, val)\n    else:\n        # \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    # 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    else\n        return node;    // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode insertHelper(TreeNode node, int val) {\n    if (node == null)\n        return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val)\n        node.left = insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = insertHelper(node.right, val);\n    else\n        return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int val) {\n    root = InsertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? InsertHelper(TreeNode? node, int val) {\n    if (node == null) return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val)\n        node.left = InsertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = InsertHelper(node.right, val);\n    else\n        return node;     // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    UpdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = Rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc (t *aVLTree) insert(val int) {\n    t.root = t.insertHelper(t.root, val)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return NewTreeNode(val)\n    }\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if val &lt; node.Val.(int) {\n        node.Left = t.insertHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.insertHelper(node.Right, val)\n    } else {\n        // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = t.rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(val: Int) {\n    root = insertHelper(node: root, val: val)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc insertHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return TreeNode(x: val)\n    }\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if val &lt; node!.val {\n        node?.left = insertHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = insertHelper(node: node?.right, val: val)\n    } else {\n        return node // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    updateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node: node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(val) {\n    this.root = this.#insertHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#insertHelper(node, val) {\n    if (node === null) return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val) node.left = this.#insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#insertHelper(node.right, val);\n    else return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    this.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.#rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(val: number): void {\n    this.root = this.insertHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\ninsertHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val) {\n        node.left = this.insertHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.insertHelper(node.right, val);\n    } else {\n        return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    this.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n  root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val) {\n  if (node == null) return TreeNode(val);\n  /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n  if (val &lt; node.val)\n    node.left = insertHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = insertHelper(node.right, val);\n  else\n    return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n  updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n  node = rotate(node);\n  // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfn insert(&amp;mut self, val: i32) {\n    self.root = Self::insert_helper(self.root.clone(), val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn insert_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n            match {\n                let node_val = node.borrow().val;\n                node_val\n            }\n            .cmp(&amp;val)\n            {\n                Ordering::Greater =&gt; {\n                    let left = node.borrow().left.clone();\n                    node.borrow_mut().left = Self::insert_helper(left, val);\n                }\n                Ordering::Less =&gt; {\n                    let right = node.borrow().right.clone();\n                    node.borrow_mut().right = Self::insert_helper(right, val);\n                }\n                Ordering::Equal =&gt; {\n                    return Some(node); // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n                }\n            }\n            Self::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n\n            /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n            node = Self::rotate(Some(node)).unwrap();\n            // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(node)\n        }\n        None =&gt; Some(TreeNode::new(val)),\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(AVLTree *tree, int val) {\n    tree-&gt;root = insertHelper(tree-&gt;root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == NULL) {\n        return newTreeNode(val);\n    }\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    } else {\n        // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node;\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfun insert(value: Int) {\n    root = insertHelper(root, value)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfun insertHelper(n: TreeNode?, value: Int): TreeNode {\n    if (n == null)\n        return TreeNode(value)\n    var node = n\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (value &lt; node.value) node.left = insertHelper(node.left, value)\n    else if (value &gt; node.value) node.right = insertHelper(node.right, value)\n    else return node // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n\n    updateHeight(node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{insert}\n\n[class]{AVLTree}-[func]{insert_helper}\n</code></pre> avl_tree.zig<pre><code>// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, val: T) !void {\n    self.root = (try self.insertHelper(self.root, val)).?;\n}\n\n// \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {\n    var node = node_;\n    if (node == null) {\n        var tmp_node = try self.mem_allocator.create(inc.TreeNode(T));\n        tmp_node.init(val);\n        return tmp_node;\n    }\n    // 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\u5e76\u63d2\u5165\u8282\u70b9\n    if (val &lt; node.?.val) {\n        node.?.left = try self.insertHelper(node.?.left, val);\n    } else if (val &gt; node.?.val) {\n        node.?.right = try self.insertHelper(node.?.right, val);\n    } else {\n        return node;            // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    self.updateHeight(node);    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    // 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    node = self.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2_2","title":"2. \u00a0 \u5220\u9664\u8282\u70b9","text":"<p>\u7c7b\u4f3c\u5730\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5220\u9664\u8282\u70b9\u65b9\u6cd5\u7684\u57fa\u7840\u4e0a\uff0c\u9700\u8981\u4ece\u5e95\u81f3\u9876\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig avl_tree.py<pre><code>def remove(self, val: int):\n    \"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n    self._root = self.remove_helper(self._root, val)\n\ndef remove_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode | None:\n    \"\"\"\u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\n    if node is None:\n        return None\n    # 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664\n    if val &lt; node.val:\n        node.left = self.remove_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.remove_helper(node.right, val)\n    else:\n        if node.left is None or node.right is None:\n            child = node.left or node.right\n            # \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if child is None:\n                return None\n            # \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else:\n                node = child\n        else:\n            # \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            temp = node.right\n            while temp.left is not None:\n                temp = temp.left\n            node.right = self.remove_helper(node.right, temp.val)\n            node.val = temp.val\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    # 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return nullptr;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    else {\n        if (node-&gt;left == nullptr || node-&gt;right == nullptr) {\n            TreeNode *child = node-&gt;left != nullptr ? node-&gt;left : node-&gt;right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == nullptr) {\n                delete node;\n                return nullptr;\n            }\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else {\n                delete node;\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != nullptr) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode removeHelper(TreeNode node, int val) {\n    if (node == null)\n        return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val)\n        node.left = removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = removeHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode child = node.left != null ? node.left : node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null)\n                return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else\n                node = child;\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int val) {\n    root = RemoveHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? RemoveHelper(TreeNode? node, int val) {\n    if (node == null) return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val)\n        node.left = RemoveHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = RemoveHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode? child = node.left ?? node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null)\n                return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else\n                node = child;\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode? temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = RemoveHelper(node.right, temp.val!.Value);\n            node.val = temp.val;\n        }\n    }\n    UpdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = Rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc (t *aVLTree) remove(val int) {\n    t.root = t.removeHelper(t.root, val)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return nil\n    }\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if val &lt; node.Val.(int) {\n        node.Left = t.removeHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.removeHelper(node.Right, val)\n    } else {\n        if node.Left == nil || node.Right == nil {\n            child := node.Left\n            if node.Right != nil {\n                child = node.Right\n            }\n            if child == nil {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n                return nil\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                node = child\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            temp := node.Right\n            for temp.Left != nil {\n                temp = temp.Left\n            }\n            node.Right = t.removeHelper(node.Right, temp.Val.(int))\n            node.Val = temp.Val\n        }\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = t.rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc remove(val: Int) {\n    root = removeHelper(node: root, val: val)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc removeHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return nil\n    }\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if val &lt; node!.val {\n        node?.left = removeHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = removeHelper(node: node?.right, val: val)\n    } else {\n        if node?.left == nil || node?.right == nil {\n            let child = node?.left ?? node?.right\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if child == nil {\n                return nil\n            }\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else {\n                node = child\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            var temp = node?.right\n            while temp?.left != nil {\n                temp = temp?.left\n            }\n            node?.right = removeHelper(node: node?.right, val: temp!.val)\n            node?.val = temp!.val\n        }\n    }\n    updateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node: node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(val) {\n    this.root = this.#removeHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#removeHelper(node, val) {\n    if (node === null) return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val) node.left = this.#removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#removeHelper(node.right, val);\n    else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child === null) return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else node = child;\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.#removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.#rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(val: number): void {\n    this.root = this.removeHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nremoveHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node.val) {\n        node.left = this.removeHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.removeHelper(node.right, val);\n    } else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child === null) {\n                return null;\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n  root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val) {\n  if (node == null) return null;\n  /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n  if (val &lt; node.val)\n    node.left = removeHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = removeHelper(node.right, val);\n  else {\n    if (node.left == null || node.right == null) {\n      TreeNode? child = node.left ?? node.right;\n      // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n      if (child == null)\n        return null;\n      // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n      else\n        node = child;\n    } else {\n      // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n      TreeNode? temp = node.right;\n      while (temp!.left != null) {\n        temp = temp.left;\n      }\n      node.right = removeHelper(node.right, temp.val);\n      node.val = temp.val;\n    }\n  }\n  updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n  node = rotate(node);\n  // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfn remove(&amp;self, val: i32) {\n    Self::remove_helper(self.root.clone(), val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn remove_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n            if val &lt; node.borrow().val {\n                let left = node.borrow().left.clone();\n                node.borrow_mut().left = Self::remove_helper(left, val);\n            } else if val &gt; node.borrow().val {\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, val);\n            } else if node.borrow().left.is_none() || node.borrow().right.is_none() {\n                let child = if node.borrow().left.is_some() {\n                    node.borrow().left.clone()\n                } else {\n                    node.borrow().right.clone()\n                };\n                match child {\n                    // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n                    None =&gt; {\n                        return None;\n                    }\n                    // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                    Some(child) =&gt; node = child,\n                }\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n                let mut temp = node.borrow().right.clone().unwrap();\n                loop {\n                    let temp_left = temp.borrow().left.clone();\n                    if temp_left.is_none() {\n                        break;\n                    }\n                    temp = temp_left.unwrap();\n                }\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);\n                node.borrow_mut().val = temp.borrow().val;\n            }\n            Self::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n\n            /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n            node = Self::rotate(Some(node)).unwrap();\n            // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(node)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(AVLTree *tree, int val) {\n    TreeNode *root = removeHelper(tree-&gt;root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    TreeNode *child, *grandChild;\n    if (node == NULL) {\n        return NULL;\n    }\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    } else {\n        if (node-&gt;left == NULL || node-&gt;right == NULL) {\n            child = node-&gt;left;\n            if (node-&gt;right != NULL) {\n                child = node-&gt;right;\n            }\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == NULL) {\n                return NULL;\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != NULL) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfun remove(value: Int) {\n    root = removeHelper(root, value)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfun removeHelper(n: TreeNode?, value: Int): TreeNode? {\n    var node = n ?: return null\n    /* 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664 */\n    if (value &lt; node.value) node.left = removeHelper(node.left, value)\n    else if (value &gt; node.value) node.right = removeHelper(node.right, value)\n    else {\n        if (node.left == null || node.right == null) {\n            val child = if (node.left != null) node.left else node.right\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null) return null\n            else node = child\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            var temp = node.right\n            while (temp!!.left != null) {\n                temp = temp.left\n            }\n            node.right = removeHelper(node.right, temp.value)\n            node.value = temp.value\n        }\n    }\n    updateHeight(node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>[class]{AVLTree}-[func]{remove}\n\n[class]{AVLTree}-[func]{remove_helper}\n</code></pre> avl_tree.zig<pre><code>// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, val: T) void {\n   self.root = self.removeHelper(self.root, val).?;\n}\n\n// \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {\n    var node = node_;\n    if (node == null) return null;\n    // 1. \u67e5\u627e\u8282\u70b9\u5e76\u5220\u9664\n    if (val &lt; node.?.val) {\n        node.?.left = self.removeHelper(node.?.left, val);\n    } else if (val &gt; node.?.val) {\n        node.?.right = self.removeHelper(node.?.right, val);\n    } else {\n        if (node.?.left == null or node.?.right == null) {\n            var child = if (node.?.left != null) node.?.left else node.?.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null) {\n                return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            } else {\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            var temp = node.?.right;\n            while (temp.?.left != null) {\n                temp = temp.?.left;\n            }\n            node.?.right = self.removeHelper(node.?.right, temp.?.val);\n            node.?.val = temp.?.val;\n        }\n    }\n    self.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    // 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    node = self.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3_1","title":"3. \u00a0 \u67e5\u627e\u8282\u70b9","text":"<p>AVL \u6811\u7684\u8282\u70b9\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u53c9\u641c\u7d22\u6811\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002</p>"},{"location":"chapter_tree/avl_tree/#754-avl","title":"7.5.4 \u00a0 AVL \u6811\u5178\u578b\u5e94\u7528","text":"<ul> <li>\u7ec4\u7ec7\u548c\u5b58\u50a8\u5927\u578b\u6570\u636e\uff0c\u9002\u7528\u4e8e\u9ad8\u9891\u67e5\u627e\u3001\u4f4e\u9891\u589e\u5220\u7684\u573a\u666f\u3002</li> <li>\u7528\u4e8e\u6784\u5efa\u6570\u636e\u5e93\u4e2d\u7684\u7d22\u5f15\u7cfb\u7edf\u3002</li> <li>\u7ea2\u9ed1\u6811\u4e5f\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\u3002\u76f8\u8f83\u4e8e AVL \u6811\uff0c\u7ea2\u9ed1\u6811\u7684\u5e73\u8861\u6761\u4ef6\u66f4\u5bbd\u677e\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u6240\u9700\u7684\u65cb\u8f6c\u64cd\u4f5c\u66f4\u5c11\uff0c\u8282\u70b9\u589e\u5220\u64cd\u4f5c\u7684\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002</li> </ul>"},{"location":"chapter_tree/binary_search_tree/","title":"7.4 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811","text":"<p>\u5982\u56fe 7-16 \u6240\u793a\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\uff08binary search tree\uff09\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\u3002</p> <ol> <li>\u5bf9\u4e8e\u6839\u8282\u70b9\uff0c\u5de6\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c \\(&lt;\\) \u6839\u8282\u70b9\u7684\u503c \\(&lt;\\) \u53f3\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c\u3002</li> <li>\u4efb\u610f\u8282\u70b9\u7684\u5de6\u3001\u53f3\u5b50\u6811\u4e5f\u662f\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5373\u540c\u6837\u6ee1\u8db3\u6761\u4ef6 <code>1.</code> \u3002</li> </ol> <p></p> <p> \u56fe 7-16 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811 </p>"},{"location":"chapter_tree/binary_search_tree/#741","title":"7.4.1 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u64cd\u4f5c","text":"<p>\u6211\u4eec\u5c06\u4e8c\u53c9\u641c\u7d22\u6811\u5c01\u88c5\u4e3a\u4e00\u4e2a\u7c7b <code>BinarySearchTree</code> \uff0c\u5e76\u58f0\u660e\u4e00\u4e2a\u6210\u5458\u53d8\u91cf <code>root</code> \uff0c\u6307\u5411\u6811\u7684\u6839\u8282\u70b9\u3002</p>"},{"location":"chapter_tree/binary_search_tree/#1","title":"1. \u00a0 \u67e5\u627e\u8282\u70b9","text":"<p>\u7ed9\u5b9a\u76ee\u6807\u8282\u70b9\u503c <code>num</code> \uff0c\u53ef\u4ee5\u6839\u636e\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u6765\u67e5\u627e\u3002\u5982\u56fe 7-17 \u6240\u793a\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u8282\u70b9 <code>cur</code> \uff0c\u4ece\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9 <code>root</code> \u51fa\u53d1\uff0c\u5faa\u73af\u6bd4\u8f83\u8282\u70b9\u503c <code>cur.val</code> \u548c <code>num</code> \u4e4b\u95f4\u7684\u5927\u5c0f\u5173\u7cfb\u3002</p> <ul> <li>\u82e5 <code>cur.val &lt; num</code> \uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728 <code>cur</code> \u7684\u53f3\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c <code>cur = cur.right</code> \u3002</li> <li>\u82e5 <code>cur.val &gt; num</code> \uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728 <code>cur</code> \u7684\u5de6\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c <code>cur = cur.left</code> \u3002</li> <li>\u82e5 <code>cur.val = num</code> \uff0c\u8bf4\u660e\u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\u5e76\u8fd4\u56de\u8be5\u8282\u70b9\u3002</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-17 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u67e5\u627e\u8282\u70b9\u793a\u4f8b </p> <p>\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u4e00\u81f4\uff0c\u90fd\u662f\u6bcf\u8f6e\u6392\u9664\u4e00\u534a\u60c5\u51b5\u3002\u5faa\u73af\u6b21\u6570\u6700\u591a\u4e3a\u4e8c\u53c9\u6811\u7684\u9ad8\u5ea6\uff0c\u5f53\u4e8c\u53c9\u6811\u5e73\u8861\u65f6\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def search(self, num: int) -&gt; TreeNode | None:\n    \"\"\"\u67e5\u627e\u8282\u70b9\"\"\"\n    cur = self._root\n    # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur is not None:\n        # \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        elif cur.val &gt; num:\n            cur = cur.left\n        # \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else:\n            break\n    return cur\n</code></pre> binary_search_tree.cpp<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(int num) {\n    TreeNode *cur = root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur-&gt;val &gt; num)\n            cur = cur-&gt;left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode search(int num) {\n    TreeNode cur = root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? Search(int num) {\n    TreeNode? cur = root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur =\n            cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\n    node := bst.root\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    for node != nil {\n        if node.Val.(int) &lt; num {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            node = node.Right\n        } else if node.Val.(int) &gt; num {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            node = node.Left\n        } else {\n            // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            break\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return node\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfunc search(num: Int) -&gt; TreeNode? {\n    var cur = root\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur != nil {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if cur!.val &gt; num {\n            cur = cur?.left\n        }\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else {\n            break\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nsearch(num) {\n    let cur = this.root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num) cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nsearch(num: number): TreeNode | null {\n    let cur = this.root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num) cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? search(int _num) {\n  TreeNode? cur = _root;\n  // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n  while (cur != null) {\n    // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n    else if (cur.val &gt; _num)\n      cur = cur.left;\n    // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n    else\n      break;\n  }\n  // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n  return cur;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u67e5\u627e\u8282\u70b9 */\npub fn search(&amp;self, num: i32) -&gt; OptionTreeNodeRc {\n    let mut cur = self.root.clone();\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            Ordering::Greater =&gt; cur = node.borrow().right.clone(),\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            Ordering::Less =&gt; cur = node.borrow().left.clone(),\n            // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            Ordering::Equal =&gt; break,\n        }\n    }\n\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    cur\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(BinarySearchTree *bst, int num) {\n    TreeNode *cur = bst-&gt;root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != NULL) {\n        if (cur-&gt;val &lt; num) {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            cur = cur-&gt;right;\n        } else if (cur-&gt;val &gt; num) {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            cur = cur-&gt;left;\n        } else {\n            // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfun search(num: Int): TreeNode? {\n    var cur = root\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        cur = if (cur.value &lt; num) cur.right\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.value &gt; num) cur.left\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else break\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur\n}\n</code></pre> binary_search_tree.rb<pre><code>[class]{BinarySearchTree}-[func]{search}\n</code></pre> binary_search_tree.zig<pre><code>// \u67e5\u627e\u8282\u70b9\nfn search(self: *Self, num: T) ?*inc.TreeNode(T) {\n    var cur = self.root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        } else if (cur.?.val &gt; num) {\n            cur = cur.?.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        } else {\n            break;\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_tree/binary_search_tree/#2","title":"2. \u00a0 \u63d2\u5165\u8282\u70b9","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u5f85\u63d2\u5165\u5143\u7d20 <code>num</code> \uff0c\u4e3a\u4e86\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 &lt; \u6839\u8282\u70b9 &lt; \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u63d2\u5165\u64cd\u4f5c\u6d41\u7a0b\u5982\u56fe 7-18 \u6240\u793a\u3002</p> <ol> <li>\u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff1a\u4e0e\u67e5\u627e\u64cd\u4f5c\u76f8\u4f3c\uff0c\u4ece\u6839\u8282\u70b9\u51fa\u53d1\uff0c\u6839\u636e\u5f53\u524d\u8282\u70b9\u503c\u548c <code>num</code> \u7684\u5927\u5c0f\u5173\u7cfb\u5faa\u73af\u5411\u4e0b\u641c\u7d22\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\uff08\u904d\u5386\u81f3 <code>None</code> \uff09\u65f6\u8df3\u51fa\u5faa\u73af\u3002</li> <li>\u5728\u8be5\u4f4d\u7f6e\u63d2\u5165\u8282\u70b9\uff1a\u521d\u59cb\u5316\u8282\u70b9 <code>num</code> \uff0c\u5c06\u8be5\u8282\u70b9\u7f6e\u4e8e <code>None</code> \u7684\u4f4d\u7f6e\u3002</li> </ol> <p></p> <p> \u56fe 7-18 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u63d2\u5165\u8282\u70b9 </p> <p>\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u4e24\u70b9\u3002</p> <ul> <li>\u4e8c\u53c9\u641c\u7d22\u6811\u4e0d\u5141\u8bb8\u5b58\u5728\u91cd\u590d\u8282\u70b9\uff0c\u5426\u5219\u5c06\u8fdd\u53cd\u5176\u5b9a\u4e49\u3002\u56e0\u6b64\uff0c\u82e5\u5f85\u63d2\u5165\u8282\u70b9\u5728\u6811\u4e2d\u5df2\u5b58\u5728\uff0c\u5219\u4e0d\u6267\u884c\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\u3002</li> <li>\u4e3a\u4e86\u5b9e\u73b0\u63d2\u5165\u8282\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u8282\u70b9 <code>pre</code> \u4fdd\u5b58\u4e0a\u4e00\u8f6e\u5faa\u73af\u7684\u8282\u70b9\u3002\u8fd9\u6837\u5728\u904d\u5386\u81f3 <code>None</code> \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u83b7\u53d6\u5230\u5176\u7236\u8282\u70b9\uff0c\u4ece\u800c\u5b8c\u6210\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def insert(self, num: int):\n    \"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n    # \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if self._root is None:\n        self._root = TreeNode(num)\n        return\n    # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    cur, pre = self._root, None\n    while cur is not None:\n        # \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if cur.val == num:\n            return\n        pre = cur\n        # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else:\n            cur = cur.left\n    # \u63d2\u5165\u8282\u70b9\n    node = TreeNode(num)\n    if pre.val &lt; num:\n        pre.right = node\n    else:\n        pre.left = node\n</code></pre> binary_search_tree.cpp<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == nullptr) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode *cur = root, *pre = nullptr;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur-&gt;val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur-&gt;left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode *node = new TreeNode(num);\n    if (pre-&gt;val &lt; num)\n        pre-&gt;right = node;\n    else\n        pre-&gt;left = node;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode node = new TreeNode(num);\n    if (pre.val &lt; num)\n        pre.right = node;\n    else\n        pre.left = node;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode? cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode node = new(num);\n    if (pre != null) {\n        if (pre.val &lt; num)\n            pre.right = node;\n        else\n            pre.left = node;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc (bst *binarySearchTree) insert(num int) {\n    cur := bst.root\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if cur == nil {\n        bst.root = NewTreeNode(num)\n        return\n    }\n    // \u5f85\u63d2\u5165\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\n    var pre *TreeNode = nil\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    for cur != nil {\n        if cur.Val == num {\n            return\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            cur = cur.Right\n        } else {\n            cur = cur.Left\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    node := NewTreeNode(num)\n    if pre.Val.(int) &lt; num {\n        pre.Right = node\n    } else {\n        pre.Left = node\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if root == nil {\n        root = TreeNode(x: num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur != nil {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if cur!.val == num {\n            return\n        }\n        pre = cur\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else {\n            cur = cur?.left\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    let node = TreeNode(x: num)\n    if pre!.val &lt; num {\n        pre?.right = node\n    } else {\n        pre?.left = node\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur = this.root,\n        pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val === num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    const node = new TreeNode(num);\n    if (pre.val &lt; num) pre.right = node;\n    else pre.left = node;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(num: number): void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val === num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    const node = new TreeNode(num);\n    if (pre!.val &lt; num) pre!.right = node;\n    else pre!.left = node;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int _num) {\n  // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n  if (_root == null) {\n    _root = TreeNode(_num);\n    return;\n  }\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n  while (cur != null) {\n    // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n    if (cur.val == _num) return;\n    pre = cur;\n    // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n    else\n      cur = cur.left;\n  }\n  // \u63d2\u5165\u8282\u70b9\n  TreeNode? node = TreeNode(_num);\n  if (pre!.val &lt; _num)\n    pre.right = node;\n  else\n    pre.left = node;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\npub fn insert(&amp;mut self, num: i32) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if self.root.is_none() {\n        self.root = Some(TreeNode::new(num));\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n            Ordering::Equal =&gt; return,\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    let pre = pre.unwrap();\n    let node = Some(TreeNode::new(num));\n    if num &gt; pre.borrow().val {\n        pre.borrow_mut().right = node;\n    } else {\n        pre.borrow_mut().left = node;\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(BinarySearchTree *bst, int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (bst-&gt;root == NULL) {\n        bst-&gt;root = newTreeNode(num);\n        return;\n    }\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != NULL) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur-&gt;val == num) {\n            return;\n        }\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            cur = cur-&gt;right;\n        } else {\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            cur = cur-&gt;left;\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode *node = newTreeNode(num);\n    if (pre-&gt;val &lt; num) {\n        pre-&gt;right = node;\n    } else {\n        pre-&gt;left = node;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfun insert(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == null) {\n        root = TreeNode(num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode? = null\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.value == num) return\n        pre = cur\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        cur = if (cur.value &lt; num) cur.right\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur.left\n    }\n    // \u63d2\u5165\u8282\u70b9\n    val node = TreeNode(num)\n    if (pre?.value!! &lt; num) pre.right = node\n    else pre.left = node\n}\n</code></pre> binary_search_tree.rb<pre><code>[class]{BinarySearchTree}-[func]{insert}\n</code></pre> binary_search_tree.zig<pre><code>// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, num: T) !void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (self.root == null) {\n        self.root = try self.mem_allocator.create(inc.TreeNode(T));\n        return;\n    }\n    var cur = self.root;\n    var pre: ?*inc.TreeNode(T) = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.?.val == num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        } else {\n            cur = cur.?.left;\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    var node = try self.mem_allocator.create(inc.TreeNode(T));\n    node.init(num);\n    if (pre.?.val &lt; num) {\n        pre.?.right = node;\n    } else {\n        pre.?.left = node;\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>\u4e0e\u67e5\u627e\u8282\u70b9\u76f8\u540c\uff0c\u63d2\u5165\u8282\u70b9\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002</p>"},{"location":"chapter_tree/binary_search_tree/#3","title":"3. \u00a0 \u5220\u9664\u8282\u70b9","text":"<p>\u5148\u5728\u4e8c\u53c9\u6811\u4e2d\u67e5\u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u518d\u5c06\u5176\u5220\u9664\u3002\u4e0e\u63d2\u5165\u8282\u70b9\u7c7b\u4f3c\uff0c\u6211\u4eec\u9700\u8981\u4fdd\u8bc1\u5728\u5220\u9664\u64cd\u4f5c\u5b8c\u6210\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u201c\u5de6\u5b50\u6811 &lt; \u6839\u8282\u70b9 &lt; \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\u4ecd\u7136\u6ee1\u8db3\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u6839\u636e\u76ee\u6807\u8282\u70b9\u7684\u5b50\u8282\u70b9\u6570\u91cf\uff0c\u5206 0\u30011 \u548c 2 \u4e09\u79cd\u60c5\u51b5\uff0c\u6267\u884c\u5bf9\u5e94\u7684\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u3002</p> <p>\u5982\u56fe 7-19 \u6240\u793a\uff0c\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \u65f6\uff0c\u8868\u793a\u8be5\u8282\u70b9\u662f\u53f6\u8282\u70b9\uff0c\u53ef\u4ee5\u76f4\u63a5\u5220\u9664\u3002</p> <p></p> <p> \u56fe 7-19 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 0 \uff09 </p> <p>\u5982\u56fe 7-20 \u6240\u793a\uff0c\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(1\\) \u65f6\uff0c\u5c06\u5f85\u5220\u9664\u8282\u70b9\u66ff\u6362\u4e3a\u5176\u5b50\u8282\u70b9\u5373\u53ef\u3002</p> <p></p> <p> \u56fe 7-20 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 1 \uff09 </p> <p>\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(2\\) \u65f6\uff0c\u6211\u4eec\u65e0\u6cd5\u76f4\u63a5\u5220\u9664\u5b83\uff0c\u800c\u9700\u8981\u4f7f\u7528\u4e00\u4e2a\u8282\u70b9\u66ff\u6362\u8be5\u8282\u70b9\u3002\u7531\u4e8e\u8981\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 \\(&lt;\\) \u6839\u8282\u70b9 \\(&lt;\\) \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u56e0\u6b64\u8fd9\u4e2a\u8282\u70b9\u53ef\u4ee5\u662f\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\u6216\u5de6\u5b50\u6811\u7684\u6700\u5927\u8282\u70b9\u3002</p> <p>\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\uff08\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\uff09\uff0c\u5219\u5220\u9664\u64cd\u4f5c\u6d41\u7a0b\u5982\u56fe 7-21 \u6240\u793a\u3002</p> <ol> <li>\u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\u5728\u201c\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u201d\u4e2d\u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\uff0c\u8bb0\u4e3a <code>tmp</code> \u3002</li> <li>\u7528 <code>tmp</code> \u7684\u503c\u8986\u76d6\u5f85\u5220\u9664\u8282\u70b9\u7684\u503c\uff0c\u5e76\u5728\u6811\u4e2d\u9012\u5f52\u5220\u9664\u8282\u70b9 <code>tmp</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-21 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 2 \uff09 </p> <p>\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u540c\u6837\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d\u67e5\u627e\u5f85\u5220\u9664\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u540e\u7ee7\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_search_tree.py<pre><code>def remove(self, num: int):\n    \"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n    # \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if self._root is None:\n        return\n    # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    cur, pre = self._root, None\n    while cur is not None:\n        # \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if cur.val == num:\n            break\n        pre = cur\n        # \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else:\n            cur = cur.left\n    # \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur is None:\n        return\n\n    # \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if cur.left is None or cur.right is None:\n        # \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        child = cur.left or cur.right\n        # \u5220\u9664\u8282\u70b9 cur\n        if cur != self._root:\n            if pre.left == cur:\n                pre.left = child\n            else:\n                pre.right = child\n        else:\n            # \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            self._root = child\n    # \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else:\n        # \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        tmp: TreeNode = cur.right\n        while tmp.left is not None:\n            tmp = tmp.left\n        # \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        self.remove(tmp.val)\n        # \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val\n</code></pre> binary_search_tree.cpp<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == nullptr)\n        return;\n    TreeNode *cur = root, *pre = nullptr;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur-&gt;left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == nullptr)\n        return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur-&gt;left == nullptr || cur-&gt;right == nullptr) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n        TreeNode *child = cur-&gt;left != nullptr ? cur-&gt;left : cur-&gt;right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre-&gt;left == cur)\n                pre-&gt;left = child;\n            else\n                pre-&gt;right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child;\n        }\n        // \u91ca\u653e\u5185\u5b58\n        delete cur;\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != nullptr) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(tmp-&gt;val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return;\n    TreeNode cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        TreeNode child = cur.left != null ? cur.left : cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(tmp.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return;\n    TreeNode? cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        TreeNode? child = cur.left ?? cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre!.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode? tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        Remove(tmp.val!.Value);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc (bst *binarySearchTree) remove(num int) {\n    cur := bst.root\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5f85\u5220\u9664\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\n    var pre *TreeNode = nil\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    for cur != nil {\n        if cur.Val == num {\n            break\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728\u53f3\u5b50\u6811\u4e2d\n            cur = cur.Right\n        } else {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728\u5de6\u5b50\u6811\u4e2d\n            cur = cur.Left\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5b50\u8282\u70b9\u6570\u4e3a 0 \u6216 1\n    if cur.Left == nil || cur.Right == nil {\n        var child *TreeNode = nil\n        // \u53d6\u51fa\u5f85\u5220\u9664\u8282\u70b9\u7684\u5b50\u8282\u70b9\n        if cur.Left != nil {\n            child = cur.Left\n        } else {\n            child = cur.Right\n        }\n        // \u5220\u9664\u8282\u70b9 cur\n        if cur != bst.root {\n            if pre.Left == cur {\n                pre.Left = child\n            } else {\n                pre.Right = child\n            }\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            bst.root = child\n        }\n        // \u5b50\u8282\u70b9\u6570\u4e3a 2\n    } else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u5f85\u5220\u9664\u8282\u70b9 cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        tmp := cur.Right\n        for tmp.Left != nil {\n            tmp = tmp.Left\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        bst.remove(tmp.Val.(int))\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.Val = tmp.Val\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc remove(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if root == nil {\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur != nil {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if cur!.val == num {\n            break\n        }\n        pre = cur\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else {\n            cur = cur?.left\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if cur?.left == nil || cur?.right == nil {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        let child = cur?.left ?? cur?.right\n        // \u5220\u9664\u8282\u70b9 cur\n        if cur !== root {\n            if pre?.left === cur {\n                pre?.left = child\n            } else {\n                pre?.right = child\n            }\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        var tmp = cur?.right\n        while tmp?.left != nil {\n            tmp = tmp?.left\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(num: tmp!.val)\n        // \u7528 tmp \u8986\u76d6 cur\n        cur?.val = tmp!.val\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (this.root === null) return;\n    let cur = this.root,\n        pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val === num) break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur === null) return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        const child = cur.left !== null ? cur.left : cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur !== this.root) {\n            if (pre.left === cur) pre.left = child;\n            else pre.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            this.root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        let tmp = cur.right;\n        while (tmp.left !== null) {\n            tmp = tmp.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        this.remove(tmp.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(num: number): void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (this.root === null) return;\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val === num) break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur === null) return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        const child: TreeNode | null =\n            cur.left !== null ? cur.left : cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur !== this.root) {\n            if (pre!.left === cur) pre!.left = child;\n            else pre!.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            this.root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        let tmp: TreeNode | null = cur.right;\n        while (tmp!.left !== null) {\n            tmp = tmp!.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        this.remove(tmp!.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp!.val;\n    }\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int _num) {\n  // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n  if (_root == null) return;\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n  while (cur != null) {\n    // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n    if (cur.val == _num) break;\n    pre = cur;\n    // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n    else\n      cur = cur.left;\n  }\n  // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n  if (cur == null) return;\n  // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n  if (cur.left == null || cur.right == null) {\n    // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n    TreeNode? child = cur.left ?? cur.right;\n    // \u5220\u9664\u8282\u70b9 cur\n    if (cur != _root) {\n      if (pre!.left == cur)\n        pre.left = child;\n      else\n        pre.right = child;\n    } else {\n      // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n      _root = child;\n    }\n  } else {\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n    TreeNode? tmp = cur.right;\n    while (tmp!.left != null) {\n      tmp = tmp.left;\n    }\n    // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n    remove(tmp.val);\n    // \u7528 tmp \u8986\u76d6 cur\n    cur.val = tmp.val;\n  }\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u5220\u9664\u8282\u70b9 */\npub fn remove(&amp;mut self, num: i32) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if self.root.is_none() {\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            Ordering::Equal =&gt; break,\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur.is_none() {\n        return;\n    }\n    let cur = cur.unwrap();\n    let (left_child, right_child) = (cur.borrow().left.clone(), cur.borrow().right.clone());\n    match (left_child.clone(), right_child.clone()) {\n        // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n        (None, None) | (Some(_), None) | (None, Some(_)) =&gt; {\n            // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n            let child = left_child.or(right_child);\n            let pre = pre.unwrap();\n            // \u5220\u9664\u8282\u70b9 cur\n            if !Rc::ptr_eq(&amp;cur, self.root.as_ref().unwrap()) {\n                let left = pre.borrow().left.clone();\n                if left.is_some() &amp;&amp; Rc::ptr_eq(&amp;left.as_ref().unwrap(), &amp;cur) {\n                    pre.borrow_mut().left = child;\n                } else {\n                    pre.borrow_mut().right = child;\n                }\n            } else {\n                // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n                self.root = child;\n            }\n        }\n        // \u5b50\u8282\u70b9\u6570\u91cf = 2\n        (Some(_), Some(_)) =&gt; {\n            // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n            let mut tmp = cur.borrow().right.clone();\n            while let Some(node) = tmp.clone() {\n                if node.borrow().left.is_some() {\n                    tmp = node.borrow().left.clone();\n                } else {\n                    break;\n                }\n            }\n            let tmpval = tmp.unwrap().borrow().val;\n            // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n            self.remove(tmpval);\n            // \u7528 tmp \u8986\u76d6 cur\n            cur.borrow_mut().val = tmpval;\n        }\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(BinarySearchTree *bst, int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (bst-&gt;root == NULL)\n        return;\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != NULL) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u53f3\u5b50\u6811\u4e2d\n            cur = cur-&gt;right;\n        } else {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u5de6\u5b50\u6811\u4e2d\n            cur = cur-&gt;left;\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == NULL)\n        return;\n    // \u5224\u65ad\u5f85\u5220\u9664\u8282\u70b9\u662f\u5426\u5b58\u5728\u5b50\u8282\u70b9\n    if (cur-&gt;left == NULL || cur-&gt;right == NULL) {\n        /* \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1 */\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n        TreeNode *child = cur-&gt;left != NULL ? cur-&gt;left : cur-&gt;right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (pre-&gt;left == cur) {\n            pre-&gt;left = child;\n        } else {\n            pre-&gt;right = child;\n        }\n        // \u91ca\u653e\u5185\u5b58\n        free(cur);\n    } else {\n        /* \u5b50\u8282\u70b9\u6570\u91cf = 2 */\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != NULL) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        removeItem(bst, tmp-&gt;val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfun remove(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null) return\n    var cur = root\n    var pre: TreeNode? = null\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.value == num) break\n        pre = cur\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        cur = if (cur.value &lt; num) cur.right\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur.left\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null) return\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        val child = if (cur.left != null) cur.left else cur.right\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre!!.left == cur) pre.left = child\n            else pre.right = child\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child\n        }\n        // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    } else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        var tmp = cur.right\n        while (tmp!!.left != null) {\n            tmp = tmp.left\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(tmp.value)\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.value = tmp.value\n    }\n}\n</code></pre> binary_search_tree.rb<pre><code>[class]{BinarySearchTree}-[func]{remove}\n</code></pre> binary_search_tree.zig<pre><code>// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, num: T) void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (self.root == null) return;\n    var cur = self.root;\n    var pre: ?*inc.TreeNode(T) = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.?.val == num) break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        } else {\n            cur = cur.?.left;\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null) return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.?.left == null or cur.?.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        var child = if (cur.?.left != null) cur.?.left else cur.?.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (pre.?.left == cur) {\n            pre.?.left = child;\n        } else {\n            pre.?.right = child;\n        }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    } else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        var tmp = cur.?.right;\n        while (tmp.?.left != null) {\n            tmp = tmp.?.left;\n        }\n        var tmp_val = tmp.?.val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        self.remove(tmp.?.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.?.val = tmp_val;\n    }\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_tree/binary_search_tree/#4","title":"4. \u00a0 \u4e2d\u5e8f\u904d\u5386\u6709\u5e8f","text":"<p>\u5982\u56fe 7-22 \u6240\u793a\uff0c\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u9075\u5faa\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u904d\u5386\u987a\u5e8f\uff0c\u800c\u4e8c\u53c9\u641c\u7d22\u6811\u6ee1\u8db3\u201c\u5de6\u5b50\u8282\u70b9 \\(&lt;\\) \u6839\u8282\u70b9 \\(&lt;\\) \u53f3\u5b50\u8282\u70b9\u201d\u7684\u5927\u5c0f\u5173\u7cfb\u3002</p> <p>\u8fd9\u610f\u5473\u7740\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u8fdb\u884c\u4e2d\u5e8f\u904d\u5386\u65f6\uff0c\u603b\u662f\u4f1a\u4f18\u5148\u904d\u5386\u4e0b\u4e00\u4e2a\u6700\u5c0f\u8282\u70b9\uff0c\u4ece\u800c\u5f97\u51fa\u4e00\u4e2a\u91cd\u8981\u6027\u8d28\uff1a\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u662f\u5347\u5e8f\u7684\u3002</p> <p>\u5229\u7528\u4e2d\u5e8f\u904d\u5386\u5347\u5e8f\u7684\u6027\u8d28\uff0c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u83b7\u53d6\u6709\u5e8f\u6570\u636e\u4ec5\u9700 \\(O(n)\\) \u65f6\u95f4\uff0c\u65e0\u987b\u8fdb\u884c\u989d\u5916\u7684\u6392\u5e8f\u64cd\u4f5c\uff0c\u975e\u5e38\u9ad8\u6548\u3002</p> <p></p> <p> \u56fe 7-22 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217 </p>"},{"location":"chapter_tree/binary_search_tree/#742","title":"7.4.2 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387","text":"<p>\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u8003\u8651\u4f7f\u7528\u6570\u7ec4\u6216\u4e8c\u53c9\u641c\u7d22\u6811\u5b58\u50a8\u3002\u89c2\u5bdf\u8868 7-2 \uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5404\u9879\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f\u5bf9\u6570\u9636\uff0c\u5177\u6709\u7a33\u5b9a\u4e14\u9ad8\u6548\u7684\u6027\u80fd\u3002\u53ea\u6709\u5728\u9ad8\u9891\u6dfb\u52a0\u3001\u4f4e\u9891\u67e5\u627e\u5220\u9664\u6570\u636e\u7684\u573a\u666f\u4e0b\uff0c\u6570\u7ec4\u6bd4\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387\u66f4\u9ad8\u3002</p> <p> \u8868 7-2 \u00a0 \u6570\u7ec4\u4e0e\u641c\u7d22\u6811\u7684\u6548\u7387\u5bf9\u6bd4 </p> \u65e0\u5e8f\u6570\u7ec4 \u4e8c\u53c9\u641c\u7d22\u6811 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(\\log n)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) <p>\u5728\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u662f\u201c\u5e73\u8861\u201d\u7684\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5728 \\(\\log n\\) \u8f6e\u5faa\u73af\u5185\u67e5\u627e\u4efb\u610f\u8282\u70b9\u3002</p> <p>\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u4e0d\u65ad\u5730\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\uff0c\u53ef\u80fd\u5bfc\u81f4\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u56fe 7-23 \u6240\u793a\u7684\u94fe\u8868\uff0c\u8fd9\u65f6\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4f1a\u9000\u5316\u4e3a \\(O(n)\\) \u3002</p> <p></p> <p> \u56fe 7-23 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316 </p>"},{"location":"chapter_tree/binary_search_tree/#743","title":"7.4.3 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u5e38\u89c1\u5e94\u7528","text":"<ul> <li>\u7528\u4f5c\u7cfb\u7edf\u4e2d\u7684\u591a\u7ea7\u7d22\u5f15\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u3002</li> <li>\u4f5c\u4e3a\u67d0\u4e9b\u641c\u7d22\u7b97\u6cd5\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u7528\u4e8e\u5b58\u50a8\u6570\u636e\u6d41\uff0c\u4ee5\u4fdd\u6301\u5176\u6709\u5e8f\u72b6\u6001\u3002</li> </ul>"},{"location":"chapter_tree/binary_tree/","title":"7.1 \u00a0 \u4e8c\u53c9\u6811","text":"<p>\u4e8c\u53c9\u6811\uff08binary tree\uff09\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4ee3\u8868\u201c\u7956\u5148\u201d\u4e0e\u201c\u540e\u4ee3\u201d\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff0c\u4f53\u73b0\u4e86\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u4e8c\u53c9\u6811\u7684\u57fa\u672c\u5355\u5143\u662f\u8282\u70b9\uff0c\u6bcf\u4e2a\u8282\u70b9\u5305\u542b\u503c\u3001\u5de6\u5b50\u8282\u70b9\u5f15\u7528\u548c\u53f3\u5b50\u8282\u70b9\u5f15\u7528\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig <pre><code>class TreeNode:\n    \"\"\"\u4e8c\u53c9\u6811\u8282\u70b9\u7c7b\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # \u8282\u70b9\u503c\n        self.left: TreeNode | None = None  # \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n        self.right: TreeNode | None = None # \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\n    int val;          // \u8282\u70b9\u503c\n    TreeNode *left;   // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n    TreeNode *right;  // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n    TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    int val;         // \u8282\u70b9\u503c\n    TreeNode left;   // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    TreeNode right;  // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n    TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode(int? x) {\n    public int? val = x;    // \u8282\u70b9\u503c\n    public TreeNode? left;  // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    public TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\n    Val   int\n    Left  *TreeNode\n    Right *TreeNode\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc NewTreeNode(v int) *TreeNode {\n    return &amp;TreeNode{\n        Left:  nil, // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n        Right: nil, // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n        Val:   v,   // \u8282\u70b9\u503c\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    var val: Int // \u8282\u70b9\u503c\n    var left: TreeNode? // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    var right: TreeNode? // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n\n    init(x: Int) {\n        val = x\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    val; // \u8282\u70b9\u503c\n    left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n    right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n    constructor(val, left, right) {\n        this.val = val === undefined ? 0 : val;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    val: number;\n    left: TreeNode | null;\n    right: TreeNode | null;\n\n    constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n        this.left = left === undefined ? null : left; // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n        this.right = right === undefined ? null : right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n  int val;         // \u8282\u70b9\u503c\n  TreeNode? left;  // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n  TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n  TreeNode(this.val, [this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\n    val: i32,                               // \u8282\u70b9\u503c\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n\nimpl TreeNode {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct TreeNode {\n    int val;                // \u8282\u70b9\u503c\n    int height;             // \u8282\u70b9\u9ad8\u5ea6\n    struct TreeNode *left;  // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n    struct TreeNode *right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n} TreeNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode(val _val: Int) {  // \u8282\u70b9\u503c\n    val left: TreeNode? = null   // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    val right: TreeNode? = null  // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n</code></pre> <pre><code>\n</code></pre> <pre><code>\n</code></pre> <p>\u6bcf\u4e2a\u8282\u70b9\u90fd\u6709\u4e24\u4e2a\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u5206\u522b\u6307\u5411\u5de6\u5b50\u8282\u70b9\uff08left-child node\uff09\u548c\u53f3\u5b50\u8282\u70b9\uff08right-child node\uff09\uff0c\u8be5\u8282\u70b9\u88ab\u79f0\u4e3a\u8fd9\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u7236\u8282\u70b9\uff08parent node\uff09\u3002\u5f53\u7ed9\u5b9a\u4e00\u4e2a\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u65f6\uff0c\u6211\u4eec\u5c06\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u79f0\u4e3a\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u6811\uff08left subtree\uff09\uff0c\u540c\u7406\u53ef\u5f97\u53f3\u5b50\u6811\uff08right subtree\uff09\u3002</p> <p>\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u9664\u53f6\u8282\u70b9\u5916\uff0c\u5176\u4ed6\u6240\u6709\u8282\u70b9\u90fd\u5305\u542b\u5b50\u8282\u70b9\u548c\u975e\u7a7a\u5b50\u6811\u3002\u5982\u56fe 7-1 \u6240\u793a\uff0c\u5982\u679c\u5c06\u201c\u8282\u70b9 2\u201d\u89c6\u4e3a\u7236\u8282\u70b9\uff0c\u5219\u5176\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\u5206\u522b\u662f\u201c\u8282\u70b9 4\u201d\u548c\u201c\u8282\u70b9 5\u201d\uff0c\u5de6\u5b50\u6811\u662f\u201c\u8282\u70b9 4 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\uff0c\u53f3\u5b50\u6811\u662f\u201c\u8282\u70b9 5 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\u3002</p> <p></p> <p> \u56fe 7-1 \u00a0 \u7236\u8282\u70b9\u3001\u5b50\u8282\u70b9\u3001\u5b50\u6811 </p>"},{"location":"chapter_tree/binary_tree/#711","title":"7.1.1 \u00a0 \u4e8c\u53c9\u6811\u5e38\u89c1\u672f\u8bed","text":"<p>\u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed\u5982\u56fe 7-2 \u6240\u793a\u3002</p> <ul> <li>\u6839\u8282\u70b9\uff08root node\uff09\uff1a\u4f4d\u4e8e\u4e8c\u53c9\u6811\u9876\u5c42\u7684\u8282\u70b9\uff0c\u6ca1\u6709\u7236\u8282\u70b9\u3002</li> <li>\u53f6\u8282\u70b9\uff08leaf node\uff09\uff1a\u6ca1\u6709\u5b50\u8282\u70b9\u7684\u8282\u70b9\uff0c\u5176\u4e24\u4e2a\u6307\u9488\u5747\u6307\u5411 <code>None</code> \u3002</li> <li>\u8fb9\uff08edge\uff09\uff1a\u8fde\u63a5\u4e24\u4e2a\u8282\u70b9\u7684\u7ebf\u6bb5\uff0c\u5373\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\u3002</li> <li>\u8282\u70b9\u6240\u5728\u7684\u5c42\uff08level\uff09\uff1a\u4ece\u9876\u81f3\u5e95\u9012\u589e\uff0c\u6839\u8282\u70b9\u6240\u5728\u5c42\u4e3a 1 \u3002</li> <li>\u8282\u70b9\u7684\u5ea6\uff08degree\uff09\uff1a\u8282\u70b9\u7684\u5b50\u8282\u70b9\u7684\u6570\u91cf\u3002\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u5ea6\u7684\u53d6\u503c\u8303\u56f4\u662f 0\u30011\u30012 \u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u9ad8\u5ea6\uff08height\uff09\uff1a\u4ece\u6839\u8282\u70b9\u5230\u6700\u8fdc\u53f6\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002</li> <li>\u8282\u70b9\u7684\u6df1\u5ea6\uff08depth\uff09\uff1a\u4ece\u6839\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002</li> <li>\u8282\u70b9\u7684\u9ad8\u5ea6\uff08height\uff09\uff1a\u4ece\u8ddd\u79bb\u8be5\u8282\u70b9\u6700\u8fdc\u7684\u53f6\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002</li> </ul> <p></p> <p> \u56fe 7-2 \u00a0 \u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed </p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u6211\u4eec\u901a\u5e38\u5c06\u201c\u9ad8\u5ea6\u201d\u548c\u201c\u6df1\u5ea6\u201d\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u201d\uff0c\u4f46\u6709\u4e9b\u9898\u76ee\u6216\u6559\u6750\u53ef\u80fd\u4f1a\u5c06\u5176\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8282\u70b9\u7684\u6570\u91cf\u201d\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u9ad8\u5ea6\u548c\u6df1\u5ea6\u90fd\u9700\u8981\u52a0 1 \u3002</p>"},{"location":"chapter_tree/binary_tree/#712","title":"7.1.2 \u00a0 \u4e8c\u53c9\u6811\u57fa\u672c\u64cd\u4f5c","text":""},{"location":"chapter_tree/binary_tree/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u4e8c\u53c9\u6811","text":"<p>\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u9996\u5148\u521d\u59cb\u5316\u8282\u70b9\uff0c\u7136\u540e\u6784\u5efa\u5f15\u7528\uff08\u6307\u9488\uff09\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree.py<pre><code># \u521d\u59cb\u5316\u4e8c\u53c9\u6811\n# \u521d\u59cb\u5316\u8282\u70b9\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.cpp<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.java<pre><code>// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.cs<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new(1);\nTreeNode n2 = new(2);\nTreeNode n3 = new(3);\nTreeNode n4 = new(4);\nTreeNode n5 = new(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.go<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n</code></pre> binary_tree.swift<pre><code>// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.js<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.ts<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.dart<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.rs<pre><code>// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = TreeNode::new(1);\nlet n2 = TreeNode::new(2);\nlet n3 = TreeNode::new(3);\nlet n4 = TreeNode::new(4);\nlet n5 = TreeNode::new(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.borrow_mut().left = Some(n2.clone());\nn1.borrow_mut().right = Some(n3);\nn2.borrow_mut().left = Some(n4);\nn2.borrow_mut().right = Some(n5);\n</code></pre> binary_tree.c<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.kt<pre><code>// \u521d\u59cb\u5316\u8282\u70b9\nval n1 = TreeNode(1)\nval n2 = TreeNode(2)\nval n3 = TreeNode(3)\nval n4 = TreeNode(4)\nval n5 = TreeNode(5)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.rb<pre><code>\n</code></pre> binary_tree.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_tree/binary_tree/#2","title":"2. \u00a0 \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9","text":"<p>\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u53ef\u4ee5\u901a\u8fc7\u4fee\u6539\u6307\u9488\u6765\u5b9e\u73b0\u3002\u56fe 7-3 \u7ed9\u51fa\u4e86\u4e00\u4e2a\u793a\u4f8b\u3002</p> <p></p> <p> \u56fe 7-3 \u00a0 \u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree.py<pre><code># \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\np = TreeNode(0)\n# \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = p\np.left = n2\n# \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n</code></pre> binary_tree.cpp<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode* P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1-&gt;left = P;\nP-&gt;left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1-&gt;left = n2;\n</code></pre> binary_tree.java<pre><code>TreeNode P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.cs<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.go<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.Left = n2\n</code></pre> binary_tree.swift<pre><code>let P = TreeNode(x: 0)\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P\nP.left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n</code></pre> binary_tree.js<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nlet P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.ts<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nconst P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.dart<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.rs<pre><code>let p = TreeNode::new(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.borrow_mut().left = Some(p.clone());\np.borrow_mut().left = Some(n2.clone());\n// \u5220\u9664\u8282\u70b9 p\nn1.borrow_mut().left = Some(n2);\n</code></pre> binary_tree.c<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode *P = newTreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1-&gt;left = P;\nP-&gt;left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1-&gt;left = n2;\n</code></pre> binary_tree.kt<pre><code>val P = TreeNode(0)\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P\nP.left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n</code></pre> binary_tree.rb<pre><code>\n</code></pre> binary_tree.zig<pre><code>\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>Note</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u63d2\u5165\u8282\u70b9\u53ef\u80fd\u4f1a\u6539\u53d8\u4e8c\u53c9\u6811\u7684\u539f\u6709\u903b\u8f91\u7ed3\u6784\uff0c\u800c\u5220\u9664\u8282\u70b9\u901a\u5e38\u610f\u5473\u7740\u5220\u9664\u8be5\u8282\u70b9\u53ca\u5176\u6240\u6709\u5b50\u6811\u3002\u56e0\u6b64\uff0c\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u901a\u5e38\u662f\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\u7684\uff0c\u4ee5\u5b9e\u73b0\u6709\u5b9e\u9645\u610f\u4e49\u7684\u64cd\u4f5c\u3002</p>"},{"location":"chapter_tree/binary_tree/#713","title":"7.1.3 \u00a0 \u5e38\u89c1\u4e8c\u53c9\u6811\u7c7b\u578b","text":""},{"location":"chapter_tree/binary_tree/#1_1","title":"1. \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-4 \u6240\u793a\uff0c\u5b8c\u7f8e\u4e8c\u53c9\u6811\uff08perfect binary tree\uff09\u6240\u6709\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u5b8c\u5168\u586b\u6ee1\u3002\u5728\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\uff0c\u53f6\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u7684\u5ea6\u90fd\u4e3a \\(2\\) \uff1b\u82e5\u6811\u7684\u9ad8\u5ea6\u4e3a \\(h\\) \uff0c\u5219\u8282\u70b9\u603b\u6570\u4e3a \\(2^{h+1} - 1\\) \uff0c\u5448\u73b0\u6807\u51c6\u7684\u6307\u6570\u7ea7\u5173\u7cfb\uff0c\u53cd\u6620\u4e86\u81ea\u7136\u754c\u4e2d\u5e38\u89c1\u7684\u7ec6\u80de\u5206\u88c2\u73b0\u8c61\u3002</p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u5728\u4e2d\u6587\u793e\u533a\u4e2d\uff0c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u5e38\u88ab\u79f0\u4e3a\u6ee1\u4e8c\u53c9\u6811\u3002</p> <p></p> <p> \u56fe 7-4 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#2_1","title":"2. \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-5 \u6240\u793a\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\uff08complete binary tree\uff09\u53ea\u6709\u6700\u5e95\u5c42\u7684\u8282\u70b9\u672a\u88ab\u586b\u6ee1\uff0c\u4e14\u6700\u5e95\u5c42\u8282\u70b9\u5c3d\u91cf\u9760\u5de6\u586b\u5145\u3002</p> <p></p> <p> \u56fe 7-5 \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#3","title":"3. \u00a0 \u5b8c\u6ee1\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-6 \u6240\u793a\uff0c\u5b8c\u6ee1\u4e8c\u53c9\u6811\uff08full binary tree\uff09\u9664\u4e86\u53f6\u8282\u70b9\u4e4b\u5916\uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u90fd\u6709\u4e24\u4e2a\u5b50\u8282\u70b9\u3002</p> <p></p> <p> \u56fe 7-6 \u00a0 \u5b8c\u6ee1\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#4","title":"4. \u00a0 \u5e73\u8861\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-7 \u6240\u793a\uff0c\u5e73\u8861\u4e8c\u53c9\u6811\uff08balanced binary tree\uff09\u4e2d\u4efb\u610f\u8282\u70b9\u7684\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\u4e4b\u5dee\u7684\u7edd\u5bf9\u503c\u4e0d\u8d85\u8fc7 1 \u3002</p> <p></p> <p> \u56fe 7-7 \u00a0 \u5e73\u8861\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#714","title":"7.1.4 \u00a0 \u4e8c\u53c9\u6811\u7684\u9000\u5316","text":"<p>\u56fe 7-8 \u5c55\u793a\u4e86\u4e8c\u53c9\u6811\u7684\u7406\u60f3\u7ed3\u6784\u4e0e\u9000\u5316\u7ed3\u6784\u3002\u5f53\u4e8c\u53c9\u6811\u7684\u6bcf\u5c42\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u65f6\uff0c\u8fbe\u5230\u201c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u201d\uff1b\u800c\u5f53\u6240\u6709\u8282\u70b9\u90fd\u504f\u5411\u4e00\u4fa7\u65f6\uff0c\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u201c\u94fe\u8868\u201d\u3002</p> <ul> <li>\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u7406\u60f3\u60c5\u51b5\uff0c\u53ef\u4ee5\u5145\u5206\u53d1\u6325\u4e8c\u53c9\u6811\u201c\u5206\u6cbb\u201d\u7684\u4f18\u52bf\u3002</li> <li>\u94fe\u8868\u5219\u662f\u53e6\u4e00\u4e2a\u6781\u7aef\uff0c\u5404\u9879\u64cd\u4f5c\u90fd\u53d8\u4e3a\u7ebf\u6027\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002</li> </ul> <p></p> <p> \u56fe 7-8 \u00a0 \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u7ed3\u6784\u4e0e\u6700\u5dee\u7ed3\u6784 </p> <p>\u5982\u8868 7-1 \u6240\u793a\uff0c\u5728\u6700\u4f73\u7ed3\u6784\u548c\u6700\u5dee\u7ed3\u6784\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u53f6\u8282\u70b9\u6570\u91cf\u3001\u8282\u70b9\u603b\u6570\u3001\u9ad8\u5ea6\u7b49\u8fbe\u5230\u6781\u5927\u503c\u6216\u6781\u5c0f\u503c\u3002</p> <p> \u8868 7-1 \u00a0 \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u7ed3\u6784\u4e0e\u6700\u5dee\u7ed3\u6784 </p> \u5b8c\u7f8e\u4e8c\u53c9\u6811 \u94fe\u8868 \u7b2c \\(i\\) \u5c42\u7684\u8282\u70b9\u6570\u91cf \\(2^{i-1}\\) \\(1\\) \u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u6811\u7684\u53f6\u8282\u70b9\u6570\u91cf \\(2^h\\) \\(1\\) \u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u6811\u7684\u8282\u70b9\u603b\u6570 \\(2^{h+1} - 1\\) \\(h + 1\\) \u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \u7684\u6811\u7684\u9ad8\u5ea6 \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)"},{"location":"chapter_tree/binary_tree_traversal/","title":"7.2 \u00a0 \u4e8c\u53c9\u6811\u904d\u5386","text":"<p>\u4ece\u7269\u7406\u7ed3\u6784\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u6811\u662f\u4e00\u79cd\u57fa\u4e8e\u94fe\u8868\u7684\u6570\u636e\u7ed3\u6784\uff0c\u56e0\u6b64\u5176\u904d\u5386\u65b9\u5f0f\u662f\u901a\u8fc7\u6307\u9488\u9010\u4e2a\u8bbf\u95ee\u8282\u70b9\u3002\u7136\u800c\uff0c\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u8fd9\u4f7f\u5f97\u904d\u5386\u6811\u6bd4\u904d\u5386\u94fe\u8868\u66f4\u52a0\u590d\u6742\uff0c\u9700\u8981\u501f\u52a9\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u3002</p> <p>\u4e8c\u53c9\u6811\u5e38\u89c1\u7684\u904d\u5386\u65b9\u5f0f\u5305\u62ec\u5c42\u5e8f\u904d\u5386\u3001\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u7b49\u3002</p>"},{"location":"chapter_tree/binary_tree_traversal/#721","title":"7.2.1 \u00a0 \u5c42\u5e8f\u904d\u5386","text":"<p>\u5982\u56fe 7-9 \u6240\u793a\uff0c\u5c42\u5e8f\u904d\u5386\uff08level-order traversal\uff09\u4ece\u9876\u90e8\u5230\u5e95\u90e8\u9010\u5c42\u904d\u5386\u4e8c\u53c9\u6811\uff0c\u5e76\u5728\u6bcf\u4e00\u5c42\u6309\u7167\u4ece\u5de6\u5230\u53f3\u7684\u987a\u5e8f\u8bbf\u95ee\u8282\u70b9\u3002</p> <p>\u5c42\u5e8f\u904d\u5386\u672c\u8d28\u4e0a\u5c5e\u4e8e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\uff08breadth-first traversal\uff09\uff0c\u4e5f\u79f0\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\uff08breadth-first search, BFS\uff09\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u6269\u5c55\u201d\u7684\u9010\u5c42\u904d\u5386\u65b9\u5f0f\u3002</p> <p></p> <p> \u56fe 7-9 \u00a0 \u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386 </p>"},{"location":"chapter_tree/binary_tree_traversal/#1","title":"1. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u901a\u5e38\u501f\u52a9\u201c\u961f\u5217\u201d\u6765\u5b9e\u73b0\u3002\u961f\u5217\u9075\u5faa\u201c\u5148\u8fdb\u5148\u51fa\u201d\u7684\u89c4\u5219\uff0c\u800c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u5219\u9075\u5faa\u201c\u9010\u5c42\u63a8\u8fdb\u201d\u7684\u89c4\u5219\uff0c\u4e24\u8005\u80cc\u540e\u7684\u601d\u60f3\u662f\u4e00\u81f4\u7684\u3002\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree_bfs.py<pre><code>def level_order(root: TreeNode | None) -&gt; list[int]:\n    \"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n    # \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    queue: deque[TreeNode] = deque()\n    queue.append(root)\n    # \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    res = []\n    while queue:\n        node: TreeNode = queue.popleft()  # \u961f\u5217\u51fa\u961f\n        res.append(node.val)  # \u4fdd\u5b58\u8282\u70b9\u503c\n        if node.left is not None:\n            queue.append(node.left)  # \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if node.right is not None:\n            queue.append(node.right)  # \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    return res\n</code></pre> binary_tree_bfs.cpp<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nvector&lt;int&gt; levelOrder(TreeNode *root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    queue&lt;TreeNode *&gt; queue;\n    queue.push(root);\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    vector&lt;int&gt; vec;\n    while (!queue.empty()) {\n        TreeNode *node = queue.front();\n        queue.pop();              // \u961f\u5217\u51fa\u961f\n        vec.push_back(node-&gt;val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node-&gt;left != nullptr)\n            queue.push(node-&gt;left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node-&gt;right != nullptr)\n            queue.push(node-&gt;right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return vec;\n}\n</code></pre> binary_tree_bfs.java<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;Integer&gt; levelOrder(TreeNode root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    Queue&lt;TreeNode&gt; queue = new LinkedList&lt;&gt;();\n    queue.add(root);\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    List&lt;Integer&gt; list = new ArrayList&lt;&gt;();\n    while (!queue.isEmpty()) {\n        TreeNode node = queue.poll(); // \u961f\u5217\u51fa\u961f\n        list.add(node.val);           // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null)\n            queue.offer(node.left);   // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node.right != null)\n            queue.offer(node.right);  // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.cs<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;int&gt; LevelOrder(TreeNode root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    Queue&lt;TreeNode&gt; queue = new();\n    queue.Enqueue(root);\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    List&lt;int&gt; list = [];\n    while (queue.Count != 0) {\n        TreeNode node = queue.Dequeue(); // \u961f\u5217\u51fa\u961f\n        list.Add(node.val!.Value);       // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null)\n            queue.Enqueue(node.left);    // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node.right != null)\n            queue.Enqueue(node.right);   // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.go<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root *TreeNode) []any {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    queue := list.New()\n    queue.PushBack(root)\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5207\u7247\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    nums := make([]any, 0)\n    for queue.Len() &gt; 0 {\n        // \u961f\u5217\u51fa\u961f\n        node := queue.Remove(queue.Front()).(*TreeNode)\n        // \u4fdd\u5b58\u8282\u70b9\u503c\n        nums = append(nums, node.Val)\n        if node.Left != nil {\n            // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n            queue.PushBack(node.Left)\n        }\n        if node.Right != nil {\n            // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n            queue.PushBack(node.Right)\n        }\n    }\n    return nums\n}\n</code></pre> binary_tree_bfs.swift<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root: TreeNode) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    var queue: [TreeNode] = [root]\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    var list: [Int] = []\n    while !queue.isEmpty {\n        let node = queue.removeFirst() // \u961f\u5217\u51fa\u961f\n        list.append(node.val) // \u4fdd\u5b58\u8282\u70b9\u503c\n        if let left = node.left {\n            queue.append(left) // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if let right = node.right {\n            queue.append(right) // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        }\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.js<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    const queue = [root];\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    const list = [];\n    while (queue.length) {\n        let node = queue.shift(); // \u961f\u5217\u51fa\u961f\n        list.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left) queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node.right) queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.ts<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root: TreeNode | null): number[] {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    const queue = [root];\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    const list: number[] = [];\n    while (queue.length) {\n        let node = queue.shift() as TreeNode; // \u961f\u5217\u51fa\u961f\n        list.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left) {\n            queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if (node.right) {\n            queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        }\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.dart<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;int&gt; levelOrder(TreeNode? root) {\n  // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n  Queue&lt;TreeNode?&gt; queue = Queue();\n  queue.add(root);\n  // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n  List&lt;int&gt; res = [];\n  while (queue.isNotEmpty) {\n    TreeNode? node = queue.removeFirst(); // \u961f\u5217\u51fa\u961f\n    res.add(node!.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n    if (node.left != null) queue.add(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n    if (node.right != null) queue.add(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n  }\n  return res;\n}\n</code></pre> binary_tree_bfs.rs<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfn level_order(root: &amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    let mut que = VecDeque::new();\n    que.push_back(Rc::clone(&amp;root));\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    let mut vec = Vec::new();\n\n    while let Some(node) = que.pop_front() {\n        // \u961f\u5217\u51fa\u961f\n        vec.push(node.borrow().val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if let Some(left) = node.borrow().left.as_ref() {\n            que.push_back(Rc::clone(left)); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if let Some(right) = node.borrow().right.as_ref() {\n            que.push_back(Rc::clone(right)); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        };\n    }\n    vec\n}\n</code></pre> binary_tree_bfs.c<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(TreeNode *root, int *size) {\n    /* \u8f85\u52a9\u961f\u5217 */\n    int front, rear;\n    int index, *arr;\n    TreeNode *node;\n    TreeNode **queue;\n\n    /* \u8f85\u52a9\u961f\u5217 */\n    queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);\n    // \u961f\u5217\u6307\u9488\n    front = 0, rear = 0;\n    // \u52a0\u5165\u6839\u8282\u70b9\n    queue[rear++] = root;\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    /* \u8f85\u52a9\u6570\u7ec4 */\n    arr = (int *)malloc(sizeof(int) * MAX_SIZE);\n    // \u6570\u7ec4\u6307\u9488\n    index = 0;\n    while (front &lt; rear) {\n        // \u961f\u5217\u51fa\u961f\n        node = queue[front++];\n        // \u4fdd\u5b58\u8282\u70b9\u503c\n        arr[index++] = node-&gt;val;\n        if (node-&gt;left != NULL) {\n            // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n            queue[rear++] = node-&gt;left;\n        }\n        if (node-&gt;right != NULL) {\n            // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n            queue[rear++] = node-&gt;right;\n        }\n    }\n    // \u66f4\u65b0\u6570\u7ec4\u957f\u5ea6\u7684\u503c\n    *size = index;\n    arr = realloc(arr, sizeof(int) * (*size));\n\n    // \u91ca\u653e\u8f85\u52a9\u6570\u7ec4\u7a7a\u95f4\n    free(queue);\n    return arr;\n}\n</code></pre> binary_tree_bfs.kt<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfun levelOrder(root: TreeNode?): MutableList&lt;Int&gt; {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    val queue = LinkedList&lt;TreeNode?&gt;()\n    queue.add(root)\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    val list = ArrayList&lt;Int&gt;()\n    while (!queue.isEmpty()) {\n        val node = queue.poll() // \u961f\u5217\u51fa\u961f\n        list.add(node?.value!!) // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null) queue.offer(node.left) // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n\n        if (node.right != null) queue.offer(node.right) // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.rb<pre><code>[class]{}-[func]{level_order}\n</code></pre> binary_tree_bfs.zig<pre><code>// \u5c42\u5e8f\u904d\u5386\nfn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    const L = std.TailQueue(*inc.TreeNode(T));\n    var queue = L{};\n    var root_node = try mem_allocator.create(L.Node);\n    root_node.data = root;\n    queue.append(root_node); \n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    var list = std.ArrayList(T).init(std.heap.page_allocator);\n    while (queue.len &gt; 0) {\n        var queue_node = queue.popFirst().?;    // \u961f\u5217\u51fa\u961f\n        var node = queue_node.data;\n        try list.append(node.val);              // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null) {\n            var tmp_node = try mem_allocator.create(L.Node);\n            tmp_node.data = node.left.?;\n            queue.append(tmp_node);             // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if (node.right != null) {\n            var tmp_node = try mem_allocator.create(L.Node);\n            tmp_node.data = node.right.?;\n            queue.append(tmp_node);             // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        }        \n    }\n    return list;\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p>"},{"location":"chapter_tree/binary_tree_traversal/#2","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1a\u6240\u6709\u8282\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d \\(n\\) \u4e3a\u8282\u70b9\u6570\u91cf\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u6ee1\u4e8c\u53c9\u6811\u65f6\uff0c\u904d\u5386\u5230\u6700\u5e95\u5c42\u4e4b\u524d\uff0c\u961f\u5217\u4e2d\u6700\u591a\u540c\u65f6\u5b58\u5728 \\((n + 1) / 2\\) \u4e2a\u8282\u70b9\uff0c\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_tree/binary_tree_traversal/#722","title":"7.2.2 \u00a0 \u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386","text":"<p>\u76f8\u5e94\u5730\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u904d\u5386\uff08depth-first traversal\uff09\uff0c\u4e5f\u79f0\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff08depth-first search, DFS\uff09\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u5148\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u6eaf\u7ee7\u7eed\u201d\u7684\u904d\u5386\u65b9\u5f0f\u3002</p> <p>\u56fe 7-10 \u5c55\u793a\u4e86\u5bf9\u4e8c\u53c9\u6811\u8fdb\u884c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5de5\u4f5c\u539f\u7406\u3002\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5c31\u50cf\u662f\u7ed5\u7740\u6574\u68f5\u4e8c\u53c9\u6811\u7684\u5916\u56f4\u201c\u8d70\u201d\u4e00\u5708\uff0c\u5728\u6bcf\u4e2a\u8282\u70b9\u90fd\u4f1a\u9047\u5230\u4e09\u4e2a\u4f4d\u7f6e\uff0c\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u3002</p> <p></p> <p> \u56fe 7-10 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386 </p>"},{"location":"chapter_tree/binary_tree_traversal/#1_1","title":"1. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRubyZig binary_tree_dfs.py<pre><code>def pre_order(root: TreeNode | None):\n    \"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\n    if root is None:\n        return\n    # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    res.append(root.val)\n    pre_order(root=root.left)\n    pre_order(root=root.right)\n\ndef in_order(root: TreeNode | None):\n    \"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\n    if root is None:\n        return\n    # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    in_order(root=root.left)\n    res.append(root.val)\n    in_order(root=root.right)\n\ndef post_order(root: TreeNode | None):\n    \"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\n    if root is None:\n        return\n    # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    post_order(root=root.left)\n    post_order(root=root.right)\n    res.append(root.val)\n</code></pre> binary_tree_dfs.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    vec.push_back(root-&gt;val);\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root-&gt;left);\n    vec.push_back(root-&gt;val);\n    inOrder(root-&gt;right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root-&gt;left);\n    postOrder(root-&gt;right);\n    vec.push_back(root-&gt;val);\n}\n</code></pre> binary_tree_dfs.java<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.add(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left);\n    list.add(root.val);\n    inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left);\n    postOrder(root.right);\n    list.add(root.val);\n}\n</code></pre> binary_tree_dfs.cs<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.Add(root.val!.Value);\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid InOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    InOrder(root.left);\n    list.Add(root.val!.Value);\n    InOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid PostOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    PostOrder(root.left);\n    PostOrder(root.right);\n    list.Add(root.val!.Value);\n}\n</code></pre> binary_tree_dfs.go<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    nums = append(nums, node.Val)\n    preOrder(node.Left)\n    preOrder(node.Right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(node.Left)\n    nums = append(nums, node.Val)\n    inOrder(node.Right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(node.Left)\n    postOrder(node.Right)\n    nums = append(nums, node.Val)\n}\n</code></pre> binary_tree_dfs.swift<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.append(root.val)\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root: root.left)\n    list.append(root.val)\n    inOrder(root: root.right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root: root.left)\n    postOrder(root: root.right)\n    list.append(root.val)\n}\n</code></pre> binary_tree_dfs.js<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root) {\n    if (root === null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root) {\n    if (root === null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root) {\n    if (root === null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.ts<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.dart<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n  list.add(node.val);\n  preOrder(node.left);\n  preOrder(node.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n  inOrder(node.left);\n  list.add(node.val);\n  inOrder(node.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n  postOrder(node.left);\n  postOrder(node.right);\n  list.add(node.val);\n}\n</code></pre> binary_tree_dfs.rs<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfn pre_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    if let Some(node) = root {\n        // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n        result.push(node.borrow().val);\n        result.append(&amp;mut pre_order(node.borrow().left.as_ref()));\n        result.append(&amp;mut pre_order(node.borrow().right.as_ref()));\n    }\n    result\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfn in_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    if let Some(node) = root {\n        // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n        result.append(&amp;mut in_order(node.borrow().left.as_ref()));\n        result.push(node.borrow().val);\n        result.append(&amp;mut in_order(node.borrow().right.as_ref()));\n    }\n    result\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfn post_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    if let Some(node) = root {\n        // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n        result.append(&amp;mut post_order(node.borrow().left.as_ref()));\n        result.append(&amp;mut post_order(node.borrow().right.as_ref()));\n        result.push(node.borrow().val);\n    }\n    result\n}\n</code></pre> binary_tree_dfs.c<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    arr[(*size)++] = root-&gt;val;\n    preOrder(root-&gt;left, size);\n    preOrder(root-&gt;right, size);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root-&gt;left, size);\n    arr[(*size)++] = root-&gt;val;\n    inOrder(root-&gt;right, size);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root-&gt;left, size);\n    postOrder(root-&gt;right, size);\n    arr[(*size)++] = root-&gt;val;\n}\n</code></pre> binary_tree_dfs.kt<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.add(root.value)\n    preOrder(root.left)\n    preOrder(root.right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfun inOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left)\n    list.add(root.value)\n    inOrder(root.right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfun postOrder(root: TreeNode?) {\n    if (root == null) return\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left)\n    postOrder(root.right)\n    list.add(root.value)\n}\n</code></pre> binary_tree_dfs.rb<pre><code>[class]{}-[func]{pre_order}\n\n[class]{}-[func]{in_order}\n\n[class]{}-[func]{post_order}\n</code></pre> binary_tree_dfs.zig<pre><code>// \u524d\u5e8f\u904d\u5386\nfn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    try list.append(root.?.val);\n    try preOrder(T, root.?.left);\n    try preOrder(T, root.?.right);\n}\n\n// \u4e2d\u5e8f\u904d\u5386\nfn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    try inOrder(T, root.?.left);\n    try list.append(root.?.val);\n    try inOrder(T, root.?.right);\n}\n\n// \u540e\u5e8f\u904d\u5386\nfn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    try postOrder(T, root.?.left);\n    try postOrder(T, root.?.right);\n    try list.append(root.?.val);\n}\n</code></pre> \u53ef\u89c6\u5316\u8fd0\u884c <p> \u5168\u5c4f\u89c2\u770b &gt;</p> <p>Tip</p> <p>\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u4e5f\u53ef\u4ee5\u57fa\u4e8e\u8fed\u4ee3\u5b9e\u73b0\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u7814\u7a76\u3002</p> <p>\u56fe 7-11 \u5c55\u793a\u4e86\u524d\u5e8f\u904d\u5386\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b\uff0c\u5176\u53ef\u5206\u4e3a\u201c\u9012\u201d\u548c\u201c\u5f52\u201d\u4e24\u4e2a\u9006\u5411\u7684\u90e8\u5206\u3002</p> <ol> <li>\u201c\u9012\u201d\u8868\u793a\u5f00\u542f\u65b0\u65b9\u6cd5\uff0c\u7a0b\u5e8f\u5728\u6b64\u8fc7\u7a0b\u4e2d\u8bbf\u95ee\u4e0b\u4e00\u4e2a\u8282\u70b9\u3002</li> <li>\u201c\u5f52\u201d\u8868\u793a\u51fd\u6570\u8fd4\u56de\uff0c\u4ee3\u8868\u5f53\u524d\u8282\u70b9\u5df2\u7ecf\u8bbf\u95ee\u5b8c\u6bd5\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-11 \u00a0 \u524d\u5e8f\u904d\u5386\u7684\u9012\u5f52\u8fc7\u7a0b </p>"},{"location":"chapter_tree/binary_tree_traversal/#2_1","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1a\u6240\u6709\u8282\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u9012\u5f52\u6df1\u5ea6\u8fbe\u5230 \\(n\\) \uff0c\u7cfb\u7edf\u5360\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_tree/summary/","title":"7.6 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_tree/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u4e8c\u53c9\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4f53\u73b0\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u6bcf\u4e2a\u4e8c\u53c9\u6811\u8282\u70b9\u5305\u542b\u4e00\u4e2a\u503c\u4ee5\u53ca\u4e24\u4e2a\u6307\u9488\uff0c\u5206\u522b\u6307\u5411\u5176\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\u3002</li> <li>\u5bf9\u4e8e\u4e8c\u53c9\u6811\u4e2d\u7684\u67d0\u4e2a\u8282\u70b9\uff0c\u5176\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u53ca\u5176\u4ee5\u4e0b\u5f62\u6210\u7684\u6811\u88ab\u79f0\u4e3a\u8be5\u8282\u70b9\u7684\u5de6\uff08\u53f3\uff09\u5b50\u6811\u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u76f8\u5173\u672f\u8bed\u5305\u62ec\u6839\u8282\u70b9\u3001\u53f6\u8282\u70b9\u3001\u5c42\u3001\u5ea6\u3001\u8fb9\u3001\u9ad8\u5ea6\u548c\u6df1\u5ea6\u7b49\u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u521d\u59cb\u5316\u3001\u8282\u70b9\u63d2\u5165\u548c\u8282\u70b9\u5220\u9664\u64cd\u4f5c\u4e0e\u94fe\u8868\u64cd\u4f5c\u65b9\u6cd5\u7c7b\u4f3c\u3002</li> <li>\u5e38\u89c1\u7684\u4e8c\u53c9\u6811\u7c7b\u578b\u6709\u5b8c\u7f8e\u4e8c\u53c9\u6811\u3001\u5b8c\u5168\u4e8c\u53c9\u6811\u3001\u5b8c\u6ee1\u4e8c\u53c9\u6811\u548c\u5e73\u8861\u4e8c\u53c9\u6811\u3002\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u6700\u7406\u60f3\u7684\u72b6\u6001\uff0c\u800c\u94fe\u8868\u662f\u9000\u5316\u540e\u7684\u6700\u5dee\u72b6\u6001\u3002</li> <li>\u4e8c\u53c9\u6811\u53ef\u4ee5\u7528\u6570\u7ec4\u8868\u793a\uff0c\u65b9\u6cd5\u662f\u5c06\u8282\u70b9\u503c\u548c\u7a7a\u4f4d\u6309\u5c42\u5e8f\u904d\u5386\u987a\u5e8f\u6392\u5217\uff0c\u5e76\u6839\u636e\u7236\u8282\u70b9\u4e0e\u5b50\u8282\u70b9\u4e4b\u95f4\u7684\u7d22\u5f15\u6620\u5c04\u5173\u7cfb\u6765\u5b9e\u73b0\u6307\u9488\u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386\u662f\u4e00\u79cd\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u65b9\u6cd5\uff0c\u5b83\u4f53\u73b0\u4e86\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u6269\u5c55\u201d\u7684\u9010\u5c42\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u901a\u8fc7\u961f\u5217\u6765\u5b9e\u73b0\u3002</li> <li>\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\u7686\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u5b83\u4eec\u4f53\u73b0\u4e86\u201c\u5148\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u6eaf\u7ee7\u7eed\u201d\u7684\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u6765\u5b9e\u73b0\u3002</li> <li>\u4e8c\u53c9\u641c\u7d22\u6811\u662f\u4e00\u79cd\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u627e\u6570\u636e\u7ed3\u6784\uff0c\u5176\u67e5\u627e\u3001\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \u3002\u5f53\u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u5404\u9879\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u52a3\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>AVL \u6811\uff0c\u4e5f\u79f0\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5b83\u901a\u8fc7\u65cb\u8f6c\u64cd\u4f5c\u786e\u4fdd\u5728\u4e0d\u65ad\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\u540e\u6811\u4ecd\u7136\u4fdd\u6301\u5e73\u8861\u3002</li> <li>AVL \u6811\u7684\u65cb\u8f6c\u64cd\u4f5c\u5305\u62ec\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u518d\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u518d\u53f3\u65cb\u3002\u5728\u63d2\u5165\u6216\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4f1a\u4ece\u5e95\u5411\u9876\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002</li> </ul>"},{"location":"chapter_tree/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>Q\uff1a\u5bf9\u4e8e\u53ea\u6709\u4e00\u4e2a\u8282\u70b9\u7684\u4e8c\u53c9\u6811\uff0c\u6811\u7684\u9ad8\u5ea6\u548c\u6839\u8282\u70b9\u7684\u6df1\u5ea6\u90fd\u662f \\(0\\) \u5417\uff1f</p> <p>\u662f\u7684\uff0c\u56e0\u4e3a\u9ad8\u5ea6\u548c\u6df1\u5ea6\u901a\u5e38\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u201d\u3002</p> <p>Q\uff1a\u4e8c\u53c9\u6811\u4e2d\u7684\u63d2\u5165\u4e0e\u5220\u9664\u4e00\u822c\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\uff0c\u8fd9\u91cc\u7684\u201c\u4e00\u5957\u64cd\u4f5c\u201d\u6307\u4ec0\u4e48\u5462\uff1f\u53ef\u4ee5\u7406\u89e3\u4e3a\u8d44\u6e90\u7684\u5b50\u8282\u70b9\u7684\u8d44\u6e90\u91ca\u653e\u5417\uff1f</p> <p>\u62ff\u4e8c\u53c9\u641c\u7d22\u6811\u6765\u4e3e\u4f8b\uff0c\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u8981\u5206\u4e09\u79cd\u60c5\u51b5\u5904\u7406\uff0c\u5176\u4e2d\u6bcf\u79cd\u60c5\u51b5\u90fd\u9700\u8981\u8fdb\u884c\u591a\u4e2a\u6b65\u9aa4\u7684\u8282\u70b9\u64cd\u4f5c\u3002</p> <p>Q\uff1a\u4e3a\u4ec0\u4e48 DFS \u904d\u5386\u4e8c\u53c9\u6811\u6709\u524d\u3001\u4e2d\u3001\u540e\u4e09\u79cd\u987a\u5e8f\uff0c\u5206\u522b\u6709\u4ec0\u4e48\u7528\u5462\uff1f</p> <p>\u4e0e\u987a\u5e8f\u548c\u9006\u5e8f\u904d\u5386\u6570\u7ec4\u7c7b\u4f3c\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\u662f\u4e09\u79cd\u4e8c\u53c9\u6811\u904d\u5386\u65b9\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u5b83\u4eec\u5f97\u5230\u4e00\u4e2a\u7279\u5b9a\u987a\u5e8f\u7684\u904d\u5386\u7ed3\u679c\u3002\u4f8b\u5982\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\uff0c\u7531\u4e8e\u8282\u70b9\u5927\u5c0f\u6ee1\u8db3 <code>\u5de6\u5b50\u8282\u70b9\u503c &lt; \u6839\u8282\u70b9\u503c &lt; \u53f3\u5b50\u8282\u70b9\u503c</code> \uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u8981\u6309\u7167\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u4f18\u5148\u7ea7\u904d\u5386\u6811\uff0c\u5c31\u53ef\u4ee5\u83b7\u5f97\u6709\u5e8f\u7684\u8282\u70b9\u5e8f\u5217\u3002</p> <p>Q\uff1a\u53f3\u65cb\u64cd\u4f5c\u662f\u5904\u7406\u5931\u8861\u8282\u70b9 <code>node</code>\u3001<code>child</code>\u3001<code>grand_child</code> \u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u90a3 <code>node</code> \u7684\u7236\u8282\u70b9\u548c <code>node</code> \u539f\u6765\u7684\u8fde\u63a5\u4e0d\u9700\u8981\u7ef4\u62a4\u5417\uff1f\u53f3\u65cb\u64cd\u4f5c\u540e\u5c82\u4e0d\u662f\u65ad\u6389\u4e86\uff1f</p> <p>\u6211\u4eec\u9700\u8981\u4ece\u9012\u5f52\u7684\u89c6\u89d2\u6765\u770b\u8fd9\u4e2a\u95ee\u9898\u3002\u53f3\u65cb\u64cd\u4f5c <code>right_rotate(root)</code> \u4f20\u5165\u7684\u662f\u5b50\u6811\u7684\u6839\u8282\u70b9\uff0c\u6700\u7ec8 <code>return child</code> \u8fd4\u56de\u65cb\u8f6c\u4e4b\u540e\u7684\u5b50\u6811\u7684\u6839\u8282\u70b9\u3002\u5b50\u6811\u7684\u6839\u8282\u70b9\u548c\u5176\u7236\u8282\u70b9\u7684\u8fde\u63a5\u662f\u5728\u8be5\u51fd\u6570\u8fd4\u56de\u540e\u5b8c\u6210\u7684\uff0c\u4e0d\u5c5e\u4e8e\u53f3\u65cb\u64cd\u4f5c\u7684\u7ef4\u62a4\u8303\u56f4\u3002</p> <p>Q\uff1a\u5728 C++ \u4e2d\uff0c\u51fd\u6570\u88ab\u5212\u5206\u5230 <code>private</code> \u548c <code>public</code> \u4e2d\uff0c\u8fd9\u65b9\u9762\u6709\u4ec0\u4e48\u8003\u91cf\u5417\uff1f\u4e3a\u4ec0\u4e48\u8981\u5c06 <code>height()</code> \u51fd\u6570\u548c <code>updateHeight()</code> \u51fd\u6570\u5206\u522b\u653e\u5728 <code>public</code> \u548c <code>private</code> \u4e2d\u5462\uff1f</p> <p>\u4e3b\u8981\u770b\u65b9\u6cd5\u7684\u4f7f\u7528\u8303\u56f4\uff0c\u5982\u679c\u65b9\u6cd5\u53ea\u5728\u7c7b\u5185\u90e8\u4f7f\u7528\uff0c\u90a3\u4e48\u5c31\u8bbe\u8ba1\u4e3a <code>private</code> \u3002\u4f8b\u5982\uff0c\u7528\u6237\u5355\u72ec\u8c03\u7528 <code>updateHeight()</code> \u662f\u6ca1\u6709\u610f\u4e49\u7684\uff0c\u5b83\u53ea\u662f\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u4e2d\u7684\u4e00\u6b65\u3002\u800c <code>height()</code> \u662f\u8bbf\u95ee\u8282\u70b9\u9ad8\u5ea6\uff0c\u7c7b\u4f3c\u4e8e <code>vector.size()</code> \uff0c\u56e0\u6b64\u8bbe\u7f6e\u6210 <code>public</code> \u4ee5\u4fbf\u4f7f\u7528\u3002</p> <p>Q\uff1a\u5982\u4f55\u4ece\u4e00\u7ec4\u8f93\u5165\u6570\u636e\u6784\u5efa\u4e00\u68f5\u4e8c\u53c9\u641c\u7d22\u6811\uff1f\u6839\u8282\u70b9\u7684\u9009\u62e9\u662f\u4e0d\u662f\u5f88\u91cd\u8981\uff1f</p> <p>\u662f\u7684\uff0c\u6784\u5efa\u6811\u7684\u65b9\u6cd5\u5df2\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4ee3\u7801\u4e2d\u7684 <code>build_tree()</code> \u65b9\u6cd5\u4e2d\u7ed9\u51fa\u3002\u81f3\u4e8e\u6839\u8282\u70b9\u7684\u9009\u62e9\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u8f93\u5165\u6570\u636e\u6392\u5e8f\uff0c\u7136\u540e\u5c06\u4e2d\u70b9\u5143\u7d20\u4f5c\u4e3a\u6839\u8282\u70b9\uff0c\u518d\u9012\u5f52\u5730\u6784\u5efa\u5de6\u53f3\u5b50\u6811\u3002\u8fd9\u6837\u505a\u53ef\u4ee5\u6700\u5927\u7a0b\u5ea6\u4fdd\u8bc1\u6811\u7684\u5e73\u8861\u6027\u3002</p> <p>Q\uff1a\u5728 Java \u4e2d\uff0c\u5b57\u7b26\u4e32\u5bf9\u6bd4\u662f\u5426\u4e00\u5b9a\u8981\u7528 <code>equals()</code> \u65b9\u6cd5\uff1f</p> <p>\u5728 Java \u4e2d\uff0c\u5bf9\u4e8e\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c<code>==</code> \u7528\u4e8e\u5bf9\u6bd4\u4e24\u4e2a\u53d8\u91cf\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002\u5bf9\u4e8e\u5f15\u7528\u7c7b\u578b\uff0c\u4e24\u79cd\u7b26\u53f7\u7684\u5de5\u4f5c\u539f\u7406\u662f\u4e0d\u540c\u7684\u3002</p> <ul> <li><code>==</code> \uff1a\u7528\u6765\u6bd4\u8f83\u4e24\u4e2a\u53d8\u91cf\u662f\u5426\u6307\u5411\u540c\u4e00\u4e2a\u5bf9\u8c61\uff0c\u5373\u5b83\u4eec\u5728\u5185\u5b58\u4e2d\u7684\u4f4d\u7f6e\u662f\u5426\u76f8\u540c\u3002</li> <li><code>equals()</code>\uff1a\u7528\u6765\u5bf9\u6bd4\u4e24\u4e2a\u5bf9\u8c61\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002</li> </ul> <p>\u56e0\u6b64\uff0c\u5982\u679c\u8981\u5bf9\u6bd4\u503c\uff0c\u6211\u4eec\u5e94\u8be5\u4f7f\u7528 <code>equals()</code> \u3002\u7136\u800c\uff0c\u901a\u8fc7 <code>String a = \"hi\"; String b = \"hi\";</code> \u521d\u59cb\u5316\u7684\u5b57\u7b26\u4e32\u90fd\u5b58\u50a8\u5728\u5b57\u7b26\u4e32\u5e38\u91cf\u6c60\u4e2d\uff0c\u5b83\u4eec\u6307\u5411\u540c\u4e00\u4e2a\u5bf9\u8c61\uff0c\u56e0\u6b64\u4e5f\u53ef\u4ee5\u7528 <code>a == b</code> \u6765\u6bd4\u8f83\u4e24\u4e2a\u5b57\u7b26\u4e32\u7684\u5185\u5bb9\u3002</p> <p>Q\uff1a\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u5230\u6700\u5e95\u5c42\u4e4b\u524d\uff0c\u961f\u5217\u4e2d\u7684\u8282\u70b9\u6570\u91cf\u662f \\(2^h\\) \u5417\uff1f</p> <p>\u662f\u7684\uff0c\u4f8b\u5982\u9ad8\u5ea6 \\(h = 2\\) \u7684\u6ee1\u4e8c\u53c9\u6811\uff0c\u5176\u8282\u70b9\u603b\u6570 \\(n = 7\\) \uff0c\u5219\u5e95\u5c42\u8282\u70b9\u6570\u91cf \\(4 = 2^h = (n + 1) / 2\\) \u3002</p>"}]}
\ No newline at end of file
diff --git a/sitemap.xml b/sitemap.xml
index f025427fb..691443147 100644
--- a/sitemap.xml
+++ b/sitemap.xml
@@ -320,6 +320,11 @@
          <lastmod>2024-04-02</lastmod>
          <changefreq>daily</changefreq>
     </url>
+    <url>
+         <loc>https://www.hello-algo.com/chapter_hello_algo/</loc>
+         <lastmod>2024-04-02</lastmod>
+         <changefreq>daily</changefreq>
+    </url>
     <url>
          <loc>https://www.hello-algo.com/chapter_introduction/</loc>
          <lastmod>2024-04-02</lastmod>
diff --git a/sitemap.xml.gz b/sitemap.xml.gz
index 34ac1b89a..2907470a4 100644
Binary files a/sitemap.xml.gz and b/sitemap.xml.gz differ